Appendix to ``Market Failure in Kidney Exchange''

▼❛r❦❡t ❋❛✐❧✉r❡ ✐♥ ❑✐❞♥❡② ❊①❝❤❛♥❣❡∗† ◆✐❦❤✐❧ ❆❣❛r✇❛❧✱ ■t❛✐ ❆s❤❧❛❣✐✱ ❊❞✉❛r❞♦ ❆③❡✈❡❞♦✱

‡

❈❧❛②t♦♥ ❋❡❛t❤❡rst♦♥❡✱ ❛♥❞ Ö♠❡r ❑❛r❛❞✉♠❛♥

❆♣r✐❧ ✶✽✱ ✷✵✶✽

❆❜str❛❝t ❑✐❞♥❡② ❡①❝❤❛♥❣❡s ❢❛❝✐❧✐t❛t❡ ♦✈❡r ✽✵✵ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r ✐♥ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s✳ ❲❡ s❤♦✇ t❤❛t✱ ❞❡s♣✐t❡ t❤✐s s✉❝❝❡ss✱ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡ts s✉✛❡r ❢r♦♠ ♠❛r❦❡t ❢❛✐❧✉r❡s t❤❛t ❝❛✉s❡ t❤❡ ❧♦ss ♦❢ ❤✉♥❞r❡❞s ♦❢ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r ❜✉t ❝♦✉❧❞ ❜❡ ❛❞❞r❡ss❡❞ ✇✐t❤ s✐♠♣❧❡ ♣♦❧✐❝✐❡s✳ ❖✉r ❛r❣✉♠❡♥t ❤❛s t❤r❡❡ ♣❛rts✳ ❋✐rst✱ ✇❡ ❞♦❝✉♠❡♥t t❤❛t t❤❡ ♠❛r❦❡t ✐s ❤✐❣❤❧② ❢r❛❣♠❡♥t❡❞✱ ✇✐t❤ ❛♣♣r♦①✐♠❛t❡❧② ✻✵✪ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ❜❡✐♥❣ ❛rr❛♥❣❡❞ ❜② s♠❛❧❧ ♣❧❛t❢♦r♠s✱ ♦❢t❡♥ ✇✐t❤✐♥ ❤♦s♣✐t❛❧s✱ ❛s ♦♣♣♦s❡❞ t♦ ✐♥ ❧❛r❣❡✱ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s✳ ▼♦r❡♦✈❡r✱ ✇❡ ✜♥❞ ❞✐r❡❝t ❡✈✐❞❡♥❝❡ t❤❛t s♠❛❧❧ ♣❧❛t❢♦r♠s ♦❢t❡♥ ❛rr❛♥❣❡ ✐♥❡❢✲ ✜❝✐❡♥t ❡①❝❤❛♥❣❡s✳ ❙❡❝♦♥❞✱ ✇❡ ♣r♦♣♦s❡ ❛ s✐♠♣❧❡ ♠♦❞❡❧ t♦ s❤♦✇ t❤❛t ✐♥❡✣❝✐❡♥❝② ❛r✐s❡s ❢♦r t✇♦ r❡❛s♦♥s✿ ❤♦s♣✐t❛❧s ❞♦ ♥♦t ✐♥t❡r♥❛❧✐③❡ t❤❡✐r ♣❛t✐❡♥ts✬ ❜❡♥❡✜ts ❢r♦♠ ♣❛rt✐❝✐♣❛t✐♦♥ ❛♥❞ ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠s ❞♦ ♥♦t ❣✐✈❡ ❤♦s♣✐t❛❧s ❛❞❡q✉❛t❡ ✐♥❝❡♥t✐✈❡s✳ ❚❤✐r❞✱ ✇❡ ❡st✐♠❛t❡ ❛ ♣❧❛t❢♦r♠✬s ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ t♦ q✉❛♥t✐❢② t❤❡ ✐♥❡✣❝✐❡♥❝② ❛♥❞ ❞❡s✐❣♥ ♣r❛❝t✐❝❛❧ ♠❡❝❤✲ ❛♥✐s♠s✳ ❖✉r ❡st✐♠❛t❡s s❤♦✇ t❤❛t ❤♦s♣✐t❛❧s✬ ♣r♦❞✉❝t✐♦♥ s❝❛❧❡s ❛r❡ t♦♦ s♠❛❧❧ t♦ ♠❛t❝❤ ♣❛t✐❡♥ts ❡✣❝✐❡♥t❧②✳ ❊❧✐♠✐♥❛t✐♥❣ t❤✐s ✐♥❡✣❝✐❡♥❝② r❡q✉✐r❡s ❛ ❝♦♠❜✐♥❡❞ ❛♣♣r♦❛❝❤ t❤❛t ✉s❡s ♥❡✇ ♠❡❝❤❛♥✐s♠s ❛♥❞ ✐♥❝❡♥t✐✈✐③❡s ♣❛rt✐❝✐♣❛t✐♦♥✳ ❏❊▲✿ ❉✹✼✱ ❉✹✷✱ ▲✶✶ ❑❡②✇♦r❞s✿ ❑✐❞♥❡② ❊①❝❤❛♥❣❡

❋✐rst ✈❡rs✐♦♥✿ ❙❡♣t❡♠❜❡r ✷✵✱ ✷✵✶✼✳ ❲❡ t❤❛♥❦ ❆❧❡① ●❛r③❛✱ ❙❛r❛❤ ❚❛r❛♥t♦✱ ❛♥❞ ❏❡♥♥✐❢❡r ❲❛✐♥✇r✐❣❤t ♦❢ ❯◆❖❙❀ ❏♦♥❛t❤❛♥ ❑♦♣❦❡ ❛♥❞ ▼✐❝❤❛❡❧ ❘❡❡s ♦❢ ❆P❉❀ ❈❛t❤✐ ▼✉r♣❤❡② ❛♥❞ ❆❞❛♠ ❇✐♥❣❛♠❛♥ ♦❢ ▼❡t❤♦❞✐st ❛t ❙❛♥ ❆♥t♦♥✐♦❀ ❛♥❞ ●❛r❡t ❍✐❧❧ ♦❢ ◆❑❘ ❢♦r t❤❡✐r ❡①♣❡rt✐s❡ ❛♥❞ ❛❝❝❡ss t♦ ❞❛t❛✳ ❲❡ t❤❛♥❦ ●❧❡♥♥ ❊❧❧✐s♦♥✱ P❛r❛❣ P❛t❤❛❦✱ ❆❧ ❘♦t❤✱ ❏❡❛♥ ❚✐r♦❧❡✱ ❯t❦✉ Ü♥✈❡r✱ ▼✐❦❡ ❲❤✐♥st♦♥✱ ❛♥❞ s❡♠✐♥❛r ♣❛rt✐❝✐♣❛♥ts ❢♦r ❤❡❧♣❢✉❧ ❞✐s❝✉ss✐♦♥s✳ ❆❜✐❣❛✐❧ ❖str✐❦❡r ♣r♦✈✐❞❡❞ ❡①❝❡❧❧❡♥t r❡s❡❛r❝❤ ❛ss✐st❛♥❝❡✳ ❆③❡✈❡❞♦ ❛❝❦♥♦✇❧❡❞❣❡s s✉♣♣♦rt ❢r♦♠ ❲❤❛rt♦♥✬s ❉❡❛♥✬s ❘❡s❡❛r❝❤ ❋✉♥❞✱ ❆❣❛r✇❛❧ ❢r♦♠ t❤❡ ◆❙❋ ✭❙❊❙✲✶✼✷✾✵✾✵✮✱ ❛♥❞ ❆s❤❧❛❣✐ ❢r♦♠ t❤❡ ◆❙❋ ✭❙❊❙✲✶✷✺✹✼✻✽✮✳ † ❉✐s❝❧❛✐♠❡r✿ ❚❤❡ ❞❛t❛ r❡♣♦rt❡❞ ❤❡r❡ ❤❛✈❡ ❜❡❡♥ s✉♣♣❧✐❡❞ ❜② ❯◆❖❙ ❛s t❤❡ ❝♦♥tr❛❝t♦r ❢♦r t❤❡ ❖r❣❛♥ Pr♦❝✉r❡♠❡♥t ❛♥❞ ❚r❛♥s♣❧❛♥t❛t✐♦♥ ◆❡t✇♦r❦ ✭❖P❚◆✮✳ ❚❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ r❡♣♦rt✐♥❣ ♦❢ t❤❡s❡ ❞❛t❛ ❛r❡ t❤❡ r❡s♣♦♥s✐❜✐❧✐t② ♦❢ t❤❡ ❛✉t❤♦r✭s✮ ❛♥❞ ✐♥ ♥♦ ✇❛② s❤♦✉❧❞ ❜❡ s❡❡♥ ❛s ❛♥ ♦✣❝✐❛❧ ♣♦❧✐❝② ♦❢ ♦r ✐♥t❡r♣r❡t❛t✐♦♥ ❜② t❤❡ ❖P❚◆ ♦r t❤❡ ❯✳❙✳ ●♦✈❡r♥♠❡♥t✳ ‡ ❆❣❛r✇❛❧✿ ▼■❚ ❛♥❞ ◆❇❊❘✱ ❛❣❛r✇❛❧♥❅♠✐t✳❡❞✉✳ ❆s❤❧❛❣✐✿ ❙t❛♥❢♦r❞ ❯♥✐✈❡rs✐t②✱ ✐❛s❤❧❛❣✐❅st❛♥❢♦r❞✳❡❞✉✳ ❆③❡✈❡❞♦✿ ❲❤❛rt♦♥✱ ❡❛③❡✈❡❞♦❅✇❤❛rt♦♥✳✉♣❡♥♥✳❡❞✉✳ ❋❡❛t❤❡rst♦♥❡✿ ❲❤❛rt♦♥✱ ❝❧❛②t♦♥❢❅✇❤❛rt♦♥✳✉♣❡♥♥✳❡❞✉✳ ❑❛r❛❞✉♠❛♥✿ ▼■❚✱ ♦♠❡r❦❅♠✐t✳❡❞✉✳ ∗

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✶

■♥tr♦❞✉❝t✐♦♥

❑✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡ts ❡♥❛❜❧❡ ❛♣♣r♦①✐♠❛t❡❧② ✽✵✵ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r✳ P❛t✐❡♥ts ✐♥ t❤✐s ♠❛r❦❡t ♥❡❡❞ ❛ ❦✐❞♥❡② tr❛♥s♣❧❛♥t✱ ❛♥❞ ♠♦st ❝♦♠❡ ✇✐t❤ ❛ ❧✐✈✐♥❣✱ ❜✉t ✐♥❝♦♠♣❛t✐❜❧❡✱ ❞♦♥♦r✳ ❚r❛♥s♣❧❛♥ts ❛r❡ ♦r❣❛♥✐③❡❞ ✉s✐♥❣ ♣❛t✐❡♥t✲❞♦♥♦r s✇❛♣s ❛♥❞ ❝❤❛✐♥s ✐♥✐t✐❛t❡❞ ❜② ❛❧tr✉✐st✐❝ ❞♦♥♦rs✳ ❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s ♠❛r❦❡t ❤❛s ❣r♦✇♥ ❜❡❝❛✉s❡ ♦❢ ❛ s❤♦rt❛❣❡ ♦❢ ♦r❣❛♥s ❢r♦♠ ❞❡❝❡❛s❡❞ ❞♦♥♦rs ❛♥❞ ❜❡❝❛✉s❡ ❝✉rr❡♥t ❧❛✇ ♣r♦❤✐❜✐ts ✉s✐♥❣ ♠♦♥❡t❛r② ❝♦♠♣❡♥s❛t✐♦♥ t♦ ❛❝q✉✐r❡ ♦r❣❛♥s✳✶ ❚❤❡ ❡❝♦♥♦♠✐❝ ✈❛❧✉❡ ♦❢ ❛ tr❛♥s♣❧❛♥t ✐s ❡st✐♠❛t❡❞ ❛t ♠♦r❡ t❤❛♥ ❛ ♠✐❧❧✐♦♥ ❞♦❧❧❛rs✿ ❦✐❞♥❡② tr❛♥s♣❧❛♥ts s❛✈❡ ❧✐✈❡s✱ r❡❞✉❝❡ ❤❡❛❧t❤❝❛r❡ ❝♦sts✱ ❛♥❞ s❛✈❡ t❛①♣❛②❡r ❢✉♥❞s✳✷ ❚❤✐s ♣❛♣❡r ❡✈❛❧✉❛t❡s t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡t ❛♥❞ s❤♦✇s t❤❛t✱ ❞❡s♣✐t❡ s✐❣♥✐✜❝❛♥t s✉❝❝❡ss ✐♥ ♦r❣❛♥✐③✐♥❣ tr❛♥s♣❧❛♥ts✱ t❤❡ ♠❛r❦❡t s✉✛❡rs ❢r♦♠ t✇♦ s❡r✐♦✉s ❛♥❞ ✜①❛❜❧❡ ♠❛r❦❡t ❢❛✐❧✉r❡s t❤❛t t♦❣❡t❤❡r r❡s✉❧t ✐♥ ❤✉♥❞r❡❞s ♦❢ ❧♦st tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r✳ ■♥❞❡❡❞✱ ♦✉r ❞❡s❝r✐♣t✐✈❡ ❡✈✐❞❡♥❝❡ s✉❣✲ ❣❡sts t❤❛t t❤❡ ♠❛r❦❡t ✐s ❢r❛❣♠❡♥t❡❞ ❛♥❞ ♦♣❡r❛t❡s ✐♥❡✣❝✐❡♥t❧②✳ ❲❡ ✉s❡ ♣r✐❝❡✲t❤❡♦r❡t✐❝ ❛r❣✉✲ ♠❡♥ts t♦ ❡①♣❧❛✐♥ t❤❛t t❤✐s ✐♥❡✣❝✐❡♥❝② ❛r✐s❡s ❢r♦♠ t✇♦ st❛♥❞❛r❞ ♠❛r❦❡t ❢❛✐❧✉r❡s✳ ❊❛❝❤ ❢❛✐❧✉r❡ ❝♦rr❡s♣♦♥❞s t♦ ❛ s♣❡❝✐✜❝ ♣r♦❜❧❡♠ ✇✐t❤ ❝✉rr❡♥t ✐♥st✐t✉t✐♦♥s✳ ❇♦t❤ ♣r♦❜❧❡♠s ②✐❡❧❞ s✉❜♦♣t✐♠❛❧ ✐♥❝❡♥t✐✈❡s ❢♦r ❤♦s♣✐t❛❧s✱ t❤❡ ❦❡② ❞❡❝✐s✐♦♥ ♠❛❦❡rs ✭❘♦t❤ ❡t ❛❧✳✱ ✷✵✵✺❀ ❆s❤❧❛❣✐ ❛♥❞ ❘♦t❤✱ ✷✵✶✹❀ ❘❡❡s ❡t ❛❧✳✱ ✷✵✵✾✮✳ ❋✐rst✱ ❧❛r❣❡ ♥❛t✐♦♥❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s ✉s❡ ✐♥❡✣❝✐❡♥t ♠❡❝❤❛♥✐s♠s t♦ ♠❛t❝❤ ♣❛t✐❡♥ts✳ ❉✉❡ t♦ ❜✐♦❧♦❣✐❝❛❧ ❝♦♠♣❛t✐❜✐❧✐t②✱ t❤❡r❡ ✐s s✉❜st❛♥t✐❛❧ ✈❛r✐❛t✐♦♥ ✐♥ t❤❡ ❛❞✲ ❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥ts t❤❛t r❡s✉❧t ❢r♦♠ ❞✐✛❡r❡♥t ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r t②♣❡s ❜❡✐♥❣ s✉❜♠✐tt❡❞ t♦ ❛ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠✳ ❇✉t ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠s ✐❣♥♦r❡ t❤❡s❡ ❞✐✛❡r❡♥❝❡s ✐♥ s♦❝✐❛❧ ✈❛❧✉❡✳ ❚❤✐s ❧✐♠✐t❛t✐♦♥ ✐♥ t❤❡ ❞❡s✐❣♥ ♣✉s❤❡s ❤♦s♣✐t❛❧s t♦ ♠❛t❝❤ ♣❛t✐❡♥ts ✇✐t❤ ❤✐❣❤ s♦❝✐❛❧ ✈❛❧✉❡ ♦✉ts✐❞❡ t❤❡ ♣❧❛t❢♦r♠✱ ❡✈❡♥ ✇❤❡♥ ✐t ✐s ✐♥❡✣❝✐❡♥t✳ ❙❡❝♦♥❞✱ t❤❡r❡ ✐s s❝♦♣❡ ❢♦r ❛❣❡♥❝② ♣r♦❜❧❡♠s✿ ❤♦s♣✐t❛❧s ❢❛❝❡ ♠♦st ♦❢ t❤❡ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ❜✉t r❡❝❡✐✈❡ ♦♥❧② ❛ ❢r❛❝t✐♦♥ ♦❢ t❤❡ ❜❡♥❡✜ts✳ ❲❡ s❤♦✇ t❤❛t ✜①✐♥❣ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠s ❝❛♥ r❡s✉❧t ✐♥ ❤✉♥❞r❡❞s ♠♦r❡ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r✳ ❚❤✐s ❛r❣✉♠❡♥t ❤❛s t❤r❡❡ ♣❛rts✳ ❋✐rst✱ ✇❡ ✉s❡ ❛❞♠✐♥✐str❛t✐✈❡ ❞❛t❛s❡ts ❛♥❞ ✐♥st✐t✉t✐♦♥❛❧ ❞❡t❛✐❧ t♦ ❞✐❛❣♥♦s❡ ❛♥❞ ❞♦❝✉♠❡♥t t❤❡ ❝❛✉s❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡s❡ ♠❛r❦❡t ❢❛✐❧✉r❡s✳ ❙❡❝♦♥❞✱ ✇❡ ❛r❣✉❡ t❤❛t✱ ❛❧t❤♦✉❣❤ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡ts ❧♦♦❦ ❞✐✛❡r❡♥t ❢r♦♠ ♦t❤❡r ♠❛r❦❡ts✱ t❤❡✐r ✐♥❡✣❝✐❡♥❝② ✐s ❝❛✉s❡❞ ❜② tr❛❞✐t✐♦♥❛❧ ♠❛r❦❡t ❢❛✐❧✉r❡s✳ ❚♦ ❛♥❛❧②③❡ t❤❡ ♠❛r❦❡t✱ ✐t ✐s ✉s❡❢✉❧ t♦ t❤✐♥❦ ❛❜♦✉t t❤❡ r♦❧❡ ♦❢ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ❛s ❛ ♣r♦❞✉❝❡r ♦❢ tr❛♥s♣❧❛♥ts t❤❛t t❛❦❡s ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs s✉♣♣❧✐❡❞ ❜② ❤♦s♣✐t❛❧s ❛s ✐♥♣✉ts✳ ❚❤✐s r❡✲❢r❛♠✐♥❣✱ ❜❛s❡❞ ♦♥ ♥❡♦❝❧❛ss✐✲ ❝❛❧ ♣r♦❞✉❝❡r t❤❡♦r②✱ ❛❧❧♦✇s ✉s t♦ ✉s❡ ❝❧❛ss✐❝ ♣r✐❝❡ t❤❡♦r② ❛r❣✉♠❡♥ts t❤❛t ❤❛✈❡ ❝❧❡❛r ♣♦❧✐❝② ✶ ❚❤❡r❡

❛r❡ ♦✈❡r ✾✼✱✵✵✵ ♣❛t✐❡♥ts ❝✉rr❡♥t❧② ✇❛✐t✐♥❣ ❢♦r ❛ ❦✐❞♥❡② ❢r♦♠ ❛ ❞❡❝❡❛s❡❞ ❞♦♥♦r✱ ❜✉t ❧❡ss t❤❛♥ ❛ ✜❢t❤ ❛r❡ ❡①♣❡❝t❡❞ t♦ ❜❡ tr❛♥s♣❧❛♥t❡❞ ✐♥ t❤❡ ♥❡①t ②❡❛r✳ ❇❡❝❦❡r ❛♥❞ ❊❧✐❛s ✭✷✵✵✼✮ ❛r❣✉❡ t❤❛t t❤✐s ✇❛✐t❧✐st ❝♦✉❧❞ ❜❡ ❝♦♠♣❧❡t❡❧② ❡❧✐♠✐♥❛t❡❞ ✐❢ t❤❡r❡ ✇❛s ♠♦♥❡t❛r② ❝♦♠♣❡♥s❛t✐♦♥ ❢♦r ❧✐✈❡ ❞♦♥♦rs ❛♥❞ ❛❞✈♦❝❛t❡ ❢♦r t❤❡ ❝r❡❛t✐♦♥ ♦❢ t❤✐s ♠❛r❦❡t✳ ❍♦✇❡✈❡r✱ t❤✐s t②♣❡ ♦❢ tr❛♥s❛❝t✐♦♥ ✐s ✇✐❞❡❧② ♣❛♥♥❡❞ ❜② ❜✐♦❡t❤✐❝✐sts✱ ❛♥❞ ❛❧♠♦st ❛❧❧ ❝♦✉♥tr✐❡s ❢♦r❜✐❞ ♠♦♥❡t❛r② ❝♦♠♣❡♥s❛t✐♦♥ ❢♦r ♦r❣❛♥s✳ ❚❤❡ ◆❛t✐♦♥❛❧ ❖r❣❛♥ ❚r❛♥s♣❧❛♥t❛t✐♦♥ ❆❝t ✭◆❖❚❆✮ ♣r♦❤✐❜✐ts ❝♦♠♣❡♥s❛t✐♥❣ ❞♦♥♦rs ✐♥ ❡①❝❤❛♥❣❡ ❢♦r ❛❝q✉✐r✐♥❣ ♦r❣❛♥s ✐♥ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s✳ ❚❤❡ ♠♦t✐✈❛t✐♦♥ ❢♦r ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ✐s t♦ ✉s❡ ❞♦♥♦r s✇❛♣s t♦ ❤❡❧♣ ♣❛t✐❡♥ts ✜♥❞ ❛♥ ♦r❣❛♥ ✐♥ ❛♥ ❡t❤✐❝❛❧❧② ❛♥❞ ❧❡❣❛❧❧② ❛❝❝❡♣t❛❜❧❡ ✇❛② ✭❘♦t❤✱ ✷✵✵✼✮✳ ✷ ❚r❛♥s♣❧❛♥t❛t✐♦♥ r♦✉❣❤❧② ❞♦✉❜❧❡s t❤❡ ❧✐❢❡ ❡①♣❡❝t❛♥❝② ♦❢ ♣❛t✐❡♥ts ✇✐t❤ ❡♥❞✲st❛❣❡ ❦✐❞♥❡② ❞✐s❡❛s❡✱ ❛♥❞ ✐s ❝❤❡❛♣❡r t❤❛♥ t❤❡ ❛❧t❡r♥❛t✐✈❡ tr❡❛t♠❡♥t ♦❢ ❞✐❛❧②s✐s✳ ▼❡❞✐❝❛r❡ ♣r♦✈✐❞❡s ♥❡❛r❧② ✉♥✐✈❡rs❛❧ ❝♦✈❡r❛❣❡✱ ✐rr❡s♣❡❝t✐✈❡ ♦❢ ❛❣❡✱ ❢♦r ♣❛t✐❡♥ts ✇✐t❤ ❊♥❞✲❙t❛❣❡ ❘❡♥❛❧ ❉✐s❡❛s❡ ✭❊❙❘❉✮✳ ❚❤❡ ❊❙❘❉ ♣r♦❣r❛♠ ❝♦♠♣r✐s❡s ♦❢ ❛❜♦✉t ✼✪ ♦❢ ▼❡❞✐❝❛r❡✬s ❛♥♥✉❛❧ ❜✉❞❣❡t ✭s❡❡ ❯❙❘❉❙✱ ❯♥✐t❡❞ ❙t❛t❡s ❘❡♥❛❧ ❉❛t❛ ❙②st❡♠✱ ✷✵✶✻✮✳ ❚❤❡ ❝♦st s❛✈✐♥❣s ♦❢ tr❛♥s♣❧❛♥t❛t✐♦♥ r❡❧❛t✐✈❡ t♦ ❞✐❛❧②s✐s ❛❧♦♥❡ ❤❛✈❡ ❜❡❡♥ ❡st✐♠❛t❡❞ t♦ ❜❡ ♦✈❡r ✩✷✼✵✱✵✵✵ ✭s❡❡ ❙❡❝t✐♦♥ ✷✮✳

✸ ✐♠♣❧✐❝❛t✐♦♥s✳ ❚❤✐r❞✱ ✇❡ ❡st✐♠❛t❡ t❤❡ ❦❡② ♣r✐♠✐t✐✈❡ ♦❢ t❤❡ ♠♦❞❡❧✱ ❛ tr❛♥s♣❧❛♥t ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✱ ✉s✐♥❣ ❞❛t❛ ❢r♦♠ t❤❡ ❧❛r❣❡st ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳ ❚❤❡ ♣r✐♠✐t✐✈❡ ❛❧❧♦✇s ✉s t♦ ❡st✐♠❛t❡ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✐♥❡✣❝✐❡♥❝✐❡s✱ ❞❡s✐❣♥ ♣r❛❝t✐❝❛❧ ❛❧t❡r♥❛t✐✈❡ ♠❡❝❤❛♥✐s♠s✱ ❛♥❞ s✉❣❣❡st ❛♣♣r♦♣r✐❛t❡ ♣♦❧✐❝② r❡s♣♦♥s❡s✳ ■♥ t❤❡ ✜rst ♣❛rt✱ ✇❡ ❞♦❝✉♠❡♥t ❦❡② ❢❛❝ts ❛❜♦✉t t❤❡ ♠❛r❦❡t t♦ ♠♦t✐✈❛t❡ ♦✉r ♠♦❞❡❧✳ ❲❡ ♠❡r❣❡ ❛❞♠✐♥✐str❛t✐✈❡ ❞❛t❛ ♦♥ t❤❡ ✉♥✐✈❡rs❡ ♦❢ tr❛♥s♣❧❛♥ts ✐♥ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ❛♥❞ ♣r♦♣r✐❡t❛r② ❞❛t❛ ❢r♦♠ t❤❡ t❤r❡❡ ❧❛r❣❡st ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s✳ ❚❤❡ ❞❛t❛ r❡✈❡❛❧ s❡✈❡r❛❧ s✐❣♥s ♦❢ ✐♥❡✣❝✐❡♥❝②✳ ❘❛t❤❡r t❤❛♥ ♠♦st tr❛♥s❛❝t✐♦♥s t❛❦✐♥❣ ♣❧❛❝❡ ❛t ❛ ❢❡✇ ❧❛r❣❡ ♣❧❛t❢♦r♠s✱ t❤❡ ♠❛r❦❡t ✐s ❤✐❣❤❧② ❢r❛❣♠❡♥t❡❞✳ ❉✉r✐♥❣ ♦✉r s❛♠♣❧❡ ♣❡r✐♦❞✱ ✻✷✪ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ✐♥✈♦❧✈❡❞ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❢r♦♠ t❤❡ s❛♠❡ ❤♦s♣✐t❛❧✳ ❈r✉❝✐❛❧❧②✱ ✇❡ ✜♥❞ ❞✐r❡❝t ❡✈✐❞❡♥❝❡ t❤❛t ♠❛♥② ✇✐t❤✐♥✲ ❤♦s♣✐t❛❧ ❡①❝❤❛♥❣❡s ❛r❡ ✐♥❡✣❝✐❡♥t ❢r♦♠ ❛ s♦❝✐❛❧ ♣❡rs♣❡❝t✐✈❡✳ ❆♥ ❡✣❝✐❡♥t ♠❛r❦❡t ✇♦✉❧❞ ✉s❡ ❡❛s②✲t♦✲♠❛t❝❤ ❞♦♥♦rs ♦♥❧② ❢♦r ❤❛r❞✲t♦✲♠❛t❝❤ ♣❛t✐❡♥ts✳ ❲❡ s❤♦✇ t❤❛t s♠❛❧❧ ✇✐t❤✐♥✲❤♦s♣✐t❛❧ ❡①❝❤❛♥❣❡s ♦❢t❡♥ ✈✐♦❧❛t❡ t❤✐s s✐♠♣❧❡ r✉❧❡✱ ❜✉t ❧❛r❣❡ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ❞♦ ♥♦t✳ ❲❡ ❛❧s♦ ✜♥❞ ❡✈✐❞❡♥❝❡ ❝♦♥s✐st❡♥t ✇✐t❤ ✐♥❡✣❝✐❡♥t ❤♦s♣✐t❛❧ ✐♥❝❡♥t✐✈❡s ❝❛✉s✐♥❣ t❤❡s❡ ♣r♦❜❧❡♠s✳ ▼❛♥② ❤♦s♣✐t❛❧s ❞♦ ♥♦t ♣❛rt✐❝✐♣❛t❡ ✐♥ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s✳ ❈♦♥s✐st❡♥t ✇✐t❤ ❜❛rr✐❡rs ❞✉❡ t♦ ✜①❡❞ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✱ s♠❛❧❧❡r ❤♦s♣✐t❛❧s ❛r❡ ❧❡ss ❧✐❦❡❧② t♦ ♣❛rt✐❝✐✲ ♣❛t❡✳ Pr❡✈✐♦✉s s✉r✈❡②s ❛♥❞ ❛ st✉❞② ♦❢ t❤❡ ♣❛②♠❡♥t str✉❝t✉r❡ ✐♥ t❤✐s ♠❛r❦❡t s✉❣❣❡sts t❤❛t ♣❛rt✐❝✐♣❛t✐♦♥ ❝♦sts ❛r❡ ❛ s✐❣♥✐✜❝❛♥t ❢r❛❝t✐♦♥ ♦❢ ❤♦s♣✐t❛❧ r❡✈❡♥✉❡ ❢r♦♠ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳ ❍♦✇✲ ❡✈❡r✱ t❤❡s❡ ❝♦sts ❛r❡ s♠❛❧❧ r❡❧❛t✐✈❡ t♦ t❤❡ s♦❝✐❛❧ ❜❡♥❡✜t ♦❢ tr❛♥s♣❧❛♥ts✳ ❊✈❡♥ ✇❤❡♥ ❤♦s♣✐t❛❧s ❞♦ ♣❛rt✐❝✐♣❛t❡✱ t❤❡ t②♣✐❝❛❧ ❤♦s♣✐t❛❧ ❞♦❡s ♥♦t ❝♦♥❞✉❝t ❛❧❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s t❤r♦✉❣❤ ❛ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠✳ ■♥st❡❛❞✱ ♠♦st ❤♦s♣✐t❛❧s ❝♦♥t✐♥✉❡ t♦ ♦♣❡r❛t❡ ❛s ❛ ❝♦♠♣❡t✐♥❣ s♠❛❧❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳ ❚❤❡s❡ ❢❛❝ts ♠♦t✐✈❛t❡ ♦✉r ♠♦❞❡❧ ✐♥ t❤❡ s❡❝♦♥❞ ♣❛rt✳ ❚❤❡ ❦❡② ✐♥s✐❣❤t ✉♥❞❡r❧②✐♥❣ ♦✉r ❛♣♣r♦❛❝❤ ✐s t❤❛t ❛❧t❤♦✉❣❤ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡ts ❞♦ ♥♦t ❞✐r❡❝t❧② ✉s❡ ♠♦♥❡t❛r② ✐♥❝❡♥t✐✈❡s t♦ ❛❝q✉✐r❡ ♦r❣❛♥s✱ ✐t ✐s ✉s❡❢✉❧ t♦ ❛♥❛❧②③❡ t❤❡♠ ✉s✐♥❣ ♥❡♦❝❧❛ss✐❝❛❧ ♣r♦❞✉❝❡r t❤❡♦r②✳ ❚❤❡ ♣❧❛t❢♦r♠ ♣r♦❞✉❝❡s ❛ ✜♥❛❧ ❣♦♦❞ ✭tr❛♥s♣❧❛♥ts✮ ❢r♦♠ ✐♥t❡r♠❡❞✐❛t❡ ❣♦♦❞s ✭s✉❜♠✐ss✐♦♥s ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✮ s✉♣♣❧✐❡❞ ❜② ❛ ❝♦♠♣❡t✐t✐✈❡ ❢r✐♥❣❡ ✭❤♦s♣✐t❛❧s✮ ❛❝❝♦r❞✐♥❣ t♦ ❛ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳ ❚❤✐s r❡✲❢r❛♠✐♥❣ ✐s ❜❛s❡❞ ♦♥ t❤r❡❡ ❦❡② ✐♥st✐t✉t✐♦♥❛❧ ❢❡❛t✉r❡s✳ ❋✐rst✱ ❤♦s♣✐t❛❧s ❛r❡ t❤❡ ❦❡② ❞❡❝✐s✐♦♥✲♠❛❦❡rs✳ ❚❤❡② s✉❜♠✐t ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs t♦ ❛ ♣❧❛t❢♦r♠ ✈♦❧✉♥t❛r✐❧② ❛♥❞ ❛r❡ ♥♦t ❢♦r❝❡❞ t♦ ♣❛rt✐❝✐♣❛t❡✳ ■♥st❡❛❞ ♦❢ s✉❜♠✐tt✐♥❣ ❛ ♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐r t♦ ❛ ❧❛r❣❡ ♣❧❛t❢♦r♠✱ ❛ ❤♦s♣✐t❛❧ ❝❛♥ ♣❡r❢♦r♠ ❛ tr❛♥s♣❧❛♥t ✇✐t❤ ♦t❤❡r ♣❛✐rs t❤❛t ✐t ✐s tr❡❛t✐♥❣✳ ❚❤✉s✱ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s ♠✉st r❡✇❛r❞ ❤♦s♣✐t❛❧s ✇✐t❤ tr❛♥s♣❧❛♥ts t♦ ♣r♦❝✉r❡ ♥❡❝❡ss❛r② ✐♥♣✉ts ✭♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✮✳ ❙❡❝♦♥❞✱ ❞✉❡ t♦ ❜✐♦❧♦❣✐❝❛❧ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥str❛✐♥ts✱ s♦♠❡ t②♣❡s ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❡♥❛❜❧❡ ♠♦r❡ tr❛♥s♣❧❛♥ts t❤❛♥ ♦t❤❡rs✱ ✐✳❡✳ s♦♠❡ ✐♥♣✉ts ❛r❡ ♠♦r❡ ♣r♦❞✉❝t✐✈❡ t❤❛♥ ♦t❤❡rs✳ ❋♦r ❡①❛♠♣❧❡✱ ❜❧♦♦❞ t②♣❡ ❖ ❞♦♥♦rs ❝❛♥ ❞♦♥❛t❡ t♦ ♣❛t✐❡♥ts ♦❢ ❛♥② ❜❧♦♦❞ t②♣❡✳ ❚❤✐s ❢❛❝t ♠❛❦❡s t❤❡♠ ❜♦t❤ s❝❛r❝❡ ❛♥❞ ✈❛❧✉❛❜❧❡ ✇❤❡♥ ♦r❣❛♥✐③✐♥❣ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s✳✸ ❇❡❝❛✉s❡ ❤♦s♣✐t❛❧s ❝❛♥ ♣❡r❢♦r♠ ❡①❝❤❛♥❣❡s ✇✐t❤✐♥ t❤❡✐r s❡t ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✱ ❛ ♣❧❛t❢♦r♠ ♠❛② ❤❛✈❡ t♦ ✐♥❝r❡❛s❡ t❤❡ r❡✇❛r❞s ❢♦r s✉❜♠✐tt✐♥❣ ✈❛❧✉❛❜❧❡ t②♣❡s ✐♥ ♦r❞❡r t♦ ♣r♦❝✉r❡ t❤❡♠✳ ❚❤✐r❞✱ tr❛♥s♣❧❛♥ts ✸ ■♥ ❛ s✐♠♣❧❡ ♠♦❞❡❧✱ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ s❤♦✇ t❤❛t ❝❡rt❛✐♥ ✏♦✈❡r✲❞❡♠❛♥❞❡❞✑ t②♣❡s ❡♥❛❜❧❡ t✇♦ ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥ts ❛t ❛ ♣❧❛t❢♦r♠ ✇❤✐❧❡ ♦t❤❡rs ❛❞❞ ♥♦ ✈❛❧✉❡ t♦ t❤❡ ♣♦♦❧✳ ❖✉r ❝❛❧❝✉❧❛t✐♦♥s ✇✐❧❧ ❣❡♥❡r❛❧✐③❡ t❤✐s ✐❞❡❛ ❜② ❝♦♠♣✉t✐♥❣ t❤❡ ❛❞❞✐t✐♦♥❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ❣❡♥❡r❛t❡❞ ❜② ❛ r✐❝❤ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ t②♣❡s ✉s✐♥❣ ❞❛t❛ ❢r♦♠ t❤❡ ❧❛r❣❡st ❡①❝❤❛♥❣❡✳

✹ ❛r❡ ❛ ♥❛t✉r❛❧ ♥✉♠❡r❛✐r❡ ❣♦♦❞ ❢♦r r❡✇❛r❞✐♥❣ ❤♦s♣✐t❛❧s✳ ❆ ♣❧❛t❢♦r♠ ♠✉st r❛t✐♦♥ tr❛♥s♣❧❛♥ts ❛♠♦♥❣st t❤❡ t②♣❡s ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs t❤❛t ❛r❡✱ ❞✉❡ ❜✐♦❧♦❣✐❝❛❧ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥str❛✐♥ts✱ ✐♥ ❛❜✉♥❞❛♥❝❡✳✹ ❆ tr❛♥s♣❧❛♥t ❝❛♥ t❤❡r❡❢♦r❡ ❜❡ tr❛♥s❢❡rr❡❞ t♦ ❛ ❤♦s♣✐t❛❧ ❜② s❡❧❡❝t✐♥❣ ✐ts ♣❛t✐❡♥t ✐♥st❡❛❞ ♦❢ ❛♥♦t❤❡r ❤♦s♣✐t❛❧✬s✳ ❚❤✐s ❛❧❧♦✇s tr❛♥s♣❧❛♥ts t♦ ❛❝t ❛s ❛ ♥✉♠❡r❛✐r❡ ❣♦♦❞✱ ❡✈❡♥ t❤♦✉❣❤ ❞✐r❡❝t ♠♦♥❡t❛r② tr❛♥s❢❡rs ❢♦r ❛❝q✉✐r✐♥❣ ♦r❣❛♥s ❛r❡ ♣r♦❤✐❜✐t❡❞✳ ❚❤❡ ❦❡② ♣r✐♠✐t✐✈❡ ♦❢ t❤❡ ♠♦❞❡❧ ✐s t❤❡ tr❛♥s♣❧❛♥t ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳ ■ts s❤❛♣❡ ❞❡t❡r♠✐♥❡s ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s ♦❢ t❤❡ ❡❝♦♥♦♠✐❝s ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳ ❚❤❡ r❡t✉r♥s t♦ s❝❛❧❡ ❞❡t❡r♠✐♥❡ ❤♦✇ ❡✣❝✐❡♥t ✐t ✐s t♦ ♠❛t❝❤ ♣❛t✐❡♥ts ✐♥ ❧❛r❣❡ ❡①❝❤❛♥❣❡s✳ ❚❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ♦❢ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r s♦♠❡ s✉❜♠✐ss✐♦♥s ♣r♦❞✉❝❡ ♠♦r❡ tr❛♥s♣❧❛♥ts t❤❛♥ ♦t❤❡rs✳ ❋♦r ♦✉r ❡♠♣✐r✐❝❛❧ ❛♥❛❧②s✐s ✐♥ t❤❡ t❤✐r❞✲♣❛rt✱ ✇❡ ❝♦♥s✐❞❡r ❛ st❡❛❞②✲st❛t❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ✇❤❡r❡ ❛❧❧ ✉♥✐ts ❛r❡ ♠❡❛s✉r❡❞ ✐♥ ✢♦✇s✳ ❖✉r ♠❛✐♥ r❡s✉❧t✱ ❚❤❡♦r❡♠ ✶✱ ❢♦r♠❛❧❧② s❤♦✇s t❤❛t ♠❛r❦❡t ❢❛✐❧✉r❡ ❝❛♥ r❡s✉❧t ❢r♦♠ t✇♦ s♦✉r❝❡s ♦❢ ✐♥❡✣❝✐❡♥❝②✳ ❚❤❡ ✜rst s♦✉r❝❡ ✐s ❜❛s❡❞ ♦♥ ❛♥ ✐♥❡✣❝✐❡♥t ♠❡❝❤❛♥✐s♠✳ ▼♦st ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s r✉♥ ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠s t❤❛t s❡❧❡❝t ✇❤✐❝❤ ♣❛t✐❡♥ts ❛r❡ ♠❛t❝❤❡❞✳ ❲❤❡♥ ❛ ❤♦s✲ ♣✐t❛❧ s✉❜♠✐ts ❛ ♣❛t✐❡♥t t♦ t❤❡ ❡①❝❤❛♥❣❡✱ t❤❡ ❤♦s♣✐t❛❧ ✐s r❡✇❛r❞❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ✇✐t❤ ✇❤✐❝❤ t❤❡ ♣❛t✐❡♥t ✐s ♠❛t❝❤❡❞✳ ■♥ ❝♦♥tr❛st✱ ❚❤❡♦r❡♠ ✶ s❤♦✇s t❤❛t✱ t♦ ♠❛①✐♠✐③❡ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡✱ ❤♦s♣✐t❛❧s s❤♦✉❧❞ ❜❡ r❡✇❛r❞❡❞ ❜❛s❡❞ ♦♥ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ t❤❡✐r s✉❜♠✐ss✐♦♥s✳ ❇❡❝❛✉s❡ ❡①✐st✐♥❣ ♣❧❛t❢♦r♠s ❞♦ ♥♦t r❡✇❛r❞ ❤♦s♣✐t❛❧s ❢♦r t❤❡ s♦❝✐❛❧ ✈❛❧✉❡ ♦❢ t❤❡✐r s✉❜♠✐ss✐♦♥s✱ ❡✈❡♥ ❛ ❤♦s♣✐t❛❧ t❤❛t ♠❛①✐♠✐③❡s t❤❡ ♥✉♠❜❡r ♦❢ ✐ts ♦✇♥ ♣❛t✐❡♥ts t❤❛t ❛r❡ tr❛♥s♣❧❛♥t❡❞ ❤❛s t♦ ♣❡r❢♦r♠ s♦❝✐❛❧❧② ✐♥❡✣❝✐❡♥t ♠❛t❝❤❡s✳ ❍♦s♣✐t❛❧s ♠✉st t❤❡r❡❢♦r❡ ❞❡❝✐❞❡ ❜❡t✇❡❡♥ ❤❡❧♣✐♥❣ t❤❡✐r ♣❛t✐❡♥ts ♦r ♣❡r❢♦r♠✐♥❣ s♦❝✐❛❧❧② ❡✣❝✐❡♥t ♠❛t❝❤❡s✳ ❚❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ❛♠❡❧✐♦r❛t❡❞ ✉s✐♥❣ ♣♦✐♥t ♠❡❝❤❛♥✐s♠s t❤❛t r❡✇❛r❞ ❤♦s♣✐t❛❧s ❛❝❝♦r❞✐♥❣ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳✺ ❚❤❡ s❡❝♦♥❞ s♦✉r❝❡ ♦❢ ✐♥❡✣❝✐❡♥❝② ❛r❡ ❞✉❡ t♦ ❡①t❡r♥❛❧✐t✐❡s ✐❢ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ❞✐✛❡rs ❢r♦♠ s♦❝✐❛❧ ✇❡❧❢❛r❡✳ ❚❤❡s❡ ❡①t❡r♥❛❧✐t✐❡s ❛r✐s❡ ✐❢ ❤♦s♣✐t❛❧s ❞♦ ♥♦t ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡✐r ♦✇♥ ♣❛t✐❡♥ts t❤❛t ❛r❡ tr❛♥s♣❧❛♥t❡❞✳ ■t ❝r❡❛t❡s ✐♥❡✣❝✐❡♥❝②✱ ❢♦r ❡①❛♠♣❧❡✱ ✐❢ ❛ ❤♦s♣✐t❛❧ ♣❡r❢♦r♠s t♦♦ ♠❛♥② ✐♥t❡r♥❛❧ ♠❛t❝❤❡s t♦ ❛✈♦✐❞ t❤❡ ❛❞♠✐♥✐str❛t✐✈❡ ❝♦sts ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡s❡ ♠❛t❝❤❡s ❤❛♣♣❡♥ ❛t ❛♥ ✐♥❡✣❝✐❡♥t❧② s♠❛❧❧ s❝❛❧❡✳ ❲❡ ❛r❣✉❡ t❤❛t t❤❡s❡ ❛❞♠✐♥✐str❛t✐✈❡ ❝♦sts ❛r❡ s♠❛❧❧ r❡❧❛t✐✈❡ t♦ t❤❡ ✈❛❧✉❡ ♦❢ ❛ tr❛♥s♣❧❛♥t t♦ ❛ ♣❛t✐❡♥t ❛♥❞ ❤✐s ♦r ❤❡r ❤❡❛❧t❤ ✐♥s✉r❡r✳ ❇✉t✱ t❤❡s❡ ❝♦sts ❛r❡ ❛ s✐❣♥✐✜❝❛♥t ❢r❛❝t✐♦♥ ♦❢ ❤♦s♣✐t❛❧ r❡✈❡♥✉❡ ❢r♦♠ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡s❡ ❡①t❡r♥❛❧✐t✐❡s ❛r❡ ❧✐❦❡❧② ❞✉❡ t♦ ❛❣❡♥❝② ♣r♦❜❧❡♠s ❝r❡❛t❡❞ ❜② t❤❡ r❡✐♠❜✉rs❡♠❡♥t s②st❡♠ ❢♦r ❤♦s♣✐t❛❧s✳ ❚❤❡♦r❡♠ ✶ s❤♦✇s t❤❛t t❤❡r❡ ✐s ♥♦ ✐♥❡✣❝✐❡♥❝② ✐❢ ♥❡✐t❤❡r ♦❢ t❤❡s❡ t✇♦ s♦✉r❝❡s ♦❢ ♠❛r❦❡t ❢❛✐❧✉r❡ ✐s ♣r❡s❡♥t✳ ❚❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ♠❛r❦❡t ❢❛✐❧✉r❡ s♦✉r❝❡s ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❧♦♥❣✲st❛♥❞✐♥❣ ❝♦♥❝❡r♥s ♦❢ s✉r❣❡♦♥s✱ ✐♥s✉r❡rs✱ ♣❧❛t❢♦r♠s✱ ❛♥❞ r❡s❡❛r❝❤❡rs✱ ❛♥❞ ✇✐t❤ r❡❝❡♥t ♣♦❧✐❝② ❝❤❛♥❣❡s✳✻ ✹❆

♣❛t✐❡♥t ✇✐t❤ ❜❧♦♦❞ t②♣❡ ❖ ❝❛♥ ♦♥❧② r❡❝❡✐✈❡ ❛ ❦✐❞♥❡② ❢r♦♠ ❛ ❜❧♦♦❞ t②♣❡ ❖ ❞♦♥♦r✳ ❚❤✐s ❢❡❛t✉r❡ ❝r❡❛t❡s ❛♥ ❛❜✉♥❞❛♥❝❡ ♦❢ ❖ ♣❛t✐❡♥ts ✇✐t❤ ❞♦♥♦rs ♦❢ ♦t❤❡r ❜❧♦♦❞ t②♣❡s✳ ❙✐♠✐❧❛r❧②✱ ♣❛t✐❡♥ts ✇✐t❤♦✉t ❛ r❡❧❛t❡❞ ❧✐✈✐♥❣ ❞♦♥♦r ❛r❡ ❛❧s♦ ✐♥ ❛❜✉♥❞❛♥❝❡✳ ▼❛♥② ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✐♥ s✉❝❤ s✉❜♠✐ss✐♦♥s ✇✐❧❧ ✐♥❡✈✐t❛❜❧② r❡♠❛✐♥ ✉♥♠❛t❝❤❡❞✳ ❙❡❡ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ ❢♦r ❛r❣✉♠❡♥ts ❜❛s❡❞ ♦♥ ❛ ❧✐♠✐t ❡❝♦♥♦♠②✳ ✺ ❇❡❝❛✉s❡ ♦✉r s✐♠♣❧✐✜❡❞ st❡❛❞②✲st❛t❡ ♠♦❞❡❧ ✐s ♥♦t ❞②♥❛♠✐❝✱ ✐t ❞♦❡s ♥♦t ♣✐♥ ❞♦✇♥ t❤❡ ❣❛♠❡ ❢♦r♠ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❞②♥❛♠✐❝ ♠❡❝❤❛♥✐s♠✳ ❊①✐st✐♥❣ t❤❡♦r❡t✐❝❛❧ ✇♦r❦ s❤♦✇s t❤❛t ♦♣t✐♠❛❧ ❞②♥❛♠✐❝ ♠❡❝❤❛♥✐s♠s ✐♥ r❡❧❛t❡❞ s❡tt✐♥❣s ❛r❡ ❝♦♠♣❧✐❝❛t❡❞✳ ❍♦✇❡✈❡r✱ s✐♠♣❧❡ t♦❦❡♥ ♠❡❝❤❛♥✐s♠s t❤❛t ❦❡❡♣ tr❛❝❦ ♦❢ ♣♦✐♥t ❜❛❧❛♥❝❡s ❛r❡ ❜♦t❤ s✐♠♣❧❡ ❛♥❞ ❤✐❣❤❧② ❡✣❝✐❡♥t✱ ♠❛❦✐♥❣ t❤❡s❡ ♠❡❝❤❛♥✐s♠s ❣♦♦❞ ❝❛♥❞✐❞❛t❡s ❢♦r ♣r❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥✳ ❲❡ ❞✐s❝✉ss t❤❡s❡ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♣r❛❝t✐❝❛❧ ✐ss✉❡s ✐♥ ❙❡❝t✐♦♥ ✻✳ ✻ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✺✮ ❛♥❞ ❆s❤❧❛❣✐ ❛♥❞ ❘♦t❤ ✭✷✵✶✹✮ r❡❝♦❣♥✐③❡❞ t❤❛t ❤♦s♣✐t❛❧s ♠❛② ❤❛✈❡ ✐♥❝❡♥t✐✈❡s t♦ ♠❛t❝❤

✺ ❖✉r ❛♥❛❧②s✐s ♣✐♥♣♦✐♥ts t❤❡ s♦✉r❝❡s ♦❢ ♠❛r❦❡t ❢❛✐❧✉r❡ t♦ s♣❡❝✐✜❝ ✇❡❞❣❡s✱ ❛ r❡s✉❧t t❤❛t ♣♦✐♥ts t♦ ♣r❛❝t✐❝❛❧ ♣♦❧✐❝② ✐♠♣❧✐❝❛t✐♦♥s✳ ❚❤❡ t❤✐r❞ ♣❛rt ♦❢ ♦✉r ❛r❣✉♠❡♥t ✉s❡s t❤❡ ❞❛t❛ t♦ q✉❛♥t✐❢② t❤❡s❡ ✐ss✉❡s✬ ✐♠♣♦rt❛♥❝❡ ❛♥❞ t♦ ❞❡s✐❣♥ r❡s♣♦♥s❡s✳ ❲❡ ❡st✐♠❛t❡ t❤❡ ♣❧❛t❢♦r♠✬s tr❛♥s♣❧❛♥t ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ❛❞♠✐♥✐s✲ tr❛t✐✈❡ ❞❛t❛ ❢r♦♠ t❤❡ ❧❛r❣❡st ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✱ t❤❡ ◆❛t✐♦♥❛❧ ❑✐❞♥❡② ❘❡❣✐str② ✭◆❑❘✮✱ ❛♥❞ ❞❡t❛✐❧❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ ♦♣❡r❛t✐♦♥❛❧ ♣r♦❝❡❞✉r❡s ✉s❡❞ ✐♥ t❤❡ ♣❧❛t❢♦r♠✳ ❚❤✐s ✐♥❢♦r♠❛t✐♦♥ ❛❧❧♦✇s ✉s t♦ ❜✉✐❧❞ ❛♥ ❡♠♣✐r✐❝❛❧ ♠♦❞❡❧ ♦❢ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳✼ ❚❤❡ ♠♦❞❡❧ ✜ts t❤❡ ❞❛t❛ ✇❡❧❧ ❛♥❞ ❛❧❧♦✇s ✉s t♦ ❡st✐♠❛t❡ t❤❡ ✢♦✇ ♦❢ tr❛♥s♣❧❛♥ts t❤❛t ✇♦✉❧❞ ❜❡ ♣r♦❞✉❝❡❞ ❣✐✈❡♥ ❛ ✢♦✇ ♦❢ s✉❜♠✐ss✐♦♥s✳ ❚❤❡ ❡st✐♠❛t❡❞ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ②✐❡❧❞s t❤r❡❡ s❡ts ♦❢ r❡s✉❧ts✳ ❋✐rst✱ ✇❡ ♠❡❛s✉r❡ t❤❡ r❡t✉r♥s t♦ s❝❛❧❡ ♦❢ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❛♥❞ ❡st✐♠❛t❡ t❤❡ ✐♥❡✣❝✐❡♥❝② ❢r♦♠ ♠❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥✳ ❲❡ ✜♥❞ t❤❛t t❤❡ ❧❛r❣❡st ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ✐s ✇❡❧❧ ❛❜♦✈❡ t❤❡ ♠✐♥✐♠✉♠ ❡✣❝✐❡♥t s❝❛❧❡✳✽ ❆t t❤❡ s❛♠❡ t✐♠❡✱ ❛❧♠♦st ❛❧❧ s✐♥❣❧❡✲❤♦s♣✐t❛❧ ❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s ❛r❡ ❢❛r ❜❡❧♦✇ t❤❡ ❡✣❝✐❡♥t s❝❛❧❡✳ ❚❤✐s ❞✐✛❡r❡♥❝❡ ✐♥ ♣r♦❞✉❝t✐✈✐t② s✉❣❣❡sts ❝♦♥s✐❞❡r❛❜❧❡ ✐♥❡✣❝✐❡♥❝② ❞✉❡ t♦ ❢r❛❣♠❡♥t❛t✐♦♥✳ ❲❡ ❡st✐♠❛t❡ t❤❛t t❤❡ ❣❛✐♥s ❢r♦♠ ♠♦✈✐♥❣ ❛❧❧ t❤❡ ♣r♦❞✉❝t✐♦♥ t♦ t❤❡ ❡✣❝✐❡♥t s❝❛❧❡ ✐s ❛t ❧❡❛st ✷✵✵✱ ❛♥❞ ❧✐❦❡❧② ❝❧♦s❡r t♦ ✹✵✵ tr❛♥s♣❧❛♥ts✳ ❚❤✉s✱ ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ❞❡s❝r✐♣t✐✈❡ ❡✈✐❞❡♥❝❡ ❛♥❞ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✱ ❢r❛❣♠❡♥t❛t✐♦♥ ❤❛s ❛ ❧❛r❣❡ ❡✣❝✐❡♥❝② ❝♦st✳ ❙❡❝♦♥❞✱ ✇❡ ❛ss❡ss t❤❡ ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠ ❛♥❞ ❝❤❛r❛❝t❡r✐③❡ r❡✇❛r❞s ✐♥ ❛♥ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠✳✾ ❚❤❡♦r❡♠ ✶ s❤♦✇s t❤❛t ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠s r❡✇❛r❞ s✉❜♠✐ss✐♦♥s ❛♣♣r♦①✐♠❛t❡❧② ❛❝❝♦r❞✐♥❣ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✱ ✇❤✐❧❡ ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠s r❡✇❛r❞ s✉❜♠✐ss✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛t❝❤✐♥❣✳ ▼♦t✐✈❛t❡❞ ❜② t❤✐s r❡s✉❧t✱ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛♥❞ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛t❝❤✐♥❣ ❢♦r ❡❛❝❤ t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥ ✉s✐♥❣ t❤❡ ❡st✐♠❛t❡❞ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳ ❲❡ ✜♥❞ t❤❛t ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛r❡ ❝♦♥s✐❞❡r❛❜❧② ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛t❝❤✐♥❣✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t ❡①✐st✐♥❣ ♠❡❝❤❛♥✐s♠s ❛r❡ ❢❛r ❢r♦♠ ♦♣t✐♠❛❧✳ ❲❡ t❤❡♥ ❞❡✈❡❧♦♣ ♣♦✐♥t ♠❡❝❤❛♥✐s♠s t❤❛t ❣✐✈❡ ❛♣♣r♦①✐♠❛t❡❧② ♦♣t✐♠❛❧ ✐♥❝❡♥t✐✈❡s t♦ ❤♦s♣✐t❛❧s✳ ❚❤✐r❞✱ ✇❡ st✉❞② t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ t✇♦ s♦✉r❝❡s ♦❢ ♠❛r❦❡t ❢❛✐❧✉r❡✳ ■♥ ♦✉r ♣r✐❝❡✲t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤✱ t❤❡ ❧♦ss ✐♥ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ❞✉❡ t♦ t❤❡ ✐♥❡✣❝✐❡♥t ♠❡❝❤❛♥✐s♠ ❞❡♣❡♥❞s ♦♥ t❤❡ ✇❡❞❣❡ ❜❡t✇❡❡♥ ❝✉rr❡♥t ❛♥❞ ♦♣t✐♠❛❧ r❡✇❛r❞s✱ ❛♥❞ ♦♥ t❤❡ ❡❧❛st✐❝✐t② ♦❢ s✉♣♣❧② ❢r♦♠ ❤♦s♣✐t❛❧s✳ ❲❡ ❤❛✈❡ ❡st✐♠❛t❡❞ t❤❡ ✇❡❞❣❡s ❛♥❞ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✱ ❜✉t ♦✉r ❞❛t❛ ❞♦ ♥♦t ❤❛✈❡ ❡♥♦✉❣❤ ✐♥❢♦r♠❛t✐♦♥ t♦ ❝r❡❞✐❜❧② ❡st✐♠❛t❡ s✉♣♣❧② ❡❧❛st✐❝✐t✐❡s✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛❧❝✉❧❛t❡ t❤✐s ❞❡❛❞✇❡✐❣❤t ❧♦ss ✉♥❞❡r ❛ ❜r♦❛❞ r❛♥❣❡ ♦❢ ❛ss✉♠♣t✐♦♥s ♦♥ ❡❧❛st✐❝✐t✐❡s✳ ❊①❝❡♣t ✉♥❞❡r ❡①tr❡♠❡ ❛ss✉♠♣t✐♦♥s✱ ♣❛t✐❡♥ts ✐♥t❡r♥❛❧❧② ✐♥ st❛t✐❝ ♠♦❞❡❧s✳ ❙✉r❣❡♦♥s ❛♥❞ ✐♥s✉r❡rs ❤❛✈❡ ♥♦t❡❞ t❤❛t ✐t ♠❛② ❜❡ ✐♥ t❤❡ ✐♥t❡r❡st ♦❢ ✐♥s✉r❡rs t♦ s✉❜s✐❞✐③❡ ❡①❝❤❛♥❣❡s✱ ❛♥❞ ❤❛✈❡ ♣r♦♣♦s❡❞ t❤❛t t❤❡② ❞♦ s♦ ✭❘❡❡s ❡t ❛❧✳✱ ✷✵✶✷✮✳ ✼ ❙t❛♥❞❛r❞ ♠❡t❤♦❞s ♦❢ ❡st✐♠❛t✐♥❣ ❛ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❛r❡ ♥♦t s✉✐t❡❞ t♦ ♦✉r ❡♠♣✐r✐❝❛❧ s❡tt✐♥❣ ❜❡❝❛✉s❡ t❤❡② r❡❧② ♦♥ ❧♦✇✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥s✱ t②♣✐❝❛❧❧② ❈♦❜❜✲❉♦✉❣❧❛s ✇✐t❤ ❛ ❢❡✇ t②♣❡s ♦❢ ✐♥♣✉ts✱ ❛♥❞ ❞❛t❛ ♦♥ ♦❜s❡r✈❡❞ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❞❛t❛ ❢♦r ♠❛♥② ✜r♠s ✭▼❛rs❝❤❛❦ ❛♥❞ ❆♥❞r❡✇s✱ ✶✾✹✹❀ ❖❧❧❡② ❛♥❞ P❛❦❡s✱ ✶✾✾✻✮✳ ✽ ❇❡❝❛✉s❡ r❡t✉r♥s t♦ s❝❛❧❡ ❛r❡ r♦✉❣❤❧② ❝♦♥st❛♥t ❛t ❛ ❧❛r❣❡ ❡♥♦✉❣❤ s❝❛❧❡✱ ❣❛✐♥s ❞✉❡ t♦ s❝❛❧❡ ❡❝♦♥♦♠✐❡s ❛❧♦♥❡ ❢r♦♠ ♠❡r❣✐♥❣ t❤❡ ♠❛❥♦r ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s ❛r❡ ❧✐❦❡❧② s♠❛❧❧✳ ❚❤✐s ♦❜s❡r✈❛t✐♦♥ ♦♥ t❤❡ s❝❛❧❡ ❡❝♦♥♦♠✐❡s ❝❛♥ ❛❧s♦ ❡①♣❧❛✐♥ ✇❤②✱ ✐♥st❡❛❞ ♦❢ t✐♣♣✐♥❣ t♦ ❛ s✐♥❣❧❡ ♣❧❛t❢♦r♠✱ ♠✉❧t✐♣❧❡ ❧❛r❣❡ ♥❛t✐♦♥❛❧ ❡①❝❤❛♥❣❡s ❝♦✲❡①✐st ✭❊❧❧✐s♦♥ ❛♥❞ ❋✉❞❡♥❜❡r❣✱ ✷✵✵✸✮✳ ✾ ❚❤✐s ✐s ✐♥ t❤❡ s♣✐r✐t ♦❢ t❤❡ s✉✣❝✐❡♥t st❛t✐st✐❝s ❛♣♣r♦❛❝❤ ✐♥ t❤❡ ♣✉❜❧✐❝ ✜♥❛♥❝❡ ❧✐t❡r❛t✉r❡ ✭❉✐①✐t ❛♥❞ ❙❛♥❞♠♦✱ ✶✾✼✼❀ ❙❛❡③✱ ✷✵✵✶❀ ❈❤❡tt②✱ ✷✵✵✾✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ♦✉r ❛♥❛❧②s✐s s✉✛❡rs ❢r♦♠ t❤❡ st❛♥❞❛r❞ ❝❛✈❡❛t t❤❛t ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❞❡♣❡♥❞ ♦♥ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ s✉❜♠✐ss✐♦♥s✱ ✇❤✐❝❤ ✐s ❡♥❞♦❣❡♥♦✉s✳

✻ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s s✐❣♥✐✜❝❛♥t ❜✉t ❝♦♥s✐❞❡r❛❜❧② ❧♦✇❡r t❤❛♥ t❤❡ ✐♥❡✣❝✐❡♥❝② ❞✉❡ t♦ ♠❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ ❡①❝❡♣t ✐♥ ❡①tr❡♠❡ ❝❛s❡s✱ ❜♦t❤ t❤❡ ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠ ❛♥❞ ❛❣❡♥❝② ♣r♦❜❧❡♠s ❝❛✉s❡ s✐❣♥✐✜❝❛♥t ✐♥❡✣❝✐❡♥❝② ✐♥ t❤❡ ♠❛r❦❡t✳ ❚❛❦❡♥ t♦❣❡t❤❡r✱ t❤❡s❡ r❡s✉❧ts ♠♦t✐✈❛t❡ ❛ ❝♦♠❜✐♥❡❞ ❛♣♣r♦❛❝❤ t❤❛t ✐♠♣r♦✈❡s t❤❡ ♠❡❝❤❛♥✐s♠ ❞❡s✐❣♥ ❛♥❞ ✐♠♣❧❡♠❡♥ts ♣♦❧✐❝✐❡s t❤❛t ❡♥❝♦✉r❛❣❡ ❤♦s♣✐t❛❧ ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ t❤❡ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s✳ ❆❢t❡r ♣r❡s❡♥t✐♥❣ ♦✉r ♠❛✐♥ ❛r❣✉♠❡♥t✱ ✇❡ ❞✐s❝✉ss ✐♠♣♦rt❛♥t ❧✐♠✐t❛t✐♦♥s ♦❢ ♦✉r ❛♥❛❧②s✐s ❛♥❞ s♦♠❡ ❡①t❡♥s✐♦♥s✳ ❋✐rst✱ ✇❡ ❞✐s❝✉ss ✉♥s♦❧✈❡❞ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♣r❛❝t✐❝❛❧ ✐ss✉❡s ✐♥✈♦❧✈❡❞ ✐♥ ✉s✐♥❣ ♦✉r r❡s✉❧ts ✐♥ t❤❡ r❡❛❧ ✇♦r❧❞✳ ❙❡❝♦♥❞✱ ✇❡ ❝♦♥s✐❞❡r ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❡❝❤❛♥✐s♠s t❤❛t ❛tt❡♠♣t t♦ s♦❧✈❡ ❜♦t❤ ♠❛r❦❡t ❢❛✐❧✉r❡s s✐♠✉❧t❛♥❡♦✉s❧②✳ ❚❤✐r❞✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❡✛❡❝t ♦❢ ✐♠♣❡r❢❡❝t ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ ♦❧✐❣♦♣♦❧✐st✐❝ ♣❧❛t❢♦r♠s ♦♥ ❡✣❝✐❡♥❝②✳ ❋♦✉rt❤✱ ✇❡ ❞✐s❝✉ss r❡❝❡♥t ❝❤❛♥❣❡s ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡ts t❤❛t ❛tt❡♠♣t t♦ ❛❞❞r❡ss t❤❡ t✇♦ ♠❛r❦❡t ❢❛✐❧✉r❡s t❤❛t ✇❡ ❞❡s❝r✐❜❡✳ ❖✉r ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❙❡❝t✐♦♥ ✷ ❞❡s❝r✐❜❡s ✐♥st✐t✉t✐♦♥❛❧ ❞❡t❛✐❧s ❛♥❞ ❞❛t❛ s♦✉r❝❡s✳ ❚❤❡ t❤r❡❡ ♣❛rts ♦❢ ♦✉r ❛r❣✉♠❡♥t ❛r❡ ❞❡✈❡❧♦♣❡❞ ✐♥ ❙❡❝t✐♦♥s ✸✱ ✹✱ ❛♥❞ ✺✱ r❡s♣❡❝t✐✈❡❧②✳ ❙❡❝t✐♦♥ ✻ ❞✐s❝✉ss❡s t❤❡ ❧✐♠✐t❛t✐♦♥s ❛♥❞ ❡①t❡♥s✐♦♥s✱ ❛♥❞ ❙❡❝t✐♦♥ ✼ ❝♦♥❝❧✉❞❡s✳

✷ ✷✳✶

❇❛❝❦❣r♦✉♥❞ ❛♥❞ ❉❛t❛ ❇❛s✐❝s ♦❢ ❑✐❞♥❡② ❊①❝❤❛♥❣❡

❚❤✐s s✉❜s❡❝t✐♦♥ ❞❡s❝r✐❜❡s t❤❡ ❜❛s✐❝s ♦❢ ❦✐❞♥❡② tr❛♥s♣❧❛♥t❛t✐♦♥ ❛♥❞ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❛♥❞ ❝❛♥ ❜❡ s❦✐♣♣❡❞ ❜② r❡❛❞❡rs ✇❤♦ ❛r❡ ❢❛♠✐❧✐❛r ✇✐t❤ t❤✐s ❧✐t❡r❛t✉r❡✳ ❊♥❞✲❙t❛❣❡ ❘❡♥❛❧ ❉✐s❡❛s❡ ✭❊❙❘❉✮ ❛✤✐❝ts ♠♦r❡ t❤❛♥ ❤❛❧❢ ❛ ♠✐❧❧✐♦♥ ❆♠❡r✐❝❛♥s✳ ❚❤❡ ❞✐s❡❛s❡ ✐s ❛❧♠♦st ✉♥✐✈❡rs❛❧❧② ❝♦✈❡r❡❞ ❜② ▼❡❞✐❝❛r❡✱ ✐♥❝❧✉❞✐♥❣ ❢♦r ♣❛t✐❡♥ts ✉♥❞❡r t❤❡ ❛❣❡ ♦❢ ✻✺✳ ❚❤❡ ▼❡❞✐❝❛r❡ ❊❙❘❉ ♣r♦❣r❛♠ ❛❝❝♦✉♥ts ❢♦r ✼✪ ♦❢ ✐ts ❜✉❞❣❡t✱ ♠♦st❧② s♣❡♥t ♦♥ ♣❛t✐❡♥ts ✉♥❞❡r❣♦✐♥❣ ❞✐❛❧②s✐s ✭❯❙❘❉❙✱ ❯♥✐t❡❞ ❙t❛t❡s ❘❡♥❛❧ ❉❛t❛ ❙②st❡♠✱ ✷✵✶✻✮✳ ❚❤❡ ♣r❡❢❡rr❡❞ tr❡❛t♠❡♥t ❢♦r ❊❙❘❉ ♣❛t✐❡♥ts ✐s tr❛♥s♣❧❛♥t❛t✐♦♥✱ ✇❤✐❝❤ ✐♥❝r❡❛s❡s t❤❡ q✉❛❧✐t② ❛♥❞ ❧❡♥❣t❤ ♦❢ ❧✐❢❡ ❜② s❡✈❡r❛❧ ②❡❛rs ❛♥❞ ✐s ❝❤❡❛♣❡r t❤❛♥ ❞✐❛❧②s✐s✳ ❚r❛♥s♣❧❛♥t❛t✐♦♥ s❛✈❡s s❡✈❡r❛❧ ❤✉♥❞r❡❞ t❤♦✉s❛♥❞ ❞♦❧❧❛rs ♣❡r ▼❡❞✐❝❛r❡ ❜❡♥❡✜❝✐❛r② ❛♥❞ ❡✈❡♥ ♠♦r❡ ❢♦r ♣r✐✈❛t❡❧② ✐♥s✉r❡❞ ♣❛t✐❡♥ts ✭❲♦❧❢❡ ❡t ❛❧✳✱ ✶✾✾✾❀ ■r✇✐♥ ❡t ❛❧✳✱ ✷✵✶✷❀ ❍❡❧❞ ❡t ❛❧✳✱ ✷✵✶✻✮✳ ▼♦r❡♦✈❡r✱ t❤❡ ❤❡❛❧t❤ r✐s❦s t♦ s✉✐t❛❜❧② s❝r❡❡♥❡❞ ❧✐✈✐♥❣ ❞♦♥♦rs ❛r❡ s♠❛❧❧✳ ❚❛❦❡♥ t♦❣❡t❤❡r✱ t❤❡s❡ ❢❛❝ts ✐♥❞✐❝❛t❡ t❤❛t ❧✐✈✐♥❣ ❞♦♥♦r ❦✐❞♥❡② tr❛♥s♣❧❛♥ts ❤❛✈❡ ❧❛r❣❡ ❡❝♦♥♦♠✐❝ ✈❛❧✉❡✳ ❍❡❧❞ ❡t ❛❧✳ ✭✷✵✶✻✮ ♣❧❛❝❡s t❤❡ ❡❝♦♥♦♠✐❝ ✈❛❧✉❡ ♦❢ ❛♥ ❛✈❡r❛❣❡ ❦✐❞♥❡② tr❛♥s♣❧❛♥t ❛t ✩✶✳✶ ♠✐❧❧✐♦♥✳✶✵ ❚❤❡r❡ ✐s ❛ s❡✈❡r❡ s❤♦rt❛❣❡ ♦❢ ♦r❣❛♥s ❢♦r tr❛♥s♣❧❛♥t❛t✐♦♥✳ ❊❛❝❤ ②❡❛r✱ ❛♣♣r♦①✐♠❛t❡❧② ✶✸✱✵✵✵ ♣❛t✐❡♥ts ❛r❡ tr❛♥s♣❧❛♥t❡❞ ✉s✐♥❣ ♦r❣❛♥s ❢r♦♠ ❞❡❝❡❛s❡❞ ❞♦♥♦rs ❛♥❞ ❛♥♦t❤❡r ✺✱✺✵✵ ❢r♦♠ ❧✐✈✐♥❣ ✶✵ ❍❡❧❞ ❡t ❛❧✳ ✭✷✵✶✻✮ ❝♦♥❞✉❝t ❛ ❝♦♠♣r❡❤❡♥s✐✈❡ ❝♦st✲❜❡♥❡✜t ❛♥❛❧②s✐s ✇✐t❤ s❡♥s✐t✐✈✐t② t♦ ❛ r❛♥❣❡ ♦❢ ❛ss✉♠♣t✐♦♥s✳ ❚❤❡② ✈❛❧✉❡ ♦♥❡ ②❡❛r ♦❢ ♣❡r❢❡❝t ❤❡❛❧t❤ ❛t ✩✷✵✵✱✵✵✵ ❛♥❞ ❛❞❥✉st ❢♦r ❞✐✛❡r❡♥❝❡s ✐♥ q✉❛❧✐t② ♦❢ ❧✐❢❡ ❜❡t✇❡❡♥ ❞✐❛❧②s✐s ❛♥❞ ❛ tr❛♥s♣❧❛♥t✳ ❚❤❡ ❝♦sts ✐♥❝❧✉❞❡ ❞✐✛❡r❡♥❝❡s ✐♥ ❡①♣❡❝t❡❞ ♠❡❞✐❝❛❧ ❝♦sts ✐♥❝✉rr❡❞ ♦✈❡r t❤❡ ❧✐❢❡t✐♠❡ ♦❢ ❛ ❞✐❛❧②③❡❞ ♣❛t✐❡♥t ❛♥❞ ❛ tr❛♥s♣❧❛♥t❡❞ ♣❛t✐❡♥t✱ ❞✐s❝♦✉♥t❡❞ ❛t t❤❡ r❛t❡ ♦❢ ✸✪ ♣❡r ②❡❛r✳ ❚❤❡ ❝♦st s❛✈✐♥❣s ♦♥ ❞✐❛❧②s✐s ❛❧♦♥❡ ❛r❡ s✐❣♥✐✜❝❛♥t✳ ■♥ ✷✵✶✹✱ ▼❡❞✐❝❛r❡ ♣❛✐❞ ✩✽✼✱✻✸✽ ♣❡r ②❡❛r ♣❡r ❞✐❛❧②s✐s ♣❛t✐❡♥t ❜✉t ♦♥❧② ✩✸✷✱✺✽✻ ✐♥ ♣♦st✲tr❛♥s♣❧❛♥t ❝♦sts ♣❡r ②❡❛r ♣❡r ♣❛t✐❡♥t ✭❯❙❘❉❙✱ ❯♥✐t❡❞ ❙t❛t❡s ❘❡♥❛❧ ❉❛t❛ ❙②st❡♠✱ ✷✵✶✻✱ ❈❤❛♣t❡rs ✼ ❛♥❞ ✶✶✮✳

✼ ❞♦♥♦rs✳ ❉❡♠❛♥❞ ❢❛r ♦✉tstr✐♣s t❤✐s s✉♣♣❧② ✇✐t❤ ❛♣♣r♦①✐♠❛t❡❧② ✸✺✱✵✵✵ ♣❛t✐❡♥ts ❛❞❞❡❞ t♦ t❤❡ ❞❡❝❡❛s❡❞ ❞♦♥♦r ❦✐❞♥❡② ✇❛✐t❧✐st ✐♥ ❡❛❝❤ ♦❢ t❤❡ r❡❝❡♥t ❢❡✇ ②❡❛rs✳ ❚❤❡ s❤♦rt❛❣❡ ❤❛s r❡s✉❧t❡❞ ✐♥ t❤❡ ❦✐❞♥❡② ✇❛✐t❧✐st ❣r♦✇✐♥❣ t♦ ❛❧♠♦st ✶✵✵✱✵✵✵ ♣❛t✐❡♥ts ❛♥❞ ❛❜♦✉t ✽✱✵✵✵ ♣❛t✐❡♥ts ♦♥ t❤❡ ❧✐st ❞②✐♥❣ ♦r ❜❡✐♥❣ ❝❛t❡❣♦r✐③❡❞ ❛s t♦♦ s✐❝❦ t♦ tr❛♥s♣❧❛♥t ✇❤✐❧❡ ✇❛✐t✐♥❣ ♦♥ t❤❡ ❧✐st✳✶✶ ▼♦♥❡t❛r② ❝♦♠♣❡♥s❛t✐♦♥ ❝❛♥♥♦t ❜❡ ✉s❡❞ t♦ ❛❞❞r❡ss t❤✐s s❤♦rt❛❣❡ ❜❡❝❛✉s❡ ♦❢ ❡t❤✐❝❛❧ ❛♥❞ ❧❡❣❛❧ r❡❛s♦♥s✱ ❛♥❞ ❝♦♠♣❡♥s❛t✐♦♥ ✐s ❢♦r❜✐❞❞❡♥ ✐♥ ❛❧♠♦st ❡✈❡r② ❝♦✉♥tr② ✭❇❡❝❦❡r ❛♥❞ ❊❧✐❛s✱ ✷✵✵✼✮✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❯❙✳ ❑✐❞♥❡② ❡①❝❤❛♥❣❡ ✐s ❛♥ ✐♥♥♦✈❛t✐✈❡ ✇❛② t♦ ❛♠❡❧✐♦r❛t❡ t❤✐s s❤♦rt❛❣❡✳ ❚❤✐s ❢♦r♠ ♦❢ tr❛♥s♣❧❛♥✲ t❛t✐♦♥ s❡r✈❡s ♣❛t✐❡♥ts ✇❤♦ ❤❛✈❡ ❛ ✇✐❧❧✐♥❣ ❧✐✈❡ ❞♦♥♦r ✇✐t❤ ✇❤♦♠ t❤❡② ❛r❡ ♥♦t ❜✐♦❧♦❣✐❝❛❧❧② ❝♦♠♣❛t✐❜❧❡✳ ❙✉❝❤ ♣❛t✐❡♥ts ❝❛♥ s✇❛♣ ❞♦♥♦rs ✇✐t❤ ♦t❤❡rs ✐♥ t❤❡ s❛♠❡ s✐t✉❛t✐♦♥✱ ❡♥❛❜❧✐♥❣ tr❛♥s♣❧❛♥ts ❢♦r ♠❛♥② ♣❛t✐❡♥ts✳ ❚❤❡s❡ s✇❛♣s ❛r❡ ♦r❣❛♥✐③❡❞ ❜② ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s t❤❛t ♠❛t❝❤ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs r❡❣✐st❡r❡❞ ✇✐t❤ t❤❡♠✳ ❚❤❡ ♣❧❛t❢♦r♠s r❡❝❡✐✈❡ t❤r❡❡ t②♣❡s ♦❢ s✉❜♠✐ss✐♦♥s✳ ❚❤❡ ♠♦st ❝♦♠♠♦♥ t②♣❡ ✐s ❛ ♣❛✐r✱ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ♣❛t✐❡♥t ❛♥❞ ❛ ❧✐✈✐♥❣ ❜✉t ✐♥❝♦♠♣❛t✐❜❧❡ ❞♦♥♦r✳ ❚❤❡ s❡❝♦♥❞ t②♣❡ ✐s ❛♥ ❛❧tr✉✐st✐❝ ❞♦♥♦r✱ ✇❤♦ ✐s ✇✐❧❧✐♥❣ t♦ ❞♦♥❛t❡ ❛ ❦✐❞♥❡② t♦ ❛ str❛♥❣❡r✳ ❋✐♥❛❧❧②✱ t❤❡r❡ ❛r❡ s♦♠❡ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥ts✱ ✇❤♦ ❞♦ ♥♦t ❤❛✈❡ ❛ ✇✐❧❧✐♥❣ ❧✐✈❡ ❞♦♥♦r✳ ❚❤❡ ♣❧❛t❢♦r♠ ♦r❣❛♥✐③❡s tr❛♥s♣❧❛♥ts ✐♥ t✇♦ ✇❛②s✳ ❚❤❡ ✜rst✱ ❝❛❧❧❡❞ ❛ ❝②❝❧❡✱ ✐♥✈♦❧✈❡s ❛ s❡t ♦❢ ♣❛✐rs✳ ❚❤❡ ❦✐❞♥❡② ❢r♦♠ ♦♥❡ ♦❢ t❤❡ ♣❛✐r✬s ❞♦♥♦r ✐s tr❛♥s♣❧❛♥t❡❞ ✐♥t♦ t❤❡ ♣❛t✐❡♥t ✐♥ t❤❡ ♥❡①t ♣❛✐r ✉♥t✐❧ t❤❡ ❝②❝❧❡ ✐s ❝❧♦s❡❞✳ ❆❧❧ tr❛♥s♣❧❛♥ts ❛r❡ ❝❛rr✐❡❞ ♦✉t s✐♠✉❧t❛♥❡♦✉s❧② t♦ r❡❞✉❝❡ t❤❡ r✐s❦ t❤❛t ❛ ♣❛✐r ❞♦♥❛t❡s ❛ ❦✐❞♥❡② ✇✐t❤♦✉t ❛❧s♦ r❡❝❡✐✈✐♥❣ ♦♥❡✳ ❈②❝❧❡s ❛r❡ ✉s✉❛❧❧② ❧✐♠✐t❡❞ t♦ ❛t ♠♦st t❤r❡❡ ♣❛✐rs ❞✉❡ t♦ ❧♦❣✐st✐❝❛❧ ❝♦♥str❛✐♥ts✳ ❚❤❡ s❡❝♦♥❞ t②♣❡✱ ❝❛❧❧❡❞ ❛ ❝❤❛✐♥✱ ✐s ✐♥✐t✐❛t❡❞ ✇❤❡♥ ❛♥ ❛❧tr✉✐st✐❝ ❞♦♥♦r ❞♦♥❛t❡s t♦ ❛ ♣❛t✐❡♥t ✐♥ ❛♥ ✐♥❝♦♠♣❛t✐❜❧❡ ♣❛✐r✳ ❚❤❡ ❞♦♥♦r ❢r♦♠ t❤✐s ♣❛✐r ❝❛♥ t❤❡♥ ❝♦♥t✐♥✉❡ t❤❡ ❝❤❛✐♥ ❜② ❞♦♥❛t✐♥❣ t♦ t❤❡ ♥❡①t ♣❛✐r ❛♥❞ s♦ ♦♥ ✉♥t✐❧ t❤❡ ❝❤❛✐♥ t❡r♠✐♥❛t❡s ✇✐t❤ ❛♥ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥t✳ ❈❤❛✐♥s ❝❛♥ ❜❡ ✈❡r② ❧♦♥❣ ✐♥ ♣r✐♥❝✐♣❧❡ ❜❡❝❛✉s❡ tr❛♥s♣❧❛♥ts ❞♦ ♥♦t ❤❛✈❡ t♦ ❜❡ ♣❡r❢♦r♠❡❞ s✐♠✉❧t❛♥❡♦✉s❧②✳ ❍♦✇❡✈❡r✱ ♦✉r ❞❛t❛ ❢r♦♠ t❤❡ ◆❛t✐♦♥❛❧ ❑✐❞♥❡② ❘❡❣✐str② ✭◆❑❘✮ ✐♥❞✐❝❛t❡ t❤❛t ♠♦st ❝❤❛✐♥s ✐♥✈♦❧✈❡ ❢♦✉r t♦ ✜✈❡ tr❛♥s♣❧❛♥ts✳ ■♥✐t✐❛❧❧②✱ ❝②❝❧❡s ✇❡r❡ t❤❡ ♠♦st ❝♦♠♠♦♥ t②♣❡ ♦❢ tr❛♥s❛❝t✐♦♥✱ ❜✉t ❝❤❛✐♥s ❜❡❝❛♠❡ ♠♦r❡ ✐♠♣♦rt❛♥t ♦✈❡r t✐♠❡ ❛♥❞ ♥♦✇ ❛❝❝♦✉♥t ❢♦r ❛❜♦✉t ✾✵✪ ♦❢ t❤❡ tr❛♥s♣❧❛♥ts ✐♥ ♦✉r ❞❛t❛✳ ❚❤❡r❡ ❛r❡ t✇♦ t②♣❡s ♦❢ ❜✐♦❧♦❣✐❝❛❧ ❝♦♠♣❛t✐❜✐❧✐t② ❝♦♥str❛✐♥ts ♦♥ ❦✐❞♥❡② tr❛♥s♣❧❛♥ts✿ ❆❇❖ ❜❧♦♦❞✲t②♣❡ ❛♥❞ t✐ss✉❡✲t②♣❡ ❝♦♠♣❛t✐❜✐❧✐t② ✭❉❛♥♦✈✐t❝❤✱ ✷✵✵✾✮✳ ❆ ❞♦♥♦r ✐s ❜❧♦♦❞✲t②♣❡ ✐♥✲ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❛ ♣❛t✐❡♥t ✐❢ t❤❡ ❞♦♥♦r ❤❛s ❛ ❜❧♦♦❞ ❛♥t✐❣❡♥ t❤❛t t❤❡ ♣❛t✐❡♥t ❧❛❝❦s✳ ❚❤❡r❡ ❛r❡ t✇♦ ❜❧♦♦❞ ❛♥t✐❣❡♥s✱ ❦♥♦✇♥ ❛s ❆ ❛♥❞ ❇✳ ❇❧♦♦❞ t②♣❡ ✐s ❡✐t❤❡r ❆ ♦r ❇ ✐❢ ✐t ❤❛s ♦♥❡ ♦❢ t❤❡s❡ ❛♥t✐❣❡♥s✱ ❆❇ ✇✐t❤ ❜♦t❤✱ ❛♥❞ ❖ ✇✐t❤ ♥❡✐t❤❡r✳ ❆ ❞♦♥♦r ✐s t✐ss✉❡✲t②♣❡ ✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❛ ♣❛t✐❡♥t ✐❢ t❤❡ ❞♦♥♦r ❤❛s ❝❡rt❛✐♥ ❝❡❧❧✲s✉r❢❛❝❡ ♣r♦t❡✐♥s ✭❛♥t✐❣❡♥s✮ t♦ ✇❤✐❝❤ t❤❡ ♣❛t✐❡♥t ❤❛s ❛♥ ✐♠♠✉♥❡ r❡s♣♦♥s❡✳✶✷ P❛t✐❡♥t s❡♥s✐t✐③❛t✐♦♥ ❧❡✈❡❧s ❞✐✛❡r ❝♦♥s✐❞❡r❛❜❧②✳ ❚❤❡ ♠♦st ❝♦♠♠♦♥ ♠❡❛s✉r❡ ♦❢ s❡♥s✐t✐③❛t✐♦♥ ✐s t❤❡ P❛♥❡❧ ❘❡❛❝t✐✈❡ ❆♥t✐❜♦❞② ✭P❘❆✮ s❝♦r❡✳ ❆ ♣❛t✐❡♥t✬s P❘❆ ✶✶ ❙t❛t✐st✐❝s

♥❛t✐♦♥❛❧✲❞❛t❛✴ ✶✷ ❚❤❡

t❛❦❡♥

❢r♦♠

❤tt♣s✿✴✴♦♣t♥✳tr❛♥s♣❧❛♥t✳❤rs❛✳❣♦✈✴❞❛t❛✴✈✐❡✇✲❞❛t❛✲r❡♣♦rts✴

✭❛❝❝❡ss❡❞ ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✼✮✳

✐♠♠✉♥❡ s②st❡♠ r❡❝♦❣♥✐③❡s ❢♦r❡✐❣♥ ❝❡❧❧s ❜❛s❡❞ ♦♥ ❝❡rt❛✐♥ ❝❡❧❧✲s✉r❢❛❝❡ ♣r♦t❡✐♥s✱ ❦♥♦✇♥ ❛s ❛♥t✐❣❡♥s✳

❚❤❡ ♦r❣❛♥✐s♠ ❤❛s ❛♥t✐❜♦❞✐❡s t❤❛t ❜✐♥❞ t♦ t❤❡s❡ ❛♥t✐❣❡♥s✱ t❛❣❣✐♥❣ ❢♦r❡✐❣♥ ❝❡❧❧s ✇❤✐❝❤ ❛r❡ t❤❡♥ ❛tt❛❝❦❡❞✳ ❍❡♥❝❡✱ ✐❢ ✇❡ ♣✉t ❛ ❝❡❧❧ ✇✐t❤ ❛♥ ❛♥t✐❣❡♥ ✐♥ t❤❡ ❜♦❞② ♦❢ ❛ ♣❡rs♦♥ ✇❤♦ ❤❛s ❛♥t✐❜♦❞✐❡s ❢♦r t❤❛t ❛♥t✐❣❡♥✱ t❤❡ ✐♠♠✉♥❡ s②st❡♠ ✇✐❧❧ ❛tt❛❝❦ ✐t✳ ♦✉t ♦❢ ❛ ❧✐st ♦❢ ❤✉♥❞r❡❞s✳ t❤❡ ❍▲❆ ❛♥t✐❣❡♥s✳

❊❛❝❤ ❞♦♥♦r ❤❛s ✉♣ t♦ ✻ ♣♦ss✐❜❧❡ ❤✉♠❛♥ ❧❡✉❦♦❝②t❡ ❛♥t✐❣❡♥ ✭❍▲❆✮ ♣r♦t❡✐♥s

❙✐♠✐❧❛r❧②✱ ❛ r❡❝✐♣✐❡♥t ❤❛s ❛ ❧✐st ♦❢ ❛♥t✐❜♦❞✐❡s t♦ s♦♠❡✱ ♣♦ss✐❜❧② ❧❛r❣❡✱ s✉❜s❡t ♦❢

■❢ t❤❡ r❡❝✐♣✐❡♥t ❤❛s ❛♥ ❛♥t✐❜♦❞② t♦ ♦♥❡ ♦❢ t❤❡ ❞♦♥♦r ❦✐❞♥❡②✬s ❛♥t✐❣❡♥s✱ t❤❡ r❡❝✐♣✐❡♥t✬s

✽ ✐s ❜❡t✇❡❡♥ ✵ ❛♥❞ ✶✵✵ ❛♥❞ ❞❡♥♦t❡s t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ ❛ r❡♣r❡s❡♥t❛t✐✈❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❞♦♥♦rs ✇✐t❤ ✇❤♦♠ ❛ ♣❛t✐❡♥t ✐s t✐ss✉❡✲t②♣❡ ✐♥❝♦♠♣❛t✐❜❧❡✳ ✷✳✷

❑❡② ■♥st✐t✉t✐♦♥❛❧ ❋❡❛t✉r❡s ❛♥❞ t❤❡ ❊❝♦♥♦♠✐❝s ♦❢ ❑✐❞♥❡② ❊①✲ ❝❤❛♥❣❡

❚❤❡r❡ ❛r❡ t❤r❡❡ ❦❡② ✐♥st✐t✉t✐♦♥❛❧ ❢❡❛t✉r❡s t❤❛t ❞r✐✈❡ ♦✉t❝♦♠❡s ❛♥❞ ✇✐❧❧ ❢❛❝t♦r ✐♥ ♦✉r ❛♥❛❧②s✐s✳ ❋✐rst✱ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ t❛❦❡s ♣❧❛❝❡ ❜♦t❤ ✐♥ ❧❛r❣❡✱ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ❛♥❞ ✇✐t❤✐♥ ✐♥❞✐✈✐❞✉❛❧ ❤♦s♣✐t❛❧s✳ ❚❤❡r❡ ❛r❡ t❤r❡❡ ♠❛❥♦r ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ❝✉rr❡♥t❧② ♦♣❡r❛t✐♥❣ ✐♥ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s✿ t❤❡ ◆❛t✐♦♥❛❧ ❑✐❞♥❡② ❘❡❣✐str② ✭◆❑❘✮✱ ✇❤✐❝❤ ✐s t❤❡ ❧❛r❣❡st❀ t❤❡ ❆❧❧✐❛♥❝❡ ❢♦r P❛✐r❡❞ ❑✐❞♥❡② ❉♦♥❛t✐♦♥ ✭❆P❉✮❀ ❛♥❞ t❤❡ ❯♥✐t❡❞ ◆❡t✇♦r❦ ❢♦r ❖r❣❛♥ ❙❤❛r✐♥❣ ✭❯◆❖❙✮ ❑P❉ P✐❧♦t Pr♦❣r❛♠✳ ❚❤❡s❡ ❧❛r❣❡ ♣❧❛t❢♦r♠s ♠❛t❝❤ ♣❛t✐❡♥ts ✉s✐♥❣ ♦♣t✐♠✐③❛t✐♦♥ s♦❢t✇❛r❡ t❤❛t ♠❛①✐♠✐③❡s ❛ ✇❡✐❣❤t❡❞ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts✳ ❚❤❡② ❞✐✛❡r ✐♥ t❡r♠s ♦❢ ❡①❛❝t ❛❧❣♦r✐t❤♠s ❛♥❞ ♦♣❡r❛t✐♦♥❛❧ ❞❡t❛✐❧s✳✶✸ ❇❡s✐❞❡s t❤❡s❡ ♠❛❥♦r ♣❧❛t❢♦r♠s✱ t❤❡r❡ ❛r❡ s♠❛❧❧ r❡❣✐♦♥❛❧ ♣❧❛t❢♦r♠s ❛♥❞ ✐♥❞✐✈✐❞✉❛❧ ❤♦s♣✐t❛❧s t❤❛t ❛❧s♦ ♦r❣❛♥✐③❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s✳ ❆s ✇❡ ✇✐❧❧ s❡❡ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ ♠♦st ❤♦s♣✐t❛❧s t❤❛t ♣❛rt✐❝✐♣❛t❡ ✐♥ ❧❛r❣❡ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ❛❧s♦ ♠❛t❝❤ ♣❛t✐❡♥ts ♦✉ts✐❞❡ t❤♦s❡ ♣❧❛t❢♦r♠s✳ ▼♦r❡♦✈❡r✱ ❤♦s♣✐t❛❧s ❛r❡ ♥♦t ❢♦r❝❡❞ t♦ ♣❛rt✐❝✐♣❛t❡ ✐♥ ♣❧❛t❢♦r♠s✳ P❧❛t❢♦r♠s ❡✛❡❝t✐✈❡❧② r❡✇❛r❞ ❤♦s♣✐t❛❧s ✇✐t❤ tr❛♥s♣❧❛♥ts ✐♥ ♦r❞❡r t♦ r❡❝❡✐✈❡ s✉❜♠✐ss✐♦♥s ✭❘♦t❤ ❡t ❛❧✳ ✷✵✵✺❀ ❆s❤❧❛❣✐ ❛♥❞ ❘♦t❤ ✷✵✶✹✮ ❛s ❤♦s♣✐t❛❧s ♣❡r❢♦r♠ t❤❡ tr❛♥s♣❧❛♥ts ♦♥ ♣❛t✐❡♥ts t❤❡② s✉❜♠✐t t♦ ❛ ♣❧❛t❢♦r♠✳ ❘❡✇❛r❞s ❝❛♥ ❛❧s♦ ❜❡ ❡①♣❧✐❝✐t✳ ❋♦r ❡①❛♠♣❧❡✱ ♠♦st ♣❧❛t❢♦r♠s r❡✇❛r❞ ❤♦s♣✐t❛❧s t❤❛t s✉❜♠✐t ❛❧tr✉✐st✐❝ ❞♦♥♦rs ❜② ♠❛t❝❤✐♥❣ ♦♥❡ ♦❢ t❤❡✐r ✉♥♣❛✐r❡❞ ♣❛t✐❡♥ts✳ ❙❡❝♦♥❞✱ t❤❡r❡ ✐s s✉❜st❛♥t✐❛❧ ✈❛r✐❛t✐♦♥ ✐♥ t❤❡ s♦❝✐❛❧ ✈❛❧✉❡ ♦❢ ❞✐✛❡r❡♥t s✉❜♠✐ss✐♦♥s ❞✉❡ t♦ ❜✐♦❧♦❣✐❝❛❧ ❝♦♠♣❛t✐❜✐❧✐t②✳ ❖♥❡ r❡❛s♦♥ ❢♦r t❤✐s ✈❛r✐❛t✐♦♥ ✐s ❜❧♦♦❞✲t②♣❡ ❝♦♠♣❛t✐❜✐❧✐t②✳ ❚♦ s✐♠♣❧✐❢② ❡①♣♦s✐t✐♦♥✱ ❛ss✉♠❡ t❤❛t t❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ❜❧♦♦❞ t②♣❡s✱ ❖ ❛♥❞ ❆✳ ❚❤❡s❡ t✇♦ t②♣❡s t♦❣❡t❤❡r ❛r❡ t❤❡ ✈❛st ♠❛❥♦r✐t② ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✐♥ t❤❡ ❯✳❙✳ ❉❡♥♦t❡ ❛ ♣❛✐r ✇✐t❤ ♣❛t✐❡♥t ❜❧♦♦❞ t②♣❡ ❳ ❛♥❞ ❞♦♥♦r ❜❧♦♦❞ t②♣❡ ❨ ❛s ❳✲❨✱ ❛♥❞ ❧❡t q❳✲❨ ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ♣❛✐rs ✐♥ ❛ ♣♦♦❧✳ ❆ss✉♠❡ t❤❛t q❆✲❖ < q❖✲❆ ✱ ✇❤✐❝❤ ✐s t❤❡ ❡♠♣✐r✐❝❛❧❧② r❡❧❡✈❛♥t ❝❛s❡✳ ❋♦r t❤✐s s✐♠♣❧✐✜❡❞ ❝❛s❡✱ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ s❤♦✇❡❞ t❤❛t t❤❡ ♥✉♠❜❡r f (q) ♦❢ tr❛♥s♣❧❛♥ts t❤❛t ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ ✐s ❛♣♣r♦①✐♠❛t❡❧② f (q) = 2 · q❆✲❖ + 1 · (q❆✲❆ + q❖✲❖ ) + 0 · q❖✲❆ ✳ ✭✶✮ ❚❤✐s r❡s✉❧t ❢♦❧❧♦✇s ❜❡❝❛✉s❡ ❆✲❆ ❛♥❞ ❖✲❖ ♣❛✐rs ❝❛♥ ❜❡ ♠❛t❝❤❡❞ ✇✐t❤ ♣❛✐rs ♦❢ t❤❡ s❛♠❡ t②♣❡✳ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ ❝❛❧❧ t❤❡s❡ ♣❛✐rs s❡❧❢✲❞❡♠❛♥❞❡❞✳ ❙❡❧❢✲❞❡♠❛♥❞❡❞ ♣❛✐rs ❤❛✈❡ ❛ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ 1✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡② ❣❡♥❡r❛t❡ 1 ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥t ✇❤❡♥ t❤❡② ❥♦✐♥ t❤❡ ♣♦♦❧✳ ❍♦✇❡✈❡r✱ ❛♥ ❖✲❆ ♣❛✐r ❝❛♥ ♦♥❧② ❜❡ tr❛♥s♣❧❛♥t❡❞ ✉s✐♥❣ ❛ ❝②❝❧❡ ✇✐t❤ ♦♥❡ ♦❢ t❤❡ ✈❛❧✉❛❜❧❡ ❆✲❖ ♣❛✐rs✳ ❚❤✉s✱ t❤❡r❡ ✇✐❧❧ ❜❡ ♠❛♥② ❧❡❢t♦✈❡r ❖✲❆ ♣❛✐rs✱ ❛♥❞ t❤❡② ❝❛♥ ♦♥❧② ❜❡ tr❛♥s♣❧❛♥t❡❞ ✐❢ ♠♦r❡ ❆✲❖ ♣❛✐rs ❥♦✐♥ t❤❡ ♣♦♦❧✳ ❆✲❖ ♣❛✐rs ❛r❡ ❝❛❧❧❡❞ ♦✈❡r✲❞❡♠❛♥❞❡❞ ❛♥❞ ❤❛✈❡ ❛ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ 2✳ ❖✲❆ ♣❛✐rs ❛r❡ ❝❛❧❧❡❞ ✉♥❞❡r✲❞❡♠❛♥❞❡❞ ❛♥❞ ❤❛✈❡ ❛ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ 0✳ ❆♥ ✐♠♠✉♥❡ s②st❡♠ ✇✐❧❧ ❛tt❛❝❦ t❤❡ ❦✐❞♥❡②✱ ❧❡❛❞✐♥❣ t♦ r❡❥❡❝t✐♦♥✳

❆ r❡❝✐♣✐❡♥t ✐s t✐ss✉❡✲t②♣❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❛

❞♦♥♦r✬s ❦✐❞♥❡② ✐❢ s❤❡ ❤❛s ♥♦ ❛♥t✐❜♦❞✐❡s ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡ ❛♥t✐❣❡♥s ♦❢ t❤❡ ❞♦♥♦r✬s ❦✐❞♥❡② ✭❉❛♥♦✈✐t❝❤✱ ✷✵✵✾✮✳ ◆♦t❡ t❤❛t ❛ tr❛♥s♣❧❛♥t ❜❡t✇❡❡♥ ❝❡rt❛✐♥ ✐♥❝♦♠♣❛t✐❜❧❡ ♣❛t✐❡♥t✲❞♦♥♦r ❤❛s ❜❡❝♦♠❡ ♣♦ss✐❜❧❡ ❞✉❡ t♦ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❞❡s❡♥s✐t✐③❛t✐♦♥ t❡❝❤♥♦❧♦❣✐❡s ✭❖r❛♥❞✐ ❡t ❛❧✳✱ ✷✵✶✹✮✳

✶✸ ❙❡❡

❆❜r❛❤❛♠ ❡t ❛❧✳ ✭✷✵✵✼✮❀ ❆s❤❧❛❣✐ ❡t ❛❧✳ ✭✷✵✶✻✮❀ ❆♥❞❡rs♦♥ ❡t ❛❧✳ ✭✷✵✶✹✮❀ ❉✐❝❦❡rs♦♥ ❡t ❛❧✳ ✭✷✵✶✷✮✳

✾ ✉♥❞❡r✲❞❡♠❛♥❞❡❞ ♣❛✐r ❝♦♠♣❡t❡s ✇✐t❤ ❛♥♦t❤❡r ✉♥❞❡r✲❞❡♠❛♥❞❡❞ ♣❛✐r ❛♥❞ ❛❞❞s ♥♦ ✈❛❧✉❡ t♦ t❤❡ ♣♦♦❧✳ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ s❤♦✇❡❞ t❤❛t t❤✐s q✉❛❧✐t❛t✐✈❡ ♣❛tt❡r♥ ❤♦❧❞s ❡✈❡♥ ✐♥ ❛ ♠♦❞❡❧ ✇✐t❤ ❛❧❧ ♣♦ss✐❜❧❡ ❜❧♦♦❞ t②♣❡s✳ ❈✉rr❡♥t ♣❧❛t❢♦r♠ r✉❧❡s ❧❛r❣❡❧② ✐❣♥♦r❡ s✉❝❤ ✈❛r✐❛t✐♦♥ ✐♥ s♦❝✐❛❧ ✈❛❧✉❡ ♦❢ s✉❜♠✐ss✐♦♥s✱ t❤❡r❡❜② ✐♥❞✉❝✐♥❣ ❤♦s♣✐t❛❧s t♦ ♣❡r❢♦r♠ s♦❝✐❛❧❧② ✐♥❡✣❝✐❡♥t ✐♥t❡r♥❛❧ ♠❛t❝❤❡s✳ ❈♦♥s✐❞❡r ❛ ❤♦s♣✐t❛❧ ✇✐t❤ t✇♦ ♦✈❡r✲❞❡♠❛♥❞❡❞ ♣❛✐rs✳ ❚❤❡ ❤♦s♣✐t❛❧ ❝♦✉❧❞ ♠❛t❝❤ ♣❡r❢♦r♠ ❛ ♣❛✐r✇✐s❡ ❡①❝❤❛♥❣❡ t♦ ❝♦♥❞✉❝t t✇♦ tr❛♥s♣❧❛♥ts✳ ❍♦✇❡✈❡r✱ ✐❢ t❤❡ ❤♦s♣✐t❛❧ s✉❜♠✐ts t❤❡ ♣❛✐rs t♦ t❤❡ ♣❧❛t❢♦r♠✱ t❤❡ ❤♦s♣✐t❛❧ r❡❝❡✐✈❡s t✇✐❝❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ♣❛✐r ✐s ♠❛t❝❤❡❞✳ ❆❝❝♦r❞✐♥❣ t♦ ♦✉r ❞❛t❛✱ t❤❡ ❤♦s♣✐t❛❧ ❡①♣❡❝ts ♦♥❧② ✶✳✻ tr❛♥s♣❧❛♥ts ❢r♦♠ s✉❜♠✐tt✐♥❣ t❤❡s❡ t✇♦ ♣❛✐rs t♦ t❤❡ ♣❧❛t❢♦r♠ ❜❡❝❛✉s❡ t❤❡ ♣❧❛t❢♦r♠ ✉s❡s ❡❛❝❤ ♦❢ t❤❡s❡ s♣❡❝✐✜❝ ♣❛✐rs ✐♥ ❛♥ ❡①❝❤❛♥❣❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✵✳✽✳ ❚❤✐s ❡①♣❡❝t❛t✐♦♥ ♣✉s❤❡s t❤❡ ❤♦s♣✐t❛❧ t♦ ♠❛t❝❤ ✐ts ♣❛t✐❡♥ts ♦✉ts✐❞❡ t❤❡ ♣❧❛t❢♦r♠✳ ■❢ t❤❡ ❤♦s♣✐t❛❧ s✉❜♠✐ts t❤❡ ♣❛✐rs t♦ t❤❡ ♣❧❛t❢♦r♠✱ t❤❡ t✇♦ ♣❛✐rs t♦❣❡t❤❡r ❣❡♥❡r❛t❡ t✇✐❝❡ t❤❡✐r ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✳ ❚❤❡ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ ♠♦❞❡❧ s✉❣❣❡sts t❤❡ ♣❧❛t❢♦r♠ ❝♦✉❧❞ ❣❡♥❡r❛t❡ ❢♦✉r ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥ts ✉s✐♥❣ t❤❡s❡ ♣❛✐rs✳ ❯s✐♥❣ ❛ ♠♦r❡ r❡❛❧✐st✐❝ ❡♠♣✐r✐❝❛❧ ♠♦❞❡❧✱ ✇❡ ❡st✐♠❛t❡ ♦♥❧② t❤r❡❡ ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥ts ✭❙❡❝t✐♦♥ ✺✮✳ ❊✐t❤❡r ✇❛②✱ ♠❛t❝❤✐♥❣ t❤❡s❡ t✇♦ ♣❛✐rs ✇✐t❤✐♥ t❤❡ ❤♦s♣✐t❛❧ ✐s s♦❝✐❛❧❧② ✐♥❡✣❝✐❡♥t ❞❡s♣✐t❡ t❤❡ ❤♦s♣✐t❛❧✬s ❞❡s✐r❡ t♦ ❤❡❧♣ ✐ts ♣❛t✐❡♥ts✳ ❖✉r t❤❡♦r❡t✐❝❛❧ ❛♥❛❧②s✐s ✇✐❧❧ s✉♠♠❛r✐③❡ ❛❧❧ t❤❡ ✇❛②s ✐♥ ✇❤✐❝❤ s✉❝❤ ✐♥❡✣❝✐❡♥❝✐❡s ❝❛♥ ❤❛♣♣❡♥✱ ❛♥❞ ❝❧❛r✐❢② ❤♦✇ ♣❧❛t❢♦r♠ r✉❧❡s ❝❛♥ ❜❡ r❡❞❡s✐❣♥❡❞ t♦ ❡❧✐♠✐♥❛t❡ t❤✐s ♣r♦❜❧❡♠✳ ❆♥ ✐♠♣♦rt❛♥t ❝♦r♦❧❧❛r② ♦❢ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮✬s r❡s✉❧ts ✐s t❤❛t tr❛♥s♣❧❛♥ts ❛r❡ ❛ ♥❛t✉r❛❧ ♥✉✲ ♠❡r❛✐r❡ ✐♥ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳ ❇❡❝❛✉s❡ ❤♦s♣✐t❛❧s ❤❛✈❡ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ✉♥❞❡r✲ ❞❡♠❛♥❞❡❞ ♣❛✐rs✱ ✐t ✐s ❡❛s② ❢♦r ❛ ♣❧❛t❢♦r♠ t♦ tr❛♥s❢❡r tr❛♥s♣❧❛♥ts ❢r♦♠ ♦♥❡ ❤♦s♣✐t❛❧ t♦ ❛♥♦t❤❡r ❜② ❝❤♦♦s✐♥❣ ✇❤✐❝❤ ✉♥❞❡r✲❞❡♠❛♥❞❡❞ ♣❛✐rs t♦ ♠❛t❝❤✱ ✇✐t❤♦✉t ❝♦♠♣r♦♠✐s✐♥❣ ❡✣❝✐❡♥❝②✳ ❚❤✐r❞✱ ❤♦s♣✐t❛❧s ❞♦ ♥♦t ♥❡❝❡ss❛r✐❧② ♠❛①✐♠✐③❡ ❛ ✉t✐❧✐t❛r✐❛♥ ♠❡❛s✉r❡ ♦❢ ♣❛t✐❡♥t ✇❡❧❢❛r❡✱ ❝❛✉s✐♥❣ ❡①t❡r♥❛❧✐t✐❡s✳ ❲❡ r❡❢❡r t♦ ❛♥② s✉❝❤ ❞✐✈❡r❣❡♥❝❡ ❛s ❛♥ ❛❣❡♥❝② ♣r♦❜❧❡♠ ❜❡t✇❡❡♥ ❤♦s♣✐t❛❧s ❛♥❞ ♣❛t✐❡♥ts ❜❡❝❛✉s❡ t❤❡② ♣r✐♠❛r✐❧② ❛r✐s❡ ❢r♦♠ ❛ r❡✐♠❜✉rs❡♠❡♥t s②st❡♠ t❤❛t ❞♦❡s ♥♦t ✐♥❝❡♥t✐✈✐③❡ ❡✣❝✐❡♥t ❛❝t✐♦♥s ❢r♦♠ t❤❡ ♣❛t✐❡♥ts✬ ♣❡rs♣❡❝t✐✈❡✳ ❍♦s♣✐t❛❧s ✐♥❝✉r ♠♦st ♦❢ t❤❡ ❝♦sts ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❜✉t r❡❝❡✐✈❡ ♦♥❧② ❛ ❢r❛❝t✐♦♥ ♦❢ t❤❡ s♦❝✐❛❧ ❜❡♥❡✜ts✳ ❲❤✐❧❡ t❤❡ s♦❝✐❛❧ ✈❛❧✉❡ ❢r♦♠ ♦♥❡ tr❛♥s♣❧❛♥t ✐s ♠♦r❡ t❤❛♥ ✩✶✱✵✵✵✱✵✵✵✱ ❤♦s♣✐t❛❧ r❡✈❡♥✉❡s ❛r❡ ❜❡t✇❡❡♥ ✩✶✵✵✱✵✵✵ t♦ ✩✶✻✵✱✵✵✵ ♣❡r tr❛♥s♣❧❛♥t✳✶✹ ❱❛r✐❛❜❧❡ ♣r♦✜ts ❛r❡ ❧✐❦❡❧② ♠✉❝❤ s♠❛❧❧❡r✳ ❚❤✉s✱ ❡✈❡♥ s♦❝✐❛❧❧② ✐♥s✐❣♥✐✜❝❛♥t ❝♦sts ♦❢ ♣❡r❢♦r♠✐♥❣ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ t❤r♦✉❣❤ ❛ ♣❧❛t❢♦r♠ ❝❛♥ ❜❡ ✐♠♣♦rt❛♥t ❢♦r ❤♦s♣✐t❛❧s✳ ❈♦♥✈❡rs❛t✐♦♥s ✇✐t❤ ❤♦s♣✐t❛❧ st❛✛ ✐♥❞✐❝❛t❡ t❤❛t ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s ✐♥✈♦❧✈❡s ❧♦❣✐st✐❝❛❧ ❛♥❞ ❛❞♠✐♥✐str❛t✐✈❡ ❤❛ss❧❡ ✐♥ ❛❞❞✐t✐♦♥ t♦ ❞✐r❡❝t ❝♦sts ❛r✐s✐♥❣ ❢r♦♠ ❜✐♦❧♦❣✐❝❛❧ t❡st✐♥❣ ❛♥❞ ♣❧❛t❢♦r♠ ❢❡❡s✳✶✺ Pr❡✈✐♦✉s s✉r✈❡②s ❛♥❞ ✐♥t❡r✈✐❡✇s ❤❛✈❡ ❢♦✉♥❞ t❤❛t t❤❡s❡ ❧♦❣✐st✐❝❛❧ ❛♥❞ ✜♥❛♥❝✐❛❧ ❝♦sts ❛r❡ ❝♦♠♠♦♥❧② ❝✐t❡❞ ❜❛rr✐❡rs t♦ ♣❛rt✐❝✐♣❛t✐♦♥ ✭❊❧❧✐s♦♥✱ ✷✵✶✹❀ ❆♠❡r✐❝❛♥ ❙♦❝✐❡t② ♦❢ ❚r❛♥s♣❧❛♥t ❙✉r❣❡♦♥s✱ ✷✵✶✻✮✳ ❇❡s✐❞❡s ❝♦sts✱ ❤♦s♣✐t❛❧s ♠❛② ❤❛✈❡ ❜❡❤❛✈✐♦r❛❧ r❡❛s♦♥s ❢♦r ♥♦t ♣❡r❢❡❝t❧② ♠❛①✐♠✐③✐♥❣ ♣❛t✐❡♥t ✇❡❧❢❛r❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛♥❡❝❞♦t❛❧ ❡✈✐❞❡♥❝❡ ✐♥❞✐❝❛t❡s ✶✹ ❙❡❡ ❍❡❧❞ ❡t ❛❧✳ ✭✷✵✶✻✮❀ ❯❙❘❉❙✱ ❯♥✐t❡❞ ❙t❛t❡s ❘❡♥❛❧ ❉❛t❛ ❙②st❡♠ ✭✷✵✶✸✮✳ ❚❤✐s ❛♠♦✉♥t ✐♥❝❧✉❞❡s ♣❛②♠❡♥ts ❢♦r s✉r❣❡r② t❡❛♠s✱ ❞r✉❣s✱ ❡q✉✐♣♠❡♥t✱ ❛♥❞ ❝❛♣✐t❛❧✳ ✶✺ P❧❛t❢♦r♠s r❡q✉✐r❡ ❡①t❡♥s✐✈❡ ❜✐♦❧♦❣✐❝❛❧ t❡st✐♥❣✱ ✇❤✐❝❤ ✐s ♣❛rt✐❝✉❧❛r❧② ❝♦♠♣❧✐❝❛t❡❞ ❜❡❝❛✉s❡ ❞♦♥♦rs ❛♥❞ ♣❛t✐❡♥ts ❛r❡ ✐♥ ❞✐✛❡r❡♥t ❤♦s♣✐t❛❧s✳ P❧❛t❢♦r♠s ❛❧s♦ ❝❤❛r❣❡ ❢❡❡s✱ ✇❤✐❝❤ ❛r❡ ♣❛✐❞ ❜② ❤♦s♣✐t❛❧s✳ ◆❑❘ ❝❤❛r❣❡s ❛♥♥✉❛❧ ❢❡❡s ♦❢ ❛❜♦✉t ✩✶✵✱✵✵✵ ♣❧✉s ❛❜♦✉t ✩✹✱✵✵✵ ♣❡r tr❛♥s♣❧❛♥t✳ ❙❡❡ ◆❛t✐♦♥❛❧ ❑✐❞♥❡② ❘❡❣✐str② ✭✷✵✶✻✮ ❢♦r ◆❑❘✬s ❢❡❡s✱ ❛♥❞ ❘❡❡s ❡t ❛❧✳ ✭✷✵✶✷✮ ❛♥❞ ❲❛❧❧ ❡t ❛❧✳ ✭✷✵✶✼✮ ❢♦r ❛ ❜r♦❛❞❡r ❞✐s❝✉ss✐♦♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❝♦sts ❜♦r♥❡ ❜② ❤♦s♣✐t❛❧s✳

✶✵ ❝♦♥s✐❞❡r❛❜❧❡ ❤❡t❡r♦❣❡♥❡✐t② r❡❣❛r❞✐♥❣ ❤♦s♣✐t❛❧ s♦♣❤✐st✐❝❛t✐♦♥✿ s♦♠❡ ❤♦s♣✐t❛❧s r❡♣♦rt ✉s✐♥❣ ♦♣t✐♠✐③❛t✐♦♥ s♦❢t✇❛r❡ t♦ ♠❛t❝❤ ♣❛t✐❡♥ts ✇❤✐❧❡ ♦t❤❡rs ♠❛♥✉❛❧❧② s❡❛r❝❤ ❢♦r ♠❛t❝❤❡s✳ ✷✳✸

❉❛t❛

❲❡ ❛ss❡♠❜❧❡❞ t✇♦ ❞❛t❛s❡ts ❢♦r t❤✐s ♣❛♣❡r✳ ❚❤❡ ✜rst✱ t❤❡ tr❛♥s♣❧❛♥t ❞❛t❛s❡t✱ r❡❝♦r❞s ❛❧❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ✐♥ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s✳ ❲❡ ✉s❡ t❤✐s ❞❛t❛s❡t t♦ ❞♦❝✉♠❡♥t ❢r❛❣♠❡♥t❛t✐♦♥✱ ✐♥❡✣❝✐❡♥❝② ❛♥❞ ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ t❤❡ ♠❛r❦❡t ❢♦r ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳ ❚❤❡ s❡❝♦♥❞✱ t❤❡ ◆❑❘ ❞❛t❛s❡t✱ r❡❝♦r❞s ❛❧❧ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs t❤❛t r❡❣✐st❡r❡❞ ✇✐t❤ t❤❡ ❧❛r❣❡st ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✱ t❤❡ ◆❛t✐♦♥❛❧ ❑✐❞♥❡② ❘❡❣✐str② ✭◆❑❘✮✳ ❲❡ ✉s❡ t❤✐s ❞❛t❛s❡t t♦ ❡st✐♠❛t❡ ❛ tr❛♥s♣❧❛♥t ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳ ❚❤❡ tr❛♥s♣❧❛♥t ❞❛t❛s❡t ❝♦♥s✐sts ♦❢ ❛♥♦♥②♠✐③❡❞ r❡❝♦r❞s ♦❢ ❡✈❡r② ❦✐❞♥❡② tr❛♥s♣❧❛♥t ❝♦♥❞✉❝t❡❞ ✐♥ t❤❡ ❯❙ ❜❡t✇❡❡♥ ❏❛♥✉❛r② ✶✱ ✷✵✵✽ ❛♥❞ ❉❡❝❡♠❜❡r ✹✱ ✷✵✶✹✳ ❲❡ ♦❜t❛✐♥❡❞ t❤✐s ❞❛t❛s❡t ❢r♦♠ t❤❡ ❖r❣❛♥ Pr♦❝✉r❡♠❡♥t ❛♥❞ ❚r❛♥s♣❧❛♥t❛t✐♦♥ ◆❡t✇♦r❦ ✭❖P❚◆✮✱ ❛ ❝♦♥tr❛❝t♦r ❢♦r t❤❡ ❯✳❙✳ ❉❡♣❛rt♠❡♥t ♦❢ ❍❡❛❧t❤ ❛♥❞ ❍✉♠❛♥ ❙❡r✈✐❝❡s✳✶✻ ❚❤❡ ❖P❚◆ ❞❛t❛s❡t ✐♥❝❧✉❞❡s ❡❛❝❤ tr❛♥s♣❧❛♥t✬s ❞❛t❡ ❛♥❞ ❧♦❝❛t✐♦♥❀ ✇❤❡t❤❡r ✐t ✐s ♣❛rt ♦❢ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡❀ t❤❡ ❛❣❡✱ s❡①✱ ✇❡✐❣❤t✱ ❤❡✐❣❤t✱ ❜♦❞② ♠❛ss ✐♥❞❡① ✭❇▼■✮✱ ❜❧♦♦❞✲t②♣❡✱ ❛♥❞ ❍▲❆ ❛♥t✐❣❡♥s ♦❢ t❤❡ ❞♦♥♦r ❛♥❞ r❡❝✐♣✐❡♥t❀ ❛♥❞ t❤❡ ✉♥❛❝❝❡♣t❛❜❧❡ ❛♥t✐❣❡♥s ❛♥❞ ♥✉♠❜❡r ♦❢ ❞❛②s ♦♥ ❞✐❛❧②s✐s ♦❢ t❤❡ r❡❝✐♣✐❡♥t✳ ❙❡❡ ❆♣♣❡♥❞✐① ❇ ❢♦r ❞❡t❛✐❧s✳ ❆❧t❤♦✉❣❤ ❛ ❝♦♠♣r❡❤❡♥s✐✈❡ s♦✉r❝❡ ❢♦r ❞❛t❛ ♦♥ tr❛♥s♣❧❛♥ts ❝♦♥❞✉❝t❡❞✱ t❤❡ ♦♥❧② ✜❡❧❞ ✐♥ t❤❡ ❖P❚◆ ❞❛t❛s❡t t❤❛t s♣❡❝✐✜❝❛❧❧② ♣❡rt❛✐♥s t♦ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ✐s ❛♥ ✐♥❞✐❝❛t♦r ❢♦r ✇❤✐❝❤ tr❛♥s✲ ♣❧❛♥ts ✇❡r❡ ♣❛rt ♦❢ s✉❝❤ ❛♥ ❡①❝❤❛♥❣❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❖P❚◆ ❞❛t❛s❡t ❞♦❡s ♥♦t ✐❞❡♥t✐❢② ✇❤✐❝❤✱ ✐❢ ❛♥②✱ ♠✉❧t✐✲❤♦s♣✐t❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ♦r❣❛♥✐③❡❞ ❛ ❣✐✈❡♥ tr❛♥s♣❧❛♥t✳ ❚♦ ❛❞❞r❡ss t❤✐s ❧✐♠✐t❛t✐♦♥✱ ✇❡ s❡♣❛r❛t❡❧② ♦❜t❛✐♥❡❞ ❛♥♦♥②♠✐③❡❞ r❡❝♦r❞s ♦❢ ❛❧❧ tr❛♥s♣❧❛♥ts ♦r❣❛♥✐③❡❞ ❜② ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ❧❛r❣❡st ♣❧❛t❢♦r♠s ❢♦r ♠✉❧t✐✲❤♦s♣✐t❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s ✐♥ t❤❡ ❯❙✿ ◆❑❘✱ ❆P❉✱ ❛♥❞ ❯◆❖❙✳ ❚❤❡s❡ ♣❧❛t❢♦r♠ ❞❛t❛ ✐♥❝❧✉❞❡ t❤❡ s❛♠❡ ✐♥❢♦r♠❛t✐♦♥ ❛s t❤❡ ❖P❚◆ ❞❛t❛s❡t✳ ❇② ♠❡r❣✐♥❣ t❤❡ ❞❛t❛ ❢r♦♠ t❤❡s❡ ♣❧❛t❢♦r♠s ✇✐t❤ t❤❡ ❖P❚◆ ❞❛t❛✱ ✇❡ ✐❞❡♥t✐✜❡❞ ✇❤✐❝❤ tr❛♥s♣❧❛♥ts ✇❡r❡ ♦r❣❛♥✐③❡❞ t❤r♦✉❣❤ ◆❑❘✱ ❆P❉✱ ❯◆❖❙✱ ♦r t❤r♦✉❣❤ ♦t❤❡r ❛✈❡♥✉❡s✳ ❚❤✐s ♠❡r❣❡ ✐s ♥♦t str❛✐❣❤t❢♦r✇❛r❞ ❜❡❝❛✉s❡ ❛❧❧ ♦❢ ♦✉r ❞❛t❛s❡ts ❛r❡ ❛♥♦♥②♠✐③❡❞✳ ❋♦rt✉♥❛t❡❧②✱ t❤❡ r✐❝❤ ❜✐♦❧♦❣✐❝❛❧ ❞❛t❛ ❛❧❧♦✇s ✉s t♦ ♠❛t❝❤ tr❛♥s♣❧❛♥ts ❛❝r♦ss ❞❛t❛s❡ts ♦♥ t❤❡ ❜❧♦♦❞ t②♣❡✱ s❡①✱ ❛♥❞ ❍▲❆ ❛♥t✐❣❡♥s ♦❢ t❤❡ r❡❝✐♣✐❡♥t ❛♥❞ ❞♦♥♦r✱ ❛s ✇❡❧❧ ❛s t❤❡ ❞❛t❡ ❛♥❞ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ tr❛♥s♣❧❛♥t✳ ▼♦r❡ ❞❡t❛✐❧s ♦♥ t❤❡ ♠❡r❣❡ ♣r♦❝❡❞✉r❡ ❛r❡ ♣r♦✈✐❞❡❞ ✐♥ ❆♣♣❡♥❞✐① ❇✳ ❲❡ ✇❡r❡ ❛❜❧❡ t♦ ♠❛t❝❤ ❛♣♣r♦①✐♠❛t❡❧② ✾✹✪ ♦❢ tr❛♥s♣❧❛♥ts ❛t t❤❡s❡ ♣❧❛t❢♦r♠s t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❖P❚◆ ❞❛t❛ ✇✐t❤ ❛ ✈❡r② ❤✐❣❤ ❞❡❣r❡❡ ♦❢ ❝❡rt❛✐♥t②✳✶✼

❚❤✐s st✉❞② ✉s❡s ❞❛t❛ ❢r♦♠ t❤❡ ❖r❣❛♥ Pr♦❝✉r❡♠❡♥t ❛♥❞ ❚r❛♥s♣❧❛♥t❛t✐♦♥ ◆❡t✇♦r❦ ✭❖P❚◆✮✳ ❚❤❡ ❖P❚◆ ❞❛t❛ s②st❡♠ ✐♥❝❧✉❞❡s ❞❛t❛ ♦♥ ❛❧❧ ❞♦♥♦r✱ ✇❛✐t✲❧✐st❡❞ ❝❛♥❞✐❞❛t❡s✱ ❛♥❞ tr❛♥s♣❧❛♥t r❡❝✐♣✐❡♥ts ✐♥ t❤❡ ❯❙✱ s✉❜♠✐tt❡❞ ❜② ♠❡♠❜❡rs ♦❢ t❤❡ ❖r❣❛♥ Pr♦❝✉r❡♠❡♥t ❛♥❞ ❚r❛♥s♣❧❛♥t❛t✐♦♥ ◆❡t✇♦r❦ ✭❖P❚◆✮✳ ❚❤❡ ❍❡❛❧t❤ ❘❡s♦✉r❝❡s ❛♥❞ ❙❡r✈✐❝❡s ❆❞♠✐♥✐str❛t✐♦♥ ✭❍❘❙❆✮✱ ❯❙ ❉❡♣❛rt♠❡♥t ♦❢ ❍❡❛❧t❤ ❛♥❞ ❍✉♠❛♥ ❙❡r✈✐❝❡s ♣r♦✈✐❞❡s ♦✈❡rs✐❣❤t t♦ t❤❡ ❛❝t✐✈✐t✐❡s ♦❢ t❤❡ ❖P❚◆ ❝♦♥tr❛❝t♦r✳ ✶✼ ■♥ ❢❛❝t✱ ✾✵✪ ♦❢ t❤❡ ♠❛t❝❤❡s ✇❡r❡ ✇✐t❤✐♥ ✶ ❞❛② ♦♥ t❤❡ tr❛♥s♣❧❛♥t ❞❛t❡✱ ✇✐t❤✐♥ ✺ ②❡❛rs ♦♥ ❞♦♥♦r ❛♥❞ r❡❝✐♣✐❡♥t ❛❣❡✱ ❛♥❞ ❛❣r❡❡❞ ♦♥ t❤❡ ❤♦s♣✐t❛❧ ✇❤❡r❡ t❤❡ tr❛♥s♣❧❛♥t ✇❛s ❝♦♥❞✉❝t❡❞ ❛s ✇❡❧❧ ❛s t❤❡ ❜❧♦♦❞ t②♣❡✱ s❡①✱ ❛♥❞ ❛❧❧ s✐① ♠❛❥♦r ❍▲❆ ❛❧❧❡❧❡s ✭✷ ❛❧❧❡❧❡s ❡❛❝❤ ❛t t❤❡ ❍▲❆✲❆✱ ❇ ❛♥❞ ❉❘ ❧♦❝✐✮ ♦❢ ❜♦t❤ t❤❡ ❞♦♥♦r ❛♥❞ r❡❝✐♣✐❡♥t✳ ✶✻

✶✶ ■♥ ♦r❞❡r t♦ ❡st✐♠❛t❡ ❛ ♣❧❛t❢♦r♠✬s tr❛♥s♣❧❛♥t ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✱ ✇❡ ✇✐❧❧ r❡q✉✐r❡ ♠♦r❡ t❤❛♥ ❥✉st ✐♥❢♦r♠❛t✐♦♥ ♦♥ tr❛♥s♣❧❛♥ts ♦r❣❛♥✐③❡❞ ❜② ❛ ♣❧❛t❢♦r♠ ✭t❤❡ ♦✉t♣✉t✮ ✕ ✇❡ ❛❧s♦ r❡q✉✐r❡ ✐♥✲ ❢♦r♠❛t✐♦♥ ❛❜♦✉t ✇❤✐❝❤ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❛r❡ s✉❜♠✐tt❡❞ t♦ t❤❡ ♣❧❛t❢♦r♠ ✭t❤❡ ✐♥♣✉ts✮✳ ❚❤❡ ❖P❚◆ ❞❛t❛s❡t ✐s ♥♦t ✇❡❧❧ s✉✐t❡❞ ❢♦r t❤✐s t❛s❦ ❜❡❝❛✉s❡ ✐t ❞♦❡s ♥♦t ✐❞❡♥t✐❢② ❛ ♣❛t✐❡♥ts✬ r❡❧❛t❡❞ ✐♥❝♦♠♣❛t✐❜❧❡ ❞♦♥♦r✱ ✐❢ ❛♥②✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❞❛t❛ ❛❧❧♦✇ ✉s t♦ ❞❡t❡r♠✐♥❡ ♥❡✐t❤❡r t❤❡ ♣❛✐rs t❤❛t ♣❛rt✐❝✐♣❛t❡❞ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♥♦r t❤❡ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs t❤❛t r❡♠❛✐♥ ✉♥♠❛t❝❤❡❞✳ ❚♦ ❛❞❞r❡ss t❤✐s ❧✐♠✐t❛t✐♦♥✱ ✇❡ ❛ss❡♠❜❧❡❞ t❤❡ ◆❑❘ ❞❛t❛s❡t✳ ■t r❡❝♦r❞s ❛❧❧ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs t❤❛t r❡❣✐st❡r❡❞ ✇✐t❤ t❤❡ ◆❑❘ ❜❡t✇❡❡♥ ❆♣r✐❧ ✷✱ ✷✵✶✷ t♦ ❉❡❝❡♠❜❡r ✹✱ ✷✵✶✹✳ ❚❤❡s❡ ❞❛t❛ ❛r❡ s♦✉r❝❡❞ ❢r♦♠ t❤❡ ❛❞♠✐♥✐str❛t✐✈❡ r❡❝♦r❞s t❤❡ ◆❑❘ ✉s❡s t♦ ♦r❣❛♥✐③❡ tr❛♥s♣❧❛♥ts✳ ■t ✐♥❝❧✉❞❡s t❤❡ r❡❣✐str❛t✐♦♥ ❞❛t❡✱ ❜❧♦♦❞ t②♣❡✱ ❛❣❡✱ s❡①✱ ❍▲❆ ❛♥t✐❣❡♥s ❢♦r ❜♦t❤ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✳ ■t ❛❧s♦ r❡❝♦r❞s ✇❤❡t❤❡r t❤❡ ♣❛t✐❡♥t ♦r ❞♦♥♦r ❧❡❢t ◆❑❘✬s s②st❡♠✱ ❛♥❞ t❤❡ ❞❛t❡ ❛♥❞ r❡❛s♦♥ ❢♦r ❞❡♣❛rt✉r❡ ✭tr❛♥s♣❧❛♥t❛t✐♦♥ ♦r ♦t❤❡r✇✐s❡✮✳ ■♥ ❛❞❞✐t✐♦♥✱ ✐t ✐♥❝❧✉❞❡s ✐♥❢♦r♠❛t✐♦♥ ♦♥ ✇❤✐❝❤ ❞♦♥♦r ✐s ♣❛✐r❡❞ ✇✐t❤ ✇❤✐❝❤ ♣❛t✐❡♥t ✭✐❢ ❛♥②✮✱ ✉♥❛❝❝❡♣t❛❜❧❡ ❛♥t✐❣❡♥s✱ ❛♥❞ ❛❧❧ t❤❡ r❡str✐❝t✐♦♥s ❛ ♣❛t✐❡♥t ♣❧❛❝❡s ♦♥ ✇❤✐❝❤ ♦r❣❛♥s ❛r❡ ❛❝❝❡♣t❛❜❧❡✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❞❡t❡r♠✐♥❡ t❤❡ s❡t ♦❢ tr❛♥s♣❧❛♥ts t❤❡ ◆❑❘ ❝♦♥s✐❞❡rs ❛❝❝❡♣t❛❜❧❡ ❛♥❞ ♠❡❞✐❝❛❧❧② ❢❡❛s✐❜❧❡✳ ❲❡ ❛❧s♦ ❤❛✈❡ ❞❡t❛✐❧❡❞ ❞❛t❛ ♦♥ ❤♦✇ t❤❡ tr❛♥s♣❧❛♥ts ✇❡r❡ ♦r❣❛♥✐③❡❞✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❞♦♥♦rs ❛♥❞ ♣❛t✐❡♥ts ✐♥✈♦❧✈❡❞✱ ❛♥❞ t❤❡ ❝❤❛✐♥ ♦r ❝②❝❧❡ ❝♦♥✜❣✉r❛t✐♦♥✳ ❆♣♣❡♥❞✐① ❇ ♣r♦✈✐❞❡s ❞❡t❛✐❧s ♦♥ ❤♦✇ ✇❡ ❛ss❡♠❜❧❡❞ t❤❡ ◆❑❘ ❞❛t❛s❡t✳

✸

❉❡s❝r✐♣t✐✈❡ ❊✈✐❞❡♥❝❡

❲❡ ♥♦✇ ✉s❡ ❞❡t❛✐❧❡❞ ❛❞♠✐♥✐str❛t✐✈❡ ❞❛t❛s❡t t♦ ❞❡s❝r✐❜❡ t❤r❡❡ ❦❡② ❢❛❝ts✿ t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡t ✐s ❤✐❣❤❧② ❢r❛❣♠❡♥t❡❞❀ t❤✐s ❢r❛❣♠❡♥t❛t✐♦♥ ❧❡❛❞s t♦ ✐♥❡✣❝✐❡♥❝②❀ ❛♥❞ t❤❡r❡ ✐s ❡✈✐❞❡♥❝❡ ♦❢ ❜r♦❛❞❧② ❞❡✜♥❡❞ ❛❣❡♥❝② ♣r♦❜❧❡♠s ❜❡t✇❡❡♥ ❤♦s♣✐t❛❧s ❛♥❞ ♣❛t✐❡♥ts✳ ✸✳✶

❋r❛❣♠❡♥t❛t✐♦♥

❲❡ ❞♦❝✉♠❡♥t ♠❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥ ❜② s❤♦✇✐♥❣ t❤❛t ❛ ❧❛r❣❡ ❢r❛❝t✐♦♥ ♦❢ tr❛♥s♣❧❛♥ts ❛r❡ ❢❛❝✐❧✲ ✐t❛t❡❞ ❜② ✐♥❞✐✈✐❞✉❛❧ ❤♦s♣✐t❛❧s ✐♥st❡❛❞ ♦❢ ❧❛r❣❡ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s✳ ❆ ♠❛r❦❡t ❝♦♦r❞✐♥❛t❡❞ ♦♥ ❛ s✐♥❣❧❡ ♣❧❛t❢♦r♠ ✇♦✉❧❞ ❤❛✈❡ ❛❧❧ tr❛♥s♣❧❛♥ts ❢❛❝✐❧✐t❛t❡❞ t❤r♦✉❣❤ t❤❛t ♣❧❛t❢♦r♠✳ ■♥ ❝♦♥tr❛st✱ ❛ ❝♦♠♣❧❡t❡❧② ❢r❛❣♠❡♥t❡❞ ♠❛r❦❡t ✇♦✉❧❞ ❤❛✈❡ ♥♦ tr❛♥s♣❧❛♥ts ❢❛❝✐❧✐t❛t❡❞ t❤r♦✉❣❤ ❛ ♠✉❧t✐✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠✳ ■♥ t❤✐s ❝❛s❡✱ ❛❧❧ tr❛♥s♣❧❛♥ts ✇♦✉❧❞ ❜❡ ✇✐t❤✐♥ ❤♦s♣✐t❛❧✱ ♠❡❛♥✐♥❣ ❡❛❝❤ tr❛♥s♣❧❛♥t❡❞ ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r ❜❡❧♦♥❣ t♦ t❤❡ s❛♠❡ ❤♦s♣✐t❛❧✳ ❚❤❡r❡❢♦r❡✱ t♦ ♠❡❛s✉r❡ ❢r❛❣♠❡♥t❛t✐♦♥✱ ✇❡ ✜rst ✐❞❡♥t✐❢② ✇✐t❤✐♥ ❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts ❜② ❝❤❡❝❦✐♥❣ ✇❤❡t❤❡r t❤❡ ❤♦s♣✐t❛❧ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ♣❛t✐❡♥t tr❛♥s♣❧❛♥t❡❞ t❤r♦✉❣❤ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❤♦s♣✐t❛❧ t❤❛t r❡❝♦✈❡r❡❞ t❤❡ ♦r❣❛♥✳✶✽ ❋r♦♠ t❤✐s s❡t✱ ✇❡ ❡①❝❧✉❞❡ ❛❧❧ tr❛♥s♣❧❛♥ts ❢❛❝✐❧✐t❛t❡❞ ❜② ♦♥❡ ♦❢ t❤r❡❡ ❧❛r❣❡ ❡①❝❤❛♥❣❡s✿ t❤❡ ◆❑❘✱ t❤❡ ❆P❉ ❛♥❞ ❯◆❖❙✳ ✶✽ ❚❤❡

❝♦♠♠♦♥ ♣r❛❝t✐❝❡ ✐s t♦ tr❛♥s♣♦rt t❤❡ ♦r❣❛♥ ❛❢t❡r r❡❝♦✈❡r② ✐♥st❡❛❞ ♦❢ tr❛♥s♣♦rt✐♥❣ t❤❡ ❞♦♥♦r ❛♥❞

r❡❝♦✈❡r✐♥❣ t❤❡ ♦r❣❛♥ ❡❧s❡✇❤❡r❡✳ ❆ ♣r✐♠❛r② ♠♦t✐✈❛t✐♦♥ ❢♦r t❤✐s ♣r❛❝t✐❝❡ ✐s t♦ s❛❢❡❣✉❛r❞ t❤❡ ❞♦♥♦r✬s ✐♥t❡r❡sts ❛♥❞ ❜❡❝❛✉s❡✱ ❜② t❤❡ t✐♠❡ ♦❢ t❤❡ tr❛♥s♣❧❛♥t✱ t❤❡ ❞♦♥♦r ❤❛s ❜✉✐❧t ❛ r❡❧❛t✐♦♥s❤✐♣ ✇✐t❤ ❤✐s ♦r ❤❡r ❤♦s♣✐t❛❧ ❛♥❞ s✉r❣❡♦♥✳ ❚❤❡ s✉r❣❡r② ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ ❞♦♥♦r r❡q✉✐r❡s ❡①t❡♥s✐✈❡ ♣r❡✲♣❧❛♥♥✐♥❣ ❛♥❞ ❢♦❧❧♦✇✲✉♣ ❝❛r❡✳ ❈♦♥✈❡rs❛t✐♦♥s ✇✐t❤ s✉r❣❡♦♥s s✉❣❣❡st t❤❛t t❤❡s❡ ❢❛❝t♦rs s❡✈❡r❡❧② ❧✐♠✐t ✇✐❧❧✐♥❣♥❡ss t♦ tr❛♥s♣♦rt t❤❡ ❞♦♥♦r ❛♥❞ ❝♦♥❞✉❝t s✉r❣❡r② ✐♥ ❛♥♦t❤❡r ❤♦s♣✐t❛❧✳

✶✷

Number of Transplants

800

600

400

200

0 2008

2010

2012

2014

Year

◆♦t❡s ✿

NKR across hospitals

APD/UNOS across hospitals

Other platforms across hospitals

NKR within hospital

APD/UNOS within hospital

Other platforms within hospital

❋✐❣✉r❡ ✶✿ ▼❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥ ❛♥❞ tr❡♥❞s ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡

❚❤❡ ✜❣✉r❡ ❞✐s♣❧❛②s t❤❡ ♥✉♠❜❡r ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ✐♥ ❞✐✛❡r❡♥t ❝❛t❡❣♦r✐❡s✳ ◆❑❘✱ ❆P❉✴❯◆❖❙ ❛♥❞ ❖t❤❡r ❝❧❛ss✐✜❡s ❛ tr❛♥s♣❧❛♥t ✐♥t♦ t❤❡ ♣❧❛t❢♦r♠ t❤❛t ❢❛❝✐❧✐t❛t❡❞ t❤❡ ❡①❝❤❛♥❣❡✱ ✐♥❝❧✉❞✐♥❣ s✐♥❣❧❡✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s✳ ❲✐t❤✐♥ ❤♦s♣✐t❛❧ ❛♥❞ ❛❝r♦ss ❤♦s♣✐t❛❧ ❝❧❛ss✐❢② ❛ tr❛♥s♣❧❛♥t ✐♥t♦ ✇❤❡t❤❡r t❤❡ ❞♦♥♦r ❤♦s♣✐t❛❧ ✇❛s t❤❡ s❛♠❡ ❛s t❤❡ ♣❛t✐❡♥t ❤♦s♣✐t❛❧✳

❋✐❣✉r❡ ✶ s❤♦✇s t❤❛t t❤❡ ♠❛r❦❡t ✐s ❢r❛❣♠❡♥t❡❞✳ ❚❤❡ t❤r❡❡ ❧❛r❣❡st ♠✉❧t✐✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s t♦❣❡t❤❡r ♦♥❧② ❛❝❝♦✉♥t ❢♦r ❛ ♠✐♥♦r✐t② s❤❛r❡ ♦❢ t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡t✳ ✻✷✪ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ❛r❡ ✇✐t❤✐♥ ❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts t❤❛t ❛r❡ ♥♦t ❢❛❝✐❧✐t❛t❡❞ ❜② t❤❡ ◆❑❘✱ ❆P❉ ♦r ❯◆❖❙✳ ▼♦r❡♦✈❡r✱ ♦✈❡r ✶✵✵ ❤♦s♣✐t❛❧s ♣❡r❢♦r♠❡❞ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s ♦✉ts✐❞❡ t❤❡s❡ t❤r❡❡ ♣❧❛t❢♦r♠s ❞✉r✐♥❣ t❤✐s ♣❡r✐♦❞✳ ❯♥❧✐❦❡ t❤❡ ❞♦♠✐♥❛♥❝❡ ♦❢ ✇✐t❤✐♥ ❤♦s♣✐t❛❧ ❡①❝❤❛♥❣❡s ✐♥ t❤❡ ♦✈❡r❛❧❧ ♠❛r❦❡t✱ t❤❡ ✈❛st ♠❛❥♦r✐t② ♦❢ t❤❡ tr❛♥s♣❧❛♥ts ❢❛❝✐❧✐t❛t❡❞ ❜② ♠✉❧t✐✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s ♦❝❝✉r ❛❝r♦ss ❤♦s♣✐t❛❧s✱ ✐♥✈♦❧✈✐♥❣ ❛ ❧✐✈✐♥❣ ❞♦♥♦r ❛♥❞ ❛ ♣❛t✐❡♥t ❢r♦♠ ❞✐✛❡r❡♥t ❤♦s♣✐t❛❧s✳ ❚❤✐s ❝♦♥tr❛st ❜❡t✇❡❡♥ t❤❡ ♦✈❡r❛❧❧ ♠❛r❦❡t ❛♥❞ t❤❡ ♣❧❛t❢♦r♠s ✐s str✐❦✐♥❣ ❜❡❝❛✉s❡ t❤❡ ♣❧❛t❢♦r♠s t♦ ♥♦t ♣r✐♦r✐t✐③❡ ❛❝r♦ss ❤♦s♣✐t❛❧ ❡①❝❤❛♥❣❡s ❛s ❛ r✉❧❡✳ ❘❛t❤❡r✱ t❤❡ ♣r❡❞♦♠✐♥❛♥❝❡ ♦❢ ❛❝r♦ss ❤♦s♣✐t❛❧ ❡①❝❤❛♥❣❡s ✐♥ t❤❡ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ✐s ❛ ❜②✲♣r♦❞✉❝t ♦❢ ♠❛①✐♠✐③✐♥❣ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts✳ ❚❤✐s s✉❣❣❡sts t❤❛t ❝♦✲♦r❞✐♥❛t✐♥❣ ❛❝r♦ss ❤♦s♣✐t❛❧s ❤❛s ♣♦t❡♥t✐❛❧ ❣❛✐♥s✳ ❲❡ ✇✐❧❧ ❢♦r♠❛❧❧② ❛♥❛❧②③❡ ❛♥❞ q✉❛♥t✐❢② t❤❡s❡ ❣❛✐♥s ✐♥ t❤❡ s✉❜s❡q✉❡♥t s❡❝t✐♦♥s✳ ❋✐❣✉r❡ ✶ ❛❧s♦ s❤♦✇s t❤❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ❣r❡✇ ❢r♦♠ ❛❜♦✉t ✹✵✵ tr❛♥s♣❧❛♥ts ✐♥ ✷✵✵✽ t♦ ❛❜♦✉t ✽✵✵ ✐♥ ✷✵✶✹✳✶✾ ❍♦✇❡✈❡r✱ ♦✈❡r❛❧❧ ♠❛r❦❡t ❣r♦✇t❤ s❡❡♠s t♦ ❤❛✈❡ s❧♦✇❡❞ ✐♥ r❡❝❡♥t ②❡❛rs✳ ❚❤❡ t♦t❛❧ ♥✉♠❜❡r r❡♠❛✐♥s ❛t ❛r♦✉♥❞ ✽✵✵✱✷✵ ✇❡❧❧ ❜❡❧♦✇ s♦♠❡ ✶✾ ❖✉r

❞❛t❛ ❢♦r t❤❡ ◆❑❘ ❡①t❡♥❞ ✉♥t✐❧ ❉❡❝❡♠❜❡r ✹✱ ✷✵✶✹✳ ❚❤✐s ❝❡♥s♦r✐♥❣ ♠❛② ❛❝❝♦✉♥t ❢♦r t❤❡ s❧✐❣❤t ❞r♦♣ ✐♥ tr❛♥s♣❧❛♥ts ✐♥ t❤❡ ❧❛st ②❡❛r ♦❢ t❤✐s ✜❣✉r❡✳ ✷✵ ❙♦✉r❝❡✿ ❤tt♣s✿✴✴♦♣t♥✳tr❛♥s♣❧❛♥t✳❤rs❛✳❣♦✈✴❞❛t❛✴✈✐❡✇✲❞❛t❛✲r❡♣♦rts✴♥❛t✐♦♥❛❧✲❞❛t❛✴ ✭❛❝❝❡ss❡❞ ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✼✮✳

✶✸ ❡st✐♠❛t❡s ♦❢ t❤❡ ♣♦t❡♥t✐❛❧ s✐③❡ ♦❢ t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡t ✭❇✐♥❣❛♠❛♥ ❡t ❛❧✳✱ ✷✵✶✷❀ ▼❛ss✐❡ ❡t ❛❧✳✱ ✷✵✶✸✮✳ ❚❤❡ ❣r♦✇t❤ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ✷✵✶✵ ❛♥❞ ✷✵✶✹ ✐s ❝♦♥❝✉rr❡♥t ✇✐t❤ t❤❡ ◆❑❘ ❜❡❝♦♠✐♥❣ t❤❡ ❞♦♠✐♥❛♥t ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳ ■t ❛❝❝♦✉♥t❡❞ ❢♦r ✸✸✳✶✪ ♦❢ ❛❧❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ✐♥ ✷✵✶✹✱ ❛♥❞ ❢❛❝✐❧✐t❛t❡❞ ♠♦r❡ t❤❛♥ ✺ t✐♠❡s ❛s ♠❛♥② tr❛♥s♣❧❛♥ts ❛s t❤❡ ❆P❉ ❛♥❞ ❯◆❖❙ ❝♦♠❜✐♥❡❞✳✷✶ ❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ ◆❑❘ ❞✉r✐♥❣ ♦✉r s❛♠♣❧❡ ♣❡r✐♦❞ ♠♦t✐✈❛t❡s ♦✉r ❢♦❝✉s ♦♥ t❤❡ ♣❧❛t❢♦r♠ ✐♥ t❤❡ s✉❜s❡q✉❡♥t s❡❝t✐♦♥s✳

✸✳✷ ❊✈✐❞❡♥❝❡ ♦❢ ■♥❡✣❝✐❡♥❝② ▼❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥ ♣♦✐♥ts t♦ ✐♥❡✣❝✐❡♥❝② ✐❢ t❤❡ r❡t✉r♥s t♦ s❝❛❧❡ ✐♥ ♠❛t❝❤✐♥❣ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❛r❡ ❤✐❣❤✳ ❍♦✇❡✈❡r✱ ✐t ✐s ♣♦ss✐❜❧❡✱ ❛t ❧❡❛st ✐♥ ♣r✐♥❝✐♣❧❡✱ t❤❛t t❤❡ ♠❛r❦❡t ✐s ❤✐❣❤❧② ❢r❛❣♠❡♥t❡❞ ❜✉t ❝❧♦s❡ t♦ ❡✣❝✐❡♥t✳ ❋♦r ❡①❛♠♣❧❡✱ ✐t ♠❛② ❜❡ t❤❛t ❤♦s♣✐t❛❧s ♦♣❡r❛t❡ ❛t ❛♥ ❛♣♣r♦①✐♠❛t❡❧② ❡✣❝✐❡♥t s❝❛❧❡✱ s♦ t❤❛t t❤❡ ❧♦ss ❞✉❡ t♦ ♠❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥ ✐s s♠❛❧❧✳ ❲❡ ♥♦✇ ♣r❡s❡♥t ❞✐r❡❝t ❡✈✐❞❡♥❝❡ t❤❛t t❤✐s ✐s ♥♦t t❤❡ ❝❛s❡ ❜② s❤♦✇✐♥❣ t❤❛t ❤♦s♣✐t❛❧s ♦❢t❡♥ ❝♦♥❞✉❝t ❡①❝❤❛♥❣❡s t❤❛t ❛r❡ ✐♥❡✣❝✐❡♥t ❢r♦♠ ❛ s♦❝✐❛❧ ♣❡rs♣❡❝t✐✈❡✳ ❖♥❡ ❡❛s✐❧② ❞❡t❡❝t❛❜❧❡ ✐♥❡✣❝✐❡♥❝② ✐s ❛ tr❛♥s♣❧❛♥t ❜❡t✇❡❡♥ ❛♥ ❖ ❜❧♦♦❞ t②♣❡ ❞♦♥♦r ❛♥❞ ❛ ♥♦♥✲❖ ❜❧♦♦❞ t②♣❡ ♣❛t✐❡♥t✳ ❆s ❡①♣❧❛✐♥❡❞ ✐♥ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ ❛♥❞ ✐♥ ❙❡❝t✐♦♥ ✷✱ ❖ ❞♦♥♦rs ❛r❡ s❝❛r❝❡ ✇❤✐❧❡ ❖ ♣❛t✐❡♥ts ❛r❡ ❛❜✉♥❞❛♥t✳ ■❢ ❛❧❧ tr❛♥s♣❧❛♥ts ❛r❡ ♦❢ ❡q✉❛❧ s♦❝✐❛❧ ✈❛❧✉❡✱ ♦♣t✐♠❛❧ ♠❛t❝❤❡s ✐♥ ❛ ❧❛r❣❡ ♠❛r❦❡t s❤♦✉❧❞ ♦♥❧② tr❛♥s♣❧❛♥t ♦r❣❛♥s ❢r♦♠ ❖ ❞♦♥♦rs t♦ ❖ ♣❛t✐❡♥ts ❜❡❝❛✉s❡ ❖ ♣❛t✐❡♥ts ❝❛♥♥♦t ❛❝❝❡♣t ♦t❤❡r ❜❧♦♦❞ t②♣❡s✳✷✷ ❚❤❡ ❡①❝❡♣t✐♦♥ t♦ t❤✐s r✉❧❡ ✐s ❢♦r ❛ ❤✐❣❤❧② s❡♥s✐t✐③❡❞ ♣❛t✐❡♥t✱ t❤❛t ✐s✱ ♦♥❡ ✇✐t❤ ❛ ✈❡r② ❤✐❣❤ P❘❆✳ ❚❤❡ ♣❧❛t❢♦r♠ ♠❛② ✇❛♥t t♦ ✉s❡ ❛♥ ❖ ❞♦♥♦r t♦ tr❛♥s♣❧❛♥t t❤✐s ♣❛t✐❡♥t ✐❢ ✐t ✐s t❤❡ ♦♥❧② ✇❛② t♦ ❣❡t t❤✐s ♣❛t✐❡♥t tr❛♥s♣❧❛♥t❡❞✳ ❋✐❣✉r❡ ✷ ❞✐s♣❧❛②s t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❖ ❞♦♥♦rs t❤❛t ❛r❡ ✉s❡❞ t♦ tr❛♥s♣❧❛♥t ♥♦♥✲❖ ♣❛t✐❡♥ts✱ ❝❛t❡✲ ❣♦r✐③❡❞ ✐♥t♦ ◆❑❘ tr❛♥s♣❧❛♥ts✱ ❆P❉✴❯◆❖❙ tr❛♥s♣❧❛♥ts✱ ❛❝r♦ss ❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts ❛t ♦t❤❡r ♣❧❛t❢♦r♠s✱ ❛♥❞ ✇✐t❤✐♥ ❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts ❛t ♦t❤❡r ♣❧❛t❢♦r♠s✳ ❆♠♦♥❣ ◆❑❘ tr❛♥s♣❧❛♥ts✱ ♦♥❧② ✻✳✺✪ ♦❢ ❖ ❞♦♥♦rs ❛r❡ ✉s❡❞ ❢♦r ♥♦♥✲❖ ♣❛t✐❡♥ts✳ ■♥ ❝♦♥tr❛st✱ ❛♠♦♥❣ ✇✐t❤✐♥ ❤♦s♣✐t❛❧ tr❛♥s✲ ♣❧❛♥ts ♦✉ts✐❞❡ t❤❡ t❤r❡❡ ♣❧❛t❢♦r♠s✱ t❤✐s ♣❡r❝❡♥t❛❣❡ ✐s ✷✷✳✽✪✳ ❚❤✐s ❞✐✛❡r❡♥❝❡ ✐s st❛t✐st✐❝❛❧❧② s✐❣♥✐✜❝❛♥t ✭p < 0.01✮ ❛♥❞ ❝♦♥st✐t✉t❡s str♦♥❣ ❡✈✐❞❡♥❝❡ t❤❛t ❤♦s♣✐t❛❧s ♦❢t❡♥ ♣❡r❢♦r♠ ✐♥❡✣❝✐❡♥t ♠❛t❝❤❡s ♦✉ts✐❞❡ t❤❡ ♣❧❛t❢♦r♠✳ ❚r❛♥s♣❧❛♥ts ❛t ❆P❉✱ ❯◆❖❙ ❛♥❞ ❛❝r♦ss✲❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts ❛t ♦t❤❡r ♣❧❛t❢♦r♠s ❛r❡ ✐♥ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❝❛t❡❣♦r✐❡s ❜✉t ❛r❡ ♠✉❝❤ ❝❧♦s❡r t♦ t❤❡ ◆❑❘✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ❡①♣❧❛♥❛t✐♦♥ ❢♦r ✐♥❡✣❝✐❡♥t ♠❛t❝❤✐♥❣ ✐s t❤❛t ✇✐t❤✐♥ ❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts ✉s❡ ❖ ❞♦♥♦rs t♦ ❤❡❧♣ ❤✐❣❤❧② s❡♥s✐t✐③❡❞ ♣❛t✐❡♥ts ✇❤♦ ✇♦✉❧❞ ♦t❤❡r✇✐s❡ r❡♠❛✐♥ ✉♥tr❛♥s♣❧❛♥t❡❞✳ ❍♦✇❡✈❡r✱ ❋✐❣✉r❡ ✷ s❤♦✇s t❤❛t ❛❧♠♦st ♥♦♥❡ ♦❢ t❤❡ ♣♦t❡♥t✐❛❧❧② ✐♥❡✣❝✐❡♥t tr❛♥s♣❧❛♥ts ✐♥ t❤❡ ❖t❤❡r ✭✇✐t❤✐♥ ❤♦s♣✐t❛❧✮ ❝❛t❡❣♦r② ✐♥✈♦❧✈❡ ❤✐❣❤❧② s❡♥s✐t✐③❡❞ ♣❛t✐❡♥ts✳ ■♥ ❝♦♥tr❛st✱ ❛❜♦✉t ❤❛❧❢ ♦❢ t❤❡ ♣♦t❡♥t✐❛❧❧② ✐♥❡✣❝✐❡♥t ◆❑❘ tr❛♥s♣❧❛♥ts ✐♥✈♦❧✈❡ ❤✐❣❤❧② s❡♥s✐t✐③❡❞ ♣❛t✐❡♥ts✳ ✷✶ ❚❤❡

❆P❉ ❤❛s ❣r♦✇♥ ✐♥ r❡❝❡♥t ②❡❛rs ❛♥❞ ❤❛s s✐❣♥✐✜❝❛♥t❧② ❝❧♦s❡❞ t❤❡ ❣❛♣✳ s♣❡❛❦✐♥❣✱ ❡✣❝✐❡♥❝② ❛s ❞✐s❝✉ss❡❞ ❤❡r❡ ♠❡❛♥s ♠❛①✐♠✐③✐♥❣ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts✳ ❍♦✇✲ ❡✈❡r✱ tr❛♥s♣❧❛♥t✐♥❣ ❛♥ ❖ ❞♦♥♦r t♦ ❛ ♥♦♥✲❖ ♣❛t✐❡♥t ✐s ❛❧s♦ ❧✐❦❡❧② t♦ ❜❡ P❛r❡t♦ ✐♥❡✣❝✐❡♥t✳ ❚♦ s❡❡ t❤✐s✱ ❝♦♥s✐❞❡r ❛ ♣❛✐r✇✐s❡ ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ t✇♦ ♦✈❡r❞❡♠❛♥❞❡❞ ❆✲❖ ♣❛✐rs✳ ❚❤✐s ❡①❝❤❛♥❣❡ r❡s✉❧ts ✐♥ t✇♦ tr❛♥s♣❧❛♥ts✳ ■t ✇♦✉❧❞ ❜❡ ♠♦r❡ ❡✣❝✐❡♥t t♦ tr❛♥s♣❧❛♥t ❡❛❝❤ ♦❢ t❤❡ ❆✲❖ ♣❛✐rs t♦ ❛♥ ✉♥❞❡r❞❡♠❛♥❞❡❞ ❖✲❆ ♣❛✐r✱ ✇❤✐❝❤ ♦t❤❡r✇✐s❡ ✇♦✉❧❞ ❜❡ ❧❡❢t ✉♥♠❛t❝❤❡❞✳ ✷✷ ❙tr✐❝t❧②

✶✹

❚❛❜❧❡ ✶✿ ❙✉♠♠❛r② ❙t❛t✐st✐❝s ❢♦r ❑✐❞♥❡② ❊①❝❤❛♥❣❡ ❚r❛♥s♣❧❛♥ts

NKR

APD / UNOS

Other platforms Across Hospital Within Hospital

1118

198

341

2719

34.7% 19.0% 5.7% 40.6%

36.4% 21.2% 3.5% 38.9%

37.2% 17.6% 7.0% 38.1%

37.1% 17.0% 5.7% 40.2%

36.8% 18.2% 3.9% 41.1%

35.4% 20.2% 1.5% 42.9%

37.5% 14.7% 6.7% 41.1%

33.4% 13.8% 2.9% 49.9%

35.0 39.7 16.4%

43.0 40.8 20.6%

30.4 37.5 12.0%

17.6 30.8 5.1%

Donor Age Mean Standard deviation

44.1 11.8

44.6 11.1

44.1 11.3

43.2 11.8

Donor Body Mass Index (BMI) Mean Standard deviation

26.5 4.0

27.0 4.0

26.6 4.1

26.5 4.2

Donor Height (cm) Mean Standard deviation

169.4 9.8

168.0 9.6

169.6 10.3

169.3 9.8

Donor Weight (kg) Mean Standard deviation

76.3 15.1

76.3 13.9

76.9 15.4

76.3 15.1

4.2 1.3

4.2 1.4

4.2 1.2

4.4 1.2

1026.6 1088.1

1048.4 848.1

1063.1 1269.5

969.1 990.9

N Patient Blood Type A B AB O Donor Blood Type A B AB O Panel Reactive Antibody (PRA) (Sensitization) Mean Standard deviation Percent >90 Transplant Ourcomes and Quality Measures

Tissue Type Mismatch (0-6) Mean Standard deviation Mean Days on Dialysis Mean Standard deviation

◆♦t❡s

✿ ❙❛♠♣❧❡ ♦❢ ❛❧❧ ❑✐❞♥❡② ❊①❝❤❛♥❣❡ ❚r❛♥s♣❧❛♥ts ❜❡t✇❡❡♥ ❏❛♥✉❛r② ✶✱ ✷✵✵✽ ❛♥❞ ❉❡❝❡♠❜❡r ✹✱ ✷✵✶✹✳

✶✺

% O donors matched to non-O patients

25%

20%

15%

10%

5%

0% NKR

APD/UNOS

Recipient PRA ≥ 90

◆♦t❡s ✿

Other platforms (across hospitals)

Other platforms (within-hospital)

Recipient PRA < 90

❋✐❣✉r❡ ✷✿ ❙♠♦❦✐♥❣✲❣✉♥ ❡✈✐❞❡♥❝❡ ♦❢ ❤♦s♣✐t❛❧s ♣❡r❢♦r♠✐♥❣ ✐♥❡✣❝✐❡♥t ♠❛t❝❤❡s

❚❤❡ ❜❛rs ❞✐s♣❧❛② t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ tr❛♥s♣❧❛♥t❡❞ ❖ ❞♦♥♦rs ✇❤♦s❡ ❦✐❞♥❡②s ✇❡r❡ tr❛♥s♣❧❛♥t❡❞ ✐♥t♦ ♥♦♥✲❖ ♣❛t✐❡♥ts ❢♦r ❞✐✛❡r❡♥t ❝❛t❡❣♦r✐❡s ♦❢ tr❛♥s♣❧❛♥ts✳ ◆❑❘✱ ❆P❉✴❯◆❖❙ ❛♥❞ ❖t❤❡r ❝❧❛ss✐✜❡s ❛ tr❛♥s♣❧❛♥t ✐♥t♦ t❤❡ ♣❧❛t❢♦r♠ t❤❛t ❢❛❝✐❧✐t❛t❡❞ t❤❡ ❡①❝❤❛♥❣❡✱ ✐♥❝❧✉❞✐♥❣ s✐♥❣❧❡✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s✳ ❲✐t❤✐♥ ❤♦s♣✐t❛❧ ❛♥❞ ❛❝r♦ss ❤♦s♣✐t❛❧ ❝❧❛ss✐❢② ❛ tr❛♥s♣❧❛♥t ✐♥t♦ ✇❤❡t❤❡r t❤❡ ❞♦♥♦r ❤♦s♣✐t❛❧ ✇❛s t❤❡ s❛♠❡ ❛s t❤❡ ♣❛t✐❡♥t ❤♦s♣✐t❛❧✳ ❚❤❡ ❝♦❧♦rs ❞❡❝♦♠♣♦s❡ t❤✐s t♦t❛❧ ✐♥t♦ ❤✐❣❤❧② s❡♥s✐t✐③❡❞ ♣❛t✐❡♥ts ✭P❘❆ ❃✾✵✮ ❛♥❞ ♥♦♥✲❤✐❣❤❧② s❡♥s✐t✐③❡❞ ♣❛t✐❡♥ts✳ ❚❤❡ ❡rr♦r ❜❛rs ❞❡♣✐❝t ✾✺✪ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ❢♦r t❤❡ t♦t❛❧s✳

❚❤✐s ❡①❡r❝✐s❡ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ✈❛❧✉❡ ♦❢ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❤♦✇ ✐t ✇❛s ♦r❣❛♥✐③❡❞✳ ❙❡❝t✐♦♥ ✷✳✶ ❛r❣✉❡s t❤❛t t❤❛t ❧♦❣✐st✐❝❛❧ ❝♦sts ♦❢ ❝♦♥❞✉❝t✐♥❣ tr❛♥s♣❧❛♥ts t❤r♦✉❣❤ ❛ ♣❧❛t❢♦r♠ ❛r❡ ♥❡❣❧✐❣✐❜❧❡ t♦ t❤❡ ✈❛❧✉❡ ♦❢ tr❛♥s♣❧❛♥ts ❧♦st ❜② ✉s✐♥❣ ♦r❣❛♥s ❢r♦♠ ❖ ❞♦♥♦rs t♦ tr❛♥s♣❧❛♥t ♥♦♥✲❖ ♣❛t✐❡♥ts✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ♠❛② ❜❡ ❞✐♠❡♥s✐♦♥s ♦♥ ✇❤✐❝❤ ✇✐t❤✐♥ ❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts ❛r❡ s✉♣❡r✐♦r t♦ tr❛♥s♣❧❛♥ts ♦r❣❛♥✐③❡❞ ❜② ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ tr❛♥s♣❧❛♥t t❤r♦✉❣❤ ❛ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠ ❝♦✉❧❞ ✐♥✈♦❧✈❡ ❛ ❧♦♥❣❡r ✇❛✐t ♦♥ ❞✐❛❧②s✐s ♦r ❛ ❧♦✇❡r✲q✉❛❧✐t② ❞♦♥♦r✳ ❍♦✇❡✈❡r✱ ❚❛❜❧❡ ✶ s❤♦✇s t❤❛t ♣❛t✐❡♥ts ✇❤♦ r❡❝❡✐✈❡ ❛ tr❛♥s♣❧❛♥t t❤r♦✉❣❤ ❛ ♣❧❛t❢♦r♠ t②♣✐❝❛❧❧② s♣❡♥❞ ♦♥❧② t✇♦ ♠♦r❡ ♠♦♥t❤s ♦♥ ❞✐❛❧②s✐s t❤❛♥ ♣❛t✐❡♥ts ✇❤♦ r❡❝❡✐✈❡ ❛ ✇✐t❤✐♥ ❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥t ♦✉ts✐❞❡ t❤❡s❡ t❤r❡❡ ♣❧❛t❢♦r♠s✳ ●✐✈❡♥ t❤❛t t❤❡ ❛✈❡r❛❣❡ ♣❛t✐❡♥t ✇❛✐t ✐s ❛❜♦✉t ✸✷ ♠♦♥t❤s✱ t❤✐s ❞✐✛❡r❡♥❝❡ r❡♣r❡s❡♥ts ❛♥ ✽✪ ❧♦♥❣❡r ✇❛✐t✐♥❣ t✐♠❡✳ ❚❤❡ ❧♦♥❣❡r ✇❛✐t✐♥❣ t✐♠❡ ❛t t❤❡ ♣❧❛t❢♦r♠s s❤♦✉❧❞ ❜❡ ❡①♣❡❝t❡❞ ❜❡❝❛✉s❡✱ ❛s ✇❡ ❞✐s❝✉ss ❜❡❧♦✇✱ ♣❛t✐❡♥ts tr❛♥s♣❧❛♥t❡❞ t❤r♦✉❣❤ t❤❡ ♣❧❛t❢♦r♠ ❛r❡✱ ♦♥ ❛✈❡r❛❣❡✱ ❤❛r❞❡r t♦ ♠❛t❝❤✳ ❋✉rt❤❡r✱ t❤❡r❡ ❞♦ ♥♦t s❡❡♠ t♦ ❜❡ ❞✐✛❡r❡♥❝❡s ✐♥ ❤♦✇ ❞❡s✐r❛❜❧❡ ❞♦♥♦rs ♠✐❣❤t ❜❡ t♦ ♣❛t✐❡♥ts✳ ❉♦♥♦r q✉❛❧✐t② ✐♥❞✐❝❛t♦rs s✉❝❤ ❛s ❛❣❡✱ ✇❡✐❣❤t✱ ❤❡✐❣❤t✱ ❛♥❞ ❇▼■ ❛r❡ s✐♠✐❧❛r ❛❝r♦ss ♣❧❛t❢♦r♠s✳ ❖♥❡ r❡❛s♦♥ ✇❤② ♣❛t✐❡♥ts ❝♦♥s✐❞❡r✐♥❣ ❛ ♠✉❧t✐✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠ ♥❡❡❞ ♥♦t ✇♦rr② ❛❜♦✉t ❞♦♥♦r q✉❛❧✐t② ✐s t❤❛t t❤❡ ♣❧❛t❢♦r♠s t②♣✐❝❛❧❧② ❛❧❧♦✇ ♣❛t✐❡♥ts ❛♥❞ ❞♦❝t♦rs t♦ s♣❡❝✐❢② ❞♦♥♦r ❛❝❝❡♣t❛❜✐❧✐t② ❝r✐t❡r✐❛✳ ❚❤❡② ❛❧s♦ ❛❧❧♦✇ ♣❛t✐❡♥ts t♦ r❡❢✉s❡ ♣r♦♣♦s❡❞ tr❛♥s♣❧❛♥ts ✐❢ t❤❡ ❞♦♥♦r ✐s ✉♥s✉✐t❛❜❧❡✳ ■❢ ❡❛❝❤ ♦❢ t❤❡s❡ ✐♥❡✣❝✐❡♥t tr❛♥s♣❧❛♥ts ❝♦♠❡s ❛t t❤❡ ❝♦st ♦❢ ♦♥❡ ♦t❤❡r tr❛♥s♣❧❛♥t✱ ❛s ✐♥ t❤❡

✶✻ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ ♠♦❞❡❧✱ t❤❡♥ ❛❝❤✐❡✈✐♥❣ t❤❡ ❧❡✈❡❧ ♦❢ ❡✣❝✐❡♥❝② ♦❜t❛✐♥❡❞ ❜② t❤❡ ◆❑❘ ✇♦✉❧❞ ❤❛✈❡ r❡s✉❧t❡❞ ✐♥ ❛❜♦✉t ✷✺✵ ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥ts ❜❡t✇❡❡♥ ✷✵✵✽ ❛♥❞ ✷✵✶✹✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❝♦♥s✐❞❡r✐♥❣ ♦♥❧② t❤❡ ❝❧❡❛r❧② ✐♥❡✣❝✐❡♥t tr❛♥s♣❧❛♥ts ✐s t❤❛t t❤❡ r❡s✉❧ts ♣r♦✈✐❞❡ tr❛♥s♣❛r❡♥t ❡✈✐❞❡♥❝❡ ♦❢ ✐♥❡✣❝✐❡♥❝②✳ ❚❤❡ t♦t❛❧ ✐♥❡✣❝✐❡♥❝②✱ ♦❢ ❝♦✉rs❡✱ ❝❛♥ ❜❡ ♠✉❝❤ ❧❛r❣❡r✳ ✸✳✸

❍♦s♣✐t❛❧ P❛rt✐❝✐♣❛t✐♦♥ ❇❡❤❛✈✐♦r ❛♥❞ ❊✈✐❞❡♥❝❡ ♦❢ ❆❣❡♥❝② Pr♦❜✲ ❧❡♠s

❲❡ ❤❛✈❡ s❤♦✇♥ t❤❛t t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡t ✐s ❤✐❣❤❧② ❢r❛❣♠❡♥t❡❞✱ ✇✐t❤ ♠❛♥② ❤♦s♣✐t❛❧s ♣❡r❢♦r♠✐♥❣ ♠❛t❝❤❡s ♦✉ts✐❞❡ t❤❡ t❤r❡❡ ♠✉❧t✐✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s✳ ▼♦r❡♦✈❡r✱ t❤✐s ❢r❛❣♠❡♥t❛t✐♦♥ ❧❡❛❞s t♦ r❡❛❧ ❡✣❝✐❡♥❝② ❧♦ss❡s ❜❡❝❛✉s❡ ✐♥t❡r♥❛❧ ♠❛t❝❤❡s ♣❡r❢♦r♠❡❞ ❜② ❤♦s♣✐t❛❧s ❛r❡ ♦❢t❡♥ s♦❝✐❛❧❧② ✐♥❡✣❝✐❡♥t✳ ❚❤❡s❡ r❡s✉❧ts ❧❡❛❞ ✉s t♦ ❛s❦ ✇❤② ❤♦s♣✐t❛❧s ❞♦ ♥♦t ♣❛rt✐❝✐♣❛t❡ ♠♦r❡ ✐♥ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s✳ ❚♦ ❛❞❞r❡ss t❤✐s ✐ss✉❡✱ ✇❡ ✜rst ❞♦❝✉♠❡♥t ❦❡② ❢❛❝ts ❛❜♦✉t ❤♦s♣✐t❛❧ ❜❡❤❛✈✐♦r ❛♥❞ t❤❡♥ ❞✐s❝✉ss ❞✐✛❡r❡♥t ❤②♣♦t❤❡s❡s t❤❛t ❝❛♥ ❡①♣❧❛✐♥ ✐t✳ ❖✉r r❡s✉❧ts ✐♥❞✐❝❛t❡ t❤❛t ❤♦s♣✐t❛❧s ❞♦ ♥♦t ♣✉r❡❧② ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥t❡❞ ♣❛t✐❡♥ts✳ ■♥st❡❛❞✱ ❤♦s♣✐t❛❧s s❡❡♠ t♦ ♠❛①✐♠✐③❡ ❝♦♠♣❧❡① ❛♥❞ ❤❡t❡r♦❣❡♥❡♦✉s ♦❜❥❡❝t✐✈❡s✱ ✐♥❝❧✉❞✐♥❣ ❜✉t ♥♦t ❧✐♠✐t❡❞ t♦ ♣r♦✜ts ❛♥❞ ♣❛t✐❡♥t ✇❡❧❢❛r❡✳ ✸✳✸✳✶

❉❡s❝r✐♣t✐✈❡ ❊✈✐❞❡♥❝❡

❲❡ ❢♦❝✉s ♦♥ ♣❛rt✐❝✐♣❛t✐♦♥ ❜❡❤❛✈✐♦r ❛t t❤❡ ◆❑❘ ❜❡❝❛✉s❡ ✐t ✐s t❤❡ ♣r✐♠❛r② ♠✉❧t✐✲❤♦s♣✐t❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ❞✉r✐♥❣ ♦✉r s❛♠♣❧❡ ♣❡r✐♦❞ ✭❚❛❜❧❡ ✶✮✳ ❚❤❡ r❡s✉❧ts ❛r❡ q✉❛❧✐t❛t✐✈❡❧② s✐♠✐❧❛r ✐❢ ✇❡ ❝♦♥s✐❞❡r ♣❛rt✐❝✐♣❛t✐♦♥ ❛t ❛♥② ♦❢ t❤❡ t❤r❡❡ ❧❛r❣❡st ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s✳ ❲❡ st❛rt ❜② ❞❡s❝r✐❜✐♥❣ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ❤♦s♣✐t❛❧ s✐③❡ ❛♥❞ t❤❡ ❡①t❡♥s✐✈❡ ❛♥❞ ✐♥t❡♥s✐✈❡ ♠❛r❣✐♥s ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ t❤❡ ◆❑❘✳ ❲❡ ♠❡❛s✉r❡ ❤♦s♣✐t❛❧ s✐③❡ ❛s t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ❝♦♥❞✉❝t❡❞ ♣❡r ②❡❛r✱ ✐♥❝❧✉❞✐♥❣ ❞❡❝❡❛s❡❞ ❞♦♥♦r ❛♥❞ ❞✐r❡❝t ❧✐✈✐♥❣ ❞♦♥♦r ❦✐❞♥❡② tr❛♥s♣❧❛♥ts✳✷✸ ❋✐❣✉r❡ ✸ ❞❡♣✐❝ts t❤❡ ❡①t❡♥s✐✈❡ ♠❛r❣✐♥ ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ ❛♠♦♥❣st ❤♦s♣✐t❛❧s ❝♦♥❞✉❝t✐♥❣ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts✳ ❆ ❤♦s♣✐t❛❧ ✐s ❝♦♥s✐❞❡r❡❞ ❛♥ ◆❑❘ ♣❛rt✐❝✐♣❛♥t ✐❢ ✐t ❤❛s ❡✈❡r s✉❜♠✐tt❡❞ ❛ ♣❛t✐❡♥t ♦r ❞♦♥♦r t♦ t❤❡ ◆❑❘✳ ❚❤❡ ✜❣✉r❡ ✐s ❛ ❜✐♥♥❡❞ s❝❛tt❡r♣❧♦t ♦❢ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❤♦s♣✐t❛❧s t❤❛t ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ◆❑❘ ✈❡rs✉s ❤♦s♣✐t❛❧ s✐③❡✳ ❋✐❣✉r❡ ✹ ❞❡♣✐❝ts t❤❡ ✐♥t❡♥s✐✈❡ ♠❛r❣✐♥ ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥✳ ❚❤❡ ✈❡rt✐❝❛❧ ❛①✐s ✐♥ t❤✐s s❝❛tt❡r♣❧♦t ✐s t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts t❤❛t ❛ ❤♦s♣✐t❛❧ ♣❡r❢♦r♠s t❤r♦✉❣❤ t❤❡ ◆❑❘✳ ❚❤❡ ✜❣✉r❡s r❡✈❡❛❧ ❢♦✉r ❦❡② ❢❛❝ts ❛❜♦✉t ♣❛rt✐❝✐♣❛t✐♦♥✳ ❋✐rst✱ ❜♦t❤ t❤❡ ❡①t❡♥s✐✈❡ ❛♥❞ ✐♥t❡♥s✐✈❡ ♠❛r❣✐♥s ❛r❡ ✐♠♣♦rt❛♥t ❞r✐✈❡rs ♦❢ ♠❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥✳ ❖♥❧② ✹✻✳✸✪ ♦❢ ❤♦s♣✐t❛❧s ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ◆❑❘✳ ❲✐t❤✐♥ t❤♦s❡ ♣❛rt✐❝✐♣❛t✐♥❣ ❤♦s♣✐t❛❧s✱ ♦♥❧② ✺✷✳✾✪ ♦❢ tr❛♥s♣❧❛♥ts ❛r❡ ❝♦♥❞✉❝t❡❞ t❤r♦✉❣❤ t❤❡ ◆❑❘✳ ❙❡❝♦♥❞✱ ❧❛r❣❡r ❤♦s♣✐t❛❧s ❛r❡ ❝♦♥s✐❞❡r❛❜❧② ♠♦r❡ ❧✐❦❡❧② t♦ ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ◆❑❘✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ❛t ❛❧❧ ✐s ❛❜♦✉t ✽✵✪ ❢♦r ❛ ❤♦s♣✐t❛❧ t❤❛t ♣❡r❢♦r♠s ❛♣♣r♦①✲ ✐♠❛t❡❧② ✷✺✵ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r ❜✉t ♦♥❧② ❛❜♦✉t ✸✺✪ ❢♦r ❛ ❤♦s♣✐t❛❧ t❤❛t ♣❡r❢♦r♠s ❛❜♦✉t ✺✵ ✷✸ ❚❤✐s

❜r♦❛❞ ♠❡❛s✉r❡ ♦❢ s✐③❡ ❧✐♠✐ts t❤❡ ❡♥❞♦❣❡♥♦✉s ❡✛❡❝t ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ t❤❡ ✉✐◆❑❘ ♦♥ ❤♦s♣✐t❛❧ s✐③❡ ❜❡❝❛✉s❡ ❞❡❝❡❛s❡❞ ❞♦♥♦r ❛♥❞ ❞✐r❡❝t ❧✐✈✐♥❣ ❞♦♥♦r tr❛♥s♣❧❛♥ts ❢♦r♠ t❤❡ ❜✉❧❦ ♦❢ ❦✐❞♥❡② tr❛♥s♣❧❛♥ts ❝♦♥❞✉❝t❡❞ ❜② ❤♦s♣✐t❛❧s✳ ▼♦r❡♦✈❡r✱ ❞✉r✐♥❣ ♦✉r s❛♠♣❧❡ ♣❡r✐♦❞✱ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❦✐❞♥❡② tr❛♥s♣❧❛♥ts ❤❛s r❡♠❛✐♥❡❞ st❛❜❧❡ r❡❧❛t✐✈❡ t♦ t❤❡ ❣r♦✇t❤ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳

0

Fraction of Hospitals in the NKR .2 .4 .6 .8

1

✶✼

0

100 200 Hospital Size (# Transplants per year)

300

Live Exchanges Facilitated through the NKR 0% 25% 50% 75% 100%

❋✐❣✉r❡ ✸✿ ❍❡t❡r♦❣❡♥❡✐t② ✐♥ ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ t❤❡ ◆❑❘

UCLA

Wisconsin

UCSF

Cornell

Hopkins

UC Davis Methodist

0

Jackson Memorial

100 200 300 Hospital Size (# Transplants per year) All centers

NKR Participants

❋✐❣✉r❡ ✹✿ ❘❡❧✐❛♥❝❡ ♦♥ t❤❡ ◆❑❘ ❢♦r ❧✐✈❡✲❞♦♥♦r ❡①❝❤❛♥❣❡s

400

✶✽ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r ✭❋✐❣✉r❡ ✸✮✳ ❚❤✐r❞✱ ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ♣❛rt✐❝✐♣❛t✐♥❣✱ ❧❛r❣❡ ❤♦s♣✐t❛❧s ❝♦♥❞✉❝t ♠♦r❡ ♦❢ t❤❡✐r ♠❛t❝❤❡s ♦✉ts✐❞❡ t❤❡ ♣❧❛t❢♦r♠ ✭❋✐❣✉r❡ ✹✮✳ ❆❧t❤♦✉❣❤ s✐③❡ ♣♦s✐t✐✈❡❧② ❝♦rr❡❧❛t❡s ✇✐t❤ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ♣❡r❢♦r♠❡❞ ✐♥ t❤❡ ◆❑❘✱ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ✐s ♥❡❣❛t✐✈❡ ✐❢ ✇❡ ❢♦❝✉s ❡①❝❧✉s✐✈❡❧② ♦♥ ❤♦s♣✐t❛❧s t❤❛t ♣❛rt✐❝✐♣❛t❡ ❛t ❛❧❧ ✭❋✐❣✉r❡ ✹✮✳ ❋♦✉rt❤✱ t❤❡r❡ ✐s ❛ ❤✐❣❤ ❞❡❣r❡❡ ♦❢ ❤❡t❡r♦❣❡♥❡✐t② ✐♥ ✐♥t❡♥s✐✈❡ ♠❛r❣✐♥ ♣❛rt✐❝✐♣❛t✐♦♥✳ ❊✈❡♥ ❛♠♦♥❣ ❤♦s♣✐t❛❧s ✇✐t❤ s✐♠✐❧❛r s✐③❡✱ ♣❛rt✐❝✐♣❛t✐♦♥ ✈❛r✐❡s ❝♦♥s✐❞❡r❛❜❧② ✭❋✐❣✉r❡ ✹✮✳ ❋♦r ❡①❛♠♣❧❡✱ ❛♠♦♥❣st t❤❡ ✜✈❡ tr❛♥s♣❧❛♥t ❤♦s♣✐t❛❧s t❤❛t ♣❡r❢♦r♠ ♠♦r❡ t❤❛♥ ✸✵✵ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r✱ ♦♥❡ ❞♦❡s ♥♦t ♣❛rt✐❝✲ ✐♣❛t❡ ❛t ❛❧❧ ✭❏❛❝❦s♦♥ ▼❡♠♦r✐❛❧✮✱ ♦♥❡ ♣❛rt✐❝✐♣❛t❡s ❝❧♦s❡ t♦ ③❡r♦ ♣❡r❝❡♥t ✭❯❈ ❉❛✈✐s ▼❡❞✐❝❛❧ ❈❡♥t❡r✮✱ t✇♦ ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ✺✵✲✻✵✪ r❛♥❣❡ ✭❯❈❙❋ ▼❡❞✐❝❛❧ ❈❡♥t❡r ❛♥❞ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❲✐s❝♦♥s✐♥ ❍♦s♣✐t❛❧✮✱ ❛♥❞ ♦♥❡ ♣❛rt✐❝✐♣❛t❡s ♠♦r❡ t❤❛♥ ✽✵✪ ✭❯❈▲❆ ▼❡❞✐❝❛❧ ❈❡♥t❡r✮✳ ■♥ ❛❞❞✐t✐♦♥ t♦ r❡✈❡❛❧✐♥❣ t❤❡ ❞❡❝✐s✐♦♥ t♦ ♣❛rt✐❝✐♣❛t❡✱ t❤❡ ❞❛t❛ ♣r♦✈✐❞❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ ❝❤❛r✲ ❛❝t❡r✐st✐❝s ♦❢ ♣❛t✐❡♥ts s✉❜♠✐tt❡❞ t♦ t❤❡ ◆❑❘ ❛♥❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ♣❛t✐❡♥ts tr❛♥s♣❧❛♥t❡❞ ❜② ❡❛❝❤ ❤♦s♣✐t❛❧ ❝❛t❡❣♦r✐③❡❞ ❜② ❤♦✇ t❤❡ tr❛♥s♣❧❛♥t ✇❛s ❢❛❝✐❧✐t❛t❡❞✳ ❚❛❜❧❡s ✶ ❛♥❞ ✷ r❡✈❡❛❧ t❤r❡❡ ♠❛✐♥ ❢❛❝ts✳ ❋✐rst✱ t❤❡ ◆❑❘ r❡❝❡✐✈❡s s✉❜♠✐ss✐♦♥s t❤❛t ❛r❡ ✈❡r② ❤❛r❞ t♦ ♠❛t❝❤ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❣❡♥❡r❛❧ ♣♦♣✉❧❛t✐♦♥ ✭❚❛❜❧❡ ✷✮✳ ❚❤❡ ❜❧♦♦❞ t②♣❡s ♦❢ ❜♦t❤ ❛❧tr✉✐st✐❝ ❛♥❞ ♣❛✐r❡❞ ❞♦♥♦rs s❦❡✇ ❛✇❛② ❢r♦♠ ❖ ❞♦♥♦rs ❛♥❞ t♦✇❛r❞ ❆ ❞♦♥♦rs r❡❧❛t✐✈❡ t♦ t❤❡ ❯❙ ♣♦♣✉❧❛t✐♦♥✳ ❚❤❡ ❞❡❝❡❛s❡❞ ❞♦♥♦r ♣♦♣✉❧❛t✐♦♥ ❤❛s ❛❜♦✉t ✹✺✪ ❖ ❞♦♥♦rs ❛♥❞ ✹✵✪ ❆ ❞♦♥♦rs✳ ■♥ ❝♦♥tr❛st✱ ♣❛t✐❡♥ts ✐♥ ♣❛✐rs ❛r❡ ❞✐s♣r♦♣♦rt✐♦♥❛t❡❧② ❧✐❦❡❧② t♦ ❤❛✈❡ ❜❧♦♦❞ t②♣❡ ❖ ✭✺✽✳✻✪✮✱ ❛♥❞ t❤❡✐r r❡❧❛t❡❞ ❞♦♥♦rs ❛r❡ ✉♥❧✐❦❡❧② t♦ ❤❛✈❡ ❜❧♦♦❞ t②♣❡ ❖ ✭✸✶✳✾✪✮✳ ❖♥❧② ❛ s♠❛❧❧ ❢r❛❝t✐♦♥ ♦❢ ♣❛✐rs ✭✶✸✳✽✪✮ ❛r❡ ♦✈❡r❞❡♠❛♥❞❡❞✳ ■♥t❡r❡st✐♥❣❧②✱ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥ts ❛r❡ ♠✉❝❤ ♠♦r❡ ❧✐❦❡❧② t♦ ❤❛✈❡ ❛♥ ❡❛s②✲t♦✲♠❛t❝❤ ❜❧♦♦❞ t②♣❡✱ ✇✐t❤ t❤❡ ♠❛❥♦r✐t② ❤❛✈✐♥❣ ❜❧♦♦❞ t②♣❡ ❆✳ ❚❤❡ ❛✈❡r❛❣❡ P❘❆ ❢♦r ♣❛t✐❡♥ts r❡❣✐st❡r❡❞ ✇✐t❤ t❤❡ ◆❑❘ ✐s ❤✐❣❤✳ ❆t ❛ ♠❡❛♥ P❘❆ ♦❢ ✹✽✳✽✪✱ t❤❡ ❛✈❡r❛❣❡ ♣❛t✐❡♥t ✐♥ t❤❡ ◆❑❘ ✐s t✐ss✉❡✲t②♣❡ ✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❛♣♣r♦①✐♠❛t❡❧② ❤❛❧❢ t❤❡ r❡❢❡r❡♥❝❡ ❞♦♥♦r ♣♦♣✉❧❛t✐♦♥✳ ❙❡❝♦♥❞✱ t❤❡ ◆❑❘ tr❛♥s♣❧❛♥ts ♣❛t✐❡♥ts ✇❤♦ ❛r❡ ❝♦♥s✐❞❡r❛❜❧② ❤❛r❞❡r t♦ ♠❛t❝❤ t❤❛♥ ♣❛t✐❡♥ts tr❛♥s♣❧❛♥t❡❞ ❜② s✐♥❣❧❡ ❤♦s♣✐t❛❧s ✭❚❛❜❧❡ ✶✮✳ ❆♣♣r♦①✐♠❛t❡❧② ✹✵✪ ♦❢ t❤❡ ♣❛t✐❡♥ts tr❛♥s♣❧❛♥t❡❞ t❤r♦✉❣❤ t❤❡ ◆❑❘ ❜❧♦♦❞ t②♣❡ ❖✱ ❛♥❞ ✹✶✪ ♦❢ ❞♦♥♦rs ✇❡r❡ ❜❧♦♦❞ t②♣❡ ❖✳ ❚❤❡ P❘❆ ♦❢ t❤❡ ♣❛t✐❡♥ts tr❛♥s♣❧❛♥t❡❞ t❤r♦✉❣❤ t❤❡ ◆❑❘ ✐s ❛♣♣r♦①✐♠❛t❡❧② ✸✺✪✱ ❛♥❞ ❛❜♦✉t ♦♥❡ ✐♥ s✐① ♣❛t✐❡♥ts ❤❛✈❡ ❛ P❘❆ ❛❜♦✈❡ ✾✵✪✳ ❚❤❡s❡ st❛t✐st✐❝s ❛r❡ s✐♠✐❧❛r ❢♦r ❛❝r♦ss ❤♦s♣✐t❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s ♥♦t ❢❛❝✐❧✐t❛t❡❞ ❜② t❤❡ ◆❑❘ ❛♥❞ tr❛♥s♣❧❛♥ts ❢❛❝✐❧✐t❛t❡❞ ❜② ❆P❉ ♦r ❯◆❖❙✳ ■♥ ❝♦♥tr❛st✱ ❛♠♦♥❣ ✇✐t❤✐♥ ❤♦s♣✐t❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s ♥♦t ❝♦♥❞✉❝t❡❞ ❜② ❛ ❧❛r❣❡ ♣❧❛t❢♦r♠✱ ❛❧♠♦st ✺✵✪ ♦❢ t❤❡ ❞♦♥♦rs ❛r❡ ❜❧♦♦❞ t②♣❡ ❖✱ ❜✉t ♦♥❧② ✹✵✪ ♦❢ t❤❡ ♣❛t✐❡♥ts ❛r❡ ❜❧♦♦❞ t②♣❡ ❖✳ ❚❤❡ ❛✈❡r❛❣❡ P❘❆ ♦❢ ♣❛t✐❡♥ts tr❛♥s♣❧❛♥t❡❞ t❤r♦✉❣❤ ✇✐t❤✐♥✲❤♦s♣✐t❛❧ ❡①❝❤❛♥❣❡s ✐s ♦♥❧② ✶✽✪✳ ❚❤✐s ✐s ❛❧♠♦st ❤❛❧❢ t❤❡ ♠❡❛♥ P❘❆ ❢♦r ♣❛t✐❡♥ts tr❛♥s♣❧❛♥t❡❞ t❤r♦✉❣❤ ♦♥❡ ♦❢ t❤❡ t❤r❡❡ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s✳ ❚❤✐r❞✱ tr❛♥s♣❧❛♥ts ♦♥ ❛❧❧ ♣❧❛t❢♦r♠s ❧♦♦❦ s✐♠✐❧❛r ✐♥ ❞♦♥♦r q✉❛❧✐t② ♠❡❛s✉r❡s t❤❛t ❞♦ ♥♦t ❛✛❡❝t ❝♦♠♣❛t✐❜✐❧✐t②✱ s✉❝❤ ❛s ✇❡✐❣❤t✱ ❜♦❞② ♠❛ss ✐♥❞❡①✱ ❛♥❞ ❛❣❡ ✭❚❛❜❧❡ ✶✮✳ ❚❤✐s ❢❛❝t s✉♣♣♦rts ♦✉r ❡q✉❛❧ tr❡❛t♠❡♥t ♦❢ ❛❧❧ tr❛♥s♣❧❛♥ts ❢♦r ✇❡❧❢❛r❡ ❝❛❧❝✉❧❛t✐♦♥s ✐rr❡s♣❡❝t✐✈❡ ♦❢ ✇❤❡t❤❡r t❤❡② ❛r❡ ❢❛❝✐❧✐t❛t❡❞ t❤♦✉❣❤ ❛ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠✳

✶✾

❚❛❜❧❡ ✷✿ ❙✉♠♠❛r② ❙t❛t✐st✐❝s ❢♦r ◆❑❘ ❙✉❜♠✐ss✐♦♥s

N

Altruistic Donors

Pairs

Unpaired Patients

164

1265

501

– – – –

23.8% 15.0% 2.6% 58.6%

51.1% 16.0% 19.0% 14.0%

44.5% 14.0% 3.7% 37.8%

44.4% 18.5% 5.2% 31.9%

– – – –

– –

0.218 0.210

0.431 0.392

0.279 0.162

0.258 0.159

– –

– –

48.8 41.1

44.4 45.1

– –

13.8% 41.9%

– –

Patient Blood Type A B AB O Donor Blood Type A B AB O Match Power Recipient/Pair Mean Standard deviation Donor Mean Standard deviation Panel Reactive Antibody (PRA) (Sensitization) Mean Standard deviation Pair Type Overdemanded Underdemanded

Note: A pair is overdemanded if the patient is blood-type compatible with the related donor. Underdemanded pairs either are O-patients without O-donors or are AB-donors without AB-patients. Sample of all patients and donors registered in the NKR between April 4, 2012 and December 1, 2014.

◆♦t❡s

NKR

✿ ❆ ♣❛✐r ✐s ♦✈❡r❞❡♠❛♥❞❡❞ ✐❢ t❤❡ ♣❛t✐❡♥t ✐s ❜❧♦♦❞ t②♣❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ r❡❧❛t❡❞ ❞♦♥♦r✱ ❜✉t ♥♦t ♦❢ t❤❡ s❛♠❡ ❜❧♦♦❞ t②♣❡✳

❯♥❞❡r❞❡♠❛♥❞❡❞ ♣❛✐rs ❛r❡ ❡✐t❤❡r ❜❧♦♦❞ t②♣❡ ❖ ♣❛t✐❡♥ts ✇✐t❤♦✉t ❜❧♦♦❞ t②♣❡ ❖ ❞♦♥♦rs ♦r ❛r❡ ❜❧♦♦❞ t②♣❡ ❆❇ ❞♦♥♦rs ✇✐t❤♦✉t ❜❧♦♦❞ t②♣❡ ❆❇ ♣❛t✐❡♥ts✳ ❙❛♠♣❧❡ ♦❢ ❛❧❧ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs r❡❣✐st❡r❡❞ ✐♥ t❤❡ ◆❑❘ ❜❡t✇❡❡♥ ❆♣r✐❧ ✷✱ ✷✵✶✷ ❛♥❞ ❉❡❝❡♠❜❡r ✹✱ ✷✵✶✹✳

✷✵ ✸✳✸✳✷

■♠♣❧✐❝❛t✐♦♥s ❢♦r ❍♦s♣✐t❛❧ ❇❡❤❛✈✐♦r

❚❤❡ ❢❛❝ts ❞❡t❛✐❧❡❞ ❛❜♦✈❡ ❤❛✈❡ ✐♠♣♦rt❛♥t ✐♠♣❧✐❝❛t✐♦♥s ❢♦r ❞✐✛❡r❡♥t ❤②♣♦t❤❡s❡s ❛❜♦✉t ❤♦s♣✐t❛❧ ❜❡❤❛✈✐♦r✳ ■♥ t❤❡ ❞✐s❝✉ss✐♦♥ t❤❛t ❢♦❧❧♦✇s✱ ✇❡ ❛♣♣r♦①✐♠❛t❡ t♦t❛❧ ♣❛t✐❡♥t ✇❡❧❢❛r❡ ✇✐t❤ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts✳ ❆s ✇❡ ❛r❣✉❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✱ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❝♦sts ❛r❡ s♠❛❧❧ r❡❧❛t✐✈❡ t♦ t❤❡ ❜❡♥❡✜ts ♦❢ tr❛♥s♣❧❛♥t❛t✐♦♥✳ ❚❤❡ ✜rst ❤②♣♦t❤❡s✐s ✐s t❤❛t t❤❛t ❤♦s♣✐t❛❧s ♠❛①✐♠✐③❡ t❤❡ t♦t❛❧ ✇❡❧❢❛r❡ ♦❢ ❛❧❧ ♣❛t✐❡♥ts ✐♥ t❤❡ s②st❡♠✱ r❡❣❛r❞❧❡ss ♦❢ ✇❤✐❝❤ ❤♦s♣✐t❛❧ ❛ ♣❛t✐❡♥t ❜❡❧♦♥❣s t♦✳ ❚❤✐s ❤②♣♦t❤❡s✐s ✐s str♦♥❣❧② r❡❥❡❝t❡❞ ❜② s❡✈❡r❛❧ ❢❡❛t✉r❡s ♦❢ t❤❡ ❞❛t❛✳ ▼♦st ❝❧❡❛r❧②✱ t❤✐s ❤②♣♦t❤❡s✐s ✐s ✐♥❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ❡✈✐❞❡♥❝❡ ♦❢ s♦❝✐❛❧❧② ✐♥❡✣❝✐❡♥t ♠❛t❝❤❡s ✭❋✐❣✉r❡ ✷✮✳ ❆ s❡❝♦♥❞ ❤②♣♦t❤❡s✐s ✐s t❤❛t ❤♦s♣✐t❛❧s ♦♥❧② ♠❛①✐♠✐③❡ t❤❡ ✇❡❧❢❛r❡ ♦❢ t❤❡✐r ♦✇♥ ♣❛t✐❡♥ts✳ ❚❤✐s ❤②♣♦t❤❡s✐s ✇❛s ✐♥✈❡st✐❣❛t❡❞ t❤❡♦r❡t✐❝❛❧❧② ❜② ❆s❤❧❛❣✐ ❛♥❞ ❘♦t❤ ✭✷✵✶✹✮✳ ❚❤❡✐r ♠❛✐♥ ♣r❡❞✐❝t✐♦♥ ✐s t❤❛t ❤♦s♣✐t❛❧s ✇✐❧❧ tr② t♦ ♠❛t❝❤ ❛s ♠❛♥② ♦❢ t❤❡✐r ♣❛t✐❡♥ts ✐♥t❡r♥❛❧❧② ❛s ♣♦ss✐❜❧❡ ❛♥❞ ♦♥❧② s✉❜♠✐t t❤❡ r❡♠❛✐♥✐♥❣ ♣❛t✐❡♥ts t♦ ❛ ♠✉❧t✐✲❤♦s♣✐t❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳ ❚❤✐s ❤②♣♦t❤❡s✐s ✜ts s♦♠❡ q✉❛❧✐t❛t✐✈❡ ♣❛tt❡r♥s ✐♥ t❤❡ ❞❛t❛✳ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ♣❛rt✐❝✐♣❛t✐♥❣✱ ❧❛r❣❡r ❤♦s♣✐t❛❧s ♣❡r❢♦r♠ ❢❡✇❡r tr❛♥s♣❧❛♥ts t❤r♦✉❣❤ t❤❡ ◆❑❘ ❜❡❝❛✉s❡ t❤❡s❡ ❤♦s♣✐t❛❧s ❤❛✈❡ ♠♦r❡ ♦♣♣♦rt✉♥✐t✐❡s t♦ ♠❛t❝❤ ♣❛t✐❡♥ts ♦✉ts✐❞❡ t❤❡ ♣❧❛t❢♦r♠ ✭❋✐❣✉r❡ ✹✮✳ ❍♦✇❡✈❡r✱ t❤❡ ❤②♣♦t❤❡s✐s ❞♦❡s ♥♦t ❡①♣❧❛✐♥ ♠❛♥② ♦t❤❡r ✐♠♣♦rt❛♥t ❛s♣❡❝ts ♦❢ t❤❡ ❞❛t❛✳ ❋♦r ❡①❛♠♣❧❡✱ ♠❛♥② ❤♦s♣✐t❛❧s ❞♦ ♥♦t ♣❛rt✐❝✐♣❛t❡ ✐♥ ❛ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠ ❛t ❛❧❧✱ ❡✈❡♥ t❤♦✉❣❤ ❛❧❧ ❤♦s♣✐t❛❧s ❧✐❦❡❧② ❤❛✈❡ ♣❛t✐❡♥ts ✇❤♦ ❝❛♥♥♦t ❜❡ ♠❛t❝❤❡❞✳✷✹ ▼♦r❡♦✈❡r✱ ♠❛♥② s♠❛❧❧ ❤♦s♣✐t❛❧s ❞♦ ♥♦t ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ◆❑❘✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡s❡ ❤♦s♣✐t❛❧s✱ ❞✉❡ t♦ t❤❡✐r s✐③❡✱ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ♦♥❡s ❧❡❛st ❧✐❦❡❧② t♦ ✜♥❞ ♠❛t❝❤❡s ♦✉ts✐❞❡ t❤❡ ♣❧❛t❢♦r♠✳ ❚❤❡s❡ ♣❛tt❡r♥s s✉❣❣❡st t❤❛t ❤♦s♣✐t❛❧s r❡s♣♦♥❞ t♦ ✜①❡❞ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡s❡ ❝♦sts ❛r❡ s♠❛❧❧ r❡❧❛t✐✈❡ t♦ ❜❡♥❡✜ts t♦ ♣❛t✐❡♥ts ❛♥❞ ❝♦st s❛✈✐♥❣s ❢r♦♠ ❞✐❛❧②s✐s t♦ ❤❡❛❧t❤ ✐♥s✉r❡rs✳ ❆ t❤✐r❞ ❤②♣♦t❤❡s✐s ✐s t❤❛t ❤♦s♣✐t❛❧s ❛r❡ ♣r♦✜t ♠❛①✐♠✐③❡rs✳ ❚❤✐s ❤②♣♦t❤❡s✐s ✐s ❝♦♥s✐st❡♥t✱ ❢♦r ❡①❛♠♣❧❡✱ ✇✐t❤ t❤❡ ❢❛❝t t❤❛t s♠❛❧❧ ❤♦s♣✐t❛❧s ❛r❡ ❧❡ss ❧✐❦❡❧② t♦ ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ◆❑❘ ✭❋✐❣✉r❡ ✸✮ ❜❡❝❛✉s❡ t❤❡ ✜①❡❞ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ ♠❛② ♥♦t ❝♦♠♣❡♥s❛t❡ ❢♦r t❤❡ ❣❛✐♥s ✐♥ ♣r♦✜ts ❢r♦♠ ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥ts✳ ❍♦✇❡✈❡r✱ t❤✐s t❤❡♦r② ❛❧♦♥❡ ❝❛♥♥♦t ❢✉❧❧② ❡①♣❧❛✐♥ t❤❡ ❧❛r❣❡ ✈❛r✐❛t✐♦♥ ✐♥ t❤❡ ❞❡❣r❡❡ ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥✱ ❡s♣❡❝✐❛❧❧② ❛♠♦♥❣ ❧❛r❣❡ ❤♦s♣✐t❛❧s✳ ❋♦r ❡①❛♠♣❧❡✱ ❈♦r♥❡❧❧ ▼❡❞✐❝❛❧ ❈❡♥t❡r ✐s ❛ ❧❛r❣❡ ❤♦s♣✐t❛❧ ✇✐t❤ ❛ ❤✐❣❤ r❛t❡ ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ t❤❡ ◆❑❘✳ ■ts ❜❡❤❛✈✐♦r ✜ts ♥❡✐t❤❡r t❤❡ ♣r❡❞✐❝t✐♦♥ ♦❢ ❛ ♣✉r❡ ♣r♦✜t ♠❛①✐♠✐③❛t✐♦♥ ♠♦t✐✈❡ ♥♦r t❤❡ ♠♦t✐✈❡ t♦ ♣✉r❡❧② ♠❛①✐♠✐③❡ t❤❡ ✇❡❧❢❛r❡ ♦❢ t❤❡ ❤♦s♣✐t❛❧✬s ♦✇♥ ♣❛t✐❡♥ts✳ ■♥t❡r✈✐❡✇s ✇✐t❤ tr❛♥s♣❧❛♥t ❝♦♦r❞✐♥❛t♦rs ❛t ❈♦r♥❡❧❧✱ r❡♣♦rt❡❞ ✐♥ ❊❧❧✐s♦♥ ✭✷✵✶✹✮✱ s✉❣❣❡st t❤❛t ❛ ♣r✐♠❛r② r❡❛s♦♥ ❢♦r ♣❛rt✐❝✐♣❛t✐♥❣ ✐s t❤❡ ✈✐❡✇ t❤❛t ❝♦♥tr✐❜✉t✐♥❣ t♦ ❛ ♥❛t✐♦♥❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ✐s ✐♠♣♦rt❛♥t✳ ❚❛❦❡♥ t♦❣❡t❤❡r✱ t❤❡ ❡✈✐❞❡♥❝❡ ♦♥ ❤♦s♣✐t❛❧ ♣❛rt✐❝✐♣❛t✐♦♥ s✉❣❣❡sts t❤❛t ❤♦s♣✐t❛❧s ♠❛①✐♠✐③❡ ❝♦♠✲ ♣❧❡① ❛♥❞ ❤❡t❡r♦❣❡♥❡♦✉s ♦❜❥❡❝t✐✈❡s✳ ❚❤✐s ✜♥❞✐♥❣ ✐s ❝♦♥s✐st❡♥t ❜♦t❤ ✇✐t❤ t❤❡ ❛♥❡❝❞♦t❛❧ ❡✈✐❞❡♥❝❡ ♦♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ r❡✈✐❡✇❡❞ ✐♥ ❙❡❝t✐♦♥ ✷ ❛♥❞ ✇✐t❤ t②♣✐❝❛❧ ✜♥❞✐♥❣s ❛❜♦✉t t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❤❡❛❧t❤❝❛r❡ ♣r♦✈✐❞❡rs✳ ❋♦r ❡①❛♠♣❧❡✱ ❈❧❡♠❡♥s ❛♥❞ ●♦tt❧✐❡❜ ✭✷✵✶✹✮ ✐♥✈❡st✐❣❛t❡ ❤♦✇ ♣r♦✈✐❞❡rs r❡s♣♦♥❞ t♦ ❝❤❛♥❣❡s ✐♥ ▼❡❞✐❝❛r❡ r❡✐♠❜✉rs❡♠❡♥t r❛t❡s✱ ❛♥❞ ❑♦❧st❛❞ ✭✷✵✶✸✮ ✐♥✈❡st✐❣❛t❡s ❤♦✇ s✉r❣❡♦♥s r❡s♣♦♥❞ t♦ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ♣✉❜❧✐❝❧② ♦❜s❡r✈❛❜❧❡ r❡♣♦rt ❝❛r❞s✳ ❚❤❡s❡ st✉❞✐❡s ✜♥❞

❘❡❝❛❧❧ t❤❛t ♦✈❡r❞❡♠❛♥❞❡❞ ♣❛✐rs ❛r❡ t②♣✐❝❛❧❧② s❝❛r❝❡✳ ❲❡ ✇✐❧❧ s❡❡ ✐♥ ❙❡❝t✐♦♥ ✺ t❤❛t ❡✈❡♥ t❤❡ ◆❑❘ ✐s ❛❜❧❡ t♦ ♠❛t❝❤ ♦♥❧② ❛♣♣r♦①✐♠❛t❡❧② ✺✵✪ ♦❢ ✐ts ❞♦♥♦rs✳ ✷✹

✷✶ t❤❛t ♣r♦✈✐❞❡rs r❡s♣♦♥❞ t♦ ✐♥❝❡♥t✐✈❡s ❜✉t ❛r❡ ♥♦t ♣✉r❡❧② ♠♦t✐✈❛t❡❞ ❜② ♣r♦✜ts ❛♥❞ ❛❧s♦ t❛❦❡ ♣❛t✐❡♥t ✇❡❧❢❛r❡ ✐♥t♦ ❛❝❝♦✉♥t✳ ❚❤❡ ❢❛❝ts ❛❜♦✉t s❡❧❡❝t✐♦♥ ✐♥t♦ ✇❤✐❝❤ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❛r❡ s✉❜♠✐tt❡❞ t♦ t❤❡ ◆❑❘ ❛❧s♦ ✐♥❞✐❝❛t❡ t❤❛t t❤❡s❡ t✇♦ t❤❡♦r✐❡s✱ ♠❛①✐♠✐③✐♥❣ ♣r♦✜ts ❛♥❞ ♠❛①✐♠✐③✐♥❣ t❤❡✐r ♦✇♥ ♣❛t✐❡♥ts✬ ✇❡❧❢❛r❡✱ ❝❛♥ ❡①♣❧❛✐♥ ♠❛♥② ❤♦s♣✐t❛❧s✬ ❜❡❤❛✈✐♦r✳ ❚❤❡s❡ t❤❡♦r✐❡s✬ s❤❛r❡❞ ✐♠♣❧✐❝❛t✐♦♥ ✐s t❤❛t ♣❛✐rs s✉❜♠✐tt❡❞ t♦ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ❛r❡ ♥❡❣❛t✐✈❡❧② s❡❧❡❝t❡❞✱ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❜❡✐♥❣ ❤❛r❞ t♦ ♠❛t❝❤✳ ■♥ ❜♦t❤ ❝❛s❡s✱ ❛ ❤♦s♣✐t❛❧ ♦♥❧② s✉❜♠✐ts ❛ ♣❛✐r t♦ ❛ ♣❧❛t❢♦r♠ ✐❢ ❛♥ ✐♥t❡r♥❛❧ ♠❛t❝❤ ✐s ♥♦t ♣♦ss✐❜❧❡✳ ❯♥❞❡r ♠♦st ♠❛t❝❤✐♥❣ t❡❝❤♥♦❧♦❣✐❡s✱ s✉❜♠✐tt❡❞ ♣❛t✐❡♥ts ✇✐❧❧ ❜❡ ❤❛r❞❡r t♦ ♠❛t❝❤ t❤❛♥ t❤♦s❡ ♥♦t s✉❜♠✐tt❡❞✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ✇❡ ❝❛♥♥♦t ❞✐r❡❝t❧② t❡st t❤✐s ♣r❡❞✐❝t✐♦♥ ❜❡❝❛✉s❡ ✇❡ ❞♦ ♥♦t ❤❛✈❡ ❞❛t❛ ♦♥ t❤❡ ❡♥t✐r❡ ♣♦♦❧ ♦❢ ♣❛t✐❡♥ts ❛✈❛✐❧❛❜❧❡ t♦ ✐♥❞✐✈✐❞✉❛❧ ❤♦s♣✐t❛❧s✳ ◆♦♥❡t❤❡❧❡ss✱ ❛s ✇❡ ❞✐s❝✉ss❡❞ ❛❜♦✈❡✱ t❤❡ ❞❛t❛ s❤♦✇ t❤❛t s✉❜♠✐tt❡❞ ♣❛✐rs ❛r❡ ♠✉❝❤ ❤❛r❞❡r t♦ ♠❛t❝❤ t❤❛♥ t❤❡ ❣❡♥❡r❛❧ ♣♦♣✉❧❛t✐♦♥✳ ▼♦r❡♦✈❡r✱ t❤❡ ◆❑❘ tr❛♥s♣❧❛♥ts ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs t❤❛t ❛r❡ ❤❛r❞❡r t♦ ♠❛t❝❤✳ ■❢ t❤❡ ◆❑❘ ❛♥❞ ❤♦s♣✐t❛❧s ❛r❡ ❡q✉❛❧❧② ❝❛♣❛❜❧❡ ♦❢ ♠❛t❝❤✐♥❣ ❤❛r❞✲t♦✲♠❛t❝❤ ♣❛t✐❡♥ts✱ t❤❡♥ t❤❡ ❞❛t❛ s✉❣❣❡st t❤❛t ❤♦s♣✐t❛❧s s❡❧❡❝t ❤❛r❞✲t♦✲♠❛t❝❤ t②♣❡s ❢♦r s✉❜♠✐ss✐♦♥ t♦ t❤❡ ◆❑❘✳ ❍♦✇❡✈❡r✱ s♦♠❡ s❡❧❡❝t✐♦♥ ♠❛② ❜❡ ❞r✐✈❡♥ ❜② t❤❡ ◆❑❘ ❜❡✐♥❣ s✐❣♥✐✜❝❛♥t❧② ❜❡tt❡r ❛t ✜♥❞✐♥❣ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s ❢♦r ❤❛r❞✲t♦✲♠❛t❝❤ ♣❛t✐❡♥ts ✇✐t❤ ❛ ❣✐✈❡♥ ♣♦♦❧✳ ■t ✐s r❡❛ss✉r✐♥❣ t❤❛t t❤❡ r❡s✉❧ts ♦♥ s❡❧❡❝t✐♦♥ ❞♦ ♥♦t ❢❛❧s✐❢② t❤❡ t✇♦ t❤❡♦r✐❡s t❤❛t ❜❡st ✜t t❤❡ ♣❛rt✐❝✐♣❛t✐♦♥ ❜❡❤❛✈✐♦r✳ ❚♦ s✉♠♠❛r✐③❡✱ t❤❡s❡ ✜♥❞✐♥❣s ❤❛✈❡ t✇♦ ✐♠♣♦rt❛♥t ✐♠♣❧✐❝❛t✐♦♥s✳ ❋✐rst✱ t❤❡r❡ ✐s ❝❧❡❛r ❡✈✐❞❡♥❝❡ ♦❢ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ ❛s ✇❡ ❞❡✜♥❡❞ ❜r♦❛❞❧② ✐♥ ❙❡❝t✐♦♥ ✷✳ ❚❤❡ ❞❛t❛ ❞✐s♣r♦✈❡ t❤❡ ❤②♣♦t❤❡s✐s t❤❛t ❤♦s♣✐t❛❧s ♣✉r❡❧② ❛♥❞ r❛t✐♦♥❛❧❧② ♠❛①✐♠✐③❡ t❤❡✐r ♦✇♥ ♣❛t✐❡♥ts✬ ✇❡❧❢❛r❡✳ ❙❡❝♦♥❞✱ t❤❡r❡ ✐s ♥♦ s✐♠♣❧❡ ♠♦❞❡❧ t❤❛t ❞❡s❝r✐❜❡s t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❛❧❧ ❤♦s♣✐t❛❧s✳ ❚❤❡s❡ t✇♦ ✜♥❞✐♥❣s r❡♣r❡s❡♥t ❛ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ❡♠♣✐r✐❝❛❧ ❝❤❛❧❧❡♥❣❡ ❜❡❝❛✉s❡ ✇❡ ❝❛♥♥♦t r❡❧② ♦♥ ❛ s✐♠♣❧❡ st②❧✐③❡❞ ♠♦❞❡❧ ♦❢ ❤♦s♣✐t❛❧ ❜❡❤❛✈✐♦r t♦ ❡✐t❤❡r ♣r♦♣♦s❡ ♣r❛❝t✐❝❛❧ ♠❛r❦❡t ❞❡s✐❣♥ s♦❧✉t✐♦♥s ♦r ♣❡r❢♦r♠ ✇❡❧❢❛r❡ ❛♥❛❧②s❡s✳ ❇✉t✱ t❤❡s❡ r❡s✉❧ts ❞♦ ✐♥❞✐❝❛t❡ t❤❛t ❛❣❡♥❝② ♣r♦❜❧❡♠s ❛r❡ ❛ ♣♦t❡♥t✐❛❧❧② ✐♠♣♦rt❛♥t ❞r✐✈❡r ♦❢ ✐♥❡✣❝✐❡♥❝② ❛♥❞ s❤♦✉❧❞ t❤❡r❡❢♦r❡ ♥♦t ❥✉st ❜❡ ❛ss✉♠❡❞ ❛✇❛②✳ ❇♦t❤ ✐♠♣❧✐❝❛t✐♦♥s ❛r❡ ✐♠♣♦rt❛♥t ❡❧❡♠❡♥ts ✐♥ t❤❡ ❛♥❛❧②s✐s ✇❡ ❞❡✈❡❧♦♣ ❜❡❧♦✇✳

✹

❚❤❡♦r②

❚❤❡ ❡✈✐❞❡♥❝❡ ❛❜♦✈❡ s❤♦✇s t❤❛t ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡ts ❛r❡ ❢r❛❣♠❡♥t❡❞ ❛♥❞ t❤❛t t❤✐s ❢r❛❣✲ ♠❡♥t❛t✐♦♥ ❧❡❛❞s t♦ r❡❛❧ ❡✣❝✐❡♥❝② ❧♦ss❡s✳ ❲❡ ♥♦✇ ❞❡✈❡❧♦♣ ❛ ♠♦❞❡❧ t♦ ❡①♣❧❛✐♥ ❤♦✇ ✐♥❡✣❝✐❡♥❝② ❛r✐s❡s✳ ❚❤❡ ♠♦❞❡❧ ❛❧s♦ s❤♦✇s ✇❤❛t st❛t✐st✐❝s ♠✉st ❜❡ ❡st✐♠❛t❡❞ t♦ ♠❡❛s✉r❡ t❤❡ ✐♥❡✣❝✐❡♥❝② ❛♥❞ t♦ ❞❡s✐❣♥ ❜❡tt❡r ♠❡❝❤❛♥✐s♠s✳ ❚❤❡ ❜❛s✐❝ ✐❞❡❛ ✐s t❤❛t t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡t ✐s ❢✉♥❞❛♠❡♥t❛❧❧② s✐♠✐❧❛r t♦ ❛ tr❛❞✐t✐♦♥❛❧ ♠❛r❦❡t✳ ◆❛♠❡❧②✱ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❝❧❡❛r✐♥❣❤♦✉s❡ ✐s ❛ ♣❧❛t❢♦r♠ t❤❛t ♣r♦❝✉r❡s s✉❜♠✐ss✐♦♥s ✭❞♦♥♦rs ❛♥❞ ♣❛t✐❡♥ts✮ ❢r♦♠ ❤♦s♣✐t❛❧s ❛♥❞ r❡✇❛r❞s t❤❡s❡ ❤♦s♣✐t❛❧s ✇✐t❤ tr❛♥s♣❧❛♥ts✳ ❚❤❡ ♣❧❛t❢♦r♠✬s ❛❜✐❧✐t② t♦ ♣r♦❞✉❝❡ tr❛♥s♣❧❛♥ts ❢r♦♠ s✉❜♠✐ss✐♦♥s ✐s ❣✐✈❡♥ ❜② ❛ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ ❞❡s❝r✐❜❡s ❤♦✇ ♠❛♥② tr❛♥s♣❧❛♥ts ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ ✇✐t❤ ❛ ❣✐✈❡♥ ♣♦♦❧ ♦❢ s✉❜♠✐ss✐♦♥s✳

✷✷ ✹✳✶

▼♦❞❡❧

❆ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ♣r♦❝✉r❡s s✉❜♠✐ss✐♦♥s ❢r♦♠ ❤♦s♣✐t❛❧s ❛♥❞ r❡✇❛r❞s ❤♦s♣✐t❛❧s ✇✐t❤ tr❛♥s♣❧❛♥ts✳ ❚❤❡ ♣❧❛t❢♦r♠✬s ❛❜✐❧✐t② t♦ ♣r♦❞✉❝❡ tr❛♥s♣❧❛♥ts ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ f ✳ ❲❡ ❝♦♥s✐❞❡r t②♣❡s ♦❢ s✉❜♠✐ss✐♦♥s ✐♥❞❡①❡❞ ❜② i = 1, . . . , I ✳ ❆ ✈❡❝t♦r ♦❢ q✉❛♥t✐t✐❡s q = (qi )i=1,...,I ✐♥ R+ s♣❡❝✐✜❡s ❛ q✉❛♥t✐t② qi ♦❢ ❡❛❝❤ s✉❜♠✐ss✐♦♥ t②♣❡ ❛✈❛✐❧❛❜❧❡ t♦ t❤❡ ♣❧❛t❢♦r♠✳ ●✐✈❡♥ ❛ ✈❡❝t♦r ♦❢ q✉❛♥t✐t✐❡s q ✱ t❤❡ ♣❧❛t❢♦r♠ ❝❛♥ ♣r♦❞✉❝❡ f (q) tr❛♥s♣❧❛♥ts✳ ❚❤❡ ♠♦❞❡❧ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❡✐t❤❡r st❛t✐❝ ♦r ❛s ❛ st❡❛❞②✲st❛t❡ ❢r♦♠ ❛ ❞②♥❛♠✐❝ ♠♦❞❡❧✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ st❡❛❞②✲st❛t❡ ✐♥t❡r♣r❡t❛t✐♦♥ ✐♥ t❤❡ ❡♠♣✐r✐❝❛❧ ❛♥❛❧②s✐s✳ ❆❧❧ ✈❛r✐❛❜❧❡s ❛r❡ ♠❡❛s✉r❡❞ ✐♥ ✢♦✇s✱ s✉❝❤ ❛s tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r✳ ❚❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ s✉♠♠❛r✐③❡s ✇❤❛t ♠❛t❝❤❡s ❛r❡ ♣♦ss✐❜❧❡✳ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ ❝❛❧❝✉❧❛t❡❞ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ❛ s✐♠♣❧❡ ♠♦❞❡❧ t❤❛t ✇❡ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ ❛ss✉♠❡❞ t❤❛t ❛❧❧ s✉❜♠✐ss✐♦♥s ❛r❡ ♣❛✐rs ❛♥❞ t❤❛t ♦♥❧② ❜❧♦♦❞ t②♣❡ ❝♦♠♣❛t✐❜✐❧✐t② ♠❛tt❡rs✱ s♦ t❤❡✐r ♠♦❞❡❧ ❤❛s I = 16 t②♣❡s✳ ❖✉r ❛♥❛❧②s✐s ❛♣♣❧✐❡s ❜♦t❤ t♦ s✐♠♣❧❡✱ t❤❡♦r❡t✐❝❛❧❧② tr❛❝t❛❜❧❡✱ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥s✱ s✉❝❤ ❛s t❤❡ ♦♥❡ ✐♥ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮✱ ❛♥❞ t♦ s♦♣❤✐st✐❝❛t❡❞ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥s✳ ❙❡❝t✐♦♥ ✺ ✉s❡s ❛♥ ❡♠♣✐r✐❝❛❧ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ t❤❛t ❛❧❧♦✇s s✉❜♠✐ss✐♦♥s t♦ ❞✐✛❡r ❜② ✇❤❡t❤❡r t❤❡② ❛r❡ ♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐rs✱ ❛❧tr✉✐st✐❝ ❞♦♥♦rs✱ ♦r ✉♥♣❛✐r❡❞ ♣❛t✐❡♥ts✱ ❛♥❞ ❜② ❛ ❤♦st ♦❢ ✈❛r✐❛❜❧❡s ✐♥❝❧✉❞✐♥❣ ❜❧♦♦❞ t②♣❡s✱ ❛♥t✐❣❡♥s✱ ❛♥❞ ❛♥t✐❜♦❞✐❡s✳ ❚❤✉s✱ t❤❡ ♥✉♠❜❡r ♦❢ t②♣❡s I ✐s ♣♦t❡♥t✐❛❧❧② ❧❛r❣❡✳ ❚❤❡ ♣❧❛t❢♦r♠ ♣r♦❞✉❝❡s tr❛♥s♣❧❛♥ts ✉s✐♥❣ s✉❜♠✐ss✐♦♥s ♣r♦✈✐❞❡❞ ❜② ❤♦s♣✐t❛❧s ✐♥❞❡①❡❞ ❜② h = 1, . . . , H ✳ ❍♦s♣✐t❛❧s ❛r❡ r❡✇❛r❞❡❞ ❢♦r t❤❡s❡ s✉❜♠✐ss✐♦♥s ✇✐t❤ tr❛♥s♣❧❛♥ts✳ ❲❡ ❛ss✉♠❡ t❤❡s❡ r❡✇❛r❞s ❛r❡ ❧✐♥❡❛r ✐♥ s✉❜♠✐ss✐♦♥s ❛♥❞ ❛♥♦♥②♠♦✉s✳ ❚❤❛t ✐s✱ t❤❡r❡ ❡①✐sts ❛ ✈❡❝t♦r ♦❢ r❡✇❛r❞s p = (pi )i=1,...,I ✐♥ RI ✇❤❡r❡ t❤❡ i✲t❤ ❝♦♠♣♦♥❡♥t ❞❡♥♦t❡s t❤❡ ✭❡①♣❡❝t❡❞✮ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ❛✇❛r❞❡❞ t♦ t❤❡ ❤♦s♣✐t❛❧ ♣❡r s✉❜♠✐ss✐♦♥ ♦❢ t②♣❡ i✳ ❚❤❡ ✉♥✐ts ♦❢ pi ❛r❡ tr❛♥s♣❧❛♥ts ♣❡r s✉❜♠✐ss✐♦♥✳ ❆ ❤♦s♣✐t❛❧ t❤❛t s✉❜♠✐ts ❛ ✢♦✇ q h ✐♥ RI+ ♦❢ s✉❜♠✐ss✐♦♥s r❡❝❡✐✈❡s ❛ ✢♦✇ p · q h ♦❢ tr❛♥s♣❧❛♥ts✳ ❚❤✐s ❧✐♥❡❛r r❡✇❛r❞ s❝❤❡❞✉❧❡ ✐s ❛ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❝✉rr❡♥t ♣❧❛t❢♦r♠s✬ r✉❧❡s ❜❡❝❛✉s❡ t❤❡✐r ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠s ♠❛①✐♠✐③❡ ❛ ✇❡✐❣❤t❡❞ s✉♠ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♠❛t❝❤❡s ✇✐t❤♦✉t ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❡♥t✐r❡ ♣♦♦❧ ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs s✉❜♠✐tt❡❞ ❜② t❤❡ ❤♦s♣✐t❛❧ ✭❙♦♥♠❡③ ❛♥❞ ❯♥✈❡r✱ ✷✵✶✸❀ ❆♥❞❡rs♦♥ ❡t ❛❧✳✱ ✷✵✶✺✮✳ ❲❤❡♥ ❛ ❤♦s♣✐t❛❧ s✉❜♠✐ts ❛♥ ❛❞❞✐t✐♦♥❛❧ ♣❛✐r✱ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ♣❧❛t❢♦r♠ ♠❛t❝❤❡s ❛ ❞✐✛❡r❡♥t ♣❛✐r ❢r♦♠ t❤❡ s❛♠❡ ❤♦s♣✐t❛❧ ❞♦❡s ♥♦t s✐❣♥✐✜❝❛♥t❧② ❝❤❛♥❣❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝✉rr❡♥t r❡✇❛r❞ ❢♦r s✉❜♠✐tt✐♥❣ ❛ t②♣❡ i ♣❛✐r ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t②✱ pi ✱ t❤❛t t❤❡ ♣❛✐r ✐s ♠❛t❝❤❡❞✳ ❲❡ ❛ss✉♠❡ t❤❛t ❤♦s♣✐t❛❧ ♣r❡❢❡r❡♥❝❡s ❛r❡ q✉❛s✐❧✐♥❡❛r ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts t❤❡② r❡❝❡✐✈❡ ❢r♦♠ t❤❡ ♣❧❛t❢♦r♠ ❛♥❞ t❤❡ ♣r✐✈❛t❡ ❝♦st ♦❢ t❤❡✐r s✉❜♠✐ss✐♦♥s✳ ❋♦r♠❛❧❧②✱ ❞❡✜♥❡ t❤❡ ♦❜❥❡❝t✐✈❡ t❤❛t ❤♦s♣✐t❛❧ h ♠❛①✐♠✐③❡s ❛s p · q h − C h (q h )✱ ✇❤❡r❡ C h (q h ) ✐s t❤❡ ♣r✐✈❛t❡ ❝♦st ♦❢ s✉❜♠✐ss✐♦♥s ✐♥ tr❛♥s♣❧❛♥t ✉♥✐ts✳ ❖♥❡ ♣❛rt✐❝✉❧❛r ❝❛s❡ ✐s ✇❤❡♥ ❤♦s♣✐t❛❧s ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡✐r ♦✇♥ ♣❛t✐❡♥ts t❤❛t ❛r❡ tr❛♥s♣❧❛♥t❡❞✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ C h (q h ) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✇✐t❤✐♥✲❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts t❤❛t t❤❡ ❤♦s♣✐t❛❧ ✇♦✉❧❞ ❤❛✈❡ t♦ ❢♦r❢❡✐t ✐♥ ♦r❞❡r t♦ s✉❜♠✐t q h ✳ ■❢P ❛❧❧ tr❛♥s♣❧❛♥ts t❤❛t ❛r❡ ♣❡r❢♦r♠❡❞ ♠✉st ❜❡ ❛❧❧♦❝❛t❡❞ t♦ s♦♠❡ h ❤♦s♣✐t❛❧✱ ✇❡ ❤❛✈❡ t❤❛t f (q) = H h=1 p · q .

✷✸ ❲❡❧❢❛r❡ ✐s ❞❡✜♥❡❞ ♦✈❡r ❛♥ ❛❧❧♦❝❛t✐♦♥ (q h )h=1,...,H t❤❛t s♣❡❝✐✜❡s t❤❡ q✉❛♥t✐t② ♦❢ ♣❛✐rs s✉♣♣❧✐❡❞ ❜② ❡❛❝❤ ❤♦s♣✐t❛❧✳ ❲❡ ✇✐❧❧ ✉s❡ t✇♦ ✇❡❧❢❛r❡ ♥♦t✐♦♥s✳ ❇♦t❤ ✇❡❧❢❛r❡ ♥♦t✐♦♥s ✉s❡ tr❛♥s♣❧❛♥ts ❛s ❛ ♥✉♠❡r❛✐r❡ ❜❡❝❛✉s❡ ♣❧❛t❢♦r♠s ❝❛♥ ❡✛❡❝t✐✈❡❧② tr❛♥s❢❡r tr❛♥s♣❧❛♥ts ❜❡t✇❡❡♥ ❤♦s♣✐t❛❧s ❜② ❝❤♦♦s✐♥❣ ✇❤✐❝❤ ✉♥❞❡r❞❡♠❛♥❞❡❞ s✉❜♠✐ss✐♦♥s t♦ ♠❛t❝❤ ✭s❡❡ ❙❡❝t✐♦♥ ✷✮✳ ❚❤❡ ✜rst ♥♦t✐♦♥ ✐s ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ W H (q 1 , . . . , q H )✱ ✇❤✐❝❤ ✐s t❤❡ t♦t❛❧ ✇❡❧❢❛r❡ ♠❡❛s✉r❡❞ ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ ❤♦s♣✐t❛❧s✳ ❋♦r♠❛❧❧②✱ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ✐s ❞❡✜♥❡❞ ❛s H

1

H

W (q , . . . , q ) = f (q) −

H X

C h (q h )

✭✷✮

h=1

✇❤❡r❡ q ✐s t❤❡ ❛❣❣r❡❣❛t❡ q✉❛♥t✐t②✳ ❍♦s♣✐t❛❧ ✇❡❧❢❛r❡ ♠❡❛s✉r❡s ❡✣❝✐❡♥❝② ❛❝❝♦r❞✐♥❣ t♦ ❤♦s♣✐t❛❧s✬ ♦❜❥❡❝t✐✈❡s✳ ❚❤✐s ✐s ❛ ❝♦♠♣❡❧❧✐♥❣ ♥♦t✐♦♥ ♦❢ ✇❡❧❢❛r❡ ✐❢ t❤❡ ❣♦❛❧ ✐s t♦ ❤❡❧♣ ❦❡② ♠❛r❦❡t ♣❛rt✐❝✐♣❛♥ts ✭❤♦s♣✐t❛❧s✱ ✐♥ t❤✐s ❝❛s❡✮ ❛❝❤✐❡✈❡ t❤❡✐r ♦❜❥❡❝t✐✈❡s✳ ❍♦s♣✐t❛❧ ✇❡❧❢❛r❡ ✐s ♥♦t ❝♦♠♣❡❧❧✐♥❣ ✐❢ t❤❡r❡ ❛r❡ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ t❤❛t ✐s✱ ✐❢ ❤♦s♣✐t❛❧s ❞♦ ♥♦t ♣✉r❡❧② ♠❛①✐♠✐③❡ ♣❛t✐❡♥t ❛♥❞ ✐♥s✉r❡r ✇❡❧❢❛r❡✳ ❆s ❞✐s❝✉ss❡❞ ✐♥ ❙❡❝t✐♦♥s ✷✳✶ ❛♥❞ ✸✱ t❤❡r❡ ✐s ❛♥❡❝❞♦t❛❧ ❛♥❞ ❡♠♣✐r✐❝❛❧ ❡✈✐❞❡♥❝❡ ♦❢ ❛❣❡♥❝② ♣r♦❜❧❡♠s✳ ❋♦r t❤✐s r❡❛s♦♥✱ ✇❡ ❛❧s♦ ❝♦♥s✐❞❡r ❛ ✉t✐❧✐t❛r✐❛♥ ✇❡❧❢❛r❡ ♠❡❛s✉r❡✱ ✇❤✐❝❤ ✇❡ t❡r♠ t♦t❛❧ ✇❡❧❢❛r❡✳ ❉❡✜♥❡ SC h (q h ) ❛s t❤❡ s♦❝✐❛❧ ❝♦st ❢♦r ❤♦s♣✐t❛❧ h t♦ s✉♣♣❧② ❛ ✈❡❝t♦r q h s✉❜♠✐ss✐♦♥s✳ ■❢ t❤❡r❡ ❛r❡ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ t❤❡♥ s♦❝✐❛❧ ❛♥❞ ♣r✐✈❛t❡ ❝♦sts ❛r❡ ❞✐✛❡r❡♥t✱ ❛♥❞ t❤❡r❡ ✐s ❛♥ ❡①t❡r♥❛❧✐t② ❢r♦♠ ❤♦s♣✐t❛❧ h✬s s✉❜♠✐ss✐♦♥s ❣✐✈❡♥ ❜② E h (q h ) = C h (q h ) − SC h (q h )✳

❋♦r ❡①❛♠♣❧❡✱ E h ✐s ♣♦s✐t✐✈❡ ✐❢ ❤♦s♣✐t❛❧ h ❛❝ts ❛s t❤♦✉❣❤ t❤❡ ✜♥❛♥❝✐❛❧ ❛♥❞ ❧♦❣✐st✐❝❛❧ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s ❛r❡ s✐❣♥✐✜❝❛♥t r❡❧❛t✐✈❡ t♦ t❤❡✐r ♣r✐✈❛t❡ ✈❛❧✉❡ ♦❢ ❛ tr❛♥s♣❧❛♥t✳ ●✐✈❡♥ ❛♥ ❛❧❧♦❝❛t✐♦♥✱ ❧❡t E(q 1 , . . . , q H ) ❜❡ t❤❡ ❛❣❣r❡❣❛t❡ ❡①t❡r♥❛❧✐t②✱ ✇❤✐❝❤ ❡q✉❛❧s t❤❡ s✉♠ ♦❢ ❡①t❡r♥❛❧✐t✐❡s ❢r♦♠ ❛❧❧ ❤♦s♣✐t❛❧s✳ ❚❤❡ ❡①t❡r♥❛❧✐t② r❡♣r❡s❡♥ts t❤❡ ❜❡♥❡✜ts t♦ st❛❦❡❤♦❧❞❡rs ♦t❤❡r t❤❛♥ ❤♦s♣✐t❛❧s✳ ■♥ t❤❡ ♣❛rt✐❝✉❧❛r ❝❛s❡ ✇❤❡r❡ t❤❡r❡ ❛r❡ ♥♦ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ ✇❡ ❤❛✈❡ ∂E = 0✳ ❉❡✜♥❡ t♦t❛❧ ✇❡❧❢❛r❡ t♦ ❜❡ 1

H

SW (q , . . . , q ) = f (q) −

H X

SC h (q h )✳

h=1

❚❤✉s✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ s♦❝✐❛❧ ❛♥❞ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ✐s ❡q✉❛❧ t♦ t❤❡ ❛❣❣r❡❣❛t❡ ❡①t❡r♥❛❧✐t② E ✳ ❉❡✜♥❡ t❤❡ ❛❣❣r❡❣❛t❡ ❝♦st ❢✉♥❝t✐♦♥ C(q) ❛s t❤❡ ♠✐♥✐♠✉♠ ♦❢ t❤❡ s✉♠ ♦❢ ❝♦sts ♦❢ ❛❧❧ ❤♦s♣✐t❛❧s s✉❜❥❡❝t t♦ t❤❡ t♦t❛❧ q✉❛♥t✐t② s✉❜♠✐tt❡❞ ❜❡✐♥❣ ❡q✉❛❧ t♦ q ✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ♣r♦❞✉❝t✐♦♥ ❝♦st✱ s♦❝✐❛❧ ❝♦st✱ ❛♥❞ ❛❣❣r❡❣❛t❡ ❝♦st ❢✉♥❝t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ♦✈❡r ❛❧❧ ♥♦♥✲ ♥❡❣❛t✐✈❡ r❡❛❧ ✈❡❝t♦rs ❛♥❞ ❛r❡ s♠♦♦t❤✳ ❆❣❣r❡❣❛t❡ ❝♦sts ❛r❡ str✐❝t❧② ❝♦♥✈❡①✳ ❚❤❡ ♠❛①✐♠✉♠ ♦❢ ❡❛❝❤ ❤♦s♣✐t❛❧✬s ♦❜❥❡❝t✐✈❡ ✐s ❛tt❛✐♥❡❞ ❢♦r s♦♠❡ q✉❛♥t✐t② ❢♦r ❡✈❡r② ✈❡❝t♦r ♦❢ r❡✇❛r❞s✳ ◗✉❛♥t✐t✐❡s ❛r❡ ❝♦❧✉♠♥ ✈❡❝t♦rs ❛♥❞ ✈❡❝t♦rs ♦❢ r❡✇❛r❞s ❛♥❞ ❣r❛❞✐❡♥ts ❛r❡ r♦✇ ✈❡❝t♦rs✳ ❉❡✜♥❡ ✜rst✲❜❡st ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ❛s t❤❡ s✉♣r❡♠✉♠ ♦❢ W H ❛♥❞ ✜rst✲❜❡st s♦❝✐❛❧ ✇❡❧❢❛r❡ ❛s t❤❡ s✉♣r❡♠✉♠ ♦❢ SW ✳

✷✹ ❚❤✐s ❜❛s✐❝ ♠♦❞❡❧ ✜♥❡ss❡s ✐♠♣♦rt❛♥t ✐ss✉❡s✱ s✉❝❤ ❛s ❡✣❝✐❡♥❝② ❝♦sts ♦❢ tr❛♥s❢❡rr✐♥❣ tr❛♥s♣❧❛♥ts✱ t❤❡ ❝❤♦✐❝❡ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥✱ ❛♥❞ t❤❡ ❝❛s❡ ♦❢ ♠✉❧t✐♣❧❡ ❝♦♠♣❡t✐♥❣ ♣❧❛t❢♦r♠s✳ ❲❡ ✇✐❧❧ r❡t✉r♥ t♦ t❤❡s❡ ✐ss✉❡s ✐♥ ❙❡❝t✐♦♥ ✻✳ ✹✳✷

■❧❧✉str❛t✐✈❡ ❡①❛♠♣❧❡

❚❤❡ ♠♦❞❡❧ ❝❧❛r✐✜❡s t❤❛t ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ✐s✱ ✐♥ ♠❛♥② ✇❛②s✱ s✐♠✐❧❛r t♦ ❛ tr❛❞✐t✐♦♥❛❧ ♠❛r✲ ❦❡t✳ ❘❡❣✉❧❛t✐♥❣ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ✐s s✐♠✐❧❛r t♦ t❤❡ ❘❛♠s❡② ✭✶✾✷✼✮✲❇♦✐t❡✉① ✭✶✾✺✻✮ ♣r♦❜❧❡♠ ♦❢ r❡❣✉❧❛t✐♥❣ ❛ ♠✉❧t✐✲♣r♦❞✉❝t ✜r♠✳ ❚❤❡ ❦❡② ❞✐✛❡r❡♥❝❡ ✐s t❤❛t✱ ✐♥st❡❛❞ ♦❢ ❛ ✜r♠ ♣r♦❞✉❝✐♥❣ ♠❛♥② ♣r♦❞✉❝ts✱ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ♣r♦❝✉r❡s ♠✉❧t✐♣❧❡ s✉❜♠✐ss✐♦♥ t②♣❡s ❢r♦♠ ❤♦s♣✐t❛❧s✳✷✺ ❚❤✐s ❝♦♥♥❡❝t✐♦♥ r❡q✉✐r❡s s♦♠❡ ❛❜str❛❝t✐♦♥ ❜❡❝❛✉s❡ t❤❡ r❡❧❡✈❛♥t ♥✉♠❡r❛✐r❡ ❛r❡ tr❛♥s♣❧❛♥ts✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r ❛ ❝♦♥❝r❡t❡ ❡①❛♠♣❧❡ t♦ ❝❧❛r✐❢② t❤✐s ❝♦♥♥❡❝t✐♦♥✳ ❚❤❡ ❡①❛♠♣❧❡ ✐s ♦♥❧② ❢♦r ✐❧❧✉str❛t✐♦♥ ❛♥❞ s♣❡❝✐✜❝ ❛ss✉♠♣t✐♦♥s ❛r❡ ♥♦t ✉s❡❞ ✐♥ t❤❡ s✉❜s❡q✉❡♥t ❛♥❛❧②s✐s✳ ▲❡t K h (q h ) ❜❡ t❤❡ ♠♦♥❡t❛r② ❝♦sts ❜♦r♥❡ ❜② ❤♦s♣✐t❛❧ h ♦❢ s❡♥❞✐♥❣ q h s✉❜♠✐ss✐♦♥s t♦ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳ ❚❤✐s ❝♦st ❝❛♥ ✐♥❝❧✉❞❡ ♣❧❛t❢♦r♠ ❢❡❡s✱ ❝♦sts ♦❢ r❡❛rr❛♥❣✐♥❣ t❤❡ ❤♦s♣✐t❛❧ s❝❤❡❞✉❧❡ ❛r♦✉♥❞ t❤❡ ♣❧❛t❢♦r♠✱ ❛♥❞ ❢✉♥❞s ❢♦r ❤✐r✐♥❣ ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥t ❝♦♦r❞✐♥❛t♦rs ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✶✮✳ ▲❡t T h (q h ) ❜❡ t❤❡ ✢♦✇ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts t❤❛t ❤♦s♣✐t❛❧ h ❢♦r❣♦❡s ✇❤❡♥ s✉❜♠✐tt✐♥❣ q h t♦ t❤❡ ♣❧❛t❢♦r♠ ❜❡❝❛✉s❡ t❤❡ ❤♦s♣✐t❛❧ ❝❛♥♥♦t ♠❛t❝❤ t❤❡s❡ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✐♥t❡r♥❛❧❧②✳ ❍♦s♣✐t❛❧s ✈❛❧✉❡ ❡❛❝❤ tr❛♥s♣❧❛♥t ❛t v ❞♦❧❧❛rs✱ ✇❤✐❝❤ ✐♥❝❧✉❞❡s ♣r♦✜ts ❛♥❞ t❤❡ ✈❛❧✉❡ t❤❛t ❤♦s♣✐t❛❧s ♣❧❛❝❡ ♦♥ tr❛♥s♣❧❛♥t✐♥❣ t❤❡✐r ♣❛t✐❡♥ts✳ ●r♦ss r❡✈❡♥✉❡s ❢r♦♠ ❛ tr❛♥s♣❧❛♥t ❛r❡ ❛♣♣r♦①✐♠❛t❡❧② ✩✶✺✵✱✵✵✵ ✭❍❡❧❞ ❡t ❛❧✳✱ ✷✵✶✻✮✳ ❋♦r ✐❧❧✉str❛t✐✈❡ ♣✉r♣♦s❡s✱ t❛❦❡ v t♦ ❜❡ ✩✺✵✱✵✵✵✱ ✇❤✐❝❤ r❡♣r❡s❡♥ts ❛ ❣❡♥❡r♦✉s ✺✵✪ ♠❛r❦✲✉♣ ♦♥ ❝♦sts✳ ■♥ tr❛♥s♣❧❛♥t ✉♥✐ts✱ ❤♦s♣✐t❛❧ ✉t✐❧✐t② ❡q✉❛❧s t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ♠✐♥✉s t❤❡ ♠♦♥❡t❛r② ❝♦sts ❞✐✈✐❞❡❞ ❜② t❤❡ ✈❛❧✉❡ ♣❡r tr❛♥s♣❧❛♥t❀ ✐✳❡✳✱ ❤♦s♣✐t❛❧ h✬s ❝♦st ❢✉♥❝t✐♦♥ ✐s K h (q h ) ✳ C h (q h ) = T h (q h ) + v

❙♦❝✐❡t② ✈❛❧✉❡s ❡❛❝❤ tr❛♥s♣❧❛♥t ❛t V ❞♦❧❧❛rs✳ ❚❤✐s s♦❝✐❛❧ ✈❛❧✉❡ ✐♥❝❧✉❞❡s t❤❡ ✈❛❧✉❡ ♦❢ tr❛♥s✲ ♣❧❛♥ts t♦ ✐♥s✉r❡rs✱ t❛①♣❛②❡rs✱ ♣❛t✐❡♥ts✱ ❛♥❞ ♦t❤❡r ♣❛t✐❡♥ts ✇❤♦ ❜❡♥❡✜t ❢r♦♠ r❡❞✉❝t✐♦♥s ✐♥ t❤❡ ❞❡❝❡❛s❡❞ ❞♦♥♦r ✇❛✐t❧✐st✳ ❍❡❧❞ ❡t ❛❧✳ ✭✷✵✶✻✮ ❡st✐♠❛t❡ t❤❛t ❤❡❛❧t❤ ❝❛r❡ s❛✈✐♥❣s ❛❧♦♥❡ ❢r♦♠ t❤❡ ❛✈❡r❛❣❡ tr❛♥s♣❧❛♥t ❛r❡ ❛r♦✉♥❞ ✩✸✵✵✱✵✵✵ ❛♥❞ t❤❡ ❣❛✐♥s ✐♥ q✉❛❧✐t② ❛♥❞ ❧❡♥❣t❤ ♦❢ ❧✐❢❡ ❛r❡ ✈❛❧✉❡❞ ❛t ♦✈❡r ✩✶✱✵✵✵✱✵✵✵✳✷✻ ❚❤✉s✱ ✐t ✐s r❡❛s♦♥❛❜❧❡ t♦ ❡st✐♠❛t❡ t❤❛t V ✐s ✐♥ t❤❡ ❜❛❧❧♣❛r❦ ♦❢ ✩✶✱✸✵✵✱✵✵✵✳ ❚❛❦❡♥ t♦❣❡t❤❡r✱ t❤❡s❡ ❡st✐♠❛t❡s s❤♦✇ ❛ s✉❜st❛♥t✐❛❧ ✇❡❞❣❡ ❜❡t✇❡❡♥ t❤❡ s♦❝✐❛❧ ❛♥❞ ♣r✐✈❛t❡ ✈❛❧✉❡s ♦❢ ❛ tr❛♥s♣❧❛♥t ❜❡❝❛✉s❡ V ✐s ♠♦r❡ t❤❛♥ ❛♥ ♦r❞❡r ♦❢ ♠❛❣♥✐t✉❞❡ ❧❛r❣❡r t❤❛♥ v ✳ ▼♦r❡♦✈❡r✱ t❤❡ ❛❞❞✐t✐♦♥❛❧ s♦❝✐❛❧ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❛ ♣❧❛t❢♦r♠ K h (q h ) ❛r❡ ♥❡❣❧✐❣✐❜❧❡ ✷✺ ❖✉r

s❡tt✐♥❣ ✐s ❛❧s♦ ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡ ♣❧❛t❢♦r♠s ❧✐t❡r❛t✉r❡✱ ✇❤❡r❡✐♥ ♣❧❛t❢♦r♠s ♠❛①✐♠✐③❡ ❛ ♣r✐✈❛t❡ ♦r s♦❝✐❛❧ ❣♦❛❧ ❜② s❡tt✐♥❣ ✐♥❝❡♥t✐✈❡s ❢♦r ♣❛rt✐❝✐♣❛♥ts ✭❘♦❝❤❡t ❛♥❞ ❚✐r♦❧❡✱ ✷✵✵✸❀ ❲❡②❧✱ ✷✵✶✵✮✳ ✷✻ ❙♦♠❡ ♣❛t✐❡♥ts ✇❤♦ r❡❝❡✐✈❡ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥t ✇♦✉❧❞ ♦t❤❡r✇✐s❡ r❡❝❡✐✈❡ ❛ ❦✐❞♥❡② ❢r♦♠ ❛ ❞❡❝❡❛s❡❞ ❞♦♥♦r✳ ❇✉t✱ ✐♥ ❡❛❝❤ ♦❢ t❤♦s❡ ❝❛s❡s✱ ❛♥♦t❤❡r ♣❛t✐❡♥t ✐♥ t❤❡ ✇❛✐t❧✐st r❡❝❡✐✈❡s t❤✐s ❦✐❞♥❡②✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s♦❝✐❛❧ ❜❡♥❡✜t ♦❢ ❡❛❝❤ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥t s❤♦✉❧❞ st✐❧❧ ❜❡ t❤❡ s❛♠❡ ❛s t❤❡ ❣❛✐♥ ❢r♦♠ ❛ s✐♥❣❧❡ tr❛♥s♣❧❛♥t✳

✷✺ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❜❡♥❡✜ts ♦❢ ♣❡r❢♦r♠✐♥❣ ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥ts✳ ❚❤✐s ✜ts ♦✉r ♠♦❞❡❧ ✇✐t❤ s♦❝✐❛❧ ❝♦sts K h (q h ) h h h h SC (q ) = T (q ) + ✳ V

❚❤❡ ❡①t❡r♥❛❧✐t② t❡r♠ ❡q✉❛❧s h

h

h

h

h

h

E (q ) = C (q ) − SC (q ) =



1 1 − v V



· K(q h )✳

❚❤❛t ✐s✱ t❤❡ ❡①t❡r♥❛❧✐t② ❡q✉❛❧s t❤❡ ❞✐✛❡r❡♥❝❡✱ ♠❡❛s✉r❡❞ ✐♥ tr❛♥s♣❧❛♥t ✉♥✐ts✱ ♦❢ ❤♦✇ ♠✉❝❤ ♠♦r❡ ❤♦s♣✐t❛❧s ❝❛r❡ ❛❜♦✉t t❤❡ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ t❤❛♥ ❞♦❡s s♦❝✐❡t②✳ ❚♦ ❞❡✈❡❧♦♣ ✐♥t✉✐t✐♦♥ ❢♦r t❤❡s❡ ✇❡❞❣❡s✬ ♠❛❣♥✐t✉❞❡✱ ❛ss✉♠❡ t❤❛t t❤❡ ♠♦♥❡t❛r② ❝♦st ✐s ❧✐♥❡❛r ✐♥ P t❤❡ ♥✉♠❜❡r ♦❢ s✉❜♠✐ss✐♦♥s✱ ✐✳❡✳ K h (q h ) = k i qih ✳ ❚❤❡ ✇❡❞❣❡ ✐s h

h

E (q ) =



k k − v V

 X k X h · qih ≈ · q ✳ v i i i

❚❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ❤♦❧❞s ❜❡❝❛✉s❡ t❤❡ s♦❝✐❛❧ ✈❛❧✉❡ ♦❢ ❛ tr❛♥s♣❧❛♥t V ✐s ♠✉❝❤ ❧❛r❣❡r t❤❛♥ t❤❡ ♠♦♥❡t❛r② ❝♦st k✳ ❚❤❡ ✇❡❞❣❡ ✐s ❧❛r❣❡ ❜❡❝❛✉s❡ ✐t ❞❡♣❡♥❞s ♦♥ t❤❡ ♣❧❛t❢♦r♠ ♣❛rt✐❝✐♣❛t✐♦♥ ❝♦sts ❜♦r♥❡ ❜② t❤❡ ❤♦s♣✐t❛❧s ❛s ❛ ❢r❛❝t✐♦♥ ♦❢ ❛ tr❛♥s♣❧❛♥t✬s ♣r✐✈❛t❡ ✈❛❧✉❡✱ ♥♦t ✐ts s♦❝✐❛❧ ✈❛❧✉❡✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ k ✐s ✩✶✵✱✵✵✵ ❛♥❞ v ✐s ✩✺✵✱✵✵✵✱ t❤❡♥ t❤❡ ✇❡❞❣❡ ♣❡r s✉❜♠✐ss✐♦♥ ✐s k/v = 0.20 tr❛♥s♣❧❛♥ts ♣❡r s✉❜♠✐ss✐♦♥✳ ❍♦s♣✐t❛❧s ❝♦♠♣❛r❡ t❤✐s ✇❡❞❣❡ t♦ t❤❡ r❡✇❛r❞s ✈❡❝t♦r p✱ ✇❤✐❝❤ ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♠❛t❝❤✐♥❣ ✈❛r✐♦✉s s✉❜♠✐ss✐♦♥s ✐♥ t❤❡ ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✇❡❞❣❡ ✐s ❧❛r❣❡ ✐♥ t❤✐s ❡①❛♠♣❧❡✱ ❛♥❞ ♣r❡❝✐s❡❧② ❜❡❝❛✉s❡ ❤♦s♣✐t❛❧s r❡❝❡✐✈❡ ❛ s♠❛❧❧ s❤❛r❡ ♦❢ tr❛♥s♣❧❛♥ts✬ s♦❝✐❛❧ ❜❡♥❡✜ts ❜✉t ❜❡❛r ♠❛♥② ♦❢ t❤❡ ❝♦sts✳ ❚❤❡r❡❢♦r❡✱ ❝♦sts t❤❛t ❛r❡ ♥❡❣❧✐❣✐❜❧❡ ❢r♦♠ ❛ s♦❝✐❛❧ ♣❡rs♣❡❝t✐✈❡ ❛r❡ ♠❛❣♥✐✜❡❞ ✇❤❡♥ ❞✐✈✐❞❡❞ ❜② v ✳ ❋✐❣✉r❡ ✺ ♣r❡s❡♥ts ❛ ❣r❛♣❤✐❝❛❧ ✐❧❧✉str❛t✐♦♥ t♦ ❝❧❛r✐❢② t❤❡ t✇♦ s♦✉r❝❡s ♦❢ ♠❛r❦❡t ❢❛✐❧✉r❡✿ ❛❣❡♥❝② ♣r♦❜❧❡♠s ❛♥❞ ✐♥❡✣❝✐❡♥t ♣❧❛t❢♦r♠ ✐♥❝❡♥t✐✈❡s✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ♣❧♦ts ❛❣❣r❡❣❛t❡ s✉♣♣❧② q ✳ ❚❤❡ ✈❡rt✐❝❛❧ ❛①✐s ♣❧♦ts ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✱ s♦❝✐❛❧ ❝♦sts✱ ❛♥❞ s♦❝✐❛❧ ❜❡♥❡✜ts✳ ❚❤❡ ❝✉rr❡♥t ✈❡❝t♦r ♦❢ r❡✇❛r❞s✱ ✇❤✐❝❤ ✐s ❡q✉❛❧ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♠❛t❝❤✐♥❣ ❡❛❝❤ ♣❛✐r✱ ✐s ❞❡♥♦t❡❞ ❜② p0 ✳ ❚❤❡ ❝✉rr❡♥t q✉❛♥t✐t② s✉♣♣❧✐❡❞ ❣✐✈❡♥ t❤❡s❡ r❡✇❛r❞s ✐s q 0 ✳ ❋♦r s✐♠♣❧✐❝✐t②✱ t❤❡ ✜❣✉r❡ ❛ss✉♠❡s t❤❛t ˜ t❤❡ ❛❣❣r❡❣❛t❡ s♦❝✐❛❧ ❝♦st ✐s ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥✱ SC(q) ✱ ♦❢ t❤❡ ❛❣❣r❡❣❛t❡ q✉❛♥t✐t②✱ ❛s ❧♦♥❣ ❛s ❡❛❝❤ ❤♦s♣✐t❛❧ ❝❤♦♦s❡s ❛♥ ♦♣t✐♠❛❧ q✉❛♥t✐t② ❣✐✈❡♥ ❛ ✈❡❝t♦r ♦❢ r❡✇❛r❞s✳✷✼ ❚❤❡ ✜❣✉r❡ s❤♦✇s t❤❛t t❤❡ ❝✉rr❡♥t ❛❧❧♦❝❛t✐♦♥ ✐s ✐♥❡✣❝✐❡♥t ❢r♦♠ ❜♦t❤ t❤❡ ❤♦s♣✐t❛❧ ❛♥❞ s♦❝✐❛❧ ♣❡rs♣❡❝t✐✈❡s✳ ❚❤❡ ❤♦s♣✐t❛❧✲♦♣t✐♠❛❧ q✉❛♥t✐t② ✐s q ∗ ❜❡❝❛✉s❡ ✐t ❡q✉❛t❡s ∇f ✇✐t❤ ♠❛r❣✐♥❛❧ ♣r✐✈❛t❡ ❝♦sts✳ ❚❤✉s✱ t❤❡ ✜rst ✐♥❡✣❝✐❡♥❝② ✐s t❤❛t t❤❡ ♣❧❛t❢♦r♠ ❣✐✈❡s ✐♥❡✣❝✐❡♥t ✐♥❝❡♥t✐✈❡s✱ p 6= ∇f ✳ ❚❤❡ s❡❝♦♥❞ ✐♥❡✣❝✐❡♥❝② ✐s t❤❛t t❤❡r❡ ❛r❡ ❛❣❡♥❝② ♣r♦❜❧❡♠s ❜❡❝❛✉s❡ ♣r✐✈❛t❡ ❝♦st ❛♥❞ s♦❝✐❛❧ ❝♦st ❞✐✛❡r✱ C h 6= SC h ♦r✱ ❛❧t❡r♥❛t✐✈❡❧②✱ ∂E 6= 0✳ ❚❤❡ s♦❝✐❛❧❧② ❡✣❝✐❡♥t q✉❛♥t✐t②✱ q ∗∗ ✱ ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ✉s✐♥❣ ❡✣❝✐❡♥t ✐♥❝❡♥t✐✈❡s✱ p = ∇f ✱ ❛♥❞ s♦❧✈✐♥❣ ❛❣❡♥❝② ♣r♦❜❧❡♠s s♦ t❤❛t ∂E = 0✳ ■♥ t❤✐s ❡①❛♠♣❧❡✱ ❛❣❡♥❝② ♣r♦❜❧❡♠s ❝❛♥ ❜❡ s♦❧✈❡❞
❜② r❡✐♠❜✉rs✐♥❣ ❤♦s♣✐t❛❧s ❢♦r t❤❡ ❝♦sts ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ t❤r♦✉❣❤ t❤❡ ♣❧❛t❢♦r♠✱ ✐✳❡✳ K h q h ✳

❙✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❛r❡ t❤❛t ❡❛❝❤ ❤♦s♣✐t❛❧✬s ♦♣t✐♠❛❧ q✉❛♥t✐t② ✐s ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ♦❢ r❡✇❛r❞s✱ ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ♠❛♣♣✐♥❣ r❡✇❛r❞s ✐♥t♦ t❤❡ ♦♣t✐♠❛❧ ❛❣❣r❡❣❛t❡ q✉❛♥t✐t② ❜② ❛❧❧ ❤♦s♣✐t❛❧s ✐s ❛ ❜✐❥❡❝t✐♦♥ ♦❢ [0, ∞)I ✐♥t♦ ✐ts❡❧❢✳ ✷✼

✷✻

∇C

∇f b b

p0

˜ ∇SC

b

q0

q∗

q ∗∗

❋✐❣✉r❡ ✺✿ ❚❤❡ ❚✇♦ ❙♦✉r❝❡s ♦❢ ▼❛r❦❡t ❋❛✐❧✉r❡

◆♦t❡s ✿

❚❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s r❡♣r❡s❡♥ts ❛❣❣r❡❣❛t❡ q✉❛♥t✐t② ♦❢ s✉❜♠✐ss✐♦♥s ✐♥t♦ t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳ ❚❤❡ ❝✉r✈❡s r❡♣r❡s❡♥t t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ s✉❜♠✐ss✐♦♥s ∇f (q)✱ t❤❡ ♠❛r❣✐♥❛❧ ♣r✐✈❛t❡ ❝♦st ♦❢ s✉❜♠✐ss✐♦♥s ˜ (q)✳ ❇♦t❤ ❛①❡s r❡♣r❡s❡♥t ∇C(q) ❢r♦♠ ❤♦s♣✐t❛❧✬s ♣❡rs♣❡❝t✐✈❡✱ ❛♥❞ t❤❡ ♠❛r❣✐♥❛❧ s♦❝✐❛❧ ❝♦st ♦❢ s✉❜♠✐ss✐♦♥s ∇SC I ✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦rs✳ ❚❤❡ ✜❣✉r❡ ❞❡♣✐❝ts t❤❡ ❝✉rr❡♥t q✉❛♥t✐t② q 0 ✱ ✇✐t❤ ❛❣❡♥❝② ♣r♦❜❧❡♠s ❛♥❞ ❛ s✉❜♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠✱ t❤❡ q✉❛♥t✐t② q ∗ ❢r♦♠ ❛ ❤♦s♣✐t❛❧✲♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ ❜✉t ✇✐t❤ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ ❛♥❞ t❤❡ ✜rst✲❜❡st q✉❛♥t✐t② q ∗∗ ✇✐t❤ ❛♥ ❡✣❝✐❡♥t ♠❡❝❤❛♥✐s♠✱ ❛♥❞ ♥♦ ❛❣❡♥❝② ♣r♦❜❧❡♠s✳ ❋♦r s✐♠♣❧✐❝✐t②✱ t❤❡ ✜❣✉r❡ ❛ss✉♠❡s t❤❛t ❛❣❣r❡❣❛t❡ s♦❝✐❛❧ ❝♦st ✐s ❛ ✉♥✐q✉❡❧② ❞❡✜♥❡❞ ❢✉♥❝t✐♦♥ ♦❢ r❡✇❛r❞s ❛♥❞ t❤❛t t❤❡ ♣❧❛t❢♦r♠ ❤❛s ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✳

❚❤✐s ✐♥t✉✐t✐✈❡ ❡①♣❧❛♥❛t✐♦♥ ❣❧♦ss❡❞ ♦✈❡r t✇♦ s✉❜t❧❡t✐❡s✱ ❛s ✇✐❧❧ ❜❡ ♠❛❞❡ ❝❧❡❛r ❜② t❤❡ ❢♦r♠❛❧ r❡✲ s✉❧ts✳ ❋✐rst✱ ❡✣❝✐❡♥t ♣❧❛t❢♦r♠ ✐♥❝❡♥t✐✈❡s ❛r❡ ♦♥❧② ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳ ❆ ♣❧❛t❢♦r♠ ❝❛♥♥♦t s❡t ✐♥❝❡♥t✐✈❡s ❡q✉❛❧ t♦ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s t♦ s❝❛❧❡✱ ❛♥❞ t❤❡r❡❢♦r❡✱ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❡①❝❡❡❞ ❛✈❡r❛❣❡ ♣r♦❞✉❝ts✳ ❍♦✇❡✈❡r✱ ❡st✐✲ ♠❛t❡s ✐♥ ❙❡❝t✐♦♥ ✺ ✇✐❧❧ s❤♦✇ t❤❛t t❤✐s ❛❞❥✉st♠❡♥t ✐s ♥❡❣❧✐❣✐❜❧❡ ❢♦r t❤❡ ◆❑❘✳ ❙❡❝♦♥❞✱ ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ r❡❛❝❤ t❤❡ ✜rst✲❜❡st s♦❝✐❛❧ ✇❡❧❢❛r❡ ❜② ♦♥❧② ✐♠♣r♦✈✐♥❣ t❤❡ ♠❡❝❤❛♥✐s♠ ✐❢ t❤❡r❡ ❛r❡ ❛❣❡♥❝② ♣r♦❜❧❡♠s✳ ❆❝❤✐❡✈✐♥❣ t❤❡ ✜rst✲❜❡st r❡q✉✐r❡s s❡tt✐♥❣ r❡✇❛r❞s ❡q✉❛❧ t♦ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ♣❧✉s t❤❡ ❡①t❡r♥❛❧✐t✐❡s✳ ❇✉t ❡✈❡♥ ✇✐t❤ ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✱ t❤❡r❡ ❛r❡ ♦♥❧② ❡♥♦✉❣❤ tr❛♥s♣❧❛♥ts t♦ ✐♠♣❧❡♠❡♥t r❡✇❛r❞ ❤♦s♣✐t❛❧s t❤❛t ❡q✉❛❧ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✱ ♠❛❦✐♥❣ ✐t ✐♠♣♦ss✐❜❧❡ t♦ r❡✇❛r❞ ❤♦s♣✐t❛❧s t♦ ❝♦rr❡❝t ❢♦r ❡①t❡r♥❛❧✐t✐❡s✳ ❚❤❡s❡ ❛r❣✉♠❡♥ts s✉❣❣❡st ❛ t✇♦✲ ♣r♦♥❣❡❞ ❛♣♣r♦❛❝❤✿ ❞❡s✐❣♥ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠s t❤❛t ❤❡❧♣ ❤♦s♣✐t❛❧s ❛❝❤✐❡✈❡ t❤❡✐r ❝♦❧❧❡❝t✐✈❡ ❣♦❛❧s✱ ❛♥❞ s✐♠✉❧t❛♥❡♦✉s❧② ✐♠♣❧❡♠❡♥t ♣♦❧✐❝✐❡s t♦ s♦❧✈❡ t❤❡ ❛❣❡♥❝② ♣r♦❜❧❡♠s✳ ✹✳✸

❖♣t✐♠❛❧ ■♥❝❡♥t✐✈❡s

❲❡ ❝❛♥ ♥♦✇ ❞❡s❝r✐❜❡ ♦♣t✐♠❛❧ r❡✇❛r❞ ✈❡❝t♦rs✱ ✇❤✐❝❤ ❝❛♥ ❣✉✐❞❡ t❤❡ ❞❡s✐❣♥ ♦❢ ♦♣t✐♠❛❧ ♠❡❝❤❛✲ ♥✐s♠s✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ❝♦❧❧❡❝ts t❤❡ ♠❛✐♥ ✐♥s✐❣❤ts✳

✷✼ ✭❖♣t✐♠❛❧ ❘❡✇❛r❞s✮✳ ❈♦♥s✐❞❡r ❛ ✈❡❝t♦r ♦❢ r❡✇❛r❞s p ❛♥❞ ❛❧❧♦❝❛t✐♦♥ (q h )h=1,...,H ✇✐t❤ str✐❝t❧② ♣♦s✐t✐✈❡ ❛❣❣r❡❣❛t❡ q✉❛♥t✐t② q t❤❛t ♠❛①✐♠✐③❡s ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ s✉❜❥❡❝t t♦ ❛❧❧ ❤♦s♣✐✲ t❛❧s ❝❤♦♦s✐♥❣ s✉♣♣❧② ♦♣t✐♠❛❧❧② ❣✐✈❡♥ p ❛♥❞ s✉❜❥❡❝t ❛❧❧♦❝❛t✐♥❣ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts t❤❛t ❛r❡ ♣r♦❞✉❝❡❞✱ f (q) = p · q ✳ ❚❤❡♥✿ ❚❤❡♦r❡♠ ✶

✶✳ ❚❤❡ ♣❧❛t❢♦r♠ r❡✇❛r❞s ❡❛❝❤ t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥ ✇✐t❤ ✐ts ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♠✐♥✉s ❛♥ ❛❞✲ ❥✉st♠❡♥t t❡r♠✱ p = ∇f (q) − A✱ ✇❤❡r❡ A=

∇f · q − f · q′ D2 C ✳ q′ · D2 C · q

✷✳ ■❢ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❤❛s ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✱ t❤❡♥ t❤❡ r❡✇❛r❞ ❢♦r ❡❛❝❤ t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥ ✐s ❡q✉❛❧ t♦ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✱ p = ∇f (q)✳ ▼♦r❡♦✈❡r✱ t❤❡ ❝❤♦s❡♥ ❤♦s♣✐t❛❧ s✉♣♣❧② (q h )h=1,...,H ❛tt❛✐♥s ✜rst✲❜❡st ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡✳ ✸✳ ■❢✱ ✐♥ ❛❞❞✐t✐♦♥✱ t❤❡r❡ ❛r❡ ♥♦ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ∂E = 0✱ t❤❡♥ t❤✐s ❛❧❧♦❝❛t✐♦♥ ❛❧s♦ ♠❛①✐♠✐③❡s t♦t❛❧ ✇❡❧❢❛r❡✳ ❚❤❡ t❤❡♦r❡♠ ❝❤❛r❛❝t❡r✐③❡s r❡✇❛r❞s ✐♥ ❛ ♠❡❝❤❛♥✐s♠ t❤❛t ♠❛①✐♠✐③❡ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡✳ ❚❤❡ ✜rst ♣❛rt s❤♦✇s t❤❛t t❤❡ r❡✇❛r❞ ❢♦r ❡❛❝❤ s✉❜♠✐ss✐♦♥ ✐♥ ❛♥ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ ✐s ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧ t♦ ✐ts ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✳ ❚❤❡ ✐♥t✉✐t✐♦♥ ❢♦r t❤✐s r❡s✉❧t ✐s s✐♠♣❧❡ ✐❢ ✇❡ ✐❣♥♦r❡ t❤❡ ❝♦♥str❛✐♥t t❤❛t t❤❡ ♣❧❛t❢♦r♠ ❝❛♥♥♦t ❛❧❧♦❝❛t❡ ♠♦r❡ tr❛♥s♣❧❛♥ts t❤❛♥ ✐t ♣r♦❞✉❝❡s✳ ❚❤❡ ♣❧❛t❢♦r♠ ✐s s✐♠✐❧❛r t♦ ❛ ✜r♠ t❤❛t ♣r♦❞✉❝❡s ❛ ❝♦♥s✉♠♣t✐♦♥ ❣♦♦❞ ✭tr❛♥s♣❧❛♥ts✮ ✉s✐♥❣ ✐♥t❡r♠❡❞✐❛t❡ ❣♦♦❞s ✭s✉❜✲ ♠✐ss✐♦♥s✮✳ ❚❤❡ s✉♣♣❧② ♦❢ ✐♥t❡r♠❡❞✐❛t❡ ❣♦♦❞s ✐s ❡✣❝✐❡♥t ✇❤❡♥ ♣r✐❝❡s p ❛r❡ ❡q✉❛❧ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ∇f ✳ ❚❤❡ ♣r♦♦❢ ✐s ✐❞❡♥t✐❝❛❧ ❢♦r ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s ❡✈❡♥ t❤♦✉❣❤ t❤❡r❡ ❛r❡ ♥♦ ♠♦♥❡t❛r② ♣r✐❝❡s ♣❛✐❞ t♦ ❛❝q✉✐r❡ s✉❜♠✐ss✐♦♥s✳ ❚❤❡ ✜rst ♦r❞❡r ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ✜rst✲❜❡st ❛❣❣r❡❣❛t❡ s✉♣♣❧② ✐s ∇C = ∇f ✳ ❚❤❡ ♠❛r❣✐♥❛❧ ❝♦st ❝✉r✈❡✱ ✇❤✐❝❤ ❣♦✈❡r♥s ❤♦s♣✐t❛❧ ✐♥❝❡♥t✐✈❡s✱ ❡q✉❛❧s t❤❡ s✉♣♣❧② ❝✉r✈❡❀ t❤❡r❡❢♦r❡✱ ♦♣t✐♠❛❧ r❡✇❛r❞s ❛r❡ p = ∇f ✳ ❚❤❡ ♦♥❧② ❝♦♠♣❧✐❝❛t✐♦♥ ✐s t❤❡ ❝♦♥str❛✐♥t t❤❛t ❛ ♣❧❛t❢♦r♠ ❝❛♥♥♦t ❛❧❧♦❝❛t❡ ♠♦r❡ tr❛♥s♣❧❛♥ts t❤❛♥ ✐t ♣r♦❞✉❝❡s✳ ❚❤✐s ❝♦♥str❛✐♥t ✐s ❜✐♥❞✐♥❣ ✐❢ f ❡①❤✐❜✐ts ✇❡❛❦❧② ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s t♦ s❝❛❧❡ ❜❡❝❛✉s❡ t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ♣r♦❞✉❝❡❞✱ f (q)✱ ✐s ❛t ❧❡❛st t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts t❤❛t ✇♦✉❧❞ ❤❛✈❡ t♦ ❜❡ ♣❛✐❞ ✐❢ r❡✇❛r❞s ✇❡r❡ ❡q✉❛❧ t♦ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✱ ∇f (q) · q ✳ ■t ✐s ❡❛s② t♦ ✈✐s✉❛❧✐③❡ t❤✐s ❝❛s❡ ✐❢ q ✐s ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❜❡❝❛✉s❡ t❤❡ t❛♥❣❡♥ts t♦ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❝r♦ss t❤❡ ✈❡rt✐❝❛❧ ❛①✐s ❜❡❧♦✇ ③❡r♦✳ ❚❤❡r❡❢♦r❡✱ ❛ ♣❧❛t❢♦r♠ ❤❛s t♦ s❤❛❞❡ ✐ts r❡✇❛r❞s t♦ ❤♦s♣✐t❛❧s r❡❧❛t✐✈❡ t♦ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳ ❚❤❡ ♦♣t✐♠❛❧ ❧❡✈❡❧ ♦❢ s❤❛❞✐♥❣ ❢♦r ❡❛❝❤ t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❛❞❥✉st♠❡♥t t❡r♠ A✳ ❚❤❡ ❛❞❥✉st♠❡♥t t❡r♠ s❛②s t❤❛t t❤❡ ♣❧❛t❢♦r♠ s❤♦✉❧❞ s❤❛❞❡ ♠♦r❡ ❛❣❣r❡ss✐✈❡❧② ♦♥ s✉❜♠✐ss✐♦♥s ✇✐t❤ ❧❡ss ❡❧❛st✐❝ s✉♣♣❧②✳ ❚♦ s❡❡ t❤✐s✱ ♥♦t❡ t❤❛t ♠❛r❣✐♥❛❧ ❝♦st ∇C ❡q✉❛❧s t❤❡ ✐♥✈❡rs❡ s✉♣♣❧② ❝✉r✈❡✱ ✉♥❞❡r s♦♠❡ r❡❣✉❧❛r✐t② ❝♦♥❞✐t✐♦♥s✱ s♦ t❤❛t D 2 C ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ✐♥✈❡rs❡ s✉♣♣❧② ❝✉r✈❡✳ ■♥ ❢❛❝t✱ ♦✉r ❢♦r♠✉❧❛ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❘❛♠s❡② ✭✶✾✷✼✮✲❇♦✐t❡✉① ✭✶✾✺✻✮ ❢♦r♠✉❧❛ ❢♦r ❛♥ ♦♣t✐♠❛❧ ❧✐♥❡❛r ❝♦♠♠♦❞✐t② t❛①✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ t❤❡ ❝r♦ss✲❡❧❛st✐❝✐t✐❡s ♦❢ s✉♣♣❧② ❛r❡ ③❡r♦✱ ✇❡ ♦❜t❛✐♥ ❛♥ ✐♥✈❡rs❡✲❡❧❛st✐❝✐t② r✉❧❡ ❢♦r t❤❡ ♦♣t✐♠❛❧ s❤❛❞✐♥❣✱ ❛s ✐♥ ❘❛♠s❡②✬s ✇♦r❦ ❛♥❞ ✐♥ t❤❡ ▲❡r♥❡r ✐♥❞❡① ❢r♦♠ ♦♣t✐♠❛❧ ♠♦♥♦♣♦❧② ♣r✐❝✐♥❣✳

✷✽ ❚❤❡ t❤❡♦r❡♠ s✉❣❣❡sts t❤❛t ❝✉rr❡♥t ♣❧❛t❢♦r♠ r✉❧❡s ❛r❡ ✐♥❡✣❝✐❡♥t✳ ■♥st❡❛❞ ♦❢ r❡✇❛r❞✐♥❣ s✉❜✲ ♠✐ss✐♦♥s ✇✐t❤ t❤❡✐r ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✱ ❝✉rr❡♥t r✉❧❡s r❡✇❛r❞ s✉❜♠✐ss✐♦♥s ✇✐t❤ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ tr❛♥s♣❧❛♥t❡❞✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ✐s ❛ ✇❡❞❣❡ ❜❡t✇❡❡♥ t❤❡ s♦❝✐❛❧ ❛♥❞ ♣r✐✈❛t❡ ❜❡♥❡✜ts ♦❢ s✉❜♠✐ss✐♦♥s✳ ❯♥❞❡r ❝✉rr❡♥t r✉❧❡s✱ ❛ ❤♦s♣✐t❛❧ ❝❤♦♦s❡s ❜❡t✇❡❡♥ s❡r✈✐♥❣ t❤❡✐r ♦✇♥ ♣❛t✐❡♥ts ♦r ♣r♦✈✐❞✐♥❣ ❛ s❡r✈✐❝❡ t♦ t❤❡ s②st❡♠ ❛s ❛ ✇❤♦❧❡✳ ❆ ❝❧❡❛r ❡①❛♠♣❧❡ ♦❢ t❤✐s ❞✐❧❡♠♠❛✱ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✷✱ ✐s ♦❢ ❛ ❤♦s♣✐t❛❧ ✇✐t❤ t✇♦ ♦✈❡r❞❡♠❛♥❞❡❞ ♣❛✐rs✳ ❚❤✐s ❤♦s♣✐t❛❧ ❝♦✉❧❞ ♠❛t❝❤ t❤❡ ♣❛✐rs ✐♥t❡r♥❛❧❧② ✐♥st❡❛❞ ♦❢ s✉❜♠✐tt✐♥❣ t❤❡♠ t♦ ❛ ♣❧❛t❢♦r♠✱ ❜✉t ❞♦✐♥❣ s♦ ✇♦✉❧❞ ❝❛✉s❡ t❤❡ t②♣❡ ♦❢ ✐♥❡✣❝✐❡♥❝② ❞♦❝✉♠❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳ ❚❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ s❤♦✇s t❤❛t✱ ✇❤❡♥ r❡t✉r♥s t♦ s❝❛❧❡ ❛r❡ ❝♦♥st❛♥t✱ t❤❡ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ r❡✇❛r❞s s✉❜♠✐ss✐♦♥s ❡①❛❝t❧② ❛❝❝♦r❞✐♥❣ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳ ❚❤❡ ❛❞❥✉st♠❡♥t t❡r♠ ✐♥ t❤✐s ❝❛s❡ ❡q✉❛❧s ③❡r♦✱ ❛♥❞ ♦♣t✐♠❛❧ r❡✇❛r❞s ❛❝❤✐❡✈❡ ✜rst✲❜❡st ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡✳ ❆s ✇❡ ✇✐❧❧ s❤♦✇ ✐♥ ❙❡❝t✐♦♥ ✺✱ t❤✐s ❝❛s❡ ✐s ❡♠♣✐r✐❝❛❧❧② r❡❧❡✈❛♥t ❜❡❝❛✉s❡ t❤❡ ◆❑❘ ✐s ✇❡❧❧ ✇✐t❤✐♥ t❤❡ r❡❣✐♦♥ ♦❢ ❛♣♣r♦①✐♠❛t❡❧② ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✳ ❚❤❡r❡❢♦r❡✱ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠s ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ✐♥ ♣r❛❝t✐❝❡ ❜② ❡st✐♠❛t✐♥❣ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❜❡❝❛✉s❡ t❤❡② ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ s✉♣♣❧② ❡❧❛st✐❝✐t✐❡s✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ♥♦♥❧✐♥❡❛r r❡✇❛r❞s ❜❡❝❛✉s❡ ✇❡ ❝❛♥ ❛❝❤✐❡✈❡ ✜rst✲❜❡st ✇❡❧❢❛r❡ ❜② r❡✇❛r❞✐♥❣ ❤♦s♣✐t❛❧s ❧✐♥❡❛r❧② ✇✐t❤ t❤❡✐r s✉❜♠✐ss✐♦♥s✳ ❖♥❡ ❛♣♣r♦❛❝❤ ❢♦r ✉s✐♥❣ t❤❡s❡ r❡s✉❧ts ✐♥ ♣r❛❝t✐❝❡ ✐s t♦ ✐♥tr♦❞✉❝❡ ❛ s✐♠♣❧❡ ❞②♥❛♠✐❝ ♣♦✐♥ts ♠❡❝❤❛♥✐s♠✳ ❋♦r ❡①❛♠♣❧❡✱ ❢♦r ❡❛❝❤ s✉❜♠✐ss✐♦♥✱ ❛ ♣❧❛t❢♦r♠ ❝❛♥ ❝r❡❞✐t ❛ ❤♦s♣✐t❛❧ ♣♦✐♥ts ❡q✉❛❧ t♦ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✳ ❚❤❡♥ ❛ ♣♦✐♥t ❝❛♥ ❜❡ s✉❜tr❛❝t❡❞ ✇❤❡♥❡✈❡r ❛ ❤♦s♣✐t❛❧ ❝♦♥❞✉❝ts ❛ tr❛♥s♣❧❛♥t✳ ❚❤❡ ♣❧❛t❢♦r♠ ♣❡r❢♦r♠s ♦♣t✐♠❛❧ ♠❛t❝❤❡s ✇✐t❤ ❛ ❝♦♥str❛✐♥t t❤❛t ♥♦ ❜❛❧❛♥❝❡ ❢❛❧❧s ❜❡❧♦✇ ❛ ❝❡rt❛✐♥ ❧❡✈❡❧✳ ◆❛t✉r❛❧❧②✱ t❤❡r❡ ❛r❡ ✐♠♣♦rt❛♥t t❤❡♦r❡t✐❝❛❧ ✐ss✉❡s r❡❧❛t❡❞ t♦ ✐♠♣❧❡♠❡♥t✐♥❣ ✐♥❝❡♥t✐✈❡s ✐♥ t❤✐s ❦✐♥❞ ♦❢ ♠❡❝❤❛♥✐s♠ ✇✐t❤♦✉t ❝♦♠♣r♦♠✐s✐♥❣ ❡✣❝✐❡♥❝② ✭❍❛✉s❡r ❛♥❞ ❍♦♣❡♥❤❛②♥✱ ✷✵✵✽✮✳ ❲❡ ✇✐❧❧ r❡t✉r♥ t♦ t❤❡s❡ ✐ss✉❡s ✐♥ ❙❡❝t✐♦♥ ✻✳ ❚❤❡ t❤✐r❞ ♣❛rt ♦❢ t❤❡ t❤❡♦r❡♠ st❛t❡s t❤❛t ✐❢ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❡①❤✐❜✐ts ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡ ❛♥❞ t❤❡r❡ ❛r❡ ♥♦ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ t❤❡♥ t❤❡ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ ❛❝❤✐❡✈❡s ✜rst✲❜❡st s♦❝✐❛❧ ✇❡❧❢❛r❡✳ ❚❤✐s r❡s✉❧t ❝❧❛r✐✜❡s t❤❛t t❤❡r❡ ❛r❡ t✇♦ ♣♦ss✐❜❧❡ s♦✉r❝❡s ♦❢ ✐♥❡✣❝✐❡♥❝② ✐♥ ♦✉r ♠♦❞❡❧✿ ✐♥❡✣❝✐❡♥t ♣❧❛t❢♦r♠ ✐♥❝❡♥t✐✈❡s ❛♥❞ ❛❣❡♥❝② ♣r♦❜❧❡♠s✳ P❧❛t❢♦r♠ ✐♥❝❡♥t✐✈❡s ❛r❡ ✐♥❡❢✲ ✜❝✐❡♥t ✐❢ r❡✇❛r❞s ❞❡✈✐❛t❡ ❢r♦♠ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✱ p 6= ∇f ✳ ■♥ t❤❡ ♣❧❛t❢♦r♠s ❧✐t❡r❛t✉r❡✱ t❤✐s ♣r♦❜❧❡♠ ✐s ✉s✉❛❧❧② ❛ttr✐❜✉t❡❞ t♦ ✇❡❞❣❡s ❜❡t✇❡❡♥ t❤❡ ♣❧❛t❢♦r♠✬s ❣♦❛❧s ❛♥❞ s♦❝✐❡t②✬s ✭❘♦✲ ❝❤❡t ❛♥❞ ❚✐r♦❧❡✱ ✷✵✵✸❀ ❆r♠str♦♥❣✱ ✷✵✵✻❀ ❲❡②❧✱ ✷✵✶✵✮✳ ❆❣❡♥❝② ♣r♦❜❧❡♠s ❡①✐st ✐❢ ❤♦s♣✐t❛❧s ❞♦ ♥♦t ❢✉❧❧② ✐♥t❡r♥❛❧✐③❡ t❤❡ ✇❡❧❢❛r❡ ♦❢ t❤❡ ♣❛rt✐❡s t❤❡② r❡♣r❡s❡♥t✱ ✐✳❡✳ ∂E 6= 0✳ ❚❤❡ ♠❛r❦❡t ❢✉♥❝t✐♦♥s ❡✣❝✐❡♥t❧② ✐❢ ♣❧❛t❢♦r♠ ✐♥❝❡♥t✐✈❡s ❛r❡ ♦♣t✐♠❛❧ ✭p = ∇f ✮ ❛♥❞ t❤❡r❡ ❛r❡ ♥♦ ❛❣❡♥❝② ♣r♦❜❧❡♠s ✭∂E = 0✮✳ ❚❤✐s s✉❣❣❡sts t❤❡ ✐♥❡✣❝✐❡♥❝② ❞♦❝✉♠❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✸ ♠❛② ❜❡ ❞✉❡ t♦ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✐♥❡✣❝✐❡♥t ♣❧❛t❢♦r♠ ✐♥❝❡♥t✐✈❡s ❛♥❞ ❛❣❡♥❝② ♣r♦❜❧❡♠s✳ ❋✐❣✉r❡ ✺ ❞❡♣✐❝ts t❤❡s❡ t✇♦ ♠❛r❦❡t ❢❛✐❧✉r❡s ✉♥❞❡r s♦♠❡ r❡❣✉❧❛r✐t② ❝♦♥❞✐t✐♦♥s✳ ❚❤❡ ❝✉rr❡♥t ❛❣❣r❡❣❛t❡ s✉♣♣❧② ✐s q 0 ✱ ✇❤✐❝❤ ✐s ❞❡t❡r♠✐♥❡❞ ❜② r❡✇❛r❞s t❤❛t ❡q✉❛❧ ♠❛t❝❤✐♥❣ ♣r♦❜❛❜✐❧✐t✐❡s✳ ■❢ ❛ ♣❧❛t❢♦r♠ s✇✐t❝❤❡s t♦ ❛♥ ❡✣❝✐❡♥t ♠❡❝❤❛♥✐s♠✱ ❛❣❣r❡❣❛t❡ s✉♣♣❧② ♠♦✈❡s t♦ q ∗ ✳ ■❢ ❛❣❡♥❝② ♣r♦❜❧❡♠s ❛r❡ ❛❧s♦ s♦❧✈❡❞✱ t❤❡ ♠❛r❦❡t ♠♦✈❡s t♦ t❤❡ ✜rst✲❜❡st ❛❣❣r❡❣❛t❡ s✉♣♣❧② q ∗∗ ✳ ❚❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❛t ❛♥② ♦❢ t❤❡s❡ ♣♦✐♥ts ✐s ❣✐✈❡♥ ❜② ❛ ✭♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧✮ ❍❛r❜❡r❣❡r tr✐❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ❛♥❞ t❤❡ ♠❛r❣✐♥❛❧ s♦❝✐❛❧ ❝♦st ❝✉r✈❡s✳ ❚❤❡ ✉♣s❤♦t ♦❢ t❤✐s ❛♥❛❧②s✐s ✐s t❤❛t✱ ♠✉❝❤ ❧✐❦❡ ✐♥ ♠♦r❡ tr❛❞✐t✐♦♥❛❧ ♠❛r❦❡ts✱ ♠❛♥② ❦❡② q✉❡st✐♦♥s

✷✾ ❛❜♦✉t ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❞❡♣❡♥❞ ♦♥ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ ✇❡ t✉r♥ t♦ ♥❡①t✳

✺

Pr♦❞✉❝t✐♦♥ ❋✉♥❝t✐♦♥ ❊st✐♠❛t❡s ❛♥❞ ❘❡s✉❧ts

❲❡ ♥♦✇ ❡st✐♠❛t❡ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ❞❛t❛ ❢r♦♠ t❤❡ ❧❛r❣❡st ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t✲ ❢♦r♠✱ t❤❡ ◆❑❘✳ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ◆❑❘ ❜❡❝❛✉s❡ ✐t ✐s t❤❡ ❞♦♠✐♥❛♥t ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ❞✉r✐♥❣ ♦✉r s❛♠♣❧❡ ♣❡r✐♦❞ ✭❚❛❜❧❡ ✶✮✳✷✽ ❲❡ ✉s❡ t❤❡s❡ ❡st✐♠❛t❡s t♦ ♠❡❛s✉r❡ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✐♥❡✣❝✐❡♥❝✐❡s✱ ❛♥❞ ❞❡✈❡❧♦♣ s✐♠♣❧❡ ♣♦❧✐❝② r❡s♣♦♥s❡s✳ ❚❤❡ ❦❡② r❡s✉❧ts ❝♦♠❡ ❢r♦♠ t❤r❡❡ ❛♥❛❧②s❡s✳ ❋✐rst✱ ✇❡ ♠❡❛s✉r❡ t♦t❛❧ ✐♥❡✣❝✐❡♥❝② ❜② ❡st✐♠❛t✐♥❣ ❜♦t❤ t❤❡ r❡t✉r♥s t♦ s❝❛❧❡ ❛♥❞ ❤♦✇ ♠❛♥② ♠♦r❡ tr❛♥s♣❧❛♥ts ✇♦✉❧❞ ❜❡ ♣❡r❢♦r♠❡❞ ✐❢ ♣r♦❞✉❝t✐♦♥ ✇❛s ♠♦✈❡❞ t♦ t❤❡ ❡✣❝✐❡♥t s❝❛❧❡✳ ❙❡❝♦♥❞✱ ✇❡ ❞❡s✐❣♥ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠s ✉s✐♥❣ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳ ❋✐♥❛❧❧②✱ ✇❡ ✉s❡ ♦✉r ♣r✐❝❡✲ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦ t♦ ♠❡❛s✉r❡ t❤❡ ❣❛✐♥s ❢r♦♠ ✐♠♣❧❡♠❡♥t✐♥❣ ♦♣t✐♠❛❧ ✐♥❝❡♥t✐✈❡s t❤r♦✉❣❤ t❤❡ ♠❡❝❤❛♥✐s♠ ❛❧♦♥❡✳ ✺✳✶

❊st✐♠❛t✐♦♥

Pr♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♠♠♦♥❧② ❡st✐♠❛t❡❞ ✉s✐♥❣ ❞❛t❛ ♦♥ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❢r♦♠ s❡✈❡r❛❧ ✜r♠s✳ ❚❤❡ ❦❡② ❡❝♦♥♦♠❡tr✐❝ ❝❤❛❧❧❡♥❣❡s ✐♥ t❤✐s ❧✐t❡r❛t✉r❡ ❛r❡ ❡♥❞♦❣❡♥❡✐t② ✐♥ t❤❡ ❝❤♦s❡♥ ✐♥♣✉ts ❛♥❞ s❡❧❡❝t✐♦♥ ✐♥ t❤❡ s❡t ♦❢ ♦♣❡r❛t✐♥❣ ✜r♠s ✭s❡❡ ▼❛rs❝❤❛❦ ❛♥❞ ❆♥❞r❡✇s✱ ✶✾✹✹❀ ❖❧❧❡② ❛♥❞ P❛❦❡s✱ ✶✾✾✻✮✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤✐s ❛♣♣r♦❛❝❤ ✐s ♥♦t ❛♣♣r♦♣r✐❛t❡ ✐♥ ♦✉r s❡tt✐♥❣ ❢♦r t❤r❡❡ r❡❛s♦♥s✳ ❋✐rst✱ t❤❡s❡ ♠❡t❤♦❞s ❛r❡ ❜❡st s✉✐t❡❞ ❢♦r ❧♦✇✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥s t❤❛t ♦♥❧② ❞❡♣❡♥❞ ♦♥ ❛ ❢❡✇ ✐♥♣✉ts✱ s✉❝❤ ❛s ❝❛♣✐t❛❧ ❛♥❞ ❧❛❜♦r✱ ❜✉t s✉✛❡r ❢r♦♠ ❛ ❝✉rs❡ ♦❢ ❞✐♠❡♥s✐♦♥❛❧✐t② ✐❢ t❤❡r❡ ❛r❡ ♠❛♥② ✐♥♣✉t t②♣❡s✳ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ✈❡❝t♦r ♦❢ ✐♥♣✉ts ✐s ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ ❜❡❝❛✉s❡ s✉❜♠✐ss✐♦♥s ❝❛♥ ✈❛r② ✐♥ ♠❛♥② ✇❛②s✳ ❙❡❝♦♥❞✱ ❝♦♠♠♦♥❧② ✉s❡❞ ❢✉♥❝t✐♦♥❛❧ ❢♦r♠s s✉❝❤ ❛s ❈♦❜❜✲ ❉♦✉❣❧❛s r❡str✐❝t ❛❧❧ ✐♥♣✉ts t♦ ❜❡ s✉❜st✐t✉t❡s✱ ❛ ♣r♦♣❡rt② t❤❛t ✐s ♥♦t ❛♣♣r♦♣r✐❛t❡ ❢♦r ❛ ♠❛t❝❤✐♥❣ ❝♦♥t❡①t✳ ❚❤✐r❞✱ t❤❡ st❛♥❞❛r❞ ♠❡t❤♦❞s ❞❡♣❡♥❞ ♦♥ ❛ ♣❛♥❡❧ ❞❛t❛s❡t ✇✐t❤ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ♦❢ ♠✉❧t✐♣❧❡ ✜r♠s ❛♥❞ ❡①♦❣❡♥♦✉s ✈❛r✐❛t✐♦♥ ♦❢ ✐♥♣✉ts✳ ❍♦✇❡✈❡r✱ ✇❡ ♦♥❧② ❤❛✈❡ ❞❛t❛ ❢r♦♠ ❛ s✐♥❣❧❡ ❧❛r❣❡ ♣❧❛t❢♦r♠✳ ❲❡ ❛r❡ ❛❜❧❡ ❝✐r❝✉♠✈❡♥t t❤❡s❡ ❡❝♦♥♦♠❡tr✐❝ ✐ss✉❡s ❜② ✉s✐♥❣ ❛♥ ❡♥❣✐♥❡❡r✐♥❣ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ ❞❡t❛✐❧❡❞ ✐♥st✐t✉t✐♦♥❛❧ ❦♥♦✇❧❡❞❣❡ ❛♥❞ ❛❞♠✐♥✐str❛t✐✈❡ ❞❛t❛ ♦♥ t❤❡ ♣r♦❝❡ss❡s ✐♥✈♦❧✈❡❞ ✐♥ ♦r❣❛♥✐③✲ ✐♥❣ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳ ❲❡ ❤❛✈❡ ❞❡t❛✐❧❡❞ ✐♥st✐t✉t✐♦♥❛❧ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ♦♣❡r❛t✐♦♥❛❧ ♣r♦❝❡❞✉r❡s ❛♥❞ ❛❧❣♦r✐t❤♠s ✉s❡❞ ❜② ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s✳ ❖♥❡ ♦❢ ✉s ✭❆s❤❧❛❣✐✮ ❤❛s ❞❡✈❡❧♦♣❡❞ t❤❡ ♠❛t❝❤✐♥❣ s♦❢t✇❛r❡ ❢♦r s❡✈❡r❛❧ ♣❧❛t❢♦r♠s✱ ❛♥❞ ❤❛s ✇♦r❦❡❞ ✇✐t❤ t❤❡ ◆❑❘✳ ▼♦r❡♦✈❡r✱ ✇❡ ❤❛✈❡ ❞❡t❛✐❧❡❞ ❞❛t❛ ♦♥ ◆❑❘ ♦♣❡r❛t✐♦♥s ❛♥❞ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❜✐♦❧♦❣✐❝❛❧ ❝♦♠♣❛t✐❜✐❧✐t② ♦❢ ✐ts ♣❛t✐❡♥t ♣♦♦❧✳ ❲❡ ✉s❡ t❤✐s ❡①♣❡r✐❡♥❝❡ ❛♥❞ ❦♥♦✇❧❡❞❣❡ t♦ ❞❡✈❡❧♦♣ ❛ ❞❡t❛✐❧❡❞ s✐♠✉❧❛t✐♦♥ ♠♦❞❡❧ ♦❢ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳ ✷✽ ❆

s✐♠✐❧❛r ❡①❡r❝✐s❡ ❝❛♥ ❜❡ ❝♦♥❞✉❝t❡❞ ✉s✐♥❣ ❞❛t❛ ❢r♦♠ t❤❡ ❆P❉ ♦r ❯◆❖❙✳ ❚❤❡r❡ ❛r❡ t✇♦ ❝❤❛❧❧❡♥❣❡s t❤❛t ♠✉st ❜❡ ♦✈❡r❝♦♠❡✳ ❋✐rst✱ t❤❡s❡ ♣❧❛t❢♦r♠s t♦❣❡t❤❡r ❤❛✈❡ r❡❧❛t✐✈❡❧② s♠❛❧❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥t❡❞ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❝r❡❛t✐♥❣ ❞✐✣❝✉❧t✐❡s ✐♥ ❛❝❝✉r❛t❡❧② ❡st✐♠❛t✐♥❣ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡❧✳ ❙❡❝♦♥❞✱ ❞✐✛❡r❡♥❝❡s ✐♥ t❤❡ ❞❛t❛ ❝♦❧❧❡❝t✐♦♥ ♣r♦t♦❝♦❧s ❛♥❞ ❧♦❣✐st✐❝s ❛❝r♦ss t❤❡s❡ ♣❧❛t❢♦r♠s ✇♦✉❧❞ r❡q✉✐r❡ ✉s t♦ ✇r✐t❡ s❡♣❛r❛t❡ ❝♦❞❡ ❢♦r t❤❡s❡ t✇♦ ♣❧❛t❢♦r♠s✳ ❲❡ t❤❡r❡❢♦r❡ ❢♦❝✉s ♦✉r ❡✛♦rts ♦♥ ❞❛t❛ ❢r♦♠ t❤❡ ◆❑❘✳

✸✵ ❲❡ s✐♠✉❧❛t❡ t❤❡ ✈❛r✐♦✉s st❡♣s ✐♥✈♦❧✈❡❞ ✐♥ ♦r❣❛♥✐③✐♥❣ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ t♦ ❡✈❛❧✉❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts✱ f (q; θ)✱ t❤❛t ❝❛♥ ❜❡ ♣r♦❞✉❝❡❞ ✇✐t❤ ❛♥② s❡t ♦❢ ✐♥♣✉ts q ❛♥❞ ♣❛r❛♠❡t❡rs θ. ❚❤❡r❡ ❛r❡ ❢♦✉r ❦❡② ❡✈❡♥ts t❤❛t t❛❦❡ ♣❧❛❝❡✱ s✉❜♠✐ss✐♦♥s t♦ t❤❡ ♣❧❛t❢♦r♠✱ tr❛♥s♣❧❛♥t ♣r♦♣♦s❛❧✱ ✜♥❛❧ r❡✈✐❡✇ ❛♥❞ tr❛♥s♣❧❛♥t❛t✐♦♥✱ ❛♥❞ ❞❡♣❛rt✉r❡ ❢r♦♠ t❤❡ ♣❧❛t❢♦r♠✳ ❊❛❝❤ st❡♣ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❞❡t❛✐❧ ❜❡❧♦✇✱ ❛♥❞ ✐s ❣♦✈❡r♥❡❞ ❜② ❛ s❡t ♦❢ ♣❛r❛♠❡t❡rs✳ ❚❤❡ ♣❛r❛♠❡t❡rs ❣♦✈❡r♥✐♥❣ t❤❡ ✜rst ❛♥❞ ❧❛st st❡♣s ❛r❡ ❞✐r❡❝t❧② ❡st✐♠❛t❡❞ ❢r♦♠ t❤❡ ◆❑❘ ❞❛t❛❀ t❤❡ ♣❛r❛♠❡t❡rs ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ s❡❝♦♥❞ st❡♣ ❛r❡ ❦♥♦✇♥❀ ❛♥❞ t❤❡ ♣❛r❛♠❡t❡rs ❢r♦♠ t❤❡ t❤✐r❞ st❡♣ ❛r❡ ❝❛❧✐❜r❛t❡❞ t♦ ✜t ♦❜s❡r✈❡❞ ❝❤❛✐♥ ❧❡♥❣t❤s ❛♥❞ tr❛♥s♣❧❛♥t❛t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ❢♦r ✈❛r✐♦✉s ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦rs✳ ❖✉r ❡st✐♠❛t✐♦♥ ❛♥❞ ❝❛❧✐❜r❛t✐♦♥ ♠❡t❤♦❞s ❛r❡ ❞❡s❝r✐❜❡❞ ❜❡❧♦✇✱ ✇✐t❤ ❞❡t❛✐❧s ♣r♦✈✐❞❡❞ ✐♥ ❆♣♣❡♥❞✐① ❈✳ ❚❤❡s❡ st❡♣s ❛♥❞ t❤❡✐r ❛ss♦❝✐❛t❡❞ ♣❛r❛♠❡t❡rs ❛r❡ ❛s ❢♦❧❧♦✇s✿ ✶✳

q ✿ ❍♦s♣✐t❛❧s s✉❜♠✐t ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✱ ❡✐t❤❡r ✐♥❞✐✈✐❞✉❛❧❧② ♦r ✐♥ ♣❛✐rs✱ t♦ t❤❡ ♣❧❛t❢♦r♠✳ ❚❤❡s❡ s✉❜♠✐ss✐♦♥s ❛r❡ ❛❞❞❡❞ t♦ t❤❡ ❝✉rr❡♥t ♣♦♦❧ ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❛❧r❡❛❞② r❡❣✐st❡r❡❞ ✇✐t❤ t❤❡ ❡①❝❤❛♥❣❡✳ P❛t✐❡♥ts ❛♥❞ ❞♦❝t♦rs✱ ❛t t❤✐s t✐♠❡✱ ❝❛♥ s✉❜♠✐t ♠✐♥✐♠❛❧ ❛❝❝❡♣t❛♥❝❡ ❝r✐t❡r✐❛ ❢♦r ❛ ❞♦♥♦r✳

❙✉❜♠✐ss✐♦♥s✱

❲❡ ❡st✐♠❛t❡ t❤❡ ✈❡❝t♦r ♦❢ ❝✉rr❡♥t s✉❜♠✐ss✐♦♥ r❛t❡s✱ q 0 ✱ ❜② r❡s❛♠♣❧✐♥❣ ❢r♦♠ t❤❡ ❛❞✲ ♠✐♥✐str❛t✐✈❡ ◆❑❘ ❞❛t❛✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ❡st✐♠❛t❡ t❤❡ t♦t❛❧ s✉❜♠✐ss✐♦♥s ♣❡r ❞❛② ❛s ❛♥ ❡♠♣✐r✐❝❛❧ ❛✈❡r❛❣❡ ❛♥❞ t❤❡♥ ❞r❛✇ s♣❡❝✐✜❝ ♣❛t✐❡♥t✴❞♦♥♦r✴♣❛✐r ❝❤❛r❛❝t❡r✐st✐❝s ❛♥❞ ♠✐♥✐✲ ♠✉♠ ❛❝❝❡♣t❛♥❝❡ ❝r✐t❡r✐❛ ❢r♦♠ t❤❡ ❞❛t❛s❡t ✇✐t❤ r❡♣❧❛❝❡♠❡♥t✳ ❚❤❡ t✐♠❡✲❛✈❡r❛❣❡ ♦❢ t❤❡s❡ s✐♠✉❧❛t✐♦♥ ❞r❛✇s ②✐❡❧❞s qˆ0,i ❢♦r ❛♥② ✜♥✐t❡ ♣❛rt✐t✐♦♥ ♦❢ ✐♥❞✐✈✐❞✉❛❧ ♣❛t✐❡♥ts✱ ❞♦♥♦rs✱ ♦r ♣❛✐rs ✐♥t♦ t②♣❡s✱ i = 1, . . . , I ✳ ❆♥ ✐❞❡♥t✐❝❛❧ r❡s❛♠♣❧✐♥❣ ♣r♦❝❡ss ❛❧❧♦✇s ✉s t♦ s✐♠✉❧❛t❡ ❛rr✐✈❛❧s ❢♦r ❛♥② ❛❧t❡r♥❛t✐✈❡ r❛t❡ ♦❢ s✉❜♠✐ss✐♦♥s q 6= q 0 ✳ ✷✳

❊❛❝❤ ❞❛②✱ t❤❡ ◆❑❘ ✐❞❡♥t✐✜❡s ❛♥ ♦♣t✐♠❛❧ ✇❡✐❣❤t❡❞ s❡t ♦❢ ♣♦✲ t❡♥t✐❛❧ ❡①❝❤❛♥❣❡s ✇✐t❤✐♥ t❤❡ st♦❝❦ ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs r❡❣✐st❡r❡❞ ✇✐t❤ t❤❡ ♣❧❛t❢♦r♠✳ ❚❤✐s ❛❧❣♦r✐t❤♠ ✐♥❝♦r♣♦r❛t❡s ❢♦✉r ❝♦♥str❛✐♥ts✳ ❋✐rst✱ ♥♦♥❡ ♦❢ t❤❡ ♣r♦♣♦s❡❞ tr❛♥s♣❧❛♥ts s❤♦✉❧❞ ❜❡ ✭❦♥♦✇♥ t♦ ❜❡✮ ❜✐♦❧♦❣✐❝❛❧❧② ✐♥❝♦♠♣❛t✐❜❧❡ ♦r r✉❧❡❞ ♦✉t ❜② ♣r❡✲s❡t ❛❝❝❡♣t❛♥❝❡ ❝r✐t❡r✐❛✳ ❚❤❡s❡ ❝♦♥str❛✐♥ts ❛r❡ ❞✐r❡❝t❧② ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❞❛t❛✳ ❙❡❝♦♥❞✱ ♥♦ ❞♦♥♦r ♦r r❡✲ ❝✐♣✐❡♥t ❝❛♥ ❜❡ ✐♥✈♦❧✈❡❞ ✐♥ ♠♦r❡ t❤❛♥ ♦♥❡ tr❛♥s♣❧❛♥t✳ ❚❤✐r❞✱ ❛ ❞♦♥♦r ✇❤♦ ✐s ♣❛rt ♦❢ ❛ ♣❛✐r ✐s ♦♥❧② ❛s❦❡❞ t♦ ❞♦♥❛t❡ ❛♥ ♦r❣❛♥ ✐❢ t❤❡ ✐♥t❡♥❞❡❞ r❡❝✐♣✐❡♥t ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ❛ tr❛♥s♣❧❛♥t✳ ❋✐♥❛❧❧②✱ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s ❧✐♠✐t t❤❡ ❝②❝❧❡ s✐③❡ ❜❡❝❛✉s❡ ♦❢ ❧♦❣✐st✐❝❛❧ ❞✐✣❝✉❧t✐❡s ✐♥ ♦r❣❛♥✐③✐♥❣ ♠❛♥② s✐♠✉❧t❛♥❡♦✉s s✉r❣❡r✐❡s✳✷✾ ❚❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤✐s ❛❧❣♦r✐t❤♠ ❛r❡ t❤❡ ✇❡✐❣❤ts wjk ✉s❡❞ ❜② t❤❡ ◆❑❘ ❢♦r ❛ tr❛♥s✲ ♣❧❛♥t ✐♥✈♦❧✈✐♥❣ ❞♦♥♦r k ❛♥❞ ♣❛t✐❡♥t j ❛♥❞ t❤❡ ♠❛①✐♠✉♠ ❝②❝❧❡ s✐③❡✳ ❈♦♥s✐st❡♥t ✇✐t❤ ❚r❛♥s♣❧❛♥t Pr♦♣♦s❛❧✿

✷✾ ❋♦r♠❛❧❧②✱

t❤❡ ◆❑❘ ♠❛①✐♠✐③❡s jk cjk wjk xjk ❜② ♣✐❝❦✐♥❣ xjk ∈ {0, 1}✱ ✇❤❡r❡ xjk = 1 ❞❡♥♦t❡s ❛ ♣r♦♣♦s❡❞ tr❛♥s♣❧❛♥t ❢r♦♠ ❞♦♥♦r k t♦ ♣❛t✐❡♥t j ❀ wjk ✐s t❤❡ ✇❡✐❣❤t ❛❝❝♦r❞❡❞ t♦ ❡❛❝❤ s✉❝❤ tr❛♥s♣❧❛♥t ❜② t❤❡ ◆❑❘❀ ❛♥❞ cjk = 1 ✐❢ ❛ tr❛♥s♣❧❛♥t ❢r♦♠ k t♦ j ✐s ❢❡❛s✐❜❧❡ ✭❜✐♦❧♦❣✐❝❛❧❧② ❝♦♠♣❛t✐❜❧❡ ❛♥❞ ❛❝❝❡♣t❛❜❧❡✮ ❛♥❞ 0 ♦t❤❡r✇✐s❡✳ ❚❤✐s ♣r♦❜❧❡♠ ✐s s✉❜❥❡❝t ❝♦♥str❛✐♥ts✳ ❋✐rst✱ ♥♦ ❞♦♥♦r ♦r ♣❛t✐❡♥t ✐s ✐♥✈♦❧✈❡❞ ✐♥ ♠♦r❡ t❤❛♥ ♦♥❡ P P t♦ t❤r❡❡ ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥t✱ ✐✳❡✳ j xjk ≤ 1 ❛♥❞ k xjk ≤ 1✳ ❙❡❝♦♥❞✱ ✐❢ ❞♦♥♦r k ❛♥❞ ♣❛t✐❡♥t j ❜❡❧♦♥❣ t♦ ❛ ♣❛✐r✱ t❤❡♥ xj ′ k = 1 ❢♦r s♦♠❡ j ′ ♦♥❧② ✐❢ xjk′ = 1 ❢♦r s♦♠❡ ❞♦♥♦r k′ ✳ ❚♦ ✇r✐t❡ t❤❡ t❤✐r❞ ❝♦♥str❛✐♥t✱ ♥♦t❡ t❤❛t ❛ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ n ✐s ❛♥ ♦r❞❡r❡❞ t✉♣❧❡✱ (j1 , j2 , . . . , jn ) ✇❤❡r❡ xjk jk+1 = 1 ❢♦r k < n ❛♥❞ xjn j1 = 1. ❲❡ ✐♠♣♦s❡ t❤❡ ❝♦♥str❛✐♥t n ≤ 3✳ ❇❡❝❛✉s❡ t❤❡r❡ ❛r❡ ❛ ✈❡r② ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❝②❝❧❡ ❧❡♥❣t❤ ❝♦♥str❛✐♥ts✱ ✇❡ ✜rst s♦❧✈❡ ❛ r❡❧❛①❡❞ ♣r♦❜❧❡♠ ✇✐t❤♦✉t t❤✐s ❧❛st ❝♦♥str❛✐♥t ❛♥❞ ✐t❡r❛t✐✈❡❧② ❛❞❞ t❤❡ ❝♦♥str❛✐♥ts t♦ ♣r♦❤✐❜✐t ❧❛r❣❡ ❝②❝❧❡s✳ ❆♣♣❡♥❞✐① ❈ ♣r♦✈✐❞❡s ❢✉rt❤❡r ❞❡t❛✐❧s ♦♥ t❤❡ ❛❧❣♦r✐t❤♠✳ P

✸✶

◆❑❘ ♣♦❧✐❝② ❛♥❞ ♦❜s❡r✈❡❞ ❞❛t❛✱ ✇❡ ♣r♦❤✐❜✐t ❛❧❧ ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ ❢♦✉r ♦r ❣r❡❛t❡r✳ ❚❤❡ ✇❡✐❣❤ts ❛r❡ ❦♥♦✇♥ t♦ ♦♥❡ ♦❢ t❤❡ ❛✉t❤♦rs ✭❆s❤❧❛❣✐✮✱ ❛♥❞ ❛r❡ ❞❡t❛✐❧❡❞ ✐♥ ❆♣♣❡♥❞✐① ❈✳ ❚❤❡② ♣r✐♦r✐t✐③❡ ✉♥❧✐❦❡❧② tr❛♥s♣❧❛♥ts ✐♥ ❛♥ ❛tt❡♠♣t t♦ ✉t✐❧✐③❡ ❤❛r❞✲t♦✲♠❛t❝❤ ❞♦♥♦rs ❛♥❞ tr❛♥s♣❧❛♥t ❤❛r❞✲t♦✲♠❛t❝❤ ♣❛t✐❡♥ts ✇❤❡♥❡✈❡r ♣♦ss✐❜❧❡✳ ❚❤❡ ✇❡✐❣❤ts t②♣✐❝❛❧❧② ♦♥❧② ❜r❡❛❦ t✐❡s ❜❡t✇❡❡♥ t✇♦ ♠❛t❝❤❡s ✇✐t❤ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ✐♥ ❢❛✈♦r ♦❢ r❡t❛✐♥✐♥❣ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✇❤♦ ❛r❡ ❧✐❦❡❧② t♦ ♠❛t❝❤ ✐♥ t❤❡ ❢✉t✉r❡✳ ✸✳ ❋✐♥❛❧ ❘❡✈✐❡✇ ❛♥❞ ❚r❛♥s♣❧❛♥t❛t✐♦♥✿ Pr♦♣♦s❡❞ tr❛♥s♣❧❛♥ts ❛r❡ r❡✈✐❡✇❡❞ ❜② ❞♦❝t♦rs✱ ♣❛t✐❡♥ts✱ ❛♥❞ ❞♦♥♦rs✱ ❛♥❞ ❛♣♣r♦✈❡❞ ❜❡❢♦r❡ ✐t ✐s ♣❡r❢♦r♠❡❞✳ ❇♦t❤ ❛♣♣r♦✈❛❧ ❛♥❞ ❜✐♦❧♦❣✐❝❛❧ t❡st✐♥❣ ❝❛♥ t❛❦❡ s❡✈❡r❛❧ ❞❛②s✳ ▼♦r❡♦✈❡r✱ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✐♥ ♣r♦♣♦s❡❞ tr❛♥s♣❧❛♥ts t❤❛t ❛r❡ ✉♥❞❡r r❡✈✐❡✇ ♦♥ ❛ ❣✐✈❡♥ ❞❛② ❛r❡ ❡①❝❧✉❞❡❞ ❢r♦♠ t❤❡ ♠❛①✐♠❛❧ ♠❛t❝❤✐♥❣ ❛❧❣♦✲ r✐t❤♠ ♦♥ t❤❛t ❞❛②✳ ❚❤✐s st❡♣ ❛❧s♦ ✐♥✈♦❧✈❡s ❛ ✜♥❛❧ s❡t ♦❢ ❜❧♦♦❞✲t❡sts t♦ ❡♥s✉r❡ ❜✐♦❧♦❣✐❝❛❧

✸✵

❝♦♠♣❛t✐❜✐❧✐t②✳

❈②❝❧❡s ✐♥ ✇❤✐❝❤ ❛♥② ♣❛t✐❡♥t r❡❢✉s❡s ♦r ✐s ❢♦✉♥❞ t♦ ❜❡ ✐♥❝♦♠♣❛t✐❜❧❡

✇✐t❤ t❤❡ ♣r♦♣♦s❡❞ ❞♦♥♦r ❛r❡ ❛❜❛♥❞♦♥❡❞✳ ◆❑❘ ✉s✉❛❧❧② ❛❜❛♥❞♦♥s ❝❤❛✐♥s ✐♥ ✇❤✐❝❤ t❤❡ s❡❝♦♥❞ ♣❛t✐❡♥t ❝❛♥♥♦t ❜❡ tr❛♥s♣❧❛♥t❡❞✳ ❋♦r ♦t❤❡r ❝❤❛✐♥s✱ ❛❧❧ ♣r♦♣♦s❛❧s ✉♥t✐❧ t❤❡ ✜rst ❢❛✐❧✉r❡ ❛r❡ ❝♦♥s✉♠♠❛t❡❞✳ ❚❤❡ ❞♦♥♦r ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ✜♥❛❧ ♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐r ✐♥ s✉❝❤ ❛ ❝❤❛✐♥ ♠❛② ✐♥✐t✐❛t❡ ♥❡✇ ❝❤❛✐♥s ✐♥ t❤❡ ❢✉t✉r❡ ♠✉❝❤ ❧✐❦❡ ❛♥ ❛❧tr✉✐st✐❝ ❞♦♥♦r✳ ❚❤✐s ❞♦♥♦r ✐s ♦❢t❡♥ r❡❢❡rr❡❞ t♦ ❛s t❤❡ ✏❜r✐❞❣❡✑ ❞♦♥♦r✳ ❈♦♥s✐st❡♥t ✇✐t❤ ◆❑❘ ♣♦❧✐❝②✱ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥ts ❛r❡ ♣r✐♦r✐t✐③❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♥❡t ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❛❧tr✉✐st✐❝ ❞♦♥♦rs ❛♥❞ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥ts ♣r❡✈✐♦✉s❧② tr❛♥s♣❧❛♥t❡❞ ❜② t❤❡ ♣❛t✐❡♥t✬s ❤♦s♣✐t❛❧✳ ❚❤✐s st❡♣ r❡s✉❧ts ✐♥ ❢r✐❝t✐♦♥s ✇✐t❤✐♥ t❤❡ s②st❡♠ t❤❛t r❡❞✉❝❡ tr❛♥s♣❧❛♥t❛t✐♦♥ r❛t❡s ✭❆❣❛r✲ ✇❛❧ ❡t ❛❧✳✱ ✷✵✶✽✮✳ ❚❤❡ ♣❛r❛♠❡t❡rs t❤❛t ❣♦✈❡r♥ t❤❡s❡ ❢r✐❝t✐♦♥s ❛r❡ t❤❡ t✐♠❡ r❡q✉✐r❡❞ ❢♦r ❡❛❝❤ ♦❢ t❤❡ t✇♦ ❛♣♣r♦✈❛❧ st❡♣s✱ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ♣r♦♣♦s❡❞ tr❛♥s♣❧❛♥t ✐s ❛❜❛♥❞♦♥❡❞ ✐♥ ❡❛❝❤ st❡♣✱ ❛♥❞ t❤❡ ❞✉r❛t✐♦♥ ❢♦r ✇❤✐❝❤ ❛ ❜r✐❞❣❡ ❞♦♥♦r ✐s r❡t❛✐♥❡❞ ✐♥ t❤❡ ♣♦♦❧ ❜❡❢♦r❡ ❞♦♥❛t✐♥❣ ❤❡r ❦✐❞♥❡② t♦ ❛ ♣❛t✐❡♥t ♦♥ t❤❡ ❞❡❝❡❛s❡❞ ❞♦♥♦r ❧✐st✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ✇❡ ❞♦ ♥♦t ❤❛✈❡ ❞❡t❛✐❧❡❞ ❞❛t❛ ♦♥ ✇❤✐❝❤ tr❛♥s♣❧❛♥ts ✇❡r❡ r❡❢✉s❡❞✱ ❤♦✇ ♦❢t❡♥ tr❛♥s♣❧❛♥ts ✇❡r❡ ❛❜♦rt❡❞ ❞✉❡ t♦ ❜✐♦❧♦❣✐❝❛❧ t❡st✐♥❣✱ ♦r ❤♦✇ ❧♦♥❣ ❡❛❝❤ r❡✈✐❡✇ ♣❤❛s❡ t❛❦❡s✳ ❆❞❞✐t✐♦♥❛❧❧②✱ t❤❡ ◆❑❘ ❞♦❡s ♥♦t s❡❡♠ t♦ ❤❛✈❡ ❝❧❡❛r✲❝✉t ❛❧❣♦r✐t❤♠✐❝ ♣♦❧✐❝✐❡s ♦♥ ❤♦✇ t♦ ✉s❡ ❜r✐❞❣❡ ❞♦♥♦rs✳

❈❤❛✐♥s ✇♦✉❧❞ ❜❡ ✐♥❞❡✜♥✐t❡❧② ❧♦♥❣ ✐❢ ❜r✐❞❣❡ ❞♦♥♦rs ✇❡r❡

❛❧❧♦✇❡❞ t♦ ✐♥✐t✐❛t❡ ♥❡✇ ❝❤❛✐♥s ❢♦r❡✈❡r ❜✉t t♦♦ s❤♦rt ✐❢ ❜r✐❞❣❡ ❞♦♥♦rs ✇❡r❡ ♥♦t ✉s❡❞✳ ❆❧t❤♦✉❣❤ ❝❛s❡s ♦❢ ❞♦♥♦rs r❡♥❡❣✐♥❣ ❛r❡ r❛r❡ ✭❈♦✇❛♥ ❡t ❛❧✳✱ ✷✵✶✼✮✱ ♣❧❛t❢♦r♠s tr② t♦ tr❛♥s✲ ♣❧❛♥t ❜r✐❞❣❡✲❞♦♥♦rs q✉✐❝❦❧②✱ t♦ ❛♥ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥t ✐❢ ♥❡❝❡ss❛r②✱ t♦ ❛✈♦✐❞ t❤❡s❡ ❝❛s❡s✳ ❲❡ ❝❛❧✐❜r❛t❡ t❤❡s❡ ♣❛r❛♠❡t❡rs ❜② s✐♠✉❧❛t✐♥❣ ♦✉r ♠♦❞❡❧ t♦ ✜♥❞ ✈❛❧✉❡s t❤❛t ♠♦st ❝❧♦s❡❧② r❡♣❧✐❝❛t❡ t❤❡ ♠❛t❝❤ ♣r♦❜❛❜✐❧✐t✐❡s✱ ❞✉r❛t✐♦♥s✱ ❝❤❛✐♥ ❧❡♥❣t❤s✱ ❛♥❞ ♣♦♦❧ s✐③❡ ♦❜s❡r✈❡❞ ✐♥ ♦✉r ❞❛t❛✳ ❲❡ ♠❛t❝❤ t❤❡s❡ st❛t✐st✐❝s ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜♠✐ss✐♦♥ t②♣❡s✿ ❛❧tr✉✐st✐❝ ❞♦♥♦r✱

✸✶

♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐rs✱ ❛♥❞ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥ts✳ t❤❛t t❤❡ ◆❑❘ ✐♠♣❧✐❝✐t❧② s❡ts ❢♦r ❤♦s♣✐t❛❧s✱

✸✵ ❚❤❡s❡

j.

✸✶ ■♥

❢❛✐❧✉r❡s ❛r❡ r❡❝♦r❞❡❞ ❜② s❡tt✐♥❣

cjk = 0

❚❤❡s❡ q✉❛♥t✐t✐❡s ❛r❡ ♣r❡❝✐s❡❧② t❤❡ r❡✇❛r❞s

p0 ✱ ❛♥❞ ❛r❡ t❤❡r❡❢♦r❡ ❦❡② ❝♦♠♣♦♥❡♥ts ♦❢ ♦✉r

❢♦r ❢✉t✉r❡ ✐t❡r❛t✐♦♥s ✐❢ t❤❡ ❞♦♥♦r

k

✇❛s r❡❢✉s❡❞ ❜② ♣❛t✐❡♥t

♣r✐♥❝✐♣❧❡✱ ✇❡ ❝♦✉❧❞ ❤❛✈❡ ❡st✐♠❛t❡❞ t❤❡s❡ ♣❛r❛♠❡t❡rs ✉s✐♥❣ s✐♠✉❧❛t❡❞ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡✳ ❍♦✇❡✈❡r✱ ❛

s✐♠✉❧❛t✐♦♥ ❢♦r ❡❛❝❤ ♣❛r❛♠❡t❡r ✈❛❧✉❡ ❝❛♥ t❛❦❡ ✇❡❡❦s✱ ♠❛❦✐♥❣ ♦♣t✐♠✐③❛t✐♦♥ ♦✈❡r t❤❡ ♣❛r❛♠❡t❡r s❡t ✐♥❢❡❛s✐❜❧❡✳

✸✷ ❡❝♦♥♦♠✐❝ ♠♦❞❡❧✳ ❖✉r s✐♠✉❧❛t✐♦♥s s✉❣❣❡st t❤❛t ❛ t✇♦✲✇❡❡❦ ♣❡r✐♦❞ ❢♦r ❜♦t❤ t❤❡ ❛❝❝❡♣t❛♥❝❡ ❛♥❞ t❤❡ ❜✐♦❧♦❣✲ ✐❝❛❧ t❡st✐♥❣ ♣❤❛s❡s ❛♥❞ ❛ ♦♥❡✲✜❢t❤ ❢❛✐❧✉r❡ r❛t❡ ❢♦r ❡❛❝❤ ♣❤❛s❡ ❜❡st ✜t t❤❡ tr❛♥s♣❧❛♥t r❛t❡s ❛♥❞ ❝❤❛✐♥ ❧❡♥❣t❤s ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❞❛t❛s❡t✳ ❘❡❞✉❝✐♥❣ t❤❡ ❢❛✐❧✉r❡ r❛t❡s ✐♥ s✐♠✉❧❛t✐♦♥s ♣r✐♠❛r✐❧② ✐♥❝r❡❛s❡s ❝❤❛✐♥ ❧❡♥❣t❤ ❛♥❞ tr❛♥s♣❧❛♥t❛t✐♦♥ r❛t❡s✱ ✇❤✐❧❡ r❡❞✉❝✐♥❣ t❤❡ ❞✉r❛t✐♦♥ ♦❢ ❡✐t❤❡r ♣❤❛s❡ ✐♥❝r❡❛s❡s t❤❡ tr❛♥s♣❧❛♥t❛t✐♦♥ r❛t❡s ✇✐t❤♦✉t ❤❛✈✐♥❣ ❛ ❧❛r❣❡ ❡✛❡❝t ♦♥ ❝❤❛✐♥ ❧❡♥❣t❤✳ ❋♦r t❤❡ ❜r✐❞❣❡ ❞♦♥♦r ♣♦❧✐❝②✱ ✇❡ ✜♥❞ t❤❛t ❛ ❤♦❧❞✲♣❡r✐♦❞ ♦❢ ✸✵ ❞❛②s ❜❡st ✜ts t❤❡ ❞❛t❛✳ ❉❡t❛✐❧s ♦♥ t❤❡ ✜t ♦❢ ♦✉r ❝❛❧✐❜r❛t❡❞ ♣❛r❛♠❡t❡rs ❛r❡ ♣r♦✈✐❞❡❞ ✐♥ ❆♣♣❡♥❞✐① ❈✳✺✳✶✳ ❋✉r✲ t❤❡r✱ ❆♣♣❡♥❞✐① ❉ r❡♣❡❛ts ❛❧❧ ♦❢ ♦✉r ❛♥❛❧②s❡s ✉♥❞❡r ❛❧t❡r♥❛t✐✈❡ ♣❛r❛♠❡t❡rs t♦ ❡①❛♠✐♥❡ r♦❜✉st♥❡ss ♦❢ ♦✉r r❡s✉❧ts✳ ✹✳

P❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ♦❢t❡♥ ❞❡♣❛rt t❤❡ ◆❑❘ ✇✐t❤♦✉t ❛ tr❛♥s♣❧❛♥t✳ ❆ ♣❛✲ t✐❡♥t ❛♥❞ ❤✐s✴❤❡r ❛ss♦❝✐❛t❡❞ ❞♦♥♦r ♠❛② ❧❡❛✈❡ t❤❡ ♣❧❛t❢♦r♠ ❜❡❝❛✉s❡ t❤❡ ♣❛t✐❡♥t ❞✐❡s✱ ❜❡❝♦♠❡s t♦♦ s✐❝❦ t♦ tr❛♥s♣❧❛♥t✱ ♦r r❡❝❡✐✈❡s ❛ ❦✐❞♥❡② tr❛♥s♣❧❛♥t ❡❧s❡✇❤❡r❡✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ♥❡❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ♣❛t✐❡♥t ♦r ❛ ❞♦♥♦r ❧❡❛✈❡s t❤❡ ◆❑❘ ✇✐t❤♦✉t ❛ tr❛♥s♣❧❛♥t✳ ❉❡♣❛rt✉r❡✿

❲❡ ❡st✐♠❛t❡ ❛ ♠♦❞❡❧ ♦❢ ❞❡♣❛rt✉r❡s ✉s✐♥❣ t❤❡ r❡❣✐str❛t✐♦♥ ❛♥❞ tr❛♥s♣❧❛♥t❛t✐♦♥ ❞❛t❡s ✭✐❢ tr❛♥s♣❧❛♥t❡❞✮ ❢♦r ❡❛❝❤ ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r✳ ❆❞❞✐t✐♦♥❛❧❧②✱ ✇❡ ✉s❡ r❡❣✉❧❛r ❞❛t❛ s♥❛♣✲ s❤♦ts ♦❢ t❤❡ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs r❡❣✐st❡r❡❞ ❛t t❤❡ ◆❑❘ t♦ ❞❡t❡r♠✐♥❡ ❤♦✇ ❧♦♥❣ t❤❡ ♣❛t✐❡♥t ♦r ❞♦♥♦r ✇❛s r❡❣✐st❡r❡❞ ✐♥ t❤❡ ◆❑❘ ✇✐t❤♦✉t ❛ tr❛♥s♣❧❛♥t✳ ❲❡ ❡st✐♠❛t❡ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❤❛③❛r❞s ♠♦❞❡❧ ❢♦r t❤❡ ❞❡♣❛rt✉r❡ ♣r♦❝❡ss ✉s✐♥❣ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞✳✸✷ ❚❤❡ ❞❡♣❛rt✉r❡ r❛t❡s ✐♥ t❤❡ ♠♦❞❡❧ ❞❡♣❡♥❞ ♦♥ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❞♦♥♦rs ✭♣❛t✐❡♥ts✮ ❡✈❡r r❡❣✐st❡r❡❞ ✇✐t❤ t❤❡ ◆❑❘ ✇❤♦ ❛r❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❛ ♣❛t✐❡♥t ✭❞♦♥♦r✮✱ ❜❧♦♦❞✲t②♣❡ ❞✉♠♠✐❡s ❢♦r t❤❡ ❞♦♥♦r ❛♥❞ t❤❡ ♣❛t✐❡♥t✱ ❛♥❞ t❤❡ ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r ❛❣❡s ❛t r❡❣✐str❛t✐♦♥✳ ❆♣♣❡♥❞✐① ❈✳✷✳✷ ♣r❡s❡♥ts t❤❡ ❡st✐♠❛t❡s ❢♦r t❤❡ ♠♦❞❡❧✳ ❚❤✐s ♣r♦❝❡❞✉r❡ ❛❧❧♦✇s ✉s t♦ ❡✈❛❧✉❛t❡ ❛ tr❛♥s♣❧❛♥t ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r ❛♥② ✈❡❝t♦r ♦❢ ✐♥♣✉ts q ❜② s✐♠✉❧❛t✐♥❣ ❡❛❝❤ ♦❢ t❤❡s❡ ❡✈❡♥ts ❢♦r ❡❛❝❤ ❝❛❧❡♥❞❛r ❞❛②✳ ●✐✈❡♥ ❛♥② ✐♥✐t✐❛❧ ♣♦♦❧ ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✐♥ t❤❡ ◆❑❘✱ t❤❡s❡ s✐♠✉❧❛t✐♦♥s ❣❡♥❡r❛t❡ ❛ ▼❛r❦♦✈ ❝❤❛✐♥ ✇✐t❤ ❛ s❡q✉❡♥❝❡ ♦❢ r❡❣✐str❛t✐♦♥s✱ tr❛♥s♣❧❛♥ts✱ ❛♥❞ ❞❡♣❛rt✉r❡s✳ ❲❡ ✐♥✐t✐❛❧✐③❡ t❤❡ ◆❑❘ ♣♦♦❧ ✇✐t❤ t❤❡ s❡t ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs r❡❣✐st❡r❡❞ ♦♥ ❆♣r✐❧ ✶✱ ✷✵✶✷✱ ❛♥❞ ❜✉r♥✲✐♥ ✷✱✵✵✵ s✐♠✉❧❛t✐♦♥ ❞❛②s ✐♥ ❡❛❝❤ ∗,t = 1 ✐❢ ❛ ❦✐❞♥❡② ❢r♦♠ r✉♥✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t❤❡ ✐♥✐t✐❛❧ ♣♦♦❧ ❡✈❡♥t✉❛❧❧② ❢❛❞❡s ❛✇❛②✳ ▲❡t yjk ❞♦♥♦r k ✐s tr❛♥s♣❧❛♥t❡❞ t♦ ♣❛t✐❡♥t j ✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ t✐♠❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✸✷ ❙♣❡❝✐✜❝❛❧❧②✱

t❤❡ ❞❡♣❛rt✉r❡ r❛t❡ ❢♦r r❡❣✐str❛t✐♦♥ j ✐s ❣✐✈❡♥ ❜② λgj exp (zj β) , ✇❤❡r❡ gj ❞❡♥♦t❡s ✇❤❡t❤❡r j ✐s ❛♥ ❛❧tr✉✐st✐❝ ❞♦♥♦r✱ ❛ ♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐r✱ ♦r ❛♥ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥t❀ λgj ✐s ❛ ❣r♦✉♣✲s♣❡❝✐✜❝ ❝♦♥st❛♥t ❞❡♣❛rt✉r❡ r✐s❦❀ zj ❞❡♥♦t❡s ❛ ✈❡❝t♦r ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s ❢♦r j ❀ ❛♥❞ β ✐s ❛ ❝♦♥❢♦r♠❛❜❧❡ ✈❡❝t♦r ♦❢ ❝♦❡✣❝✐❡♥ts✳ ❲❡ ✉s❡ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ✉s✐♥❣ t❤❡ ✭❝❡♥s♦r❡❞✮ ♦❜s❡r✈❛t✐♦♥s ♦❢ ❞❡♣❛rt✉r❡ t✐♠❡s ❢♦r ❡❛❝❤ r❡❣✐str❛t✐♦♥ ✐♥ t❤❡ ◆❑❘✳ ❈❡♥s♦r✐♥❣ ✐♥ ♦✉r ❞❛t❛s❡t ❝❛♥ ♦❝❝✉r ❜❡❝❛✉s❡ ✇❡ ♦♥❧② ♦❜s❡r✈❡ ❛ ❧♦✇❡r ❜♦✉♥❞ ❢♦r t❤❡ ❞❡♣❛rt✉r❡ t✐♠❡ ✐❢ j ✇❛s tr❛♥s♣❧❛♥t❡❞ ♦r r❡♠❛✐♥❡❞ ✐♥ t❤❡ ◆❑❘ ♣♦♦❧ ❛t t❤❡ ❡♥❞ ♦❢ ♦✉r s❛♠♣❧❡ ♣❡r✐♦❞✳

✸✸ tr❛♥s♣❧❛♥ts t♦ ❡st✐♠❛t❡ f ❢♦r ❡❛❝❤ s✐♠✉❧❛t✐♦♥ s✿ T 1 X ∗,t fˆs (ˆ q) = y , T t=1 s

✇❤❡r❡ T ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❞❛②s s✐♠✉❧❛t❡❞ ❛♥❞ |ys∗,t | ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ✐♥ ♣❡r✐♦❞ t ✐♥ s✐♠✉❧❛t✐♦♥ s. ■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ r❡♣♦rt ❡st✐♠❛t❡s ❜❛s❡❞ ♦♥ ❛♥ ❛✈❡r❛❣❡ ♦❢ ✶✵✵ s✐♠✉❧❛t✐♦♥s✳

✺✳✷ ❙❝❛❧❡✱ ■♥❡✣❝✐❡♥❝② ❛♥❞ ❖♣t✐♠❛❧ ❘❡✇❛r❞s ❚❤✐s s❡❝t✐♦♥ ✉s❡s t❤❡ ❡st✐♠❛t❡❞ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ t♦ ♠❡❛s✉r❡ t❤❡ ❛♠♦✉♥t ♦❢ ✐♥❡✣❝✐❡♥❝②✱ ❡✈❛❧✉❛t❡ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ t✇♦ s♦✉r❝❡s ♦❢ ♠❛r❦❡t ❢❛✐❧✉r❡✱ ❛♥❞ ❞❡s✐❣♥ ❜❡tt❡r ♠❡❝❤❛♥✐s♠s✳ ❲❡ ✉s❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ✭✐♥❝❧✉❞✐♥❣ ✐♥ ❜♦t❤ ♣❧❛t❢♦r♠s ❛♥❞ ❤♦s♣✐t❛❧s✮ ❛s ♦✉r ♠❡❛s✉r❡ ♦❢ s♦❝✐❛❧ ✇❡❧❢❛r❡✱ SW ✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❜❡❣✐♥ ❜② r❡♣♦rt✐♥❣ t❤❡ r❡s✉❧ts ✉♥❞❡r ♦✉r ♠❛✐♥ s♣❡❝✐✜❝❛t✐♦♥s✱ ❛♥❞ t❤❡♥ ✇❡ ❞✐s❝✉ss ❞❡t❛✐❧❡❞ r♦❜✉st♥❡ss ❛♥❛❧②s❡s✳ ✺✳✷✳✶

❘❡t✉r♥s t♦ s❝❛❧❡

❲❡ ✜rst ❞♦❝✉♠❡♥t t❤❡ ❡st✐♠❛t❡❞ r❡t✉r♥s t♦ s❝❛❧❡ ✐♥ t❤❡ tr❛♥s♣❧❛♥t ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥❀ t❤❛t ✐s✱ ❤♦✇ t❤❡ ❛✈❡r❛❣❡ ♣r♦❞✉❝t ❝❤❛♥❣❡s ✇✐t❤ ♣❧❛t❢♦r♠ s✐③❡✳ ❲❡ ❡✈❛❧✉❛t❡ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r ♣♦♦❧s ♦❢ s✉❜♠✐ss✐♦♥s q ✇✐t❤ t❤❡ s❛♠❡ ❝♦♠♣♦s✐t✐♦♥ ❛s t❤❡ ◆❑❘ ❜✉t ✇✐t❤ ❞✐✛❡r❡♥t s❝❛❧❡s ❛s ♠❡❛s✉r❡❞ ❜② t❤❡ t♦t❛❧ ✢♦✇ ♦❢ ❛❧tr✉✐sts ❛♥❞ ♣❛✐rs ♣❡r ②❡❛r✳ ❋✐❣✉r❡ ✻ ❞❡♣✐❝ts ❛✈❡r❛❣❡ ♣r♦❞✉❝ts✱ ❡q✉❛❧ t♦ f (q)/kqk1 ✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ t♦t❛❧ ✢♦✇✱ kqk1 ✳ ❚❤❡ ❡st✐♠❛t❡s s❤♦✇ ❛ r❡♠❛r❦❛❜❧❡ ♣❛tt❡r♥✳ ❆❧t❤♦✉❣❤ t❤❡ r❡t✉r♥s t♦ s❝❛❧❡ ❛❧✇❛②s ✐♥❝r❡❛s❡✱ t❤❡② r❡❛❝❤ ❛ ♣❧❛t❡❛✉ ❢❛✐r❧② q✉✐❝❦❧②✳ ❲✐t❤ ❛ s❝❛❧❡ ♦❢ ✺✸✹ ❞♦♥♦r ❛rr✐✈❛❧s ♣❡r ②❡❛r✱ t❤❡ ◆❑❘ ✐s ✇❡❧❧ ✇✐t❤✐♥ t❤❡ r❡❣✐♦♥ ♦❢ ❛♣♣r♦①✐♠❛t❡❧② ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✳ ■t ❤❛s ❛♥ ❛✈❡r❛❣❡ ♣r♦❞✉❝t ♦❢ ✵✳✺✹✱ ✇❤✐❝❤ ✈❛r✐❡s ♠❛r❣✐♥❛❧❧② ♦♥❝❡ t❤❡ s❝❛❧❡ ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡✳ ❆ ♣❧❛t❢♦r♠ t❤❛t ✐s ❤❛❧❢ t❤❡ s✐③❡ ♦❢ ◆❑❘ ❤❛s ❛♥ ❛✈❡r❛❣❡ ♣r♦❞✉❝t ♦❢ ✵✳✺✶✱ ✇❤✐❧❡ ❛ ♣❧❛t❢♦r♠ t❤❛t ✐s ❞♦✉❜❧❡ t❤❡ s✐③❡ ❤❛s ❛♥ ❛✈❡r❛❣❡ ♣r♦❞✉❝t ♦❢ ✵✳✺✼✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♠❛r❦❡t ❝❛♥ ♦♣❡r❛t❡ ❛t ❛ ❤✐❣❤ ❧❡✈❡❧ ♦❢ ❡✣❝✐❡♥❝② ❡✈❡♥ ✐❢ t❤❡r❡ ❛r❡ ❛ ❤❛♥❞❢✉❧ ♦❢ ❝♦♠♣❡t✐♥❣ ♣❧❛t❢♦r♠s✳ ❚❤❡s❡ ❡st✐♠❛t❡s s✉❣❣❡st t❤❛t ♠❡r❣❡rs ♦❢ s✉✣❝✐❡♥t❧② ❧❛r❣❡ ♣❧❛t❢♦r♠s ✇♦✉❧❞ ❤❛✈❡ s♠❛❧❧ ❡✛❡❝ts ♦♥ ❡✣❝✐❡♥❝②✳ ◆❡①t✱ ✇❡ ✉s❡ t❤❡s❡ ❡st✐♠❛t❡s t♦ ❝❛❧❝✉❧❛t❡ ✇❤❡t❤❡r ✐♥❞✐✈✐❞✉❛❧ ❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s✱ ✇❤✐❝❤ ❝♦❧❧❡❝✲ t✐✈❡❧② ❛❝❝♦✉♥t ❢♦r t❤❡ ♠❛❥♦r✐t② ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts✱ ♦♣❡r❛t❡ ❛t ❛♥ ❡✣❝✐❡♥t s❝❛❧❡✳ ❆ ❝❤❛❧❧❡♥❣❡ ✇✐t❤ t❤✐s ❡①❡r❝✐s❡ ✐s t❤❛t ✇❡ ♦❜s❡r✈❡ ♥❡✐t❤❡r t❤❡ ♥✉♠❜❡r ♥♦r t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❛✈❛✐❧❛❜❧❡ t♦ ❛ ❤♦s♣✐t❛❧✳ ❲❡ ♦♥❧② ♦❜s❡r✈❡ t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ❝♦♥❞✉❝t❡❞ ❜② ❛ ❤♦s♣✐t❛❧ ❜♦t❤ t❤r♦✉❣❤ t❤❡ ◆❑❘ ❛♥❞ ♦✉ts✐❞❡✳ ❚♦ ♠❛❦❡ ♣r♦❣r❡ss✱ ❛ss✉♠❡✱ ❢♦r t❤❡ ♠♦♠❡♥t✱ t❤❛t ❤♦s♣✐t❛❧s ❤❛✈❡ t❤❡ s❛♠❡ ♣r♦❞✉❝t✐♦♥ t❡❝❤♥♦❧♦❣② ❛♥❞ ❝♦♠♣♦s✐t✐♦♥ ❛s t❤❡ ◆❑❘✳ ❋✉rt❤❡r✱ ❛ss✉♠❡ t❤❛t ❤♦s♣✐t❛❧s ❝♦♥❞✉❝t✐♥❣ ✇✐t❤✐♥✲❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts ❞♦ ♥♦t ♣❛rt✐❝✐✲ ♣❛t❡ ✐♥ t❤❡ ◆❑❘✳ ❯♥❞❡r t❤❡s❡ ❛ss✉♠♣t✐♦♥s✱ ♦♥❡ ❝❛♥ ✉s❡ t❤❡ ♦❜s❡r✈❡❞ r❛t❡ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ❛t ✐♥❞✐✈✐❞✉❛❧ ❤♦s♣✐t❛❧s t♦ ✐♥❢❡r t❤❡ s❝❛❧❡ ❢♦r ❡❛❝❤ ❤♦s♣✐t❛❧✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❧❡t y h ❜❡

✸✹ ❋✐❣✉r❡ ✻✿ Pr♦❞✉❝t✐♦♥ ❊✣❝✐❡♥❝② ✈❡rs✉s ❙❝❛❧❡

◆♦t❡s ✿

❚❤❡ ❧✐♥❡ ♣❧♦t r❡♣r❡s❡♥ts t❤❡ ❛✈❡r❛❣❡ ♣r♦❞✉❝t ♦❢ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠ ✈❡rs✉s ✐ts s❝❛❧❡✳ ❚❤❡ ❤✐st♦❣r❛♠ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❡st✐♠❛t❡❞ s❝❛❧❡ ♦❢ ✈❛r✐♦✉s ❤♦s♣✐t❛❧s✳ ❚❤❡ ❧❡❢t ✈❡rt✐❝❛❧ ❛①✐s r❡♣r❡s❡♥ts ❛✈❡r❛❣❡ ♣r♦❞✉❝ts✱ ❞❡✜♥❡❞ ❛s t❤❡ s❤❛r❡ ♦❢ ♣❛✐rs ❛♥❞ ❛❧tr✉✐sts ✇❤♦ ❛r❡ tr❛♥s♣❧❛♥t❡❞✳ ❚❤❡ r✐❣❤t ✈❡rt✐❝❛❧ ❛①✐s ✐s t❤❡ s❝❛❧❡ ❢♦r t❤❡ ❤✐st♦❣r❛♠✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s r❡♣r❡s❡♥ts s❝❛❧❡✱ ♠❡❛s✉r❡❞ ❛s t❤❡ ②❡❛r❧② ❛rr✐✈❛❧ r❛t❡ ♦❢ ♣❛✐rs ❛♥❞ ❛❧tr✉✐sts✳ ❚❤❡ ♣❧♦t ✉s❡s t❤❡ ❜❛s❡❧✐♥❡ ♣❛r❛♠❡t❡rs ❛♥❞ t❤❡ ♣♦♦❧ ❝♦♠♣♦s✐t✐♦♥ ❢r♦♠ t❤❡ ◆❑❘✳

t❤❡ ✢♦✇ ♦❢ ✇✐t❤✐♥ ❤♦s♣✐t❛❧ ❦✐❞♥❡②
❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ❝♦♥❞✉❝t❡❞ ❛t ❤♦s♣✐t❛❧ h✳ ❚❤❡ s❝❛❧❡ ♦❢ ❤♦s♣✐t❛❧ h ✐s t❤❡♥ q h = fˆ−1 y h ✇❤❡r❡ fˆ ✐s t❤❡ ❡st✐♠❛t❡❞ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳

❚❤✐s ❡①❡r❝✐s❡ s✉❣❣❡sts t❤❛t ✐♥❞✐✈✐❞✉❛❧ ❤♦s♣✐t❛❧s ❛s ✇❡❧❧ ❛s ❯◆❖❙ ❛♥❞ ❆P❉ ♦♣❡r❛t❡ ❢❛r ❜❡❧♦✇ t❤❡ ❡✣❝✐❡♥t s❝❛❧❡✳ ❚❤❡ ❤✐st♦❣r❛♠ ✐♥ ❋✐❣✉r❡ ✻ s❤♦✇s t❤❡ ❡st✐♠❛t❡❞ ❞✐str✐❜✉t✐♦♥ ♦❢ ❤♦s♣✐t❛❧ s❝❛❧❡✳ ❚❤❡ ♠❡❞✐❛♥ ❤♦s♣✐t❛❧ ❤❛s ❛ s❝❛❧❡ ♦❢ ✾✳ ❚❤❡ ✾✵t❤ ♣❡r❝❡♥t✐❧❡ ✐s ✷✻✳ ❚❤❡ ❧❛r❣❡st✱ ▼❡t❤♦❞✐st ❍♦s♣✐t❛❧ ✐♥ ❙❛♥ ❆♥t♦♥✐♦✱ ❤❛s ❛ s❝❛❧❡ ♦❢ ✶✵✹✳ ❚❤❡ ❛✈❡r❛❣❡ ♣r♦❞✉❝t ❛t t❤❡s❡ ❡✣❝✐❡♥t s❝❛❧❡s ✐s ✵✳✶✻✱ ✵✳✸✵ ❛♥❞ ✵✳✹✹✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤✉s✱ ❛t ♦✉r ❡st✐♠❛t❡❞ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✱ ❡✈❡♥ t❤❡ ❧❛r❣❡st s✐♥❣❧❡✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s ❞♦❡s ♥♦t ♦♣❡r❛t❡ ❛t ❛♥ ❡✣❝✐❡♥t s❝❛❧❡✳ ❯◆❖❙ ❛♥❞ ❆P❉ ❤❛✈❡ ❡st✐♠❛t❡❞ ❛✈❡r❛❣❡ ♣r♦❞✉❝ts ♦❢ ✵✳✹✶ ❛♥❞ ✵✳✹✷ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ✐♠♣❧✐❡❞ ❡✣❝✐❡♥❝② ❧♦ss❡s ❛r❡ ❝♦♥s✐❞❡r❛❜❧❡ ❡✈❡♥ ❢♦r t❤❡ ❧❛r❣❡st ♣❧❛t❢♦r♠ ♦t❤❡r t❤❛♥ t❤❡ ◆❑❘✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ❡✈✐❞❡♥❝❡ ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳✷ t❤❛t ❤♦s♣✐t❛❧s ♦❢t❡♥ ♣❡r❢♦r♠ ♠❛t❝❤❡s t❤❛t ❛r❡ s♦❝✐❛❧❧② ✐♥❡✣❝✐❡♥t✱ ❛♥❞ t❤❛t ❯◆❖❙ ❛♥❞ ❆P❉ ❛r❡ ❛❧s♦ s♦♠❡✇❤❛t ❧❡ss ❡✣❝✐❡♥t t❤❛♥ t❤❡ ◆❑❘✳

✸✺

✺✳✷✳✷ ▼✐s❛❧❧♦❝❛t✐♦♥✿ ✐♥❡✣❝✐❡♥❝② ❞✉❡ t♦ s♠❛❧❧ ♣r♦❞✉❝t✐♦♥ s❝❛❧❡ ❲❡ st❛rt ❜② ✉s✐♥❣ t❤❡ ❜❛s❡❧✐♥❡ ❛♣♣r♦❛❝❤ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ t♦ ❡st✐♠❛t❡ ✐♥❡✣❝✐❡♥❝② ❞✉❡ t♦ ♠❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥✳ ❚❤❛t ✐s✱ ✇❡ ❡st✐♠❛t❡ ❤♦✇ ♠❛♥② ❛❞❞✐t✐♦♥❛❧ tr❛♥s♣❧❛♥ts ✇♦✉❧❞ ❜❡ ♣❡r❢♦r♠❡❞ ✐❢ t❤❡ ❡♥t✐r❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡t ❢✉♥❝t✐♦♥❡❞ ❛t ◆❑❘✬s ❡✣❝✐❡♥❝②✳ ❲❡ ✉s❡
h −1 h ˆ ❛ ❤♦s♣✐t❛❧✬s ❡st✐♠❛t❡❞ s❝❛❧❡✱ q = f y ✱ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ ❛✈❡r❛❣❡ ♣r♦❞✉❝t ❜❡t✇❡❡♥ t❤❡ ❤♦s♣✐t❛❧ ❛♥❞ ◆❑❘✳ ❇❡❝❛✉s❡ ◆❑❘ ♦♣❡r❛t❡s ❛t ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✱ t❤✐s ❞✐✛❡r❡♥❝❡ ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ ❤♦s♣✐t❛❧ s❝❛❧❡ ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts t❤❛t ❛r❡ ❧♦st ❞✉❡ t♦ t❤❡ ❤♦s♣✐t❛❧ ❝♦♥❞✉❝t✐♥❣ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❛t ❛♥ ✐♥❡✣❝✐❡♥t❧② s♠❛❧❧ s❝❛❧❡✳ ❚❤❡ ❛❣❣r❡❣❛t❡ ❧♦st tr❛♥s♣❧❛♥ts ❡q✉❛❧ t❤❡ t♦t❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❜❡❝❛✉s❡ ♦✉r s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢✉♥❝t✐♦♥ ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ♥❛t✐♦♥✇✐❞❡✳ ❚❤❡ ❡st✐♠❛t❡❞ ❞❡❛❞✇❡✐❣❤t ❧♦ss ♣r❡s❡♥t❡❞ ✐♥ ❚❛❜❧❡ ✸ s❤♦✇s t❤❛t ✹✹✼✳✼ tr❛♥s♣❧❛♥ts ❛r❡ ❧♦st ♣❡r ②❡❛r ❞✉❡ t♦ ♠❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥ ✭♣❛♥❡❧ ❆✱ ❝♦❧✉♠♥ ✭✶✮✮✳ ❚❤✐s ♥✉♠❜❡r ✐s ❧❛r❣❡ r❡❧❛t✐✈❡ t♦ t❤❡ ✽✵✵ tr❛♥s♣❧❛♥ts t❤❛t ❛r❡ ❝♦♥❞✉❝t❡❞ t❤r♦✉❣❤ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❡❛❝❤ ②❡❛r✳ ❚❤✐s ❜❛s❡❧✐♥❡ ❛♣♣r♦❛❝❤ ✐s s✐♠♣❧❡ ❜✉t s✉✛❡rs ❢r♦♠ ❢♦✉r ♣♦t❡♥t✐❛❧ ❜✐❛s❡s✳ ❋✐rst✱ t❤❡ ❝♦♠♣♦✲ s✐t✐♦♥ ♦❢ s✉❜♠✐ss✐♦♥s ✐♥ ❤♦s♣✐t❛❧s ♠❛② ❞✐✛❡r ❢r♦♠ t❤❛t ✐♥ t❤❡ ◆❑❘✳ ❲❡ ❛ss❡ss r♦❜✉st♥❡ss t♦ t❤✐s ❛ss✉♠♣t✐♦♥ ❜② ❡st✐♠❛t✐♥❣ ✐♥❡✣❝✐❡♥❝② ✉s✐♥❣ ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r ❝♦♠♣♦s✐t✐♦♥s ❜❛s❡❞ ♦♥ s✉❜♠✐ss✐♦♥s ❢r♦♠ t❤r❡❡ ❞✐✛❡r❡♥t ❣r♦✉♣s ♦❢ ❤♦s♣✐t❛❧s✿ ❛❧❧ ❤♦s♣✐t❛❧s✱ ❤♦s♣✐t❛❧s ✐♥ t❤❡ t♦♣ q✉❛rt✐❧❡ ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ r❛t❡✱ ❛♥❞ ❤♦s♣✐t❛❧s ✐♥ t❤❡ ❜♦tt♦♠ q✉❛rt✐❧❡✳✸✸ ❇❡❝❛✉s❡ ❤♦s♣✐t❛❧s ✐♥ t❤❡ t♦♣ q✉❛rt✐❧❡ ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ ❛r❡ ❝♦♥❞✉❝t✐♥❣ ❛ ❧❛r❣❡r ❢r❛❝t✐♦♥ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s t❤r♦✉❣❤ t❤❡ ◆❑❘✱ t❤❡② s❤♦✉❧❞ ❜❡ s✉❜♠✐tt✐♥❣ ❛ ❧❡ss s❡❧❡❝t❡❞ ♣♦♦❧ ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✳ ❊st✐♠❛t❡s ❢r♦♠ ❛ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❢r♦♠ ♦♥❧② t❤❡s❡ ❤♦s♣✐t❛❧s ❛❧❧♦✇s ✉s t♦ ❛ss❡ss r♦❜✉st♥❡ss t♦ ♣♦t❡♥t✐❛❧ ❝♦♠♣♦s✐t✐♦♥❛❧ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ s✐♥❣❧❡ ❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s ❛♥❞ t❤❡ ◆❑❘✳ ❈♦❧✉♠♥s ✭✷✮ ❛♥❞ ✭✸✮ ♣r❡s❡♥t ❡st✐♠❛t❡s ✉♥❞❡r ❛❧t❡r♥❛t✐✈❡ ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❛✈❛✐❧❛❜❧❡ t♦ t❤❡ ❤♦s♣✐t❛❧✳ ❈♦♠♣❛r✐♥❣ ❡st✐♠❛t❡s s✉❣❣❡sts t❤❛t ♦✈❡r❛❧❧ ✐♥❡✣❝✐❡♥❝② ✐s ♥♦t ♣❛rt✐❝✉❧❛r❧② s❡♥s✐t✐✈❡ t♦ t❤❡s❡ ❝♦♠♣♦s✐t✐♦♥❛❧ ❞✐✛❡r❡♥❝❡s✳ ❙❡❝♦♥❞✱ ♦✉r ❜❛s❡❧✐♥❡ ❛♣♣r♦❛❝❤ ❛ss✉♠❡s t❤❛t ❛❧❧ ✇✐t❤✐♥ ❤♦s♣✐t❛❧ tr❛♥s♣❧❛♥ts ❛r❡ ♣r♦❞✉❝❡❞ ❜② ❤♦s♣✐t❛❧s ✐♥ ✐s♦❧❛t✐♦♥ ♦❢ t❤❡ r❡st ♦❢ t❤❡ ♠❛r❦❡t✳ ❚❤❡ ❜✐❛s ❞✉❡ t♦ ❤♦s♣✐t❛❧s t❤❛t ❛❧s♦ ♣❛rt✐❝✐♣❛t❡ ✐♥ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ❞♦❡s ♥♦t ❤❛✈❡ ❛ ❝❧❡❛r ❞✐r❡❝t✐♦♥✳ ❲❡ ❛❞❞r❡ss t❤✐s ✐ss✉❡ ❜② ❞✐s❛❣❣r❡❣❛t✐♥❣ t❤❡ ❡✣❝✐❡♥❝② ❧♦ss❡s ❜② ✇❤❡t❤❡r ❛ ❤♦s♣✐t❛❧ ♣❛rt✐❝✐♣❛t❡s ✐♥ t❤❡ ◆❑❘✱ ❆P❉ ❛♥❞ ❯◆❖❙ ❛♥❞ ❜② t❤❡ ❢r❛❝t✐♦♥ ♦❢ t❤❡ ❤♦s♣✐t❛❧✬s ♣❛✐r❡❞ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s t❤❛t ❛r❡ ❝♦♥❞✉❝t❡❞ t❤r♦✉❣❤ t❤❡ ◆❑❘✳ ■❢ ✇❡ r❡str✐❝t ❛tt❡♥t✐♦♥ ♦♥❧② t♦ t❤❡ ✾✻ ❤♦s♣✐t❛❧s t❤❛t ❞♦ ♥♦t ♣❛rt✐❝✐♣❛t❡ ✐♥ ◆❑❘✱ t❤❡ ❡✣❝✐❡♥❝② ❧♦ss ✐♥ ❝♦❧✉♠♥ ✭✶✮ ✐s ✷✶✷✳✾ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r ✭♣❛♥❡❧ ❈✱ ❡①❝❧✉❞✐♥❣ t❤❡ ◆❑❘ r♦✇✮✳ ❙♦♠❡ ♦❢ t❤❡s❡ ❤♦s♣✐t❛❧s ♣❛rt✐❝✐♣❛t❡ ✐♥ ❯◆❖❙ ♦r ❆P❉ ❛♥❞ ♠❛② ❜❡ ♣r♦❞✉❝✐♥❣ tr❛♥s♣❧❛♥ts ❛t ❛ ♠♦r❡ ❡✣❝✐❡♥t s❝❛❧❡✳ ❊✈❡♥ ✐❢ ✇❡ ❛ss✉♠❡ t❤❛t ❡❛❝❤ ♦❢ t❤❡s❡ ❤♦s♣✐t❛❧s t❤❛t ♣❛rt✐❝♣❛t❡ ✐♥ ❯◆❖❙ ♦r ❆P❉ ♣r♦❞✉❝❡ tr❛♥s♣❧❛♥ts ❛t t❤❡ ❡st✐♠❛t❡❞ s❝❛❧❡s ❢♦r t❤❡ t✇♦ ♣❧❛t❢♦r♠s✱ ✇❡ ❡st✐♠❛t❡ t❤❛t t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐♥ ❝♦❧✉♠♥ ✭✶✮ ✇♦✉❧❞ ❜❡ ✶✷✼✳✵✳✸✹ ❍♦✇❡✈❡r✱ t❤✐s ❡①tr❡♠❡❧② ❝♦♥s❡r✈❛t✐✈❡ ❝❛❧❝✉❧❛t✐♦♥ ✐s ❧✐❦❡❧② s❧❛❝❦ ❢♦r t✇♦ r❡❛s♦♥s✳ ❋✐rst✱ ❡✈❡♥ ✇✐t❤✐♥ t❤❡ ♥♦♥✲◆❑❘ ❤♦s♣✐t❛❧s t❤❛t ✸✸ ❲❡

♠❡❛s✉r❡ ♣❛rt✐❝✐♣❛t✐♦♥ r❛t❡ ❛s t❤❡ ♥✉♠❜❡r ♦❢ ❞♦♥♦rs s✉❜♠✐tt❡❞ t♦ t❤❡ ◆❑❘ ❛s ❛ ❢r❛❝t✐♦♥ ♦❢ ❞♦♥♦rs

s✉❜♠✐tt❡❞ t♦ t❤❡ ◆❑❘ ♦r tr❛♥s♣❧❛♥t❡❞ ✐♥ ❛ ✇✐t❤✐♥✲❤♦s♣✐t❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳

✸✹ ❚❤❡

❞❡❛❞✇❡✐❣❤t ❧♦ss ❢r♦♠ ❤♦s♣✐t❛❧s t❤❛t ❞♦ ♥♦t ♣❛rt✐❝✐♣❛t❡ ✐♥ ❛♥② ♦❢ t❤❡ t❤r❡❡ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ❛❧♦♥❡

✐s ✶✵✻✳✾✳

■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❡st✐♠❛t❡ t❤❛t t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❢♦r ❤♦s♣✐t❛❧s t❤❛t ♣❛rt✐❝✐♣❛t❡ ♦♥❧② ✐♥ ❯◆❖❙ ♦r

❆P❉ ✐s ✷✵✳✶✱ ❛ss✉♠✐♥❣ t❤❛t ❛❧❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ❢r♦♠ t❤❡s❡ ❤♦s♣✐t❛❧s ❛r❡ ♣r♦❞✉❝❡❞ ❛t ❛ s❝❛❧❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ r❡s♣❡❝t✐✈❡ ♣❧❛t❢♦r♠✳

✸✻

♣❛rt✐❝✐♣❛t❡ ✐♥ ❡✐t❤❡r ❯◆❖❙ ♦r ❆P❉✱ t✇♦✲t❤✐r❞s ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ❛r❡ ♣❡r❢♦r♠❡❞ ✇✐t❤✐♥ t❤❡ ❤♦s♣✐t❛❧s ✭♣❛♥❡❧ ❈✮✳ ❚❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❧♦✇❡r ❜♦✉♥❞ ♦❢ ✶✷✼✳✵ ❛ss✉♠❡s t❤❛t ❛❧❧ tr❛♥s♣❧❛♥ts ❛r❡ ♣r♦❞✉❝❡❞ ❛t t❤❡ ❆P❉✴❯◆❖❙ s❝❛❧❡✳ ❙❡❝♦♥❞✱ ✐t ✐❣♥♦r❡s ❞❡❛❞✇❡✐❣❤t ❧♦ss ❢r♦♠ ❤♦s♣✐t❛❧s t❤❛t ♣❛rt✐❝✐♣❛t❡ ✐♥ ◆❑❘✳ ❲✐t❤✐♥ t❤❡ s❡t ♦❢ ◆❑❘ ♣❛rt✐❝✐♣❛♥ts✱ t❤❡ ✶✼ ❤♦s♣✐t❛❧s t❤❛t ❛r❡ ✐♥ t❤❡ ❧♦✇❡st q✉❛rt✐❧❡ ♦❢ ❢r❛❝t✐♦♥ ♦❢ tr❛♥s♣❧❛♥ts ♣❡r❢♦r♠❡❞ ✐♥ ◆❑❘ ❛❧♦♥❡ ❝♦♥tr✐❜✉t❡ t♦ ❛♥ ❡✣❝✐❡♥❝② ❧♦ss ♦❢ ✾✹✳✼ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r ✭♣❛♥❡❧ ❉✮✳ ■♥ s✉♠♠❛r②✱ ❞❡s♣✐t❡ ♣♦t❡♥t✐❛❧ ❜✐❛s ❞✉❡ t♦ s♦♠❡ ❤♦s♣✐t❛❧s ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❧❛r❣❡ ♣❧❛t❢♦r♠s✱ t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ s✉❣❣❡sts t❤❛t ❛ ❧♦ss ♦❢ ✷✵✵ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r ✐s ❛ ❝♦♥s❡r✈❛t✐✈❡ ❡st✐♠❛t❡ ❢♦r t❤❡ ❝♦sts ♦❢ ♠❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥✳ ❚❛❜❧❡ ✸✿ ❚♦t❛❧ ❊✣❝✐❡♥❝② ▲♦ss

Number of Hospitals

Kidney Exchange Transplants Per Year

Within Hospital Kidney Exchange Transplants Per Year

Efficiency Loss Additional Kidney Exchange Transplants (1)

(2)

(3)

447.7

386.1

539.7

186.2 111.4 64.6 23.9

285.2 148.1 81.6 24.8

191.9 92.7 101.5

280.8 126.9 132.1

All Hospitals

164

800.5

Panel A: All Hospitals 465.4

Top Quartile 2nd Quartile 3rd Quartile Bottom Quartile

42 48 40 34

598.8 143.2 45.7 12.7

Panel B: By hospital size (number of PKEs per year) 358.3 237.5 73.4 132.7 27.7 57.9 6.0 19.7

NKR Only UNOS or APD None

68 45 51

580.5 133.0 86.9

Top Quartile 2nd Quartile 3rd Quartile Bottom Quartile

17 17 17 17

◆♦t❡s ✿

Panel C: By Platform Membership 297.2 234.8 90.7 106.0 77.6 106.9

Panel D: By NKR Participation Rate (Fraction of PKEs facilitated through the NKR) 65.2 8.2 14.5 13.9 102.3 27.0 44.2 38.8 196.7 98.2 81.5 66.6 216.2 163.8 94.7 72.7

17.9 51.3 97.4 114.1

❈♦❧✉♠♥ ✭✶✮ ❛ss✉♠❡s t❤❛t t❤❡ t②♣✐❝❛❧ tr❛♥s♣❧❛♥t ❤♦s♣✐t❛❧ ❤❛s ❛ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐rs

❛♥❞ ❛❧tr✉✐st✐❝ ❞♦♥♦rs ❣✐✈❡♥ ❜② t❤❡ ❛✈❡r❛❣❡ r❡❣✐str❛t✐♦♥ ✐♥ t❤❡ ◆❑❘✳ ❈♦❧✉♠♥ ✭✷✮ ❛ss✉♠❡s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ✐♥ tr❛♥s♣❧❛♥t ❤♦s♣✐t❛❧s ✉s✐♥❣ ♦♥❧② t❤❡ ❤♦s♣✐t❛❧s ✇✐t❤ t❤❡ t♦♣ q✉❛rt✐❧❡ ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ r❛t❡s ✐♥ t❤❡ ◆❑❘✳ ❈♦❧✉♠♥ ✭✸✮ ❛ss✉♠❡s ❛ ❝♦♠♣♦s✐t✐♦♥ ❜❛s❡❞ ♦♥ ❤♦s♣✐t❛❧s ✇✐t❤ t❤❡ ❧♦✇❡st q✉❛rt✐❧❡ ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ r❛t❡s✳ ❚r❛♥s♣❧❛♥ts ♣❡r ②❡❛r ✐s ❝❛❧❝✉❧❛t❡❞ ✉s✐♥❣ ❞❛t❛ ❜❡t✇❡❡♥ ❆♣r✐❧ ✶✱ ✷✵✶✷ ❛♥❞ ❉❡❝❡♠❜❡r ✹✱ ✷✵✶✹✳

❚❤✐r❞✱ ❤♦s♣✐t❛❧s ♠❛② ✉s❡ ❛ ❞✐✛❡r❡♥t ♠❛t❝❤✐♥❣ t❡❝❤♥♦❧♦❣② t❤❛♥ t❤❡ ◆❑❘✳ ❋♦r ❡①❛♠♣❧❡✱ ❇✐♥❣❛✲ ♠❛♥ ❡t ❛❧✳ ✭✷✵✶✷✮ r❡♣♦rt t❤❛t ▼❡t❤♦❞✐st ❍♦s♣✐t❛❧ ✐♥ ❙❛♥ ❆♥t♦♥✐♦✱ ✇❤✐❝❤ ✐s ♥♦✇ ♣❡r❤❛♣s t❤❡ ♠♦st s♦♣❤✐st✐❝❛t❡❞ s✐♥❣❧❡✲❤♦s♣✐t❛❧ ♣r♦❣r❛♠✱ ✐♥✐t✐❛❧❧② ✉s❡❞ ❛ ▼✐❝r♦s♦❢t ❆❝❝❡ss ❉❛t❛❜❛s❡ ❛♥❞ t❤❛t t❤❡✐r ❛❧❣♦r✐t❤♠ ✇❛s ✏str❛t✐✜❡❞ ❜② ❆❇❖ ❝♦♠♣❛t✐❜✐❧✐t② ❛♥❞ t❤❡♥ ❜② ❍▲❆ ❝♦♠♣❛t✐❜✐❧✐t②✳✑

✸✺

❙✉❝❤ ❛❧❣♦r✐t❤♠s ❛r❡ ❧❡ss ❡✣❝✐❡♥t t❤❛♥ t❤❡ ❧✐♥❡❛r✲♣r♦❣r❛♠♠✐♥❣ ❛❧❣♦r✐t❤♠s ✉s❡❞ ❜② t❤❡ ◆❑❘✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ s✐♥❣❧❡✲❤♦s♣✐t❛❧ ♣r♦❣r❛♠s ❢❛❝❡ s✐♠♣❧❡r ❧♦❣✐st✐❝❛❧ ❝♦♥str❛✐♥ts✱ ✇❤✐❝❤ ♠❛②

✸✺ ■♥ ✷✵✶✸✱ ▼❡t❤♦❞✐st ❍♦s♣✐t❛❧ ✐♥ ❙❛♥ ❆♥t♦♥✐♦ ❛❞♦♣t❡❞ s♦❢t✇❛r❡ ✇r✐tt❡♥ ❜② ♦♥❡ ♦❢ ✉s ✭❆s❤❧❛❣✐✮✳

✸✼ ✐♥❝r❡❛s❡ t❤❡✐r ♣r♦❞✉❝t✐✈✐t② ✈✐s✲à✲✈✐s ♦✉r ❡st✐♠❛t❡s✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤✐s ❜✐❛s ✐s ♥♦t s✐❣♥❡❞ ✐♥ ❣❡♥❡r❛❧✱ ❜✉t ✐t ✐s ♠♦r❡ ❧✐❦❡❧② t❤❛t s✐♥❣❧❡✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s ❛r❡ ❧❡ss ❡✣❝✐❡♥t t❤❛♥ ♦✉r ❡st✐♠❛t❡❞ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳✸✻ ❋♦✉rt❤✱ t❤❡s❡ ❡①❡r❝✐s❡s ❦❡❡♣ t❤❡ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✐♥t❡r❡st❡❞ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ✜①❡❞✳ ❍♦✇❡✈❡r✱ t❤✐s ✢♦✇ ✐s ❡♥❞♦❣❡♥♦✉s ❛♥❞ ❛✛❡❝ts t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss✳ ❆❧t❤♦✉❣❤ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤✐s ❜✐❛s ✐s ❛♠❜✐❣✉♦✉s✱ ♦✉r ❜❛s❡❧✐♥❡ ❛♣♣r♦❛❝❤ ❧✐❦❡❧② ②✐❡❧❞s ❛ ❝♦♥s❡r✈❛t✐✈❡ ❡st✐♠❛t❡ ♦❢ ♦✈❡r❛❧❧ ♠❛r❦❡t ✐♥❡✣❝✐❡♥❝②✳ ❚❤❡ ♠♦st ❧✐❦❡❧② ❜✐❛s ✐s ❞✉❡ t♦ ❤♦s♣✐t❛❧s ✈❛❧✉✐♥❣ tr❛♥s♣❧❛♥ts ❛t ❧❡ss t❤❛♥ t❤❡ s♦❝✐❛❧ ✈❛❧✉❡ ❛♥❞✱ ❞✉❡ t♦ ❛❞♠✐♥✐str❛t✐✈❡ ❝♦sts✱ ❛r❡ ❧✐❦❡❧② t♦ ❡①♣❡♥❞ ✐♥❡✣❝✐❡♥t❧② ❧♦✇ ❡✛♦rt ✐♥ r❡❝r✉✐t✐♥❣ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✳ ■❢ ✐♥❝❡♥t✐✈❡s ✇❡r❡ ♦♣t✐♠❛❧✱ ❤♦s♣✐t❛❧s ♠❛② tr② t♦ r❡❝r✉✐t ♠♦r❡ ✕ ❛♥❞ ♠♦r❡ ✈❛❧✉❛❜❧❡ ✕ ❞♦♥♦rs ✐♥t♦ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳ ❖✉r ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ❛❝❝♦✉♥t ❢♦r t❤✐s ♠❛r❣✐♥ ❜❡❝❛✉s❡ ✇❡ ❞♦ ♥♦t ♦❜s❡r✈❡ r❡❝r✉✐t♠❡♥t ❡✛♦rts ❛♥❞ ✇❡ ❛r❡ t❤❡r❡❢♦r❡ ❧✐❦❡❧② t♦ ✉♥❞❡r❡st✐♠❛t❡ ♦✈❡r❛❧❧ ♠❛r❦❡t ✐♥❡✣❝✐❡♥❝②✳ ❚❛❜❧❡ ✸ ❛❧s♦ ♣♦✐♥ts t♦ ✇❤✐❝❤ t②♣❡s ♦❢ ❤♦s♣✐t❛❧s ❝♦♥❝❡♥tr❛t❡ ♠♦st ♦❢ t❤❡ ✐♥❡✣❝✐❡♥❝②✳ ❈♦♥s✐❞❡r ❝♦❧✉♠♥ ✭✶✮ ❛♥❞✱ ❢♦r t❤❡ ♣✉r♣♦s❡s ♦❢ t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✐❣♥♦r❡ t❤❡ ❜✐❛s❡s ❞✐s❝✉ss❡❞ ❛❜♦✈❡✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡② ♣❡r❢♦r♠ ✐♥t❡r♥❛❧ ❡①❝❤❛♥❣❡s ♠♦r❡ ❡✣❝✐❡♥t❧②✱ ❧❛r❣❡ ❤♦s♣✐t❛❧s ❛❝❝♦✉♥t ❢♦r ♠♦st ♦❢ t❤❡ ✐♥❡✣❝✐❡♥❝② ❜❡❝❛✉s❡ t❤❡✐r ♠❛r❦❡t s❤❛r❡ ✐s ❤✐❣❤❡r ✭♣❛♥❡❧ ❇✮✳ ■♥❞❡❡❞✱ ✺✸✳✵✪ ♦❢ t❤❡ ❧♦ss❡s ❝♦♠❡ ❢r♦♠ ❤♦s♣✐t❛❧s ✐♥ t❤❡ t♦♣ q✉❛rt✐❧❡ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥t ♥✉♠❜❡rs✳ ▼♦r❡♦✈❡r✱ ❜♦t❤ t❤❡ ✐♥t❡♥s✐✈❡ ❛♥❞ ❡①t❡♥s✐✈❡ ♠❛r❣✐♥s ♦❢ ♣❛rt✐❝✐♣❛t✐♦♥ ❛r❡ ✐♠♣♦rt❛♥t✳ ❆ ❧✐tt❧❡ ❧❡ss t❤❛♥ ❤❛❧❢ ♦❢ t❤❡ ❡✣❝✐❡♥❝② ❧♦ss❡s ❛r❡ ❞✉❡ t♦ ❤♦s♣✐t❛❧s t❤❛t ❞♦ ♥♦t ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ◆❑❘ ❛t ❛❧❧✱ ❛♥❞ ❛ q✉❛rt❡r ❢r♦♠ ❤♦s♣✐t❛❧s t❤❛t ❞♦ ♥♦t ♣❛rt✐❝✐♣❛t❡ ✐♥ ❛♥② ♦❢ t❤❡ ♥❛t✐♦♥❛❧ ♣❧❛t❢♦r♠s ✭♣❛♥❡❧ ❈✮✳ ❆♠♦♥❣ ❤♦s♣✐t❛❧s t❤❛t ❞♦ ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ◆❑❘✱ ❛ ❧❛r❣❡ s❤❛r❡ ♦❢ t❤❡ ❡✣❝✐❡♥❝② ❧♦ss ✐s ❞✉❡ t♦ t❤❡ ❤♦s♣✐t❛❧s ✇✐t❤ ❧♦✇ ♣❛rt✐❝✐♣❛t✐♦♥ ✭♣❛♥❡❧ ❉✮✳ ❚♦ s✉♠♠❛r✐③❡✱ ❛❧t❤♦✉❣❤ t❤❡ ❜❛s❡❧✐♥❡ ❡st✐♠❛t❡ ♦❢ ✹✹✼✳✼ ❧♦st tr❛♥s♣❧❛♥ts ✐s ♣♦t❡♥t✐❛❧❧② ❜✐❛s❡❞✱ ❛ ❜❛tt❡r② ♦❢ r♦❜✉st♥❡ss ❡①❡r❝✐s❡s s✉❣❣❡st t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❢r♦♠ ♠❛r❦❡t ❢r❛❣♠❡♥t❛t✐♦♥ ✐s ❧❛r❣❡✳ ❚❤❡s❡ ❧♦ss❡s ❛r✐s❡ ❢r♦♠ ❛❧❧ t②♣❡s ♦❢ ❤♦s♣✐t❛❧s✳ ❚❤❡ ♠♦st ❝♦♥s❡r✈❛t✐✈❡ ❡st✐♠❛t❡s ♣❧❛❝❡ t❤✐s ❧♦ss ❛t ♦✈❡r ✷✵✵ tr❛♥s♣❧❛♥ts ❛ ②❡❛r✳ ❆❞❞✐t✐♦♥❛❧❧②✱ t❤❡s❡ ❡st✐♠❛t❡s ❞♦ ♥♦t ❛♣♣❡❛r t♦ ❜❡ s❡♥s✐t✐✈❡ t♦ ♣♦t❡♥t✐❛❧ ❝♦♠♣♦s✐t✐♦♥❛❧ ❞✐✛❡r❡♥❝❡s ✐♥ t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣♦♦❧✳ ❚❛❜❧❡ ❉✺ ✐♥ ❆♣✲ ♣❡♥❞✐① ❉ ❢✉rt❤❡r ❡✈❛❧✉❛t❡s t❤❡s❡ r❡s✉❧ts✬ r♦❜✉st♥❡ss t♦ ❛❧t❡r♥❛t✐✈❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ♣❛r❛♠❡t❡rs t❤❛t ✇❡r❡ ❝❛❧✐❜r❛t❡❞✳ ❆❝r♦ss ✈❛r✐♦✉s s♣❡❝✐✜❝❛t✐♦♥s✱ ✇❡ ❝♦♥t✐♥✉❡ t♦ ✜♥❞ t❤❛t ❛♥ ❡st✐♠❛t❡❞ ✷✵✵ ❧♦st tr❛♥s♣❧❛♥ts ✐s ❝♦♥s❡r✈❛t✐✈❡✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ❝♦♥s✐st❡♥t ✇✐t❤ ♦✉r ❞❡s❝r✐♣t✐✈❡ ✜♥❞✐♥❣ t❤❛t ❤♦s♣✐t❛❧s ♦❢t❡♥ ♣❡r❢♦r♠ ✐♥❡✣❝✐❡♥t ♠❛t❝❤❡s✳

✺✳✷✳✸ ▼❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛♥❞ ✐♥❡✣❝✐❡♥❝② ♦❢ ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠s ❲❡ ♥♦✇ t❡st ✇❤❡t❤❡r ❝✉rr❡♥t ♣❧❛t❢♦r♠ r✉❧❡s ❛r❡ ❡✣❝✐❡♥t✳ ❚❤❡♦r❡♠ ✶ s❤♦✇s t❤❛t✱ ✐❢ t❤❡ r✉❧❡s ❛r❡ ♦♣t✐♠❛❧✱ t❤❡♥ r❡✇❛r❞s ❢♦r ❞✐✛❡r❡♥t s✉❜♠✐ss✐♦♥ t②♣❡s✱ p0 ✱ s❤♦✉❧❞ ❛♣♣r♦①✐♠❛t❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✱ p∗ = ∇f −A✳ ❲❡ ✇✐❧❧ t❡st t❤✐s ❡q✉❛❧✐t② ❛t t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ r❛t❡ ♦❢ s✉❜♠✐ss✐♦♥s✱ q0 , ❢♦r t❤❡ ◆❑❘ ❞✉r✐♥❣ ♦✉r s❛♠♣❧❡ ♣❡r✐♦❞✳ ✸✻ ❙❡❡

❆❣❛r✇❛❧ ❡t ❛❧✳ ✭✷✵✶✽✮ ❢♦r ❛♥ ❛♥❛❧②s✐s ♦❢ ❤♦✇ ✈❛r✐♦✉s ❧♦❣✐st✐❝s ✐♥✢✉❡♥❝❡ t❤❡ ♣r♦❞✉❝t✐✈✐t② ♦❢ ❛ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠✳ ◆❑❘✬s ♣r❛❝t✐❝❡s ❛r❡ ♦♣t✐♠✐③❡❞ t♦ ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ❣✐✈❡♥ t❤❡ ❛✈❛✐❧❛❜❧❡ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✳

✸✽ ❈✉rr❡♥t r❡✇❛r❞s✱ p0 , ❡q✉❛❧ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛t❝❤✐♥❣ ❡❛❝❤ ❦✐♥❞ ♦❢ s✉❜♠✐ss✐♦♥✳ ❚❤❡s❡ ♣r♦❜❛❜✐❧✐t✐❡s ❝❛♥ ❜❡ ❡❛s✐❧② ❡st✐♠❛t❡❞ ❢r♦♠ ♦✉r s✐♠✉❧❛t✐♦♥s✱ ❛♥❞ t❤❡ ❡st✐♠❛t❡❞ ♣r♦❜❛❜✐❧✐t✐❡s ❝❧♦s❡❧② ♠❛t❝❤ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ✐♥ t❤❡ ❞❛t❛ ✭s❡❡ ❆♣♣❡♥❞✐① ❈✳✺✳✶✮✳ ▼❛r❣✐♥❛❧ ♣r♦❞✉❝ts ∇f ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❜② ♥✉♠❡r✐❝❛❧❧② ❞✐✛❡r❡♥t✐❛t✐♥❣ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳ ■♥ ♣r✐♥❝✐♣❧❡✱ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❛❞❥✉st♠❡♥t t❡r♠ r❡q✉✐r❡s ❡st✐♠❛t❡s ♦❢ t❤❡ s✉♣♣❧② ❡❧❛st✐❝✐t② ♠❛✲ tr✐①✳ ■t ✐s ❞✐✣❝✉❧t t♦ ❡st✐♠❛t❡ t❤✐s ♠❛tr✐① ✉s✐♥❣ ♦❜s❡r✈❡❞ s✉❜♠✐ss✐♦♥s t♦ t❤❡ ◆❑❘ ❜❡❝❛✉s❡ ✐t ✐s ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧✳ ❇✉t✱ t❤❡ ❛❞❥✉st♠❡♥t t❡r♠ ✐s s♠❛❧❧ ❜❡❝❛✉s❡ r❡t✉r♥s t♦ s❝❛❧❡ ❛r❡ ❛♣✲ ♣r♦①✐♠❛t❡❧② ❝♦♥st❛♥t ❢♦r ◆❑❘✬s s✐③❡✳ ❚❤❡r❡❢♦r❡✱ ♦♣t✐♠❛❧ r❡✇❛r❞s ❛r❡ ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳ ❋♦r♠❛❧❧②✱ ❚❤❡♦r❡♠ ✶ ✐♠♣❧✐❡s t❤❛t t❤❡ q✉❛♥t✐t②✲✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ t❤❡ ❛❞❥✉st♠❡♥t t❡r♠ ✐s q f q = ∇f · − ✳ A· kqk1

kqk1

kqk1

❚❤❛t ✐s✱ t❤❡ ❛✈❡r❛❣❡ ❧❡✈❡❧ ♦❢ s❤❛❞✐♥❣ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❛✈❡r❛❣❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ♣r♦❞✉❝t✳ ❊✈❛❧✉❛t✐♥❣ t❤✐s ❢♦r♠✉❧❛ ✉s✐♥❣ ✉s✐♥❣ t❤❡ ❡st✐♠❛t❡❞ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❞❡r✐✈❛t❡s ❢♦r ❡❛❝❤ ♦❢ t❤❡ ✶✹✷✾ ♣❛t✐❡♥t✴❞♦♥♦r t②♣❡s ②✐❡❧❞s ❛♥ ❛✈❡r❛❣❡ s❤❛❞✐♥❣ ♦❢ ♦♥❧② 2.16 × 10−4 ✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ s✐♠♣❧② ❛♣♣r♦①✐♠❛t❡ ♦♣t✐♠❛❧ r❡✇❛r❞s ✇✐t❤ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❜❡❝❛✉s❡ s❤❛❞✐♥❣ ✐s ♥♦t ❛ ♠❛❥♦r ❝♦♥❝❡r♥✳ ❋✐❣✉r❡ ✼❛ ♣❧♦ts ❝✉rr❡♥t r❡✇❛r❞s ✭t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛t❝❤✐♥❣ p0 ✮ ✈❡rs✉s ♦♣t✐♠❛❧ r❡✇❛r❞s ✭♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ∇f ✮✳ ❙♦♠❡ ♦❢ t❤❡ ✶✹✷✾ t②♣❡s ❤❛✈❡ ♥❡❣❛t✐✈❡ ❡st✐♠❛t❡❞ ♠❛r❣✐♥❛❧ ♣r♦❞✲ ✉❝ts ❜❡❝❛✉s❡ t❤❡ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠s ❛r❡ ♠②♦♣✐❝✱ ✇❤✐❝❤ ❝❛♥ r❡s✉❧t ✐♥ ❝r♦✇❞✐♥❣ ♦✉t ♦❢ ❢✉t✉r❡ tr❛♥s♣❧❛♥ts✳ ❍♦✇❡✈❡r✱ ♥❡❣❛t✐✈❡ ♣♦✐♥t ❡st✐♠❛t❡s ❝❛♥ ❛❧s♦ r❡s✉❧t ❢r♦♠ ♥♦✐s❡ ❞✉❡ t♦ s✐♠✉❧❛t✐♦♥ ❡rr♦r✳ ❋✐❣✉r❡ ✼❜ ❛❣❣r❡❣❛t❡s t❤❡s❡ ❡st✐♠❛t❡s ✇✐t❤ ❝❛t❡❣♦r✐❡s ❝♦♥str✉❝t❡❞ ❜② ✉s✐♥❣ t❤❡ ❝❧❛ss✐✲ ✜❝❛t✐♦♥ ✐♥t♦ ✉♥❞❡r❞❡♠❛♥❞❡❞✱ ♦✈❡r❞❡♠❛♥❞❡❞ ❛♥❞ s❡❧❢❞❡♠❛♥❞❡❞ t②♣❡s ❜❛s❡❞ ♦♥ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮✱ s♣❧✐t ❜② t❤❡ ✐♠♠✉♥❡ s❡♥s✐t✐③❛t✐♦♥ ❧❡✈❡❧s ♦❢ t❤❡ ♣❛t✐❡♥t✳ ❚❤❡s❡ ❛❣❣r❡❣❛t❡❞ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛♥❞ ♠❛t❝❤ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❡st✐♠❛t❡❞ ♠♦r❡ ♣r❡❝✐s❡❧②✳ ❚❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛r❡ q✉❛❧✐t❛t✐✈❡❧② s✐♠✐❧❛r t♦ t❤❡ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ t❤❡♦r❡t✐❝❛❧ ♣r❡✲ ❞✐❝t✐♦♥s ❞✐s❝✉ss❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳ ❈♦♥s✐st❡♥t ✇✐t❤ ♦✉r ❡st✐♠❛t❡s✱ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ ❞❡❧✐✈❡r ❛ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ ✵ ❢♦r ✉♥❞❡r❞❡♠❛♥❞❡❞ ♣❛✐rs✱ ❜✉t ♦✉r ❡st✐♠❛t❡s ❞✐✛❡r ❢♦r s♦♠❡ ♦t❤❡r t②♣❡s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ♠♦❞❡❧ ✐♥ ❘♦t❤ ❡t ❛❧✳ ✭✷✵✵✼✮ ❞❡❧✐✈❡rs ❛ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ ✷ ❢♦r ♦✈❡r❞❡♠❛♥❞❡❞ ♣❛✐rs✳ ❲❡ ❡st✐♠❛t❡ t❤❛t ❛♥ ♦✈❡r❞❡♠❛♥❞❡❞ ♣❛✐r ✇✐t❤ ❧♦✇ s❡♥s✐t✐③❛t✐♦♥ ❤❛s ❛ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ ✶✳✻✹✳ ❆ r❡❛s♦♥ ❢♦r t❤✐s ❞✐✛❡r❡♥❝❡ ✐s t❤❛t t❤❡s❡ ♣❛✐rs ♦♥❧② ❛r❡ ♠❛t❝❤❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✵✳✽✷ ❛❝❝♦r❞✐♥❣ t♦ ♦✉r ❞❛t❛✳ ❖✉r ❡♠♣✐r✐❝❛❧ ♠♦❞❡❧ ❛❧s♦ r❡✜♥❡s t❤❡ ♣r❡❞✐❝t✐♦♥s ❢r♦♠ t❤❡ t❤❡♦r❡t✐❝❛❧ ♠♦❞❡❧s ❜② s❤♦✇✐♥❣ ❤♦✇ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ✈❛r② ✇✐t❤ s❡♥s✐t✐③❛t✐♦♥✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ♦❢ ♦✈❡r❞❡♠❛♥❞❡❞ ❛♥❞ s❡❧❢✲❞❡♠❛♥❞❡❞ ♣❛✐rs ❛r❡ ❝♦♥s✐❞❡r❛❜❧② ❧♦✇❡r ✐❢ t❤❡s❡ ♣❛✐rs ❛r❡ s❡♥s✐t✐③❡❞✳ ❚❤❡s❡ ✜♥❡r r❡s✉❧ts ❝❛♥ ❜❡ ✐♠♣♦rt❛♥t ✇❤❡♥ ❞❡s✐❣♥✐♥❣ ♣r❛❝t✐❝❛❧ ♠❡❝❤❛♥✐s♠s✳ ❇♦t❤ ✜❣✉r❡s s❤♦✇ ❛ ❧❛r❣❡ ✇❡❞❣❡ ❜❡t✇❡❡♥ ❝✉rr❡♥t ❛♥❞ ♦♣t✐♠❛❧ r❡✇❛r❞s✳ ■❢ ❝✉rr❡♥t r❡✇❛r❞s ✇❡r❡ ♦♣t✐♠❛❧✱ ❛❧❧ ♣♦✐♥ts ♦♥ t❤❡s❡ t✇♦ ✜❣✉r❡s ✇♦✉❧❞ ❜❡ ♦♥ t❤❡ ✹✺✲❞❡❣r❡❡ ❧✐♥❡✳ ❆❧tr✉✐st✐❝ ❞♦♥♦rs ❛♥❞ ♦✈❡r❞❡♠❛♥❞❡❞ ♣❛✐rs ✇✐t❤ ❧♦✇ P❘❆ ❛r❡ ❢❛r ❜❡❧♦✇ t❤✐s ❧✐♥❡✳ ❖✈❡r❞❡♠❛♥❞❡❞ ♣❛✐rs ✇✐t❤ ❧♦✇ s❡♥s✐t✐③❛t✐♦♥ ❤❛✈❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ♦❢ ✶✳✻✹✱ ❜✉t t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♠❛t❝❤✐♥❣ t❤❡♠ ✐s ♦♥❧② ✵✳✽✷✳ ❊✈❡♥ ♠♦r❡ ❡①tr❡♠❡✱ ❛❧tr✉✐st✐❝ ❖ ❞♦♥♦rs ❤❛✈❡ ❛ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ ✶✳✽✻✱ ❜✉t t❤❡✐r ♣r♦❜✲ ❛❜✐❧✐t② ♦❢ ♠❛t❝❤✐♥❣ ✐s ♦♥❧② ✵✳✾✹✳ ❚❤❡r❡❢♦r❡✱ ❤♦s♣✐t❛❧s ❛r❡ ♥♦t r❡✇❛r❞❡❞ ❡♥♦✉❣❤ ❢♦r s✉❜♠✐tt✐♥❣

✸✾

Matching Probability

1.00

●● ● ● ●● ● ● ● ●●● ● ●●●● ●● ●● ●● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ●●● ● ●●● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ●●● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ●● ● ●● ●● ●● ● ● ●● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ●●●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ●● ●● ●● ●●●● ● ●● ●● ●● ● ● ●● ● ● ●● ●● ●●● ●● ●● ● ● ● ● ●● ●● ●● ● ●● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●●● ● ● ● ● ● ●●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ●● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ●● ●● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ●● ●● ● ●●● ●● ●● ● ● ● ● ●● ● ● ● ●● ● ● ●● ●●● ● ●● ●● ● ●● ● ●● ●● ●● ●●● ● ● ●● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ●●●●●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ●● ●● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ●●●●● ● ● ● ●● ●●●● ●● ● ●● ● ●● ● ● ● ●● ●●● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●●●● ● ●●● ● ● ●● ● ●●● ● ● ●● ● ●● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ●● ●●● ●● ● ● ● ● ●● ●●● ● ●● ● ●● ●● ● ● ● ● ● ● ● ●●●●● ● ● ● ●●● ● ● ● ●● ● ●● ● ● ●● ● ● ● ●● ● ●●●● ● ● ● ● ● ● ●● ●● ● ● ●●●● ●●●●●● ● ●● ●● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ●●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ●● ●● ● ●● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●● ● ●● ●● ● ● ● ● ●●● ● ●●●● ● ●● ● ● ● ● ●●●● ● ●● ● ●●● ● ● ● ● ● ●● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ●● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ●●● ●● ● ● ●● ●● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●●●● ●●● ● ● ●●●●● ● ● ●● ● ● ● ● ●●●● ● ●● ● ● ●● ●● ● ● ● ●●●● ● ●● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ●●● ● ●●● ●● ● ●● ● ●● ● ●● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ●● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●

0.75

0.50

0.25

0.00 −1

0

1

Category ● ● ● ● ●

Altruist non−O Altruist O Overdemanded Selfdemanded Underdemanded

2

Marginal Product

✭❛✮

Matching Probability

1.00

PRA

0.75

(10,90]

● (90,100] [0,10] Altruist

0.50

●

Category ● Altruist non−O ● Altruist O ● Overdemanded ● Selfdemanded ● Underdemanded

●

0.25 ●

0.00 0.0

0.5

1.0

1.5

2.0

Marginal Product

✭❜✮

❋✐❣✉r❡ ✼✿ Pr✐✈❛t❡ ✈❡rs✉s ❙♦❝✐❛❧❧② ❖♣t✐♠❛❧ ❘❡✇❛r❞s ❢♦r ❙✉❜♠✐ss✐♦♥ ❚②♣❡s

◆♦t❡s ✿

❚❤❡ ✈❡rt✐❝❛❧ ❛①✐s ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ s✉❜♠✐ss✐♦♥ ❜❡✐♥❣ ♠❛t❝❤❡❞✱ ✇❤✐❝❤ ❛r❡ t❤❡ ♣r✐✈❛t❡ r❡✇❛r❞s t❤❛t ❤♦s♣✐t❛❧s r❡❝❡✐✈❡ ❛❝❝♦r❞✐♥❣ t♦ ❝✉rr❡♥t ❡①❝❤❛♥❣❡ r✉❧❡s✳ ❚❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ♣❧♦ts t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♦❢ ❛ s✉❜♠✐ss✐♦♥✱ ✇❤✐❝❤ ❡q✉❛❧s t❤❡ s♦❝✐❛❧ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤❡ s✉❜♠✐ss✐♦♥ ✐♥ t❡r♠s ♦❢ tr❛♥s♣❧❛♥ts✳ ❊❛❝❤ ♣♦✐♥t ❝♦rr❡s♣♦♥❞ t♦ ❛ s✉❜♠✐ss✐♦♥ ✐♥ t❤❡ ❞❛t❛✳ Pr♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛t❝❤✐♥❣ ❛♥❞ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛r❡ ❝❛❧❝✉❧❛t❡❞ ✐♥ t❤❡ ❜❛s❡❧✐♥❡ s✐♠✉❧❛t✐♦♥✳ ▼❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛r❡ ♠❡❛s✉r❡❞ ✇✐t❤ s✉❜st❛♥t✐❛❧ ♥♦✐s❡ ❛t t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❧❡✈❡❧ ❜❡❝❛✉s❡✱ ❞✉❡ t♦ ❝♦♠♣✉t❛t✐♦♥❛❧ r❡❛s♦♥s✱ ❡❛❝❤ ✐♥❞✐✈✐❞✉❛❧ ❞❡r✐✈❛t✐✈❡ ✉s❡s ❛ s♠❛❧❧ ♥✉♠❜❡r ♦❢ s✐♠✉❧❛t✐♦♥ ❞❛②s✳ ■♥ ❛❣❣r❡❣❛t❡❞ ✈❡rs✐♦♥ ❞✐✛❡r❡♥t ❞♦ts ♦❢ t❤❡ s❛♠❡ ❝♦❧♦r ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❞✐✛❡r❡♥t P❘❆ ❧❡✈❡❧s✳ ❋✐❣✉r❡ ✼❛ s❤r✐♥❦s t❤❡ ❡st✐♠❛t❡❞ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛♥❞ ♠❛t❝❤ ♣r♦❜❛❜✐❧✐t✐❡s t♦✇❛r❞s t♦ t❤❡ ❣r♦✉♣ ♠❡❛♥s ❢♦❧❧♦✇✐♥❣ t❤❡ ♣r♦❝❡❞✉r❡ r❡❝♦♠♠❡♥❞❡❞ ❜② ▼♦rr✐s ✭✶✾✽✸✮✳

✹✵ t❤❡s❡ t②♣❡s✱ ❛♥❞ ✐t ♠❛② ❡①♣❧❛✐♥ ✇❤② ✇❡ s❡❡ r❡❧❛t✐✈❡❧② ❢❡✇ ♦❢ t❤❡s❡ t②♣❡s ❛r❡ s✉❜♠✐tt❡❞ t♦ t❤❡ ◆❑❘✳ ❖t❤❡r s✉❜♠✐ss✐♦♥ t②♣❡s ❛r❡ ❞r❛st✐❝❛❧❧② ♦✈❡r✲♣r✐❝❡❞✳ ❯♥❞❡r❞❡♠❛♥❞❡❞ ♣❛✐rs ✇✐t❤ ❧♦✇ s❡♥s✐t✐③❛t✐♦♥ ❤❛✈❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ♦❢ ❛♣♣r♦①✐♠❛t❡❧② ✵✳✶✸ ❜✉t ❤❛✈❡ ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ ♠❛t❝❤❡❞ ♦❢ ❛r♦✉♥❞ ✵✳✸✽✱ ✇❤✐❝❤ ✐s s✉❜st❛♥t✐❛❧✳ ❚❤❡s❡ ❞✐✛❡r❡♥❝❡s s✉❣❣❡st t❤❡ ♣❧❛t❢♦r♠ ❝❛♥ ❞♦ ❝♦♥s✐❞❡r❛❜❧② ❜❡tt❡r ❜② ✐♥❝r❡❛s✐♥❣ r❡✇❛r❞s t♦ t❤❡ ♣r♦❞✉❝t✐✈❡ ❛♥❞ ✉♥❞❡r✈❛❧✉❡❞ s✉❜♠✐ss✐♦♥s ✇❤✐❧❡ r❡❞✉❝✐♥❣ r❡✇❛r❞s t♦ t❤❡ ✉♥♣r♦❞✉❝t✐✈❡ s✉❜♠✐ss✐♦♥s✳ ✺✳✷✳✹

❖♣t✐♠❛❧ ♣♦✐♥t ♠❡❝❤❛♥✐s♠s

❚❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛♥❞ t❤❡ s♠❛❧❧ ❛❞❥✉st♠❡♥t t❡r♠ A t♦❣❡t❤❡r s✉❣❣❡st t❤❛t ❝✉rr❡♥t ♣❧❛t✲ ❢♦r♠ r✉❧❡s ❛r❡ ❢❛r ❢r♦♠ ♦♣t✐♠❛❧✳ ❲❡ ♥♦✇ s❤♦✇ t❤❛t ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛r❡ ❤✐❣❤❧② ♣r❡❞✐❝t❛❜❧❡ ✉s✐♥❣ ❛ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r ❝❛t❡❣♦r✐❡s✱ ❛♥❞ ✉s❡ t❤❡s❡ ❝❛t❡❣♦r✐❡s t♦ s✉❣❣❡st ❛ ♣r❛❝t✐❝❛❧ ♣♦✐♥t ♠❡❝❤❛♥✐s♠ t❤❛t ✐s ❝❧♦s❡ t♦ ♦♣t✐♠❛❧✳ ❲❡ ✉s❡ ❛ r❡❣r❡ss✐♦♥ tr❡❡ t♦ ❝♦♥str✉❝t ❝❛t❡❣♦r✐❡s t❤❛t ❜❡st ♣r❡❞✐❝t ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳ ❚❤❡ ❛♣♣r♦❛❝❤ ❛❧❧♦✇s ❢♦r ❛ ❞❛t❛✲❞r✐✈❡♥ ❝❛t❡❣♦r✐③❛t✐♦♥ ♦❢ ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r t②♣❡s✳ ❲❡ ❛❧❧♦✇❡❞ t❤❡ tr❡❡ t♦ ❞❡♣❡♥❞ ♦♥ t❤❡ ♣❛t✐❡♥t✬s P❘❆✱ s✉❜♠✐ss✐♦♥ t②♣❡ ✭❛❧tr✉✐st✐❝✱ ♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐r✱ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥t✮✱ ❛♥❞ ❆❇❖ ❜❧♦♦❞ t②♣❡✳ ❋✐❣✉r❡ ✽ s❤♦✇s t❤❡ ❝❛t❡❣♦r✐❡s ❢♦✉♥❞ ❜② ❛ st❛♥❞❛r❞ ❛❧❣♦r✐t❤♠ ❢♦r ✜♥❞✐♥❣ t❤❡ ❜❡st ❝r♦ss✲✈❛❧✐❞❛t❡❞ ♣r❡❞✐❝t♦r ❢♦r t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳ ❚❤❡s❡ ❝❛t❡❣♦r✐❡s ❛r❡ ✐♥t✉✐t✐✈❡ ❛s t❤❡② s♣❧✐t s✉❜♠✐ss✐♦♥s ❜❛s❡❞ ♣r✐♠❛r✐❧② ♦♥ s✉❜♠✐ss✐♦♥ t②♣❡✱ ✇❤❡t❤❡r ♦r ♥♦t t❤❡ ♣❛t✐❡♥t✴❞♦♥♦r ✐s ❜❧♦♦❞ t②♣❡ ❖✱ ❛♥❞ ♦♥ ✐♠♠✉♥❡ s❡♥s✐t✐✈✐t②✳ ❚❤❡ ♣r♦❝❡❞✉r❡ ❝❤♦s❡ ❛ tr❡❡ ✇✐t❤ ❢❡✇ ❧❡❛✈❡s ✇✐t❤ ❝❛t❡❣♦r② ♠❡❛♥ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ∇f ❛♥❞ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛t❝❤✐♥❣ p0 t❤❛t ❛r❡ ❞✐s♣❡rs❡❞ r❡❧❛t✐✈❡ t♦ t❤❡ ✭❛♣♣r♦♣r✐❛t❡❧② s❤r✉♥❦✮ ✇✐t❤✐♥✲❝❛t❡❣♦r② st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✳ ❚❤✐s s✉❣❣❡sts t❤❛t ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛♥❞ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♠❛t❝❤✐♥❣ ❛r❡ ❛♣♣r♦①✐♠❛t❡❞ ✇✐t❤ ❛ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ❝❛t❡❣♦r✐❡s✳ ❆ ♠❡❝❤❛♥✐s♠ t❤❛t ❛ss✐❣♥s ♣♦✐♥ts ❜❛s❡❞ ♦♥ t❤❡s❡ ❝❛t❡❣♦r✐❡s ❝❛♥ ❜❡ ❡①♣❧❛✐♥❡❞ t♦ ♣❛rt✐❝✐♣❛♥ts ✇✐t❤ t❤✐s tr❡❡ ♦r ❛ s✐♠♣❧❡ t❛❜❧❡ ✭❢♦r ❡①❛♠♣❧❡✱ ❚❛❜❧❡ ❈✹✮✳✸✼ P♦✐♥ts ❝♦✉❧❞ ❜❡ ❛✇❛r❞❡❞ ✇❤❡♥ t❤❡ ❤♦s♣✐t❛❧ s✉❜♠✐ts ❛ ♣❛t✐❡♥t ❛♥❞✴♦r ❞♦♥♦r t♦ t❤❡ ◆❑❘✱ ♦r ❛t t❤❡ t✐♠❡ ♦❢ tr❛♥s♣❧❛♥t❛t✐♦♥✳ ❖✉r t❤❡♦r② s✉❣❣❡sts t❤❛t ♣♦✐♥ts✱ ✐❢ ❛✇❛r❞❡❞ ✉♣♦♥ s✉❜♠✐ss✐♦♥✱ s❤♦✉❧❞ ❜❡ ♦♥❧② s❧✐❣❤t❧② s❤❛❞❡❞ r❡❧❛t✐✈❡ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ∇f . ❆ ♣♦✐♥t s❤♦✉❧❞ t❤❡♥ ❜❡ s✉❜tr❛❝t❡❞ ✇❤❡♥❡✈❡r t❤❡ ❤♦s♣✐t❛❧ ❝♦♥❞✉❝ts ❛ tr❛♥s♣❧❛♥t ❢♦r ♦♥❡ ♦❢ ✐ts ♣❛t✐❡♥t ❜❡❝❛✉s❡ ✐t ✐s t❤❡ ♥✉♠❡r❛✐r❡ ✐♥ ♦✉r ♠♦❞❡❧✳ ❘❡✇❛r❞s ❛t s✉❜♠✐ss✐♦♥ r❛✐s❡ t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t ❤♦s♣✐t❛❧s ✇✐❧❧ ♠❛❦❡ s❤✐❧❧ s✉❜♠✐ss✐♦♥s✳ ❚❤✐s r❡❛s♦♥✐♥❣ ♠♦t✐✈❡s ❛✇❛r❞✐♥❣ ♣♦✐♥ts ❛t tr❛♥s♣❧❛♥t❛t✐♦♥✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts s❤♦✉❧❞ ❜❡ ❞✐✈✐❞❡❞ ❜② t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♠❛t❝❤✐♥❣ p0 ✐♥ ♦r❞❡r t♦ ✐♠♣❧❡♠❡♥t ✐❞❡♥t✐❝❛❧ ❡①♣❡❝t❡❞ r❡✇❛r❞s✳ ❆s ❜❡❢♦r❡✱ ❛ ♣♦✐♥t ✐s ❞❡❞✉❝t❡❞ ❢♦r ❡❛❝❤ tr❛♥s♣❧❛♥t t❤❡ ❤♦s♣✐t❛❧ ❝♦♥❞✉❝ts✳ ❚❤❡ ♦♣t✐♠❛❧ ♣♦✐♥ts ❛✇❛r❞❡❞ ❛t t❤❡ t✐♠❡ ♦❢ tr❛♥s♣❧❛♥t❛t✐♦♥ ✐s ❞❡♥♦t❡❞ ❛s r∗ ✐♥ ❋✐❣✉r❡ ✽✳ ❚❤❡ ♣❧❛t❢♦r♠ ❝♦✉❧❞ ✐♠♣♦s❡ ❛ ❧♦✇❡r ❜♦✉♥❞ ♦♥ t❤❡ ♣♦✐♥t ❜❛❧❛♥❝❡ ❛ ❤♦s♣✐t❛❧ ♠✉st ❤❛✈❡ ❜❡❢♦r❡ ♦♥❡ ♦❢ ✐ts s✉❜♠✐ss✐♦♥s ✐s tr❛♥s♣❧❛♥t❡❞✳ ❆ ❤♦s♣✐t❛❧ t❤❛t ❤❛s ❤✐t t❤✐s ❜♦✉♥❞ ✇♦✉❧❞ ♦♥❧② ❜❡ ❛❜❧❡ t♦ tr❛♥s♣❧❛♥t ❡✐t❤❡r ❛♥ ❛❧tr✉✐st✐❝ ❞♦♥♦r ♦r ❛ ♦✈❡r❞❡♠❛♥❞❡❞ ♣❛✐r t❤❛t ❞♦❡s ♥♦t ❤❛✈❡ ❛ ❤✐❣❤❧② s❡♥s✐t✐③❡❞ ♣❛t✐❡♥t ✭◆♦♥✲❖ P❛t✐❡♥t✱ ❖ ❉♦♥♦r✱ P❘❆❁✾✻✪✮✳ ❲❡ ♣♦st♣♦♥❡ ❛ ♠♦r❡ ❞❡t❛✐❧❡❞ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡s❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❞❡t❛✐❧s ❛❜♦✉t t❤❡ ♣♦✐♥t s②st❡♠ t♦ ❙❡❝t✐♦♥ ✻✳ ✸✼ ❊✈✐❞❡♥❝❡ ❢r♦♠ t❤❡ ♣s②❝❤♦❧♦❣② ❧✐t❡r❛t✉r❡ s✉❣❣❡sts t❤❛t ♣r♦❢❡ss✐♦♥❛❧s ❛r❡ ❧✐❦❡❧② t♦ ✉s❡ ❞❛t❛✲❞r✐✈❡♥ ❛❞✈✐❝❡ ✇❤❡♥ ✐t ✐s ♣r❡s❡♥t❡❞ ✇✐t❤ s✐♠♣❧❡ ❞❡❝✐s✐♦♥ tr❡❡s ✭●✐❣❡r❡♥③❡r ❛♥❞ ●♦❧❞st❡✐♥✱ ✶✾✾✻❀ ●✐❣❡r❡♥③❡r ❛♥❞ ❑✉r③❡♥✲ ❤❛❡✉s❡r✱ ✷✵✵✺✮✳

Pairs

Altruists

O Patient

O Donor O Donor N = 62

PRA < 96 AB Patient N = 501

Non-O Donor N = 102

PRA >= 91 N = 100

PRA < 91 N = 148

Non-O Donor N = 493

O Donor N = 108

mean  std

mean  std

mean  std

mean  std

mean  std

mean  std

 p0 0.94 0.00

 p0  0.86 0.04

 p0  0.30 0.23

 p0 0.81 0.03

 p0  0.28 0.16

 p0  0.84 0.01

∇

f

1.86 0.17

 r ∗  1.99

Unpaired Patients

Non-O Patient

∇

f

0.84 0.21

 r ∗  0.97

∇

f

0.13 0.23

 r ∗  -0.55

∇

f

0.72 0.22

 r ∗  -0.11

∇

f

0.08 0.19

 r ∗  -0.72

∇

f

1.44 0.30

 r ∗  0.72

Non-O Donor N = 292

PRA >= 96 N = 124

mean  std  p0  0.85 0.03

∇

f

 0.69 0.26

 r ∗  -0.19

mean  std

mean  std

 p0  0.28 0.21

 p0  0.24 0.13

∇

f

0.09 0.21

 r ∗  -0.66

∇

f

0.07 0.20

  r ∗  -0.71

❋✐❣✉r❡ ✽✿ ❘❡❣r❡ss✐♦♥ ❚r❡❡ ❢♦r ▼❛r❣✐♥❛❧ Pr♦❞✉❝ts

◆♦t❡s ✿ ❈❛t❡❣♦r✐❡s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② r❡❣r❡ss✐♦♥ tr❡❡ ❛♥❛❧②s✐s t♦ ♣r❡❞✐❝t ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✇❤❡t❤❡r ❛ s✉❜♠✐ss✐♦♥ ✐s ❛ ♣❛✐r ♦r ❛❧tr✉✐st✱

✹✶

❜❧♦♦❞ t②♣❡s✱ ❛♥❞ t❤❡ ♣❛t✐❡♥t✬s P❘❆✳ ❖✉r ♣r♦❝❡❞✉r❡ ❢♦❧❧♦✇❡❞ st❛♥❞❛r❞ r❡❝♦♠♠❡♥❞❛t✐♦♥s ✐♥ ❋r✐❡❞♠❛♥ ❡t ❛❧✳ ✭✷✵✵✶✮✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ✉s❡❞ ✶✵✲❢♦❧❞ ❝r♦ss✲✈❛❧✐❞❛t✐♦♥ t♦ ♣✐❝❦ t❤❡ ♣❡♥❛❧t② ♣❛r❛♠❡t❡r ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ♥♦❞❡s✱ r❡q✉✐r❡❞ ❡❛❝❤ ❧❡❛❢ t♦ ❤❛✈❡ ❛t ❧❡❛st ✷✵ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ ♣r✉♥❡❞ ❛ ❧❡❛❢ ✐❢ ✐t ❞✐❞ ♥♦t ✐♥❝r❡❛s❡ t❤❡ ♦✈❡r❛❧❧ ✜t ❜② ❛t ❧❡❛st ✷✪✳ ❙t❛♥❞❛r❞ ❡rr♦rs ❢♦r t❤❡ s✐♠✉❧❛t✐♦♥s ❛r❡ ❝❛❧❝✉❧❛t❡❞ ❜② ❢♦❧❧♦✇✐♥❣ ❈❤❛♣t❡r ✶✷ ♦❢ ❘♦❜❡rt ❛♥❞ ❈❛s❡❧❧❛ ✭✷✵✵✹✮✳ ❚❤❡ ✇✐t❤✐♥ ❝❛t❡❣♦r② st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ✐s ❡st✐♠❛t❡❞ ✉s✐♥❣ s❤r✐♥❦❛❣❡ ♠❡t❤♦❞s r❡❝♦♠♠❡♥❞❡❞ ✐♥ ▼♦rr✐s ✭✶✾✽✸✮✳

✹✷ ✺✳✷✳✺

❲❡❧❢❛r❡ ❣❛✐♥s ❢r♦♠ ♦♣t✐♠❛❧ ♣♦✐♥t ♠❡❝❤❛♥✐s♠s

❲❡ ♥♦✇ ❡st✐♠❛t❡ t❤❡ ❣❛✐♥ ✐♥ ✇❡❧❢❛r❡ ❢r♦♠ ♠♦✈✐♥❣ t♦ t❤❡ ♣♦✐♥t ♠❡❝❤❛♥✐s♠ ❞❡s❝r✐❜❡❞ ❛❜♦✈❡✳ ❚❤✐s ❣❛✐♥ ✐s ❡q✉❛❧ t♦ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss t❤❛t ❝❛♥ ❜❡ ❛✈♦✐❞❡❞ ❜② r❡✇❛r❞✐♥❣ ❤♦s♣✐t❛❧s ♦♣t✐♠❛❧❧② ❛s ✐♥ ❚❤❡♦r❡♠ ✶✳ ❲❡ ❜❡❣✐♥ ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❣❛✐♥ ✐♥ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ❛♥❞ ❧❛t❡r ❝♦♥s✐❞❡r t❤❡ ❣❛✐♥ ✐♥ s♦❝✐❛❧ ✇❡❧❢❛r❡✳ ❈❛❧❝✉❧❛t✐♥❣ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s s✐♠✐❧❛r t♦ ❛ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ✈❡rs✐♦♥ ♦❢ ❝❛❧❝✉❧❛t✐♥❣ ❞❡❛❞✲ ✇❡✐❣❤t ❧♦ss ❢r♦♠ ❧✐♥❡❛r ❝♦♠♠♦❞✐t② t❛①❛t✐♦♥✳ ❋✐❣✉r❡ ✾ ✐❧❧✉str❛t❡s ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦✲ ❥❡❝t✐♦♥ ❛♥❞ ❞❡♣✐❝ts t❤❡ ❝✉rr❡♥t ❛❣❣r❡❣❛t❡ s✉♣♣❧② q 0 ✱ t❤❡ ❝✉rr❡♥t r❡✇❛r❞s p0 ✱ t❤❡ ❝✉rr❡♥t ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ∇f 0 ✱ ❛♥❞ t❤❡ ♦♣t✐♠❛❧ ❛❣❣r❡❣❛t❡ s✉♣♣❧② q ∗ ✳ ❚❤❡ ❤♦s♣✐t❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss W H (q ∗ ) − W H (q 0 ) ❡q✉❛❧s t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ❝✉r✈❡ ∇f ❛♥❞ ♠❛r❣✐♥❛❧ ❝♦st ❝✉r✈❡ ∇C ❜❡t✇❡❡♥ q 0 ❛♥❞ q ∗ ✳ ❚❤❡r❡❢♦r❡✱ t❤✐s ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s t❤❡ ✐♥t❡❣r❛❧ ♦❢ ∇f (q) − ∇C(q) ❛s q ❣♦❡s ❢r♦♠ q 0 t♦ q ∗ ✳ ❚❤✐s ❝❛❧❝✉❧❛t✐♦♥ ✐s t❤❡ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ✈❡r✲ s✐♦♥ ♦❢ t❤❡ ❍❛r❜❡r❣❡r tr✐❛♥❣❧❡ ❢♦r♠✉❧❛✱ t❤❛t ✐s✱ t❤❡ ❛r❡❛ ❜❡t✇❡❡♥ t❤❡ ♠❛r❣✐♥❛❧ ❜❡♥❡✜t ❛♥❞ ♠❛r❣✐♥❛❧ ❝♦st ❝✉r✈❡s✳

∇f0

∇f

deadweight loss

p∗

p0

∇C q0

◆♦t❡s ✿

A

q∗

b

❋✐❣✉r❡ ✾✿ ❍♦s♣✐t❛❧✲❲❡❧❢❛r❡ ❉❡❛❞✇❡✐❣❤t ▲♦ss ❢r♦♠ t❤❡ ❈✉rr❡♥t ▼❡❝❤❛♥✐s♠

❚❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❢r♦♠ t❤❡ ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠ ✐s t❤❡ s❤❛❞❡❞ ❛r❡❛ ❜❡t✇❡❡♥ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛♥❞ t❤❡ s✉♣♣❧② ❝✉r✈❡✳ ❈✉rr❡♥t r❡✇❛r❞s ❛r❡ p0 ✱ ❡q✉❛❧ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♠❛t❝❤✐♥❣ ❡❛❝❤ t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥✱ ✇❤✐❧❡ ♦♣t✐♠❛❧ r❡✇❛r❞s p∗ ❡q✉❛❧ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳ ❈✉rr❡♥t q✉❛♥t✐t✐❡s q 0 ❛♥❞ r❡✇❛r❞s p0 ❛r❡ ♦❜s❡r✈❡❞✳ ▼❛r❣✐♥❛❧ ♣r♦❞✉❝ts ∇f ✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❝✉rr❡♥t ✈❛❧✉❡ ∇f 0 ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳ ■♥ ❝♦♥tr❛st✱ t❤❡ s✉♣♣❧② ❝✉r✈❡ ∇C ❛♥❞ ♦♣t✐♠❛❧ r❡✇❛r❞s p∗ ❛♥❞ q✉❛♥t✐t✐❡s q ∗ ❛r❡ ♥♦t ♦❜s❡r✈❡❞✱ ❛♥❞ ❞❡♣❡♥❞ ♦♥ t❤❡ ❡❧❛st✐❝✐t② ♦❢ s✉♣♣❧②✳ ❚❤❡ ✜❣✉r❡ r❡♣r❡s❡♥ts ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ♦❜❥❡❝ts✱ s♦ t❤❛t t❤✐s ❛r❡❛ ✐s ❛ ♣❛t❤ ✐♥t❡❣r❛❧ ❣♦✐♥❣ ❢r♦♠ ❝✉rr❡♥t r❡✇❛r❞s p0 t♦ ♦♣t✐♠❛❧ r❡✇❛r❞s p∗ ✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ s❤♦✇s t❤❛t ❛ s✐♠✐❧❛r ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ ❤♦s♣✐t❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❤♦❧❞s ✐♥ ♦✉r s❡tt✐♥❣✳

✹✸ Pr♦♣♦s✐t✐♦♥ ✶✳ ❈♦♥s✐❞❡r ❛ str✐❝t❧② ♣♦s✐t✐✈❡ ❛❣❣r❡❣❛t❡ s✉♣♣❧② ♦❢ ♣❛✐rs✱ ❤♦s♣✐t❛❧s ❝❤♦♦s❡ s✉♣♣❧② ♦♣t✐♠❛❧❧② ❣✐✈❡♥ r❡✇❛r❞s

p0 ✳

t❤❛t r❡s✉❧ts ✇❤❡♥

q∗

❛♥❞ r❡✇❛r❞s

❆ss✉♠❡ t❤❛t t❤❡ ♠❛tr✐①

D 2 C(q ∗ ) −

❈♦♥s✐❞❡r ❛❣❣r❡❣❛t❡ s✉♣♣❧②

p t❤❛t ♠❛①✐♠✐③❡ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ❛s ✐♥ ❚❤❡♦r❡♠ ✶✳ D 2 f (q ∗ ) ✐s ♥♦♥✲s✐♥❣✉❧❛r✳ ❚❤❡♥ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐♥ ∗

❜②

q0✱

❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ❛t

q0

✐s ❛♣♣r♦①✐♠❛t❡❞

1 (∇f 0 − p0 ) · (q ∗ − q 0 ). 2

❆❧t❡r♥❛t✐✈❡❧②✱ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜②

1 (∇f 0 − p0 )[D 2 C(q 0 ) − D 2 f (q 0 )]−1 (∇f 0 − p0 )′ ✳ 2 ❚❤❡ ❡rr♦r ✐♥ ❜♦t❤ ❛♣♣r♦①✐♠❛t✐♦♥s ✐s ❜♦✉♥❞❡❞ ❜② ❛♥ ♦❢

✭✸✮

o(kq ∗ −q 0 k2 ) t❡r♠ ♣❧✉s ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡

|A · (q − q 0 )|✳ ∗

❚❤❡s❡ ❢♦r♠✉❧❛s ❛r❡ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ✈❡rs✐♦♥ ♦❢ t❤❡ st❛♥❞❛r❞ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r t❤❡ ❍❛r✲ ❜❡r❣❡r tr✐❛♥❣❧❡ ✐♥ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝♦♠♠♦❞✐t② t❛①❛t✐♦♥✳ ❚❤❡ ✜rst ❢♦r♠✉❧❛ ✐s t❤❡ ♠✉❧✲ t✐❞✐♠❡♥s✐♦♥❛❧ ✈❡rs✐♦♥ ♦❢ t❤❡ ♦♥❡ ❤❛❧❢ ❜❛s❡ t✐♠❡s ❤❡✐❣❤t ❢♦r♠✉❧❛✳ ❚❤❡ s❡❝♦♥❞ ❢♦r♠✉❧❛ ✐s t❤❡ ❡q✉✐✈❛❧❡♥t ♦❢ t❤❡ ♦♥❡ ❤❛❧❢ ♦❢ t❤❡ t❛① ✇❡❞❣❡ sq✉❛r❡❞ t✐♠❡s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✐♥✈❡rs❡ s✉♣♣❧② ♠✐♥✉s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✐♥✈❡rs❡ ❞❡♠❛♥❞✳ ❚❤❡ s❡❝♦♥❞ ❢♦r♠✉❧❛ s❤♦✇s t❤❛t t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s ♦♥❡ ❤❛❧❢ ♦❢ ❛ q✉❛❞r❛t✐❝ ❡①♣r❡ss✐♦♥ ✐♥ t❤❡ ✇❡❞❣❡ ∇f 0 − p0 ✳ ❚❤❡ t❡r♠ D 2 C(q 0 ) ❛❝❝♦✉♥ts ❢♦r t❤❡ ❢❛❝t t❤❛t ❛ ♠♦r❡ ❡❧❛st✐❝ s✉♣♣❧② ❧❡❛❞s t♦ ❧❛r❣❡r ❞❡❛❞✇❡✐❣❤t ❧♦ss❡s✳ ❚❤❡ t❡r♠ D2 f ❛❝❝♦✉♥ts ❢♦r t❤❡ ❝❤❛♥❣❡ ✐♥ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ✐♥ r❡s♣♦♥s❡ t♦ ❛ ❝❤❛♥❣❡ ✐♥ q ✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s ❧♦✇❡r ✐❢ ✐♥❝r❡❛s✐♥❣ t❤❡ s✉♣♣❧② ♦❢ ♦✈❡r❞❡♠❛♥❞❡❞ ♣❛✐rs r❡s✉❧ts ✐♥ t❤❡s❡ ♣❛✐rs ❜❡❝♦♠✐♥❣ ❧❡ss ✉s❡❢✉❧✳ ❚❤❡ ♣r♦♣♦s✐t✐♦♥ s❤♦✇s t❤❛t ❡st✐♠❛t✐♥❣ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss r❡q✉✐r❡s ❡st✐♠❛t❡s ♦❢ ∇f 0 − p0 ❛♥❞ ❡✐t❤❡r q ∗ − q 0 ♦r D2 C(q 0 ) − D2 f (q 0 )✳ ❲❡ ❝❛♥ ❡st✐♠❛t❡ ∇f 0 ✱ p0 ✱ ❛♥❞ q 0 ✉s✐♥❣ t❤❡ ❡st✐♠❛t❡❞ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ❜❡❝❛✉s❡ ✇❡ ❞♦ ♥♦t ❤❛✈❡ ❛ ❣♦♦❞ ❡st✐♠❛t❡ ♦❢ t❤❡ ❤♦s♣✐t❛❧ s✉♣♣❧② ❝✉r✈❡✱ ✇❡ ❝❛♥♥♦t ❞✐r❡❝t❧② ❡st✐♠❛t❡ q ∗ ♦r D2 C(q 0 )✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤❡ ❧❛r❣❡ ✇❡❞❣❡ ❜❡t✇❡❡♥ t❤❡ ❝✉rr❡♥t ♣r✐✈❛t❡ ❛♥❞ s♦❝✐❛❧ ✐♥❝❡♥t✐✈❡s s✉❣❣❡sts t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s s✐❣♥✐✜❝❛♥t ✉♥❧❡ss t❤❡ s✉♣♣❧② ❡❧❛st✐❝✐t② ✐s ❡①tr❡♠❡❧② s♠❛❧❧✳ ❲❡ ❝❛♥ ✉s❡ ❡q✉❛t✐♦♥ ✭✸✮ t♦ ❢♦r♠❛❧✐③❡ t❤✐s ♣♦✐♥t ❜② q✉❛♥t✐❢②✐♥❣ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss❡s ❢♦r ❛ r❛♥❣❡ ♦❢ s✉♣♣❧② ❡❧❛st✐❝✐t✐❡s✳ ❲❡ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ ♠❡❝❤❛♥✐s♠s t❤❛t s❡t r❡✇❛r❞ ✈❡❝t♦rs ❢♦r t❤❡ ❝❛t❡❣♦r✐❡s ✐♥ t❤❡ r❡❣r❡ss✐♦♥ tr❡❡ ❛♥❛❧②s✐s ❛❜♦✈❡ ✭❋✐❣✉r❡ ✾✮✳ ❲❡ ❡st✐♠❛t❡ t❤❡ ✇❡❞❣❡ ∇f 0 − p0 ❛♥❞ t❤❡ ❝✉r✈❛t✉r❡ ♠❛tr✐① D 2 f ❢♦r t❤❡s❡ ❝❛t❡❣♦r✐❡s ✉s✐♥❣ ♦✉r ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥✳ ❚♦ ✉s❡ ❡q✉❛t✐♦♥ ✭✸✮✱ ✇❡ ♥❡❡❞ t♦ s♣❡❝✐❢② s✉♣♣❧② ❡❧❛st✐❝✐t✐❡s t❤r♦✉❣❤ t❤❡ ♠❛tr✐① D2 C(q 0 )✳ ❖♥❡ ❝❤❛❧❧❡♥❣❡ ✐♥ ❞✐r❡❝t❧② s♣❡❝✐❢②✐♥❣ t❤✐s q✉❛♥t✐t② ✐s t❤❛t ❞✐✛❡r❡♥t s✉❜♠✐ss✐♦♥ t②♣❡s ♠❛② r❡s♣♦♥❞ ❞✐✛❡r❡♥t❧② t♦ r❡✇❛r❞s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ s✉❜♠✐ss✐♦♥ ♦❢ ❤❛r❞✲t♦✲♠❛t❝❤ t②♣❡s t♦ t❤❡ s②st❡♠ ♠❛② ♥♦t s✉❜st❛♥t✐❛❧❧② ❞❡❝r❡❛s❡ ✇❤❡♥ r❡✇❛r❞s ❛r❡ ❧♦✇❡r❡❞ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ❢❡✇ ♦t❤❡r ❛✈❡♥✉❡s ❢♦r ♠❛t❝❤✐♥❣ t❤❡♠✳ ❖✉r ❛♣♣r♦❛❝❤ ✐s t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♠❛①✐♠✉♠ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✉♥❞❡r ✈❛r②✐♥❣ ❜♦✉♥❞s ♦♥ t❤❡ ♠❛①✐♠✉♠ ❡❧❛st✐❝✐t② ♦❢ ❛♥② t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥✳ ❚❤✐s ♠❡t❤♦❞ ❛❧❧♦✇s ✉s t♦ ❜❡ ❛❣♥♦st✐❝ ❛❜♦✉t t❤❡ s✉♣♣❧② ❡❧❛st✐❝✐t✐❡s ♦❢ ❞✐✛❡r❡♥t s✉❜♠✐ss✐♦♥ t②♣❡s✳ ❚❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s ③❡r♦ ✇❤❡♥ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ♠❛①✐♠✉♠ ❡❧❛st✐❝✐t② ✐s ③❡r♦ ❜❡❝❛✉s❡ t❤❡ s✉❜♠✐ss✐♦♥s ✇✐❧❧ ♥♦t r❡s♣♦♥❞ t♦ t❤❡ r❡✇❛r❞s s②st❡♠✱ r❡s✉❧t✐♥❣ ✐♥ q ∗ = q 0 ✳ ❆s ✇❡ ✐♥❝r❡❛s❡ t❤❡ ❜♦✉♥❞ ♦♥ t❤❡

✹✹ ❡❧❛st✐❝✐t②✱ s✉❜♠✐ss✐♦♥s r❡s♣♦♥❞ ❛♥❞ t❤❡ ♠❛①✐♠✉♠ ✐♠♣❧✐❡❞ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐♥❝r❡❛s❡s✳ ❋✉rt❤❡r✱ ✇❡ ❝❛♥ r❡♣❡❛t t❤✐s ❡①❡r❝✐s❡ ❢♦r ✈❛r②✐♥❣ ❛ss✉♠♣t✐♦♥s ♦♥ ❝r♦ss✲❡❧❛st✐❝✐t✐❡s✳✸✽ 120 Cross Elasticity, max ρ = -0.05 Cross Elasticity, max ρ = 0 Cross Elasticity, max ρ = 0.05

100

∆ in Number of Transplants per Year

Deadweight Loss (Transplants per Year)

120

80

60

40

20

0

Cross Elasticity, max ρ = -0.05 Cross Elasticity, max ρ = 0 Cross Elasticity, max ρ = 0.05

100

80

60

40

20

0 0

1

2

3

4

5

Maximum Own Elasticity, ǫ

✭❛✮ ❍♦s♣✐t❛❧✲❲❡❧❢❛r❡ ❉❡❛❞✇❡✐❣❤t ▲♦ss

◆♦t❡s ✿

6

0

1

2

3

4

5

Maximum Own Elasticity, ǫ

✭❜✮ ❚r❛♥s♣❧❛♥ts ▲♦st ✐♥ t❤❡ P❧❛t❢♦r♠

❋✐❣✉r❡ ✶✵✿ ▲♦ss❡s ❉✉❡ t♦ t❤❡ ❈✉rr❡♥t ▼❡❝❤❛♥✐s♠

❊st✐♠❛t❡❞ ❧♦ss❡s ❢r♦♠ t❤❡ ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠✱ ✉s✐♥❣ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❢r♦♠ Pr♦♣♦s✐t✐♦♥ ✶✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡❧❛st✐❝✐t② ♠❛tr✐① ♦❢ s✉♣♣❧②✳ ▼❛①✐♠✉♠ ♦✇♥✲❡❧❛st✐❝✐t✐❡s ❛r❡ ✐♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s✳

❋✐❣✉r❡ ✶✵❛ ♣❧♦ts t❤❡ ♠❛①✐♠✉♠ ❤♦s♣✐t❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❢♦r ❜♦✉♥❞s ♦♥ t❤❡ ♦✇♥✲♣r✐❝❡ ❡❧❛st✐❝✲ ✐t✐❡s r❛♥❣✐♥❣ ❢r♦♠ ✵ t♦ ✻✳ ❚❤❡ ❝✉r✈❡ ✐♥ t❤❡ ♠✐❞❞❧❡ ❞❡s❝r✐❜❡s t❤❡ r❡s✉❧ts ❢♦r ③❡r♦ ❝r♦ss✲♣r✐❝❡ ❡❧❛st✐❝✐t✐❡s✱ ❛♥❞ t❤❡ ♦t❤❡r t✇♦ ❝✉r✈❡s ♣r❡s❡♥t r❡s✉❧ts ❢♦r ♥♦♥✲③❡r♦ ❝r♦ss✲♣r✐❝❡ ❡❧❛st✐❝✐t✐❡s✳ ❚❤❡ ❤♦s♣✐t❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s ③❡r♦ ✐❢ s✉♣♣❧② ✐s ♣❡r❢❡❝t❧② ✐♥❡❧❛st✐❝ ❛♥❞ ✐s ✐♥❝r❡❛s✐♥❣ ✐♥ ❡❧❛st✐❝✐t②✳ ❚❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s s✐❣♥✐✜❝❛♥t ❢♦r ♠♦st ♦❢ t❤✐s r❛♥❣❡ ❛♥❞ ❛❜♦✈❡ ✹✵ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r ✐❢ t❤❡ ♠❛①✐♠✉♠ ❡❧❛st✐❝✐t② ✐s ❛t ❧❡❛st ✷✳ ❋♦r ✈❡r② ❤✐❣❤ ❡❧❛st✐❝✐t✐❡s✱ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐♥✲ ❝r❡❛s❡s ❛t ❛ s❧♦✇❡r r❛t❡ ❜❡❝❛✉s❡ ♦❢ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❝✉r✈❛t✉r❡✳ ❚❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❛t ❛♥ ❡❧❛st✐❝✐t② ♦❢ ✻ ✐s ♦♥❧② ❜❡t✇❡❡♥ ✼✺ ❛♥❞ ✶✵✺ ❜❡❝❛✉s❡ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ♦❢ t❤❡ ♣r♦❞✉❝t✐✈❡ t②♣❡s t❤❛t t❤❡ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ ❛ttr❛❝ts ❞❡❝r❡❛s❡ ✇✐t❤ s✉♣♣❧②✳ ❆❧t❤♦✉❣❤ t❤❡ r❡s✉❧ts ❢♦r ❧❛r❣❡ ❡❧❛st✐❝✐t✐❡s ❛r❡ s✉❜❥❡❝t t♦ ❣r❡❛t❡r ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✱ ✐t ✐s ✉♥❧✐❦❡❧② t❤❛t t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss❡s ❝♦♠❡ ❝❧♦s❡ t♦ t❤❡ ❡✣❝✐❡♥❝② ❧♦ss r❡❧❛t✐✈❡ t♦ t❤❡ ✜rst✲❜❡st ❛❧❧♦❝❛t✐♦♥✱ ❡✈❡♥ ❢♦r ❡❧❛st✐❝✐t✐❡s ♦❢ ❛❜♦✉t ✻✳ ❚❤❡ ❤♦s♣✐t❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss❡s ✇✐❧❧ ✉♥❞❡rst❛t❡ t❤❡ ❧♦ss ✐♥ s♦❝✐❛❧ ✇❡❧❢❛r❡ ✐❢ ❤♦s♣✐t❛❧s ✉♥❞❡r✈❛❧✉❡ ✸✽ ❙♣❡❝✐✜❝❛❧❧②✱

✇❡ s♦❧✈❡❞ t❤❡ ♣r♦❜❧❡♠ 1 (∇f 0 − p0 )[D 2 C(q 0 ) − D 2 f (q 0 )]−1 (∇f 0 − p0 )′ 2  2 −1  2 −1 ∂ C ∂ C p0,j p0,j s✳t✳ 2 ≤ ε ❛♥❞ ≥0 2 ∂ qj q0,j ∂ qj q0,j  2 −1  2 −1 ∂ C ∂ C p0,j p0,k −1  2 + p0,k ∂ C ∂ 2 qj q0,j ∂ 2 qk q0,k =ρ . ∂qj ∂qk q0,j 2 max

D 2 C(q0 )

❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ t❤❡ ❜♦✉♥❞ ♦♥ ❡❧❛st✐❝✐t✐❡s✱ ε✳

6

✹✺ tr❛♥s♣❧❛♥ts✳ ❋✐❣✉r❡ ✶✵❜ s❤♦✇s t❤❡ t♦t❛❧ ✐♥❝r❡❛s❡ ✐♥ tr❛♥s♣❧❛♥ts ❢❛❝✐❧✐t❛t❡❞ ❜② t❤❡ ◆❑❘ ✐❢ ✐t ❛❞♦♣ts t❤❡ ♦♣t✐♠❛❧ ♣♦✐♥ts s②st❡♠✳ ❚♦ ❞♦ t❤✐s✱ ✇❡ ❛❞❞❡❞ t❤❡ ❛r❡❛ ✉♥❞❡r ∇C t♦ t❤❡ ❤♦s♣✐t❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss ♥✉♠❜❡rs ❝❛❧❝✉❧❛t❡❞ ❛❜♦✈❡ ✭s❡❡ ❋✐❣✉r❡ ✾✮✳ ❇❡❝❛✉s❡ ❛ tr❛♥s♣❧❛♥t ✐♥❝r❡❛s❡ ❛t t❤❡ ◆❑❘ ✇✐❧❧ ❝♦♠❡ ❛t t❤❡ ❝♦st ♦❢ ❢❡✇❡r tr❛♥s♣❧❛♥ts ❛t ❤♦s♣✐t❛❧s✱ t❤✐s ❝❛❧❝✉❧❛t✐♦♥ ♦✈❡rst❛t❡s t❤❡ ❧♦ss ✐♥ t♦t❛❧ ✇❡❧❢❛r❡ ❢r♦♠ t❤❡ ❝✉rr❡♥t ♠❡❝❤❛♥✐s♠✳ ◆♦t s✉r♣r✐s✐♥❣❧②✱ t❤❡ ❡st✐♠❛t❡❞ ❧♦ss❡s ❛r❡ ❤✐❣❤❡r t❤❛♥ t❤❡ ♣r❡✈✐♦✉s ✜❣✉r❡✳ ❆ ❧✐tt❧❡ ♦✈❡r ✹✵ tr❛♥s♣❧❛♥ts ❛r❡ ❧♦st ✐❢ t❤❡ ♠❛①✐♠✉♠ ❡❧❛st✐❝✐t② ✐s ✶✳ ❚❤✐s ♥✉♠❜❡r ✐s ❜❡t✇❡❡♥ ✾✺ ❛♥❞ ✶✷✵ ❢♦r ❛♥ ❡❧❛st✐❝✐t② ♦❢ ✻✳ ❚❤❡r❡❢♦r❡✱ s♦❝✐❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s ❤✐❣❤❡r t❤❛♥ ❤♦s♣✐t❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss✱ ❜✉t t❤❡ t✇♦ ❛r❡ q✉❛❧✐t❛t✐✈❡❧② s✐♠✐❧❛r✳ ❚❛❦❡♥ t♦❣❡t❤❡r✱ t❤❡s❡ r❡s✉❧ts ✐♠♣❧② t❤❛t ❛❞❞r❡ss✐♥❣ t❤❡ ✐♥❡✣❝✐❡♥t ♣❧❛t❢♦r♠ ✐♥❝❡♥t✐✈❡s ❤❛s ❛ ❧❛r❣❡ ♣♦s✐t✐✈❡ ✐♠♣❛❝t ✉♥❧❡ss t❤❡ ❡❧❛st✐❝✐t② ♦❢ s✉♣♣❧② ✐s ❡①tr❡♠❡❧② ❧♦✇✳ ❲❤✐❧❡ ✇❡ ❞♦ ♥♦t ❤❛✈❡ q✉❛s✐✲❡①♣❡r✐♠❡♥t❛❧ ❡✈✐❞❡♥❝❡ ♦♥ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❡❧❛st✐❝✐t✐❡s✱ t❤❡ ❡✈✐❞❡♥❝❡ ✐♥ ❙❡❝t✐♦♥ ✸ ✐s t②♣✐❝❛❧ ♦❢ ♠❛r❦❡ts ✇✐t❤ ❡❧❛st✐❝ s✉♣♣❧②✳ ▼♦st ❤♦s♣✐t❛❧s ♦♥❧② r❡❣✐st❡r ❛ s✉❜s❡t ♦❢ t❤❡✐r ♣❛t✐❡♥ts ✇✐t❤ t❤❡ ◆❑❘✱ ❛♥❞ ♠❛♥② ♦t❤❡r ❤♦s♣✐t❛❧s ❞♦ ♥♦t ♣❛rt✐❝✐♣❛t❡✳ ❚❤❡s❡ ♦❜s❡r✈❛t✐♦♥s ❛r❡ ❝♦♥s✐st❡♥t ✇✐t❤ ♠❛♥② ❤♦s♣✐t❛❧s ❜❡✐♥❣ ♦♥ t❤❡ ♠❛r❣✐♥✱ s✉❣❣❡st✐♥❣ t❤❛t ❤♦s♣✐t❛❧s r❡s♣♦♥❞ t♦ ✐♥❝❡♥t✐✈❡s ❛♥❞ t❤❛t s✉♣♣❧② ✐s ❛t ❧❡❛st ♠♦❞❡r❛t❡❧② ❡❧❛st✐❝✳ ❚❤❡r❡❢♦r❡✱ ♦♣t✐♠❛❧ ♣♦✐♥t ♠❡❝❤❛♥✐s♠s ❛r❡ ♥♦t ♦♥❧② ❧♦✇✲❞✐♠❡♥s✐♦♥❛❧ ❜✉t ❛❧s♦ ❧✐❦❡❧② t♦ ❤❛✈❡ ❛ s✉❜st❛♥t✐❛❧ ❡✛❡❝t ♦♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts✳

✻

❉✐s❝✉ss✐♦♥ ❛♥❞ ❘♦❜✉st♥❡ss

✻✳✶ ■♠♣♦rt❛♥❝❡ ♦❢ ❛❣❡♥❝② ♣r♦❜❧❡♠s ❛♥❞ ✐♥❡✣❝✐❡♥t ♣❧❛t❢♦r♠ ✐♥❝❡♥✲ t✐✈❡s ■♥ ♦r❞❡r t♦ ❜❡tt❡r ❣✉✐❞❡ ♣♦❧✐❝②✱ ✇❡ ♥♦✇ ❞✐s❝✉ss t❤❡ q✉❛♥t✐t❛t✐✈❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ t✇♦ ♠❛r❦❡t ❢❛✐❧✉r❡s ✐❞❡♥t✐✜❡❞ ❛❜♦✈❡✳ ❲❤✐❧❡ ✇❡ ❝❛♥♥♦t ❞❡❝♦♠♣♦s❡ t❤❡ ❡✛❡❝ts ♦❢ ❡❛❝❤ ♠❛r❦❡t ❢❛✐❧✉r❡✱ t❤❡ r❡s✉❧ts ❣✐✈❡ ✉s ✉s❡❢✉❧ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ✇❤❡t❤❡r t❤❡s❡ ♠❛r❦❡t ❢❛✐❧✉r❡s ❛r❡ ✐♠♣♦rt❛♥t✳ ❋✐rst✱ t❤❡ ♠✐s❛❧❧♦❝❛t✐♦♥ ❛♥❛❧②s✐s ②✐❡❧❞s ❛ ❝♦♥s❡r✈❛t✐✈❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦r t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ♦❢ ❛❜♦✉t ✷✵✵ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r✳ ❚❤❡ ❛❝t✉❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐s ♣♦t❡♥t✐❛❧❧② ♠✉❝❤ ❧❛r❣❡r ❛s ♠♦st s♣❡❝✐✜❝❛t✐♦♥s ②✐❡❧❞ ♥✉♠❜❡rs ✐s ❛❜♦✉t t✇✐❝❡ ❛s ❧❛r❣❡✳ ❚❤❡r❡❢♦r❡✱ ✐t ♠✉st ❜❡ t❤❡ ❝❛s❡ t❤❛t ❛t ❧❡❛st ♦♥❡ ♠❛r❦❡t ❢❛✐❧✉r❡ ✐s q✉❛♥t✐t❛t✐✈❡❧② ✐♠♣♦rt❛♥t✳ ❙❡❝♦♥❞✱ t❤❡ ❍❛r❜❡r❣❡r tr✐❛♥❣❧❡ ❛♥❛❧②s✐s s❤♦✇s t❤❛t ✐♥❡✣❝✐❡♥t ♣❧❛t❢♦r♠ ✐♥❝❡♥t✐✈❡s s✐❣♥✐✜❝❛♥t❧② r❡❞✉❝❡ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ✐❢ s✉♣♣❧② ✐s ♥♦t ✐♥❡❧❛st✐❝✳ ▼♦r❡♦✈❡r✱ ✐❢ t❤❡r❡ ❛r❡ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ t❤❡ ❣❛✐♥s ✐♥ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❢r♦♠ ❛♥ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ ✇✐❧❧ ❜❡ ❡✈❡♥ ❤✐❣❤❡r ❜❡❝❛✉s❡ ❤♦s♣✐t❛❧s ✉♥❞❡r✈❛❧✉❡ tr❛♥s♣❧❛♥ts✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ❞❡❞✉❝ts t❤❡ tr❛♥s♣❧❛♥t✲❞❡♥♦♠✐♥❛t❡❞ ♣r✐✈❛t❡ ❝♦st ✐♥❝✉rr❡❞ ✇❤❡♥ ❤♦s♣✐t❛❧s ♣r♦✈✐❞❡ ♠♦r❡ s✉❜♠✐ss✐♦♥s t♦ t❤❡ ♣❧❛t❢♦r♠✳ ❲❤❡♥ t❤❡r❡ ❛r❡ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ t❤❡s❡ ♣r✐✈❛t❡ ❝♦sts ❛r❡ s✐❣♥✐✜❝❛♥t❧② ✐♥✢❛t❡❞ r❡❧❛t✐✈❡ t♦ s♦❝✐❛❧ ❝♦sts✳ ❚❛❦❡♥ t♦❣❡t❤❡r✱ t❤❡s❡ r❡s✉❧ts ✐♠♣❧② t❤❛t ❛❣❡♥❝② ♣r♦❜❧❡♠s ❛r❡ ✐♠♣♦rt❛♥t ✉♥❧❡ss ❡❧❛st✐❝✐t✐❡s ❛r❡ ❡①tr❡♠❡❧② ❤✐❣❤✳ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s t❤❛t t❤❡r❡ ❛r❡ ♥♦ ❛❣❡♥❝② ♣r♦❜❧❡♠s✱ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ ❡q✉❛❧s t♦t❛❧ ✇❡❧❢❛r❡✱ ❛♥❞ t❤❡ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ r❡❛❝❤❡s ✜rst✲❜❡st ✇❡❧❢❛r❡ ✭❚❤❡♦r❡♠ ✶✮✳ ❚❤✉s✱ t❤❡ t♦t❛❧ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐♥ t❤❡ ♠✐s❛❧❧♦❝❛t✐♦♥ ❛♥❛❧②s✐s ♠✉st ❜❡ ❝♦♠♣❧❡t❡❧② ❛❝❝♦✉♥t❡❞ ❢♦r ❜② t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ✐♥ t❤❡ ❍❛r❜❡r❣❡r tr✐❛♥❣❧❡ ❛♥❛❧②s✐s✳ ❨❡t ❡✈❡♥ ❢♦r ❛ ❤✐❣❤ ❡❧❛st✐❝✐t② ♦❢ ✻✱ t❤❡ ❍❛r❜❡r❣❡r tr✐❛♥❣❧❡ ②✐❡❧❞s ❛ ❞❡❛❞✇❡✐❣❤t ❧♦ss ♦❢ ❛t ♠♦st ✶✷✵✱ st✐❧❧ ❜❡❧♦✇ ♦✉r ❧♦✇❡r ❜♦✉♥❞ r❡s✉❧t

✹✻ ♦❢ ✷✵✵ ❢r♦♠ t❤❡ ♠✐s❛❧❧♦❝❛t✐♦♥ ❛♥❛❧②s✐s✳ ❚❤❡ ♦♥❧② ✇❛② t❤❡s❡ ❡st✐♠❛t❡s ❝❛♥ ♦✈❡r❧❛♣ ✐s ✐❢ ✇❡ ❤❛✈❡ ❤✐❣❤ ❡❧❛st✐❝✐t✐❡s ❛♥❞ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ Pr♦♣♦s✐t✐♦♥ ✶ ✐s s✐❣♥✐✜❝❛♥t❧② ❞♦✇♥✇❛r❞ ❜✐❛s❡❞✳ ❚❤❡ ❜✐❛s ✐♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡ q✉❛❞r❛t✐❝ ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥✱ s♦ t❤❛t t❤❡ ❜✐❛s ✐s ❤✐❣❤ ✐❢ ∇f ✐s ❡①tr❡♠❡❧② ❝♦♥✈❡①✳ ❚❤✉s✱ ❛ttr✐❜✉t✐♥❣ ❛❧❧ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss t♦ ✐♥❡✣❝✐❡♥t ♣❧❛t❢♦r♠ ✐♥❝❡♥t✐✈❡s r❡q✉✐r❡s t❤❛t ❡❧❛st✐❝✐t✐❡s ❛r❡ ❤✐❣❤✱ ∇f ✐s s✉✣❝✐❡♥t❧② ❝♦♥✈❡①✱ ❛♥❞ t❤❡ ❞♦✇♥✇❛r❞ ❜✐❛s ✐♥ t❤❡ ❡st✐♠❛t❡❞ ❧♦✇❡r ❜♦✉♥❞ ♦♥ ✐♥❡✣❝✐❡♥❝② ✐s s♠❛❧❧✳ ❚❤❡ ✉♣s❤♦t ✐s t❤❛t ❛ ♣♦❧✐❝② t❤❛t ❛❞❞r❡ss❡s ❡✐t❤❡r ♠❛r❦❡t ❢❛✐❧✉r❡ ✐s ❧✐❦❡❧② t♦ ❜❡ ✈❛❧✉❛❜❧❡ ❛♥❞ ❣❡♥❡r❛t❡ ❣❛✐♥s ✐♥ t❤❡ ♦r❞❡r ♦❢ ❤✉♥❞r❡❞s ♦❢ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r✳ ❊①❝❡♣t ✉♥❞❡r ❡①tr❡♠❡ ❛ss✉♠♣t✐♦♥s ❛❜♦✉t t❤❡ s✉♣♣❧② ❢✉♥❝t✐♦♥✱ t❤❡r❡ ❛r❡ s✐❣♥✐✜❝❛♥t ❣❛✐♥s ❜♦t❤ ❢r♦♠ ✐♠♣❧❡♠❡♥t✐♥❣ ♠♦r❡ ❡✣❝✐❡♥t ♠❡❝❤❛♥✐s♠s ❛♥❞ ❢r♦♠ s♦❧✈✐♥❣ ❛❣❡♥❝② ♣r♦❜❧❡♠s✳ ✻✳✷

■♠♣❧❡♠❡♥t✐♥❣ ❛ ♣♦✐♥t ♠❡❝❤❛♥✐s♠

❖✉r st❡❛❞②✲st❛t❡ ♠♦❞❡❧ s❤♦✇s t❤❛t ❛ ♠❡❝❤❛♥✐s♠ t❤❛t r❡✇❛r❞s ❤♦s♣✐t❛❧s ✇✐t❤ ♠❛r❣✐♥❛❧ ♣r♦❞✲ ✉❝ts ✐s ❡✣❝✐❡♥t✳ ▼♦r❡♦✈❡r✱ ♦✉r ❡♠♣✐r✐❝❛❧ r❡s✉❧ts s✉❣❣❡st t❤❛t ❛ ❧♦✇✲❞✐♠❡♥s✐♦♥❛❧ ♣♦✐♥t ♠❡❝❤✲ ❛♥✐s♠ ✇♦✉❧❞ ❧✐❦❡❧② ❛❝❤✐❡✈❡ s✐③❛❜❧❡ ❡✣❝✐❡♥❝② ❣❛✐♥s✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤✐s ♠♦❞❡❧ ❞♦❡s ♥♦t s♣❡❝✲ ✐❢② ❛♥ ❡①t❡♥s✐✈❡ ❢♦r♠ ❣❛♠❡ ❛♥❞ ❛❜str❛❝ts ❛✇❛② ❢r♦♠ ❞②♥❛♠✐❝ ❝♦♥s✐❞❡r❛t✐♦♥s✳ ❚❤✐s r❛✐s❡s ♣r❛❝t✐❝❛❧ ❛♥❞ t❤❡♦r❡t✐❝❛❧ q✉❡st✐♦♥s ❛❜♦✉t ❤♦✇ t♦ ❞❡s✐❣♥ ❛♥❞ ✐♠♣❧❡♠❡♥t ❛ ❞②♥❛♠✐❝ ♣♦✐♥ts ♠❡❝❤❛♥✐s♠✱ ❛ t❛s❦ t❤❛t r❡q✉✐r❡s ❞❡t❛✐❧❡❞ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ r✉❧❡s ❛♥❞ ❛♥ ❛♥❛❧②s✐s ♦❢ r❡s✉❧t✐♥❣ ✐♥❝❡♥t✐✈❡s✳ ❲❤✐❧❡ r❡s♦❧✈✐♥❣ ❛❧❧ t❤❡s❡ ❞❡t❛✐❧s ✐s ❜❡②♦♥❞ t❤❡ s❝♦♣❡ ♦❢ ♦✉r ♣❛♣❡r✱ ✇❡ ❞✐s❝✉ss s♦♠❡ ❦❡② t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♣r❛❝t✐❝❛❧ ✐ss✉❡s✳ ■♥ ❜♦t❤ t❤❡♦r② ❛♥❞ ♣r❛❝t✐❝❡✱ ❛ ♥❛t✉r❛❧ ♠❡❝❤❛♥✐s♠ ❢♦r s♦❧✈✐♥❣ t❤✐s ♣r♦❜❧❡♠ ✐s t❤❡ ♣♦✐♥t s②st❡♠ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✷✳✹✳ ❆ ♠♦t✐✈❛t✐♦♥ ❢♦r t❤✐s ❦✐♥❞ ♦❢ ♠❡❝❤❛♥✐s♠ ❝♦♠❡s ❢r♦♠ t❤❡ ❞②♥❛♠✐❝ ♠❡❝❤❛♥✐s♠ ❞❡s✐❣♥ ❧✐t❡r❛t✉r❡✳ ▼ö❜✐✉s ✭✷✵✵✶✮✱ ❍❛✉s❡r ❛♥❞ ❍♦♣❡♥❤❛②♥ ✭✷✵✵✽✮✱ ❋r✐❡❞♠❛♥ ❡t ❛❧✳ ✭✷✵✵✻✮✱ ❛♥❞ ●✉♦ ❛♥❞ ❍ör♥❡r ✭✷✵✶✺✮ ❝❛❧❧ t❤✐s ❦✐♥❞ ♦❢ ♠❡❝❤❛♥✐s♠ ❛ ❝❤✐♣s✱ s❝r✐♣s✱ ♦r t♦❦❡♥ ♠❡❝❤❛♥✐s♠✳ ▼ö❜✐✉s ✭✷✵✵✶✮✱ ❍❛✉s❡r ❛♥❞ ❍♦♣❡♥❤❛②♥ ✭✷✵✵✽✮✱ ❛♥❞ ❆❜❞✉❧❦❛❞✐r♦➜❧✉ ❛♥❞ ❇❛❣✇❡❧❧ ✭✷✵✶✸✮ ❝♦♥s✐❞❡r ❞②♥❛♠✐❝ ❢❛✈♦r ❡①❝❤❛♥❣❡✱ ❛♥❞ ●✉♦ ❛♥❞ ❍ör♥❡r ✭✷✵✶✺✮ ♣r❡s❡♥ts ♣r♦✈✐s✐♦♥ ♦❢ ❣♦♦❞s t♦ ❛ ❝♦♥s✉♠❡r ✇✐t❤ st♦❝❤❛st✐❝ ✈❛❧✉❛t✐♦♥s✳ ❚❤❡ ❣❡♥❡r❛❧ ✜♥❞✐♥❣ ♦❢ t❤✐s ❧✐t❡r❛t✉r❡ ✐s t❤❛t t♦❦❡♥ ♠❡❝❤❛♥✐s♠s✱ ❛s ♣r♦♣♦s❡❞ ❜② ▼ö❜✐✉s✱ ❞♦ ❜❡tt❡r t❤❛♥ ❛✉t❛r❦② ❜✉t ♥♦t ❛s ✇❡❧❧ ❛s ❛♥ ♦♣t✐♠❛❧ ❞②♥❛♠✐❝ ♠❡❝❤❛♥✐s♠✳ ■♥ ❢❛❝t✱ t♦❦❡♥ ♠❡❝❤❛♥✐s♠s ❛r❡ ❝❧♦s❡ t♦ ✜rst✲❜❡st ✐❢ ♣❧❛②❡rs ❛r❡ ♣❛t✐❡♥t ❛♥❞ t❤❡r❡ ❛r❡ ♠❛♥② t✐♠❡ ♣❡r✐♦❞s✳ ❘❡s✉❧ts ✐♥ ❏❛❝❦s♦♥ ❛♥❞ ❙♦♥♥❡♥s❝❤❡✐♥ ✭✷✵✵✼✮ ✐♠♣❧② t❤❛t t♦❦❡♥ ♠❡❝❤❛♥✐s♠s✬ ✐♥❡✣❝✐❡♥❝② ❞❡❝❧✐♥❡s ❛s sq✉❛r❡ r♦♦t ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♣❡r✐♦❞s ✭s❡❡ ●✉♦ ❛♥❞ ❍ör♥❡r✱ ✷✵✶✺✮✳ ❚❤✉s✱ t❤❡ t❤❡♦r❡t✐❝❛❧ ❧✐t❡r❛t✉r❡ s✉❣❣❡sts t❤❛t ♣♦✐♥t s②st❡♠s✱ ✇❤✐❧❡ ♥♦t ❡①❛❝t❧② ♦♣t✐♠❛❧✱ ❛r❡ s✐♠♣❧❡ ❛♥❞ ❛❝❤✐❡✈❡ ❛ ❤✐❣❤ ❧❡✈❡❧ ♦❢ ❡✣❝✐❡♥❝②✳✸✾ ❆♥♦t❤❡r ♠♦t✐✈❛t✐♦♥ ❢♦r ✉s✐♥❣ ❛ ♣♦✐♥t ♠❡❝❤❛♥✐s♠ ✐s ♣r❛❝t✐❝❛❧✐t②✳ ❚❤❡ s✐♠♣❧✐❝✐t② ♦❢ t❤❡ ♠❡❝❤✲ ❛♥✐s♠ ❛♥❞ s✐♠✐❧❛r✐t② t♦ ✜❛t ♠♦♥❡② ♠❛❦❡s ✐t ♣r♦♠✐s✐♥❣✳ ❙✐♠✐❧❛r ♠❡❝❤❛♥✐s♠s ❤❛✈❡ ❜❡❡♥ ♣r❡✈✐✲ ♦✉s❧② ✉s❡❞ ✐♥ ♠❛r❦❡t ❞❡s✐❣♥ ❛♣♣❧✐❝❛t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ Pr❡♥❞❡r❣❛st ✭✷✵✶✼✮ ❞❡s❝r✐❜❡s ❤♦✇ ❛ s✐♠✐❧❛r ♠❡❝❤❛♥✐s♠ ✇❛s ✉s❡❞ t♦ ✐♥❝r❡❛s❡ t❤❡ ❡✣❝✐❡♥❝② ♦❢ ❢♦♦❞ ❞✐str✐❜✉t✐♦♥ ❛❝r♦ss ❢♦♦❞ ❜❛♥❦s✳

❚❤✐s ♠❡ss❛❣❡ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ❧✐t❡r❛t✉r❡ ♦♥ ♠♦♥❡t❛r② ❡❝♦♥♦♠✐❝s✳ ❆❧t❤♦✉❣❤ ♦♣t✐♠❛❧ ❞②♥❛♠✐❝ ♠❡❝❤✲ ❛♥✐s♠s ❝❛♥ ♦❢t❡♥ ✐♠♣r♦✈❡ ♦♥ ♠♦♥❡② ✭❑♦❝❤❡r❧❛❦♦t❛✱ ✶✾✾✽✮✱ ♠♦❞❡❧s ✐♥ t❤❡ tr❛❞✐t✐♦♥ ♦❢ ❑✐②♦t❛❦✐ ❛♥❞ ❲r✐❣❤t ✭✶✾✽✾✮ s❤♦✇ t❤❛t ♠♦♥❡② ❝❛♥ ❛❝❤✐❡✈❡ ❤✐❣❤ ❧❡✈❡❧s ♦❢ ❡✣❝✐❡♥❝② ❡✈❡♥ ✇✐t❤ s✐♠♣❧❡ ✐♥st✐t✉t✐♦♥s✳ ✸✾

✹✼

❆♥ ✐♠♣♦rt❛♥t ✐ss✉❡ ✇✐t❤ ❛♣♣❧②✐♥❣ ♣♦✐♥t s②st❡♠s ✐s t❤❛t t❤❡② r❡q✉✐r❡ s❡✈❡r❛❧ ✏♣❧✉♠❜✐♥❣✑ ❞❡❝✐s✐♦♥s✳ ❙❤♦✉❧❞ t❤❡ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠ ✐♠♣♦s❡ ❛ str✐❝t ❜♦✉♥❞ ♦♥ ♥❡❣❛t✐✈❡ ❜❛❧❛♥❝❡s❄ ■❢ s♦✱ ✇❤❛t ✐s t❤❡ ♦♣t✐♠❛❧ ♠✐♥✐♠✉♠ ❜❛❧❛♥❝❡ ❝♦♥str❛✐♥t❄ ❆ t✐❣❤t ❝♦♥str❛✐♥t ♣r♦✈✐❞❡s str♦♥❣❡r ✐♥❝❡♥t✐✈❡s t♦ ❤♦s♣✐t❛❧s ❜✉t ♠❛② r❡❞✉❝❡ ❡✣❝✐❡♥❝②✳ ❙❤♦✉❧❞ ♣♦✐♥ts ❜❡ ❝r❡❞✐t❡❞ ✇❤❡♥ ♣❛✐rs ❛r❡ s✉❜♠✐tt❡❞✱ ♦r s❤♦✉❧❞ ♣♦✐♥ts ❜❡ ❝r❡❞✐t❡❞ ✇❤❡♥ ♣❛✐rs ❛r❡ tr❛♥s♣❧❛♥t❡❞❄

❍♦✇ ♦❢t❡♥ s❤♦✉❧❞

♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❜❡ r❡❝❛❧❝✉❧❛t❡❞ ❛s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✐♥ t❤❡ ♣❧❛t❢♦r♠ ❝❤❛♥❣❡s❄ ❘❡❝❛❧❝✉❧❛t✐♥❣ t❤❡♠ ♦❢t❡♥ ✐s ❝♦♠♣❧❡① ❛♥❞ r❡❞✉❝❡s tr❛♥s♣❛r❡♥❝②✱ ❜✉t r❡❝❛❧❝✉❧❛t✐♥❣ ✐♥❢r❡q✉❡♥t❧② ❝❛♥ r❡❞✉❝❡ ❡✣❝✐❡♥❝②✳

✻✳✸

▼❛①✐♠✐③✐♥❣ s♦❝✐❛❧ ✇❡❧❢❛r❡

❚❤❡♦r❡♠ ✶ ❞❡s❝r✐❜❡s ♠❡❝❤❛♥✐s♠s t❤❛t ♠❛①✐♠✐③❡ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡✳

❆ ♥❛t✉r❛❧ ❛❧t❡r♥❛t✐✈❡

✇♦✉❧❞ ❜❡ t♦ ✉s❡ ♠❡❝❤❛♥✐s♠s t❤❛t ♠❛①✐♠✐③❡ s♦❝✐❛❧ ✇❡❧❢❛r❡✳ ❘❡✇❛r❞s ✐♥ t❤❡s❡ ♠❡❝❤❛♥✐s♠s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳

❈♦♥s✐❞❡r ❛ ✈❡❝t♦r ♦❢ r❡✇❛r❞s p ❛♥❞ str✐❝t❧② ♣♦s✐t✐✈❡ ❛❣❣r❡❣❛t❡ q✉❛♥t✐t② ✈❡❝t♦r q t❤❛t ♠❛①✐♠✐③❡ s♦❝✐❛❧ ✇❡❧❢❛r❡ s✉❜❥❡❝t t♦ ❛❧❧ ❤♦s♣✐t❛❧s ❝❤♦♦s✐♥❣ s✉♣♣❧② ♦♣t✐♠❛❧❧② ❣✐✈❡♥ p ❛♥❞ s✉❜❥❡❝t t♦ ❛❧❧♦❝❛t✐♥❣ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts t❤❛t ❛r❡ ♣r♦❞✉❝❡❞✳ ❆ss✉♠❡ t❤❛t t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❤❛s ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡ ❛♥❞ t❤❛t ♣r✐✈❛t❡ ❝♦sts ❛r❡ str✐❝t❧② ❝♦♥✈❡①✳ ❉❡✜♥❡ t❤❡ ❛❣❣r❡❣❛t❡ ❡①t❡r♥❛❧✐t② ❛s ❛ ❢✉♥❝t✐♦♥ ˜ ❛s ✐♥ ❡q✉❛t✐♦♥ ✭❆✶✷✮✱ ❛♥❞ ❛ss✉♠❡ t❤❛t ✐t ✐s s♠♦♦t❤ ❛t t❤❡ ❛❣❣r❡❣❛t❡ ♦❢ ❛❣❣r❡❣❛t❡ q✉❛♥t✐t② E q✉❛♥t✐t② q ✳ ❚❤❡♥✿ Pr♦♣♦s✐t✐♦♥ ✷ ✭❖♣t✐♠❛❧ ❘❡✇❛r❞s ❢♦r ▼❛①✐♠✐③✐♥❣ ❚♦t❛❧ ❲❡❧❢❛r❡✮✳

✶✳ ❚❤❡ ♣❧❛t❢♦r♠ r❡✇❛r❞s ❡❛❝❤ t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥ ✇✐t❤ ✐ts ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ♣❧✉s ❛♥ ❛❞❥✉st✲ ♠❡♥t t❡r♠✳ p = ∇f (q) + ASW ✱ ✇❤❡r❡ ASW =

λSW 1 ˜ ∇ E(q) − q ′ D 2 C(q)✳ 1 + λSW 1 + λSW

❛♥❞ λSW =

˜ ∇E(q) ·q ✳ ′ 2 q D C(q)q

✷✳ ❚❤❡ ❛❞❥✉st♠❡♥t t❡r♠ ASW ❝❛♥ ❜❡ ♥♦♥✲③❡r♦ ❡✈❡♥ ✇✐t❤ ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✳ ✸✳ ❚❤❡ ♦♣t✐♠❛❧ r❡✇❛r❞s ❛tt❛✐♥ ✜rst✲❜❡st s♦❝✐❛❧ ✇❡❧❢❛r❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❛✈❡r❛❣❡ ❡①t❡r♥❛❧✐t② ˜ ❛t t❤❡ ♦♣t✐♠✉♠✱ ∇E(q) · q ✱ ✐s ③❡r♦✳ P❛rt ✶ s❤♦✇s t❤❡ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ r❡✇❛r❞s s✉❜♠✐ss✐♦♥s ❜② t❤❡✐r ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ♣❧✉s ❛♥ ❛❞❥✉st♠❡♥t✳ ❚❤❡ ❛❞❥✉st♠❡♥t ❡q✉❛❧s ❛♥ ❡①t❡r♥❛❧✐t② t❡r♠✱ ✇❤✐❝❤ ✐s ❣r❡❛t❡r ❢♦r s✉❜♠✐ss✐♦♥s t❤❛t ❣❡♥❡r❛t❡ ♠♦r❡ ❡①t❡r♥❛❧✐t✐❡s✱ ♠✐♥✉s ❛ s❤❛❞✐♥❣ t❡r♠ t❤❛t ❞❡♣❡♥❞s ♦♥ ❡❧❛st✐❝✐t✐❡s✳ ■♥ t❤❡ ✜rst✲❜❡st✱ ❤♦s♣✐t❛❧s ❛r❡ r❡✇❛r❞❡❞ ❢♦r t❤❡✐r ♠❛r❣✐♥❛❧ ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ ♣❧❛t❢♦r♠ ❛s ✇❡❧❧ ❛s ❛♥② ❡①t❡r♥❛❧✐t✐❡s✳ ❍♦✇❡✈❡r✱ ✐❢ t❤❡r❡ ❛r❡ ♥♦t ❡♥♦✉❣❤ tr❛♥s♣❧❛♥ts t♦ ♣❛② ❢♦r t❤❡ ❡①t❡r♥❛❧✐t✐❡s✱

✹✽ t❤❡ ♣❧❛♥♥❡r ❤❛s t♦ s❤❛❞❡ r❡✇❛r❞s✳ ❆s ✐♥ ♦♣t✐♠❛❧ ❧✐♥❡❛r ❝♦♠♠♦❞✐t② t❛①❛t✐♦♥✱ ✐t ✐s ❜❡tt❡r t♦ s❤❛❞❡ r❡✇❛r❞s ❢♦r s✉❜♠✐ss✐♦♥s ✇✐t❤ ♠♦r❡ ✐♥❡❧❛st✐❝ s✉♣♣❧②✳ P❛rt ✷ s❤♦✇s t❤❛t t❤❡ ❦❡② ❞✐✛❡r❡♥❝❡ ✐♥ t❤✐s ❝❛s❡✱ r❡❧❛t✐✈❡ t♦ ❚❤❡♦r❡♠ ✶✱ ✐s t❤❛t t❤❡ ❛❞❥✉st♠❡♥t t❡r♠ ✐s ♥♦t ③❡r♦✱ ❡✈❡♥ ❢♦r ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♦♣t✐♠❛❧ r❡✇❛r❞s ❞❡♣❡♥❞ ♦♥ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥✳ ❚♦ s❡t ♦♣t✐♠❛❧ r❡✇❛r❞s✱ ♦♥❡ ♠✉st ❦♥♦✇ t❤❡ ❡①t❡r♥❛❧✐t✐❡s ❣❡♥❡r❛t❡❞ ❜② ❡❛❝❤ t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥✳ ❙✉❝❤ ❦♥♦✇❧❡❞❣❡ r❡q✉✐r❡s ✐❞❡♥t✐❢②✐♥❣ t❤❡ s✉❜♠✐ss✐♦♥ t②♣❡s ❢♦r ✇❤✐❝❤ ❤♦s♣✐t❛❧ ♦❜❥❡❝t✐✈❡s ❞❡✈✐❛t❡ ♠♦st ❢r♦♠ s♦❝✐❛❧ ♦❜❥❡❝t✐✈❡s✳ ▼♦r❡♦✈❡r✱ ♦♥❡ ♥❡❡❞s t♦ ❦♥♦✇ t❤❡ ❡❧❛st✐❝✐t② ♠❛tr✐① ✐♥ ♦r❞❡r t♦ ♠❡❛s✉r❡ ❤♦✇ ♠✉❝❤ s❤❛❞✐♥❣ ♠✉st ❜❡ ❞♦♥❡ ❢♦r ❡❛❝❤ s✉❜♠✐ss✐♦♥ t②♣❡✳ ❊❧❛st✐❝✐t✐❡s ♠❛tt❡r s♦ ❧♦♥❣ ❛s t❤❡ ❛✈❡r❛❣❡ ❡①t❡r♥❛❧✐t② ✐s ♥♦♥✲③❡r♦ ❜❡❝❛✉s❡ ✐t r❡s✉❧ts ✐♥ t❤❡ ♠✉❧t✐♣❧✐❡r λSW ❜❡✐♥❣ ♥♦♥✲③❡r♦ ❛♥❞ ❛♥ ❛❞❥✉st♠❡♥t t❡r♠ t❤❛t ❞❡♣❡♥❞s ♦♥ ❡❧❛st✐❝✐t✐❡s✳ ❋✐♥❛❧❧②✱ ♣❛rt ✸ s❤♦✇s t❤❛t t❤❡ ♦♣t✐♠❛❧ r❡✇❛r❞ ✈❡❝t♦r ❞♦❡s ♥♦t ❛tt❛✐♥ ✜rst✲❜❡st s♦❝✐❛❧ ✇❡❧❢❛r❡✳ ❚❤❡r❡❢♦r❡✱ ❛❧❧♦❝❛t✐♦♥s t❤❛t ❛❝❤✐❡✈❡ ✜rst✲❜❡st s♦❝✐❛❧ ✇❡❧❢❛r❡ r❡q✉✐r❡ ♥♦♥✲❧✐♥❡❛r ❛♥❞ ❝♦♠♣❧❡① ✐♥❝❡♥t✐✈❡s ❢♦r ❤♦s♣✐t❛❧s✳ ❚❛❦❡♥ t♦❣❡t❤❡r✱ ✉s✐♥❣ ♦♥❧② t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❡❝❤❛♥✐s♠ t♦ ♠❛①✐♠✐③❡ s♦❝✐❛❧ ✇❡❧❢❛r❡✱ ❛s ♦♣✲ ♣♦s❡❞ t♦ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡✱ r✉♥s ✐♥t♦ ✐♠♣♦rt❛♥t ❝❤❛❧❧❡♥❣❡s✳ ❖♣t✐♠❛❧ r❡✇❛r❞s ❛r❡ ♠♦r❡ ❝♦♠♣❧❡①✱ ❞❡♣❡♥❞ ♦♥ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥✱ ❛♥❞ ❛r❡ s❡♥s✐t✐✈❡ t♦ ❝❤❛♥❣❡s ✐♥ t❤❡ ✐♥❝❡♥t✐✈❡s ❢❛❝✐♥❣ ❤♦s♣✐t❛❧s t❤❛t ❝❛♥ ❛✛❡❝t ♦✈❡r❛❧❧ ❡①t❡r♥❛❧✐t✐❡s✳ ❚❤❡s❡ r❡s✉❧ts s✉❣❣❡st t❤❛t s♦❧✈✐♥❣ ❛❣❡♥❝② ♣r♦❜❧❡♠s ✐s ❛♥ ✐♠♣♦rt❛♥t ❝♦♠♣❧❡♠❡♥t t♦ ✐♠♣r♦✈✐♥❣ t❤❡ ❞❡s✐❣♥ ♦❢ t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❡❝❤❛♥✐s♠✳ ✻✳✹

❈♦♠♣❡t✐♥❣ ♣❧❛t❢♦r♠s

❚✇♦ ♥❛t✉r❛❧ ♣♦❧✐❝② r❡s♣♦♥s❡s t♦ t❤❡ ❢r❛❣♠❡♥t❛t✐♦♥ ❛♥❞ ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s t♦ s❝❛❧❡ ❛r❡ t♦ ♠❛♥❞❛t❡ ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ ❛ s✐♥❣❧❡ ♣❧❛t❢♦r♠ ♦r t♦ ♠❡r❣❡ ♣❧❛t❢♦r♠s✳ ❚❤❡s❡ r❡❝♦♠♠❡♥❞❛t✐♦♥s r❛✐s❡ q✉❡st✐♦♥s ❛❜♦✉t t❤❡ ♦♣t✐♠❛❧ str❛t❡❣② ❢♦r ❝♦♠♣❡t✐♥❣ ♣❧❛t❢♦r♠s ❛♥❞ t❤❡ ❡✣❝✐❡♥❝② ❝♦sts ♦❢ ✐♠♣❡r❢❡❝t ❝♦♠♣❡t✐t✐♦♥✳ ❚♦ ❛❞❞r❡ss t❤❡s❡ ✐ss✉❡s✱ ❝♦♥s✐❞❡r ❛ ♣❧❛t❢♦r♠ t❤❛t ❢❛❝❡s ❛♥ ✐♥✈❡rs❡ s✉♣♣❧② ♦❢ s✉❜♠✐ss✐♦♥s PS (q)✳ ❆ss✉♠❡ t❤❛t t❤❡ ♣❧❛t❢♦r♠ ♠❛①✐♠✐③❡s t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts f (q) t❤❛t ✐t ❢❛❝✐❧✐t❛t❡s✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ❞❡s❝r✐❜❡s t❤❡ ♦♣t✐♠❛❧ r❡✇❛r❞s✳ ✭❖❧✐❣♦♣♦❧✐st✐❝ P❧❛t❢♦r♠s✮✳ ❈♦♥s✐❞❡r ❛ ♣❧❛t❢♦r♠ ❢❛❝✐♥❣ ❛ s♠♦♦t❤ ✐♥✈❡rs❡ s✉♣♣❧② ❝✉r✈❡ ♦❢ s✉❜♠✐ss✐♦♥s PS (·)✳ ❈♦♥s✐❞❡r ❛ ✈❡❝t♦r ♦❢ r❡✇❛r❞s p ❛♥❞ str✐❝t❧② ♣♦s✐t✐✈❡ ❛❣❣r❡❣❛t❡ q✉❛♥t✐t② q t❤❛t ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ✐♥ t❤❡ ♣❧❛t❢♦r♠ s✉❜❥❡❝t t♦ ❛❧❧♦❝❛t✐♥❣ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts t❤❛t ❛r❡ ♣r♦❞✉❝❡❞✳ ❆ss✉♠❡ t❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❤❛s ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✳ ❚❤❡♥✿ Pr♦♣♦s✐t✐♦♥ ✸

✶✳ ❚❤❡ ♣❧❛t❢♦r♠ r❡✇❛r❞s ❡❛❝❤ t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥ ✇✐t❤ ✐ts ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✱ ♣❧✉s ❛♥ ❛❞✲ ❥✉st♠❡♥t t❡r♠✱ p = ∇f (q) + AC ✱ ✇❤❡r❡ q ′ DPS (q)q ∇f (q) − q ′ DPS (q)✳ AC = f (q)

✷✳ ❚❤❡ ❛❞❥✉st♠❡♥t t❡r♠ AC ❝❛♥ ❜❡ ♥♦♥✲③❡r♦ ❡✈❡♥ ✇✐t❤ ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✳ ■♥ ♣❛r✲ t✐❝✉❧❛r✱ r❡✇❛r❞s ❛r❡ ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ r❡✇❛r❞s ✐♥ ❛♥ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ ❢r♦♠ ❚❤❡♦r❡♠ ✶✳

✹✾ ✸✳ ■❢ s✉♣♣❧② ✐s ♣❡r❢❡❝t❧② ❡❧❛st✐❝✱ s♦ t❤❛t t❤❡ ♠❛tr✐①

DPS

✐s ③❡r♦✱ t❤❡♥ r❡✇❛r❞s ❡q✉❛❧ ♠❛r❣✐♥❛❧

♣r♦❞✉❝ts✱ ❛s ✐♥ t❤❡ ♦♣t✐♠❛❧ r❡✇❛r❞s ✐♥ ❚❤❡♦r❡♠ ✶✳

❚❤❡ ♣r♦♣♦s✐t✐♦♥ s❤♦✇s t❤❛t ❛ ♣❧❛t❢♦r♠ t❤❛t ♠❛①✐♠✐③❡s t❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝✐❧✐t❛t❡❞ tr❛♥s♣❧❛♥ts ❞♦❡s ♥♦t s❡t s♦❝✐❛❧❧② ❡✣❝✐❡♥t r❡✇❛r❞s✳ ■♥st❡❛❞ ♦❢ s❡tt✐♥❣ r❡✇❛r❞s ❡q✉❛❧ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✱ t❤❡ ♣❧❛t❢♦r♠ s✉❜s✐❞✐③❡s s✉❜♠✐ss✐♦♥s t❤❛t ❛r❡ ✈❡r② ♣r♦❞✉❝t✐✈❡✳ ❚♦ ✉♥❞❡rst❛♥❞ t❤❡ ✐♥t✉✐t✐♦♥ ❢♦r t❤✐s✱ ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡r❡ s✉♣♣❧② ❤❛s ③❡r♦ ❝r♦ss ❡❧❛st✐❝✐t✐❡s ❛♥❞ ♦✇♥✲❡❧❛st✐❝✐t✐❡s ❞❡♥♦t❡❞ ǫi ✳ ❚❤❡♥ t❤❡ ♦♣t✐♠❛❧ r❡✇❛r❞s ❢♦r♠✉❧❛ s✐♠♣❧✐✜❡s t♦ ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ▲❡r♥❡r ✐♥❞❡① ❢♦r♠✉❧❛✿ ∂ i f − pi 1 = pi ǫi



f − ∂ i f · qi f



✳

❚❤✐s ❢♦r♠✉❧❛ ❞❡s❝r✐❜❡s t❤❡ ♦♣t✐♠❛❧ ♠❛r❦✲❞♦✇♥ ✐♥ r❡✇❛r❞s r❡❧❛t✐✈❡ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts✳ ■❢ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ t②♣❡ ♦❢ s✉❜♠✐ss✐♦♥✱ t❤❡♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ✐s tr✐✈✐❛❧❧② ❡q✉❛❧ t♦ ③❡r♦ ❜❡❝❛✉s❡ f ❡①❤✐❜✐ts ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ♦♣t✐♠❛❧ r❡✇❛r❞s ❡q✉❛❧ t❤❡ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✳ ❲❤❡♥ t❤❡r❡ ❛r❡ ♠✉❧t✐♣❧❡ s✉❜♠✐ss✐♦♥ t②♣❡s✱ t❤❡♥ t❤❡ q✉❛♥t✐t② ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ ∂i f − pi ❛❝r♦ss s✉❜♠✐ss✐♦♥s ♠✉st ❜❡ ③❡r♦ ❜❡❝❛✉s❡ t❤❡ ♣❧❛t❢♦r♠ ❝❛♥♥♦t ♣r♦♠✐s❡ r❡✇❛r❞s t❤❛t ❡①❝❡❡❞ ✐ts ♣r♦❞✉❝t✳ ❚❤❡ ❡①♣r❡ss✐♦♥ s❤♦✇s t❤❡ ♣❧❛t❢♦r♠ ❤❛s ✐♥❝❡♥t✐✈❡s t♦ s❦❡✇ t❤❡ r❡✇❛r❞s✿ ♦♣t✐♠❛❧ ♠❛r❦❞♦✇♥s ❛r❡ ❧❛r❣❡r ❢♦r s✉❜♠✐ss✐♦♥s ✇✐t❤ ❧♦✇ ❡❧❛st✐❝✐t✐❡s ❛♥❞ s✉❜♠✐ss✐♦♥ ❝❛t❡❣♦r✐❡s t❤❛t ❛r❡ ❧❡ss ♣r♦❞✉❝t✐✈❡ ♦♥ t❤❡ ♠❛r❣✐♥✳ ❚❤❡ ♣r♦♣♦s✐t✐♦♥ ✐♠♣❧✐❡s t❤❛t ❝♦♠♣❡t✐♥❣✱ ❡♠♣✐r❡✲❜✉✐❧❞✐♥❣ ♣❧❛t❢♦r♠s ❡①♣❧♦✐t t❤❡✐r ♠❛r❦❡t ♣♦✇❡r ❛♥❞ s❡t r❡✇❛r❞s ✐♥❡✣❝✐❡♥t❧②✳ ❆❞❞✐t✐♦♥❛❧❧②✱ t❤❡ ♣r♦♣♦s✐t✐♦♥ ✐♠♣❧✐❡s t❤❛t ♣❧❛t❢♦r♠s s❡t ❡✣❝✐❡♥t r❡✇❛r❞s ✐❢ t❤❡ ♠❛r❦❡t ✐s ✈❡r② ❝♦♠♣❡t✐t✐✈❡✳ ❖♣t✐♠❛❧ r❡✇❛r❞s ❛r❡ ❝❧♦s❡ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ✐❢ s✉♣♣❧② ✐s ✈❡r② ❡❧❛st✐❝✱ ✐✳❡✳ ✐❢ ǫi ✐s ❝❧♦s❡ t♦ ✐♥✜♥✐t② ♦r✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ DPS ✐s ❝❧♦s❡ t♦ ③❡r♦✳

✼

❈♦♥❝❧✉s✐♦♥

❑✐❞♥❡② ❡①❝❤❛♥❣❡ ✐♠♣r♦✈❡s ❛ ♣❛t✐❡♥t✬s q✉❛❧✐t② ♦❢ ❧✐❢❡ ❛♥❞ ❡①t❡♥❞s ❧✐❢❡ ❡①♣❡❝t❛♥❝② ✇❤✐❧❡ r❡❞✉❝✲ ✐♥❣ ❝♦sts✳ ❲❡ ❞❡♠♦♥str❛t❡ t❤❛t ❢r❛❣♠❡♥t❛t✐♦♥ ✐♥ t❤❡ ❯❙ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡t r❡s✉❧ts ✐♥ ❛♥ ❡✣❝✐❡♥❝② ❧♦ss ♦❢ ❜❡t✇❡❡♥ ✷✺ t♦ ✺✺ ♣❡r❝❡♥t ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ♣❡r ②❡❛r✳ ❚❤❡ ✐♥❡✣❝✐❡♥❝② ❛r✐s❡s ❞✉❡ t♦ t✇♦ st❛♥❞❛r❞ ♠❛r❦❡t ❢❛✐❧✉r❡s✳ ❋✐rst✱ ♣❧❛t❢♦r♠s ✉s❡ ✐♥❡✣❝✐❡♥t ♠❡❝❤❛♥✐s♠s t❤❛t ❞♦ ♥♦t r❡✇❛r❞ ❤♦s♣✐t❛❧s ❛❝❝♦r❞✐♥❣ t♦ ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ♦❢ t❤❡✐r ❝♦♥tr✐❜✉✲ t✐♦♥s✳ ❚❤✐s ♣r♦❜❧❡♠ ✐♥❞✉❝❡s ❤♦s♣✐t❛❧s t♦ ♣❡r❢♦r♠ ✐♥❡✣❝✐❡♥t ♠❛t❝❤❡s ❞❡s♣✐t❡ s❡❡❦✐♥❣ t♦ ❤❡❧♣ t❤❡✐r ♣❛t✐❡♥ts✳ ❙❡❝♦♥❞✱ t❤❡r❡ ❛r❡ ❛❣❡♥❝② ♣r♦❜❧❡♠s t❤❛t ♠❛❦❡ ❤♦s♣✐t❛❧s t♦♦ s❡♥s✐t✐✈❡ t♦ t❤❡ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s✳ ❖✉r ❛♥❛❧②s✐s s❤♦✇s t❤❛t ❜♦t❤ ♠❛r❦❡t ❢❛✐❧✉r❡s ❛r❡ ❧✐❦❡❧② ✐♠♣♦rt❛♥t ❛♥❞ t❤❛t ♣❧❛t❢♦r♠s ❝♦✉❧❞ ✉s❡ s✐♠♣❧❡ ❛❧t❡r♥❛t✐✈❡ ♠❡❝❤❛♥✐s♠s t♦ s✉❜st❛♥t✐❛❧❧② ✐♥❝r❡❛s❡ ❡✣❝✐❡♥❝②✳ ❚❤❡s❡ ✜♥❞✐♥❣s ❤❛✈❡ ❜♦t❤ s❤♦rt✲t❡r♠ ♣♦❧✐❝② ✐♠♣❧✐❝❛t✐♦♥s ❛♥❞ ❜r♦❛❞❡r ✐♠♣❧✐❝❛t✐♦♥s ❢♦r t❤❡ ❞❡s✐❣♥ ♦❢ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡ts✳ ❚❤❡r❡ ❛r❡ t✇♦ ❝❧❡❛r s❤♦rt✲t❡r♠ ♣♦❧✐❝② ✐♠♣❧✐❝❛t✐♦♥s✳ ❋✐rst✱ ❡①✐st✐♥❣ ♣❧❛t❢♦r♠s s❤♦✉❧❞ ❡①♣❡r✐♠❡♥t ✇✐t❤ ♣♦✐♥t s②st❡♠s✳ ❚❤✐s r❡❝♦♠♠❡♥❞❛t✐♦♥ ✐s ♣❛rt✐❝✉❧❛r❧② ❛❝t✐♦♥❛❜❧❡ ❜❡❝❛✉s❡ ✐t ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ❜② s✐♥❣❧❡ ♣❧❛t❢♦r♠s ❛♥❞ ✐t ✐s ✇✐❧❧ ❧✐❦❡❧②

✺✵

❤❡❧♣ t❤❡♠ ❡①♣❛♥❞✳ ❙❡❝♦♥❞✱ t❤✐r❞✲♣❛rt② ♣❛②❡rs s❤♦✉❧❞ s✉❜s✐❞✐③❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ❛t ♣❧❛t❢♦r♠s✳ ❲❡ ❛r❣✉❡❞ t❤❛t ❤♦s♣✐t❛❧s ❛r❡ ❧✐❦❡❧② r❡s♣♦♥s✐✈❡ t♦ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s✱ ❛ ❜❡❤❛✈✐♦r t❤❛t ❧❡❛❞s t♦ s✐❣♥✐✜❝❛♥t ✇❡❧❢❛r❡ ❧♦ss❡s✳

❙✉❜s✐❞✐❡s ❜② ▼❡❞✐❝❛r❡ ❛♥❞

♣r✐✈❛t❡ ♣❛②❡rs ❝❛♥ ❧✐❦❡❧② ❛❧❧❡✈✐❛t❡ t❤✐s ♣r♦❜❧❡♠✳ ▼♦r❡♦✈❡r✱ ♦✉r ❛♥❛❧②s✐s s✉❣❣❡sts t❤❛t t❤✐s t✇♦✲ ♣r♦♥❣❡❞ ❛♣♣r♦❛❝❤✱ ✇❤✐❝❤ ❛❞❞r❡ss❡s t❤❡ t✇♦ ♠❛r❦❡t ❢❛✐❧✉r❡s s❡♣❛r❛t❡❧②✱ ✐s ♠♦r❡ r♦❜✉st ❛♥❞ ❤❛s ❧♦✇❡r ❞❛t❛ r❡q✉✐r❡♠❡♥ts t❤❛♥ ♠❡❝❤❛♥✐s♠s t❤❛t ❛❞❞r❡ss ❜♦t❤ ♠❛r❦❡t ❢❛✐❧✉r❡s s✐♠✉❧t❛♥❡♦✉s❧②✳ ❈♦♥s✐st❡♥t ✇✐t❤ ♦✉r r❡s✉❧ts✱ t❤❡r❡ ❛r❡ ✐♥✐t✐❛t✐✈❡s ♠♦✈✐♥❣ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡s❡ ♣♦❧✐❝② ❝❤❛♥❣❡s✳ ❚❤❡ ◆❑❘ r❡❝❡♥t❧② st❛rt❡❞ ❡①♣❡r✐♠❡♥t✐♥❣ ✇✐t❤ ❛ ♣♦✐♥ts s②st❡♠ t❤r♦✉❣❤ t❤❡✐r ✏❈❡♥t❡r ▲✐q✉✐❞✐t② ❈♦♥tr✐❜✉t✐♦♥ Pr♦❣r❛♠✳✑ ❙♦♠❡ ♣r✐✈❛t❡ ✐♥s✉r❡rs ❤❛✈❡ st❛rt❡❞ ❝♦✈❡r✐♥❣ t❤❡ ❝♦sts ♦❢ ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣❧❛t❢♦r♠s✳ ❖✉r r❡s✉❧ts ✐♥❞✐❝❛t❡ t❤❛t t❤❡r❡ ❝❛♥ ❜❡ ❧❛r❣❡ ❣❛✐♥s ❢r♦♠ ❝♦♥t✐♥✉✐♥❣ t♦ ♠♦✈❡ ✐♥ t❤✐s ❞✐r❡❝t✐♦♥✳ ❋✉rt❤❡r✱ ❛❧❧ ♣❧❛t❢♦r♠s ❝❛♥ ✉s❡ ❞❛t❛✲❞r✐✈❡♥ r❡✇❛r❞s s②st❡♠✳ ❋✉t✉r❡ r❡s❡❛r❝❤ ❝❛♥ ❝♦♥tr✐❜✉t❡ t♦ t❤❡ ❞❡s✐❣♥ ❛♥❞ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡s❡ ♣♦❧✐❝✐❡s✳ ▼♦r❡ ❜r♦❛❞❧②✱ ♦✉r r❡s✉❧ts r❛✐s❡ t❤❡ q✉❡st✐♦♥ ♦❢ ✇❤❡t❤❡r ♦r ♥♦t t♦ ✉s❡ ❤❡❛✈②✲❤❛♥❞❡❞ r❡❣✉❧❛t✐♦♥✱ s✉❝❤ ❛s ♠❛♥❞❛t✐♥❣ ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ ❛ s✐♥❣❧❡ ♣❧❛t❢♦r♠✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❯✳❑✳✱ ◆❡t❤❡r❧❛♥❞s✱ ❛♥❞ ❈❛♥❛❞❛ ✭❉❡ ❑❧❡r❦ ❡t ❛❧✳✱ ✷✵✵✺❀ ❏♦❤♥s♦♥ ❡t ❛❧✳✱ ✷✵✵✽❀ ▼❛❧✐❦ ❛♥❞ ❈♦❧❡✱ ✷✵✶✹✮ ♠❛♥❞❛t❡ ♣❛rt✐❝✐♣❛t✐♦♥ ❛t ❛ s✐♥❣❧❡ ♥❛t✐♦♥❛❧ ♣r♦❣r❛♠✳ ❆t ❛ ✜rst ❣❧❛♥❝❡✱ t❤✐s ❛♣♣r♦❛❝❤ s❡❡♠s r❡❛s♦♥❛❜❧❡ ❜❡❝❛✉s❡ ♦❢ t❤❡ ✐♥❝r❡❛s✐♥❣ r❡t✉r♥s t♦ s❝❛❧❡ ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳

❍♦✇❡✈❡r✱ ♠❛♥❞❛t❡❞ s✐♥❣❧❡✲

♣❧❛t❢♦r♠ ♣❛rt✐❝✐♣❛t✐♦♥ ❝❛♥ r❡❞✉❝❡ ❝♦♠♣❡t✐t✐✈❡ ✐♥❝❡♥t✐✈❡s ❢♦r ♣❧❛t❢♦r♠s t❤❛t ❤❛✈❡ ❛r❣✉❛❜❧②

✹✵

❝♦♥tr✐❜✉t❡❞ t♦ ✈❛r✐♦✉s ✐♥♥♦✈❛t✐♦♥s ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✳

▼♦r❡♦✈❡r✱ ♦✉r ❡st✐♠❛t❡s ♦❢ r❡t✉r♥s

t♦ s❝❛❧❡ s✉❣❣❡st t❤❛t ✐t ✇♦✉❧❞ ♥♦t ❜❡ ✐♥❡✣❝✐❡♥t t♦ ❤❛✈❡ ❛ ❢❡✇ ❧❛r❣❡ ❡①❝❤❛♥❣❡ ✐♥ t❤❡ ❯❙ ❜❡❝❛✉s❡ ♠♦st ♦❢ t❤❡ ♣♦t❡♥t✐❛❧ ❡✣❝✐❡♥❝② ❣❛✐♥ ❝♦♠❡s ❢r♦♠ ♠♦✈✐♥❣ t❤❡ ♠❛r❦❡t ❢r♦♠ ✐♥❞✐✈✐❞✉❛❧ ❤♦s♣✐t❛❧s t♦ ♥❛t✐♦♥❛❧ ❡①❝❤❛♥❣❡s✳

❘❡❢❡r❡♥❝❡s ❆❜❞✉❧❦❛❞✐r♦➜❧✉✱ ❆t✐❧❛ ❛♥❞ ❑②❧❡ ❇❛❣✇❡❧❧✱ ✏❚r✉st✱ r❡❝✐♣r♦❝✐t②✱ ❛♥❞ ❢❛✈♦rs ✐♥ ❝♦♦♣❡r❛t✐✈❡

r❡❧❛t✐♦♥s❤✐♣s✱✑ ❆♠❡r✐❝❛♥ ❡❝♦♥♦♠✐❝ ❥♦✉r♥❛❧✿ ▼✐❝r♦❡❝♦♥♦♠✐❝s✱ ✷✵✶✸✱ ✺ ✭✷✮✱ ✷✶✸✕✷✺✾✳ ❆❜r❛❤❛♠✱ ❉❛✈✐❞ ❏✱ ❆✈r✐♠ ❇❧✉♠✱ ❛♥❞ ❚✉♦♠❛s ❙❛♥❞❤♦❧♠✱ ✏ ❈❧❡❛r✐♥❣ ❛❧❣♦r✐t❤♠s ❢♦r

❜❛rt❡r ❡①❝❤❛♥❣❡ ♠❛r❦❡ts✿ ❊♥❛❜❧✐♥❣ ♥❛t✐♦♥✇✐❞❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡s✱✑ ✐♥ ✏Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✽t❤ ❆❈▼ ❝♦♥❢❡r❡♥❝❡ ♦♥ ❊❧❡❝tr♦♥✐❝ ❝♦♠♠❡r❝❡✑ ❆❈▼ ✷✵✵✼✱ ♣♣✳ ✷✾✺✕✸✵✹✳ ❆❣❛r✇❛❧✱

◆✐❦❤✐❧✱

■t❛✐ ❆s❤❧❛❣✐✱

❊❞✉❛r❞♦ ❆③❡✈❡❞♦✱

❈❧❛②t♦♥ ❋❡❛t❤❡rst♦♥❡✱

❛♥❞

❖♠❡r ❑❛r❛❞✉♠❛♥✱ ✏❲❤❛t ▼❛tt❡rs ❢♦r t❤❡ Pr♦❞✉❝t✐✈✐t② ♦❢ ❑✐❞♥❡② ❊①❝❤❛♥❣❡❄✱✑ ▼✐♠❡♦✱ ▼■❚✱ ✷✵✶✽✳

❆♠❡r✐❝❛♥ ❙♦❝✐❡t② ♦❢ ❚r❛♥s♣❧❛♥t ❙✉r❣❡♦♥s✱ ✏ ❑✐❞♥❡② P❛✐r❡❞ ❉♦♥❛t✐♦♥✿ ❈♦♠♠✉♥✐t② P❡r✲

s♣❡❝t✐✈❡s ❛♥❞ ❇❡st Pr❛❝t✐❝❡s✱✑ ✷✵✶✻✳

✹✵ ❊①❛♠♣❧❡s ✐♥❝❧✉❞❡ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ♥♦♥✲s✐♠✉❧t❛♥❡♦✉s ❝❤❛✐♥s ✭❘❡❡s ❡t ❛❧✳✱ ✷✵✵✾✮✱ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ●❧♦❜❛❧ ❑✐❞♥❡② ❊①❝❤❛♥❣❡ ✇❤✐❝❤ ❛❧❧♦✇s ♣❛✐rs ❢r♦♠ ❞❡✈❡❧♦♣♠❡♥t ❝♦✉♥tr✐❡s t♦ ♦✈❡r❝♦♠❡ ✜♥❛♥❝✐❛❧ ❜❛rr✐❡rs ✭❘❡❡s ❡t ❛❧✳✱ ✷✵✶✼✮✱ ✈♦✉❝❤❡r ♣r♦❣r❛♠s t♦ ✐♥❝r❡❛s❡ ❞♦♥❛t✐♦♥ ❢♦r ❢✉t✉r❡ ♣r✐♦r✐t② ✭❱❡❛❧❡ ❡t ❛❧✳✱ ✷✵✶✼❀ ❲❛❧❧ ❡t ❛❧✳✱ ✷✵✶✼✮✱ ❛♥❞ ♦t❤❡r ♦♣❡r❛t✐♦♥❛❧ ✐♥♥♦✈❛t✐♦♥s t❤❛t r❡❞✉❝❡ ❢r✐❝t✐♦♥s ❛♥❞ ✐♠♣r♦✈❡ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠s✳

✺✶

❆♥❞❡rs♦♥✱ ❘♦ss✱ ■t❛✐ ❆s❤❧❛❣✐✱ ❉❛✈✐❞ ●❛♠❛r♥✐❦✱ ❛♥❞ ❨❛s❤ ❑❛♥♦r✐❛✱

✏❆ ❞②♥❛♠✐❝

♠♦❞❡❧ ♦❢ ❜❛rt❡r ❡①❝❤❛♥❣❡✱✑ ✐♥ ✏Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❚✇❡♥t②✲❙✐①t❤ ❆♥♥✉❛❧ ❆❈▼✲❙■❆▼ ❙②♠✲ ♣♦s✐✉♠ ♦♥ ❉✐s❝r❡t❡ ❆❧❣♦r✐t❤♠s✑ ❙■❆▼ ✷✵✶✹✱ ♣♣✳ ✶✾✷✺✕✶✾✸✸✳

✱

✱

✱ ▼✐❝❤❛❡❧ ❘❡❡s✱ ❆❧✈✐♥ ❊✳ ❘♦t❤✱ ❚❛②❢✉♥ ❙ö♥♠❡③✱ ❛♥❞ ▼✳ ❯t❦✉ Ü♥✈❡r✱

✏❑✐❞♥❡② ❊①❝❤❛♥❣❡ ❛♥❞ t❤❡ ❆❧❧✐❛♥❝❡ ❢♦r P❛✐r❡❞ ❉♦♥❛t✐♦♥✿ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ❈❤❛♥❣❡s

■♥t❡r❢❛❝❡s

t❤❡ ❲❛② ❑✐❞♥❡②s ❆r❡ ❚r❛♥s♣❧❛♥t❡❞✱✑

✱ ✷✵✶✺✱

❆r♠str♦♥❣✱ ▼❛r❦✱ ✏❈♦♠♣❡t✐t✐♦♥ ✐♥ t✇♦✲s✐❞❡❞ ♠❛r❦❡ts✱✑ ✷✵✵✻✱

✸✼

✭✸✮✱ ✻✻✽✕✻✾✶✳

❆s❤❧❛❣✐✱ ■t❛✐ ❛♥❞ ❆❧✈✐♥ ❊ ❘♦t❤✱ ❤♦s♣✐t❛❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡✱✑

✹✺ ❚❤❡ ❘❆◆❉ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s ✭✶✮✱ ✷✻✕✹✷✳

✱

✏❋r❡❡ r✐❞✐♥❣ ❛♥❞ ♣❛rt✐❝✐♣❛t✐♦♥ ✐♥ ❧❛r❣❡ s❝❛❧❡✱ ♠✉❧t✐✲

❚❤❡♦r❡t✐❝❛❧ ❊❝♦♥♦♠✐❝s

✱ ✷✵✶✹✱

✾

✭✸✮✱ ✽✶✼✕✽✻✸✳

✱ ▼❛①✐♠✐❧❧✐❡♥ ❇✉rq✱ P❛tr✐❝❦ ❏❛✐❧❧❡t✱ ❛♥❞ ❱❛❤✐❞❡❤ ▼❛♥s❤❛❞✐✱ t❤✐❝❦♥❡ss ✐♥ ❤❡t❡r♦❣❡♥❡♦✉s ❞②♥❛♠✐❝ ♠❛r❦❡ts✱✑

✏❖♥ ♠❛t❝❤✐♥❣ ❛♥❞

❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✶✻✵✻✳✵✸✻✷✻

✱ ✷✵✶✻✳

❇❡❝❦❡r✱ ●❛r② ❙ ❛♥❞ ❏✉❧✐♦ ❏♦r❣❡ ❊❧✐❛s✱ ✏■♥tr♦❞✉❝✐♥❣ ✐♥❝❡♥t✐✈❡s ✐♥ t❤❡ ♠❛r❦❡t ❢♦r ❧✐✈❡ ❛♥❞ ❝❛❞❛✈❡r✐❝ ♦r❣❛♥ ❞♦♥❛t✐♦♥s✱✑

❚❤❡ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝ P❡rs♣❡❝t✐✈❡s

✱ ✷✵✵✼✱

✷✶

✭✸✮✱ ✸✕✷✹✳

❇✐♥❣❛♠❛♥✱ ❆ ❲✱ ❋ ❍ ❲r✐❣❤t ❏r✱ ▼ ❑❛♣t✉r❝③❛❦✱ ▲ ❙❤❡♥✱ ❙ ❱✐❝❦✱ ❛♥❞ ❈ ▲ ▼✉r✲ ♣❤❡②✱ ✏❙✐♥❣❧❡✲❈❡♥t❡r ❑✐❞♥❡② P❛✐r❡❞ ❉♦♥❛t✐♦♥✿ ❚❤❡ ▼❡t❤♦❞✐st ❙❛♥ ❆♥t♦♥✐♦ ❊①♣❡r✐❡♥❝❡✱✑

❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ❚r❛♥s♣❧❛♥t❛t✐♦♥

✱ ✷✵✶✷✱

✶✷

✭✽✮✱ ✷✶✷✺✕✷✶✸✷✳

❇♦✐t❡✉①✱ ▼❛r❝❡❧✱ ✏❙✉r ❧❛ ❣❡st✐♦♥ ❞❡s ▼♦♥♦♣♦❧❡s P✉❜❧✐❝s ❛str❡✐♥ts ❛ ❧✬❡q✉✐❧✐❜r❡ ❜✉❞❣❡t❛✐r❡✱✑

❊❝♦♥♦♠❡tr✐❝❛

✱ ✶✾✺✻✱

❈❤❡tt②✱ ❘❛❥✱

✷✹

✭✶✮✱ ✷✷✕✹✵✳

✏❙✉✣❝✐❡♥t st❛t✐st✐❝s ❢♦r ✇❡❧❢❛r❡ ❛♥❛❧②s✐s✿

r❡❞✉❝❡❞✲❢♦r♠ ♠❡t❤♦❞s✱✑

❆♥♥✉✳ ❘❡✈✳ ❊❝♦♥✳

✱ ✷✵✵✾✱

✶

❆ ❜r✐❞❣❡ ❜❡t✇❡❡♥ str✉❝t✉r❛❧ ❛♥❞

✭✶✮✱ ✹✺✶✕✹✽✽✳

❈❧❡♠❡♥s✱ ❏❡✛r❡② ❛♥❞ ❏♦s❤✉❛ ❉ ●♦tt❧✐❡❜✱ ✏❉♦ P❤②s✐❝✐❛♥s✬ ❋✐♥❛♥❝✐❛❧ ■♥❝❡♥t✐✈❡s ❆✛❡❝t ▼❡❞✐❝❛❧ ❚r❡❛t♠❡♥t ❛♥❞ P❛t✐❡♥t ❍❡❛❧t❤❄✱✑

❆♠❡r✐❝❛♥ ❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇

✱ ✷✵✶✹✱

✶✵✹

✭✹✮✳

❈♦✇❛♥✱ ◆✐❣❡❧ ❈✳✱ ❍ ❆ ●r✐ts❝❤✱ ◆✐♠❛ ◆❛ss✐r✐✱ ❏♦s❡♣❤ ❙✐♥❛❝♦r❡✱ ❛♥❞ ❏❡✛r❡② ▲ ❱❡❛❧❡✱ ✏❇r♦❦❡♥ ❈❤❛✐♥s ❛♥❞ ❘❡♥❡❣✐♥❣✿ ❆ ❘❡✈✐❡✇ ♦❢ ✶✼✹✽ ❑✐❞♥❡② P❛✐r❡❞ ❉♦♥❛t✐♦♥ ❚r❛♥s✲

❆♠❡r✐❝❛♥ ❥♦✉r♥❛❧ ♦❢ tr❛♥s♣❧❛♥t❛t✐♦♥ ✿ ♦✣❝✐❛❧ ❥♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❙♦❝✐❡t② ♦❢ ❚r❛♥s♣❧❛♥t❛t✐♦♥ ❛♥❞ t❤❡ ❆♠❡r✐❝❛♥ ❙♦❝✐❡t② ♦❢ ❚r❛♥s♣❧❛♥t ❙✉r❣❡♦♥s ✶✼ ✾ ❉❛♥♦✈✐t❝❤✱ ●❛❜r✐❡❧ ▼ ❍❛♥❞❜♦♦❦ ♦❢ ❦✐❞♥❡② tr❛♥s♣❧❛♥t❛t✐♦♥ ♣❧❛♥ts✳✱✑

✱ ✷✵✶✼✱

✱

✱

✱ ✷✹✺✶✕✷✹✺✼✳

▲✐♣♣✐♥❝♦tt ❲✐❧❧✐❛♠s ✫

❲✐❧❦✐♥s✱ ✷✵✵✾✳

❉✐❝❦❡rs♦♥✱ ❏♦❤♥ P✱ ❆r✐❡❧ ❉ Pr♦❝❛❝❝✐❛✱ ❛♥❞ ❚✉♦♠❛s ❙❛♥❞❤♦❧♠✱ ✏❉②♥❛♠✐❝ ▼❛t❝❤✐♥❣ ✈✐❛ ❲❡✐❣❤t❡❞ ▼②♦♣✐❛ ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❑✐❞♥❡② ❊①❝❤❛♥❣❡✱✑ ✐♥ ✏❆❆❆■✑ ✷✵✶✷✳

❉✐①✐t✱ ❆✈✐♥❛s❤ ❛♥❞ ❆❣♥❛r ❙❛♥❞♠♦✱ ✏❙♦♠❡ s✐♠♣❧✐✜❡❞ ❢♦r♠✉❧❛❡ ❢♦r ♦♣t✐♠❛❧ ✐♥❝♦♠❡ t❛①❛✲ t✐♦♥✱✑

❚❤❡ ❙❝❛♥❞✐♥❛✈✐❛♥ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s

✱ ✶✾✼✼✱ ♣♣✳ ✹✶✼✕✹✷✸✳

✺✷

❊❧❧✐s♦♥✱ ❇❧❛❦❡✱ ✏❆ ❙②st❡♠❛t✐❝ ❘❡✈✐❡✇ ♦❢ ❑✐❞♥❡② P❛✐r❡❞ ❉♦♥❛t✐♦♥✿ ❆♣♣❧②✐♥❣ ▲❡ss♦♥s ❢r♦♠

❍✐st♦r✐❝ ❛♥❞ ❈♦♥t❡♠♣♦r❛r② ❈❛s❡ ❙t✉❞✐❡s t♦ ■♠♣r♦✈❡ t❤❡ ❯❙ ▼♦❞❡❧✱✑ ▼✐♠❡♦✱ ❯♥✐✈❡rs✐t② ♦❢ P❡♥♥s②❧✈❛♥✐❛✱ ✷✵✶✹✳

❊❧❧✐s♦♥✱ ●❧❡♥♥ ❛♥❞ ❉r❡✇ ❋✉❞❡♥❜❡r❣✱ ✏❑♥✐❢❡✲❊❞❣❡ ♦r P❧❛t❡❛✉✿ ❲❤❡♥ ❉♦ ▼❛r❦❡t ▼♦❞❡❧s

❚✐♣❄✱✑ ❚❤❡ ◗✉❛rt❡r❧② ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s✱ ✷✵✵✸✱ ✶✶✽ ✭✹✮✱ ✶✷✹✾✕✶✷✼✽✳ ❋r✐❡❞♠❛♥✱ ❊r✐❝ ❏✱ ❏♦s❡♣❤ ❨ ❍❛❧♣❡r♥✱ ❛♥❞ ■❛♥ ❑❛s❤✱ ✏❊✣❝✐❡♥❝② ❛♥❞ ◆❛s❤ ❡q✉✐❧✐❜r✐❛ ✐♥

❛ s❝r✐♣ s②st❡♠ ❢♦r P✷P ♥❡t✇♦r❦s✱✑ ✐♥ ✏Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✼t❤ ❆❈▼ ❝♦♥❢❡r❡♥❝❡ ♦♥ ❊❧❡❝tr♦♥✐❝ ❝♦♠♠❡r❝❡✑ ❆❈▼ ✷✵✵✻✱ ♣♣✳ ✶✹✵✕✶✹✾✳ ❋r✐❡❞♠❛♥✱ ❏❡r♦♠❡✱ ❚r❡✈♦r ❍❛st✐❡✱ ❛♥❞ ❘♦❜❡rt ❚✐❜s❤✐r❛♥✐✱ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ st❛t✐st✐❝❛❧ ❧❡❛r♥✐♥❣✱ ❱♦❧✳ ✶✱ ❙♣r✐♥❣❡r s❡r✐❡s ✐♥ st❛t✐st✐❝s ◆❡✇ ❨♦r❦✱ ✷✵✵✶✳

●✐❣❡r❡♥③❡r✱ ●❡r❞ ❛♥❞ ❉❛♥✐❡❧ ● ●♦❧❞st❡✐♥✱ ✏❘❡❛s♦♥✐♥❣ t❤❡ ❢❛st ❛♥❞ ❢r✉❣❛❧ ✇❛②✿ ♠♦❞❡❧s

♦❢ ❜♦✉♥❞❡❞ r❛t✐♦♥❛❧✐t②✳✱✑ Ps②❝❤♦❧♦❣✐❝❛❧ r❡✈✐❡✇✱ ✶✾✾✻✱ ✶✵✸ ✭✹✮✱ ✻✺✵✳ ❛♥❞ ❙t❡♣❤❛♥✐❡ ❑✉r③❡♥❤❛❡✉s❡r✱ ✏❋❛st ❛♥❞ ❢r✉❣❛❧ ❤❡✉r✐st✐❝s ✐♥ ♠❡❞✐❝❛❧ ❞❡❝✐s✐♦♥ ♠❛❦✐♥❣✱✑ ❙❝✐❡♥❝❡ ❛♥❞ ♠❡❞✐❝✐♥❡ ✐♥ ❞✐❛❧♦❣✉❡✿ ❚❤✐♥❦✐♥❣ t❤r♦✉❣❤ ♣❛rt✐❝✉❧❛rs ❛♥❞ ✉♥✐✈❡rs❛❧s✱ ✷✵✵✺✱ ♣♣✳ ✸✕

✶✺✳ ●✉♦✱ ❨ ❛♥❞ ❏ ❍ör♥❡r✱ ✏❉②♥❛♠✐❝ ♠❡❝❤❛♥✐s♠s ✇✐t❤ ♠♦♥❡②✱✑ ❚❡❝❤♥✐❝❛❧ ❘❡♣♦rt✱ ▼✐♠❡♦✱

❨❛❧❡ ❯♥✐✈❡rs✐t② ✷✵✶✺✳ ❍❛✉s❡r✱ ❈❤r✐st✐♥❡ ❛♥❞ ❍✉❣♦ ❍♦♣❡♥❤❛②♥✱ ✏❚r❛❞✐♥❣ ❢❛✈♦rs✿ ❖♣t✐♠❛❧ ❡①❝❤❛♥❣❡ ❛♥❞ ❢♦r✲

❣✐✈❡♥❡ss✱✑ ❈♦❧❧❡❣✐♦ ❈❛r❧♦ ❆❧❜❡rt♦ ❈❛r❧♦ ❆❧❜❡rt♦ ◆♦t❡❜♦♦❦s✱ ✷✵✵✽✱ ✽✽✳ ❍❡❧❞✱ P❤✐❧✐♣ ❏✱ ❋ ▼❝❈♦r♠✐❝❦✱ ❆ ❖❥♦✱ ❛♥❞ ❏♦❤♥ P ❘♦❜❡rts✱ ✏❆ ❝♦st✲❜❡♥❡✜t ❛♥❛❧②s✐s ♦❢

❣♦✈❡r♥♠❡♥t ❝♦♠♣❡♥s❛t✐♦♥ ♦❢ ❦✐❞♥❡② ❞♦♥♦rs✱✑ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ❚r❛♥s♣❧❛♥t❛t✐♦♥✱ ✷✵✶✻✱ ✶✻ ✭✸✮✱ ✽✼✼✕✽✽✺✳

■r✇✐♥✱ ❋ ❉✱ ❆ ❋ ❇♦♥❛❣✉r❛✱ ❙ ❲ ❈r❛✇❢♦r❞✱ ❛♥❞ ▼ ❋♦♦t❡✱ ✏ ❑✐❞♥❡② ♣❛✐r❡❞ ❞♦♥❛t✐♦♥✿

❛ ♣❛②❡r ♣❡rs♣❡❝t✐✈❡✱✑ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ❚r❛♥s♣❧❛♥t❛t✐♦♥✱ ✷✵✶✷✱ ✶✷ ✭✻✮✱ ✶✸✽✽✕✶✸✾✶✳ ❏❛❝❦s♦♥✱ ▼❛tt❤❡✇ ❖ ❛♥❞ ❍✉❣♦ ❋ ❙♦♥♥❡♥s❝❤❡✐♥✱ ✏❖✈❡r❝♦♠✐♥❣ ✐♥❝❡♥t✐✈❡ ❝♦♥str❛✐♥ts ❜②

❧✐♥❦✐♥❣ ❞❡❝✐s✐♦♥s✱✑ ❊❝♦♥♦♠❡tr✐❝❛✱ ✷✵✵✼✱ ✼✺ ✭✶✮✱ ✷✹✶✕✷✺✼✳ ❏♦❤♥s♦♥✱ ❘❛❝❤❡❧ ❏✱ ❏♦❛♥♥❡ ❊ ❆❧❧❡♥✱ ❙✉s❛♥ ❱ ❋✉❣❣❧❡✱ ❏ ❆♥❞r❡✇ ❇r❛❞❧❡②✱ ❈❤r✐s ❘✉❞❣❡ ❡t ❛❧✳✱ ✏❊❛r❧② ❡①♣❡r✐❡♥❝❡ ♦❢ ♣❛✐r❡❞ ❧✐✈✐♥❣ ❦✐❞♥❡② ❞♦♥❛t✐♦♥ ✐♥ t❤❡ ❯♥✐t❡❞ ❑✐♥❣❞♦♠✱✑ ❚r❛♥s♣❧❛♥t❛t✐♦♥✱ ✷✵✵✽✱ ✽✻ ✭✶✷✮✱ ✶✻✼✷✕✶✻✼✼✳

❑✐②♦t❛❦✐✱ ◆♦❜✉❤✐r♦ ❛♥❞ ❘❛♥❞❛❧❧ ❲r✐❣❤t✱ ✏❖♥ ♠♦♥❡② ❛s ❛ ♠❡❞✐✉♠ ♦❢ ❡①❝❤❛♥❣❡✱✑ ❏♦✉r♥❛❧ ♦❢ P♦❧✐t✐❝❛❧ ❊❝♦♥♦♠②✱ ✶✾✽✾✱ ✾✼ ✭✹✮✱ ✾✷✼✕✾✺✹✳

❑❧❡r❦✱

▼❛rr②

❉❡✱

❑❛r✐♥

▼

❑❡✐③❡r✱

❋r❛♥s

❍❏

❈❧❛❛s✱

▼❛r✐❛♥

❲✐t✈❧✐❡t✱

❇❡r♥❛❞❡tt❡ ❏❏▼ ❍❛❛s❡✲❑r♦♠✇✐❥❦✱ ❛♥❞ ❲✐❧❧❡♠ ❲❡✐♠❛r✱ ✏❚❤❡ ❉✉t❝❤ ♥❛t✐♦♥❛❧ ❧✐✈✲

✐♥❣ ❞♦♥♦r ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣r♦❣r❛♠✱✑ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ❚r❛♥s♣❧❛♥t❛t✐♦♥✱ ✷✵✵✺✱ ✺ ✭✾✮✱ ✷✸✵✷✕✷✸✵✺✳

✺✸

❑♦❝❤❡r❧❛❦♦t❛✱ ◆❛r❛②❛♥❛ ❘✱ ✏▼♦♥❡② ✐s ♠❡♠♦r②✱✑

❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝ ❚❤❡♦r②✱ ✶✾✾✽✱ ✽✶

✭✷✮✱ ✷✸✷✕✷✺✶✳

❑♦❧st❛❞✱ ❏♦♥❛t❤❛♥ ❚✱

✏■♥❢♦r♠❛t✐♦♥ ❛♥❞ q✉❛❧✐t② ✇❤❡♥ ♠♦t✐✈❛t✐♦♥ ✐s ✐♥tr✐♥s✐❝✿ ❊✈✐❞❡♥❝❡

❢r♦♠ s✉r❣❡♦♥ r❡♣♦rt ❝❛r❞s✱✑ ❚❤❡ ❆♠❡r✐❝❛♥ ❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇✱ ✷✵✶✸✱ ✶✵✸ ✭✼✮✱ ✷✽✼✺✕✷✾✶✵✳

▼❛❧✐❦✱ ❙❤❛✜ ❛♥❞ ❊❞✇❛r❞ ❈♦❧❡✱ ✏❋♦✉♥❞❛t✐♦♥s ❛♥❞ ♣r✐♥❝✐♣❧❡s ♦❢ t❤❡ ❈❛♥❛❞✐❛♥ ❧✐✈✐♥❣ ❞♦♥♦r ♣❛✐r❡❞ ❡①❝❤❛♥❣❡ ♣r♦❣r❛♠✱✑ ❈❛♥❛❞✐❛♥ ❥♦✉r♥❛❧ ♦❢ ❦✐❞♥❡② ❤❡❛❧t❤ ❛♥❞ ❞✐s❡❛s❡✱ ✷✵✶✹✱ ✶ ✭✶✮✱ ✻✳

▼❛rs❝❤❛❦✱ ❏❛❝♦❜ ❛♥❞ ❲✐❧❧✐❛♠ ❍ ❆♥❞r❡✇s✱ ✏❘❛♥❞♦♠ s✐♠✉❧t❛♥❡♦✉s ❡q✉❛t✐♦♥s ❛♥❞ t❤❡ t❤❡♦r② ♦❢ ♣r♦❞✉❝t✐♦♥✱✑ ❊❝♦♥♦♠❡tr✐❝❛✱ ✶✾✹✹✱ ♣♣✳ ✶✹✸✕✷✵✺✳

▼❛ss✐❡✱ ❆❧❧❛♥ ❇✱ ❙♦♠♠❡r ❊ ●❡♥tr②✱ ❘♦❜❡rt ❆ ▼♦♥t❣♦♠❡r②✱ ❆❞❛♠ ❆ ❇✐♥❣❛♠❛♥✱ ❛♥❞ ❉♦rr② ▲ ❙❡❣❡✈✱ ✏❈❡♥t❡r✲▲❡✈❡❧ ❯t✐❧✐③❛t✐♦♥ ♦❢ ❑✐❞♥❡② P❛✐r❡❞ ❉♦♥❛t✐♦♥✱✑ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ❚r❛♥s♣❧❛♥t❛t✐♦♥✱ ✷✵✶✸✱ ✶✸ ✭✺✮✱ ✶✸✶✼✕✶✸✷✷✳

▼ö❜✐✉s✱ ▼❛r❦✉s✱ ✏❚r❛❞✐♥❣ ❢❛✈♦rs✱✑

▼✐♠❡♦✱ ▼✐❝r♦s♦❢t ❘❡s❡❛r❝❤✱ ✷✵✵✶✳

▼♦rr✐s✱ ❈❛r❧ ◆✱ ✏P❛r❛♠❡tr✐❝ ❡♠♣✐r✐❝❛❧ ❇❛②❡s ✐♥❢❡r❡♥❝❡✿

t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱✑ ❏♦✉r♥❛❧

♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❙t❛t✐st✐❝❛❧ ❆ss♦❝✐❛t✐♦♥✱ ✶✾✽✸✱ ✼✽ ✭✸✽✶✮✱ ✹✼✕✺✺✳

◆❛t✐♦♥❛❧ ❑✐❞♥❡② ❘❡❣✐str②✱

✏▼❡♠❜❡r

❈❡♥t❡r

❚❡r♠s✫

❦✐❞♥❡②r❡❣✐str②✳♦r❣✴❞♦❝s✴◆❑❘❴▼❈❴❚❡r♠s❴❈♦♥❞✐t✐♦♥s✳♣❞❢

❈♦♥❞✐t✐♦♥s✱✑

❤tt♣✿✴✴✇✇✇✳

✷✵✶✻✳

❖❧❧❡②✱ ● ❙t❡✈❡♥ ❛♥❞ ❆r✐❡❧ P❛❦❡s✱ ✏❚❤❡ ❞②♥❛♠✐❝s ♦❢ ♣r♦❞✉❝t✐✈✐t② ✐♥ t❤❡ t❡❧❡❝♦♠♠✉♥✐❝❛✲ t✐♦♥s ❡q✉✐♣♠❡♥t ✐♥❞✉str②✱✑ ❊❝♦♥♦♠❡tr✐❝❛✱ ✶✾✾✻✱ ✻✹ ✭✻✮✱ ✶✷✻✸✕✶✷✾✼✳

❖r❛♥❞✐✱ ❇❏✱ ❏▼ ●❛r♦♥③✐❦✲❲❛♥❣✱ ❆❧❧❛♥ ❇ ▼❛ss✐❡✱ ❆♥❞r❡❛ ❆ ❩❛❝❤❛r②✱ ❏❘ ▼♦♥t✲ ❣♦♠❡r②✱ ❑❏ ❱❛♥ ❆r❡♥❞♦♥❦✱ ▼❛r❦ ❉ ❙t❡❣❛❧❧✱ ❙❈ ❏♦r❞❛♥✱ ❏ ❖❜❡r❤♦❧③❡r✱ ❚❇ ❉✉♥♥ ❡t ❛❧✳✱ ✏◗✉❛♥t✐❢②✐♥❣ t❤❡ r✐s❦ ♦❢ ✐♥❝♦♠♣❛t✐❜❧❡ ❦✐❞♥❡② tr❛♥s♣❧❛♥t❛t✐♦♥✿ ❛ ♠✉❧t✐✲ ❝❡♥t❡r st✉❞②✱✑ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ❚r❛♥s♣❧❛♥t❛t✐♦♥✱ ✷✵✶✹✱ ✶✹ ✭✼✮✱ ✶✺✼✸✕✶✺✽✵✳

Pr❡♥❞❡r❣❛st✱ ❈❛♥✐❝❡✱

✏❚❤❡ ❆❧❧♦❝❛t✐♦♥ ♦❢ ❋♦♦❞ t♦ ❋♦♦❞ ❇❛♥❦s✱✑ ❯♥✐✈❡rs✐t② ♦❢ ❈❤✐❝❛❣♦✱

♠✐♠❡♦✱ ✷✵✶✼✳

❘❛♠s❡②✱ ❋r❛♥❦ P✱ ✏❆ ❈♦♥tr✐❜✉t✐♦♥ t♦ t❤❡ ❚❤❡♦r② ♦❢ ❚❛①❛t✐♦♥✱✑

❚❤❡ ❊❝♦♥♦♠✐❝ ❏♦✉r♥❛❧✱

✶✾✷✼✱ ✸✼ ✭✶✹✺✮✱ ✹✼✕✻✶✳

❘❡❡s✱ ▼✐❝❤❛❡❧ ❆✱ ❏♦♥❛t❤❛♥ ❊ ❑♦♣❦❡✱ ❘♦♥❛❧❞ P P❡❧❧❡t✐❡r✱ ❉♦rr② ▲ ❙❡❣❡✈✱ ▼❛tt❤❡✇ ❊ ❘✉tt❡r✱ ❆❧❢r❡❞♦ ❏ ❋❛❜r❡❣❛✱ ❏❡✛r❡② ❘♦❣❡rs✱ ❖❧❡❤ ● P❛♥❦❡✇②❝③✱ ❏❛♥❡t ❍✐❧❧❡r✱ ❆❧✈✐♥ ❊ ❘♦t❤✱ ❛♥❞ ♦t❤❡rs✱ ✏❆ ♥♦♥s✐♠✉❧t❛♥❡♦✉s✱ ❡①t❡♥❞❡❞✱ ❛❧tr✉✐st✐❝✲ ❞♦♥♦r ❝❤❛✐♥✱✑ ◆❡✇ ❊♥❣❧❛♥❞ ❏♦✉r♥❛❧ ♦❢ ▼❡❞✐❝✐♥❡✱ ✷✵✵✾✱ ✸✻✵ ✭✶✶✮✱ ✶✵✾✻✕✶✶✵✶✳

✱ ▼❛r❦ ❆ ❙❝❤♥✐t③❧❡r✱ ❊❨ ❩❛✈❛❧❛✱ ❏❛♠❡s ❆ ❈✉t❧❡r✱ ❆❧✈✐♥ ❊ ❘♦t❤✱ ❋r❛♥❦ ❉ ■r✇✐♥✱ ❙t❡♣❤❡♥ ❲ ❈r❛✇❢♦r❞✱ ❛♥❞ ❆❧❛♥ ❇ ▲❡✐❝❤t♠❛♥✱ ✏❈❛❧❧ t♦ ❞❡✈❡❧♦♣ ❛ st❛♥❞❛r❞ ❛❝q✉✐s✐t✐♦♥ ❝❤❛r❣❡ ♠♦❞❡❧ ❢♦r ❦✐❞♥❡② ♣❛✐r❡❞ ❞♦♥❛t✐♦♥✱✑ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ tr❛♥s♣❧❛♥t❛t✐♦♥✱ ✷✵✶✷✱ ✶✷ ✭✻✮✱ ✶✸✾✷✕✶✸✾✼✳

✺✹

✱ ❚② ❇ ❉✉♥♥✱ ❈❤r✐st✐❛♥ ❙ ❑✉❤r✱ ❈❤r✐st♦♣❤❡r ▲ ▼❛rs❤✱ ❏❡✛r❡② ❘♦❣❡rs✱ ❙✉s❛♥ ❊ ❘❡❡s✱ ❆❧❡❥❛♥❞r❛ ❈✐❝❡r♦✱ ▲❛✉r✐❡ ❏ ❘❡❡❝❡✱ ❆❧✈✐♥ ❊ ❘♦t❤✱ ❖❜✐ ❊❦✇❡♥♥❛ ❡t ❛❧✳✱ ✏❑✐❞♥❡② ❡①❝❤❛♥❣❡ t♦ ♦✈❡r❝♦♠❡ ✜♥❛♥❝✐❛❧ ❜❛rr✐❡rs t♦ ❦✐❞♥❡② tr❛♥s♣❧❛♥t❛t✐♦♥✱✑ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ❚r❛♥s♣❧❛♥t❛t✐♦♥✱ ✷✵✶✼✱ ✶✼ ✭✸✮✱ ✼✽✷✕✼✾✵✳

❘♦❜❡rt✱ ❈❤r✐st✐❛♥ ❛♥❞ ●❡♦r❣❡ ❈❛s❡❧❧❛✱

✏▼♦♥t❡ ❈❛r❧♦ ❙t❛t✐st✐❝❛❧ ▼❡t❤♦❞s ❙♣r✐♥❣❡r✲

❱❡r❧❛❣✱✑ ◆❡✇ ❨♦r❦✱ ✷✵✵✹✳

❘♦❝❤❡t✱ ❏❡❛♥✲❈❤❛r❧❡s ❛♥❞ ❏❡❛♥ ❚✐r♦❧❡✱

✏P❧❛t❢♦r♠ ❝♦♠♣❡t✐t✐♦♥ ✐♥ t✇♦✲s✐❞❡❞ ♠❛r❦❡ts✱✑

❏♦✉r♥❛❧ ♦❢ t❤❡ ❡✉r♦♣❡❛♥ ❡❝♦♥♦♠✐❝ ❛ss♦❝✐❛t✐♦♥✱ ✷✵✵✸✱ ✶ ✭✹✮✱ ✾✾✵✕✶✵✷✾✳

❘♦t❤✱ ❆❧✈✐♥ ❊✱ ✏❘❡♣✉❣♥❛♥❝❡ ❛s ❛ ❈♦♥str❛✐♥t ♦♥ ▼❛r❦❡ts✱✑

❚❤❡ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝ P❡r✲

s♣❡❝t✐✈❡s✱ ✷✵✵✼✱ ✷✶ ✭✸✮✱ ✸✼✕✺✽✳

✱ ❚❛②❢✉♥ ❙ö♥♠❡③✱ ❛♥❞ ▼ ❯t❦✉ Ü♥✈❡r✱ ✏❚r❛♥s♣❧❛♥t ❈❡♥t❡r ■♥❝❡♥t✐✈❡s ✐♥ ❑✐❞♥❡② ❊①✲ ❝❤❛♥❣❡✱✑ ▼✐♠❡♦✱ ❇♦st♦♥ ❈♦❧❧❡❣❡✱ ✷✵✵✺✳

✱

✱ ❛♥❞

✱

✏❊✣❝✐❡♥t ❦✐❞♥❡② ❡①❝❤❛♥❣❡✿

❈♦✐♥❝✐❞❡♥❝❡ ♦❢ ✇❛♥ts ✐♥ ♠❛r❦❡ts ✇✐t❤

❝♦♠♣❛t✐❜✐❧✐t②✲❜❛s❡❞ ♣r❡❢❡r❡♥❝❡s✱✑ ❚❤❡ ❆♠❡r✐❝❛♥ ❡❝♦♥♦♠✐❝ r❡✈✐❡✇✱ ✷✵✵✼✱ ♣♣✳ ✽✷✽✕✽✺✶✳

❙❛❡③✱ ❊♠♠❛♥✉❡❧✱

✏❯s✐♥❣ ❡❧❛st✐❝✐t✐❡s t♦ ❞❡r✐✈❡ ♦♣t✐♠❛❧ ✐♥❝♦♠❡ t❛① r❛t❡s✱✑ ❚❤❡ r❡✈✐❡✇ ♦❢

❡❝♦♥♦♠✐❝ st✉❞✐❡s✱ ✷✵✵✶✱ ✻✽ ✭✶✮✱ ✷✵✺✕✷✷✾✳

❙♦♥♠❡③✱ ❚❛②❢✉♥ ❛♥❞ ▼ ❯t❦✉ ❯♥✈❡r✱ ✏▼❛r❦❡t ❞❡s✐❣♥ ❢♦r ❦✐❞♥❡② ❡①❝❤❛♥❣❡✱✑ ❚❤❡ ❍❛♥❞❜♦♦❦ ♦❢ ▼❛r❦❡t ❉❡s✐❣♥✱ ✷✵✶✸✱ ♣✳ ✾✸✳

❯❙❘❉❙✱ ❯♥✐t❡❞ ❙t❛t❡s ❘❡♥❛❧ ❉❛t❛ ❙②st❡♠✱

✷✵✶✸ ❯❙❘❉❙ ❛♥♥✉❛❧ ❞❛t❛ r❡♣♦rt✿

❊♣✐✲

❞❡♠✐♦❧♦❣② ♦❢ ❦✐❞♥❡② ❞✐s❡❛s❡ ✐♥ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s✱ ❱♦❧✳ ✷✱ ◆❛t✐♦♥❛❧ ■♥st✐t✉t❡s ♦❢ ❍❡❛❧t❤✱

◆❛t✐♦♥❛❧ ■♥st✐t✉t❡ ♦❢ ❉✐❛❜❡t❡s ❛♥❞ ❉✐❣❡st✐✈❡ ❛♥❞ ❑✐❞♥❡② ❉✐s❡❛s❡s✱ ❇❡t❤❡s❞❛✱ ▼❉✱ ✷✵✶✸✳ ✱ ✷✵✶✻ ❯❙❘❉❙ ❛♥♥✉❛❧ ❞❛t❛ r❡♣♦rt✿ ❊♣✐❞❡♠✐♦❧♦❣② ♦❢ ❦✐❞♥❡② ❞✐s❡❛s❡ ✐♥ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s✱ ❱♦❧✳ ✷✱ ◆❛t✐♦♥❛❧ ■♥st✐t✉t❡s ♦❢ ❍❡❛❧t❤✱ ◆❛t✐♦♥❛❧ ■♥st✐t✉t❡ ♦❢ ❉✐❛❜❡t❡s ❛♥❞ ❉✐❣❡st✐✈❡ ❛♥❞ ❑✐❞♥❡② ❉✐s❡❛s❡s✱ ❇❡t❤❡s❞❛✱ ▼❉✱ ✷✵✶✻✳

❱❡❛❧❡✱ ❏❡✛r❡② ▲✱ ❆❧❡①❛♥❞❡r ▼ ❈❛♣r♦♥✱ ◆✐♠❛ ◆❛ss✐r✐✱ ●❛❜r✐❡❧ ❉❛♥♦✈✐t❝❤✱ ❍ ❆❧❜✐♥ ●r✐ts❝❤✱ ❆♠② ❲❛t❡r♠❛♥✱ ❏♦s❡♣❤ ❉❡❧ P✐③③♦✱ ❏✐♠ ❈ ❍✉✱ ▼❛r❡❦ P②❝✐❛✱ ❙✉③❛♥♥❡ ▼❝●✉✐r❡ ❡t ❛❧✳✱ ✏❱♦✉❝❤❡rs ❢♦r ❋✉t✉r❡ ❑✐❞♥❡② ❚r❛♥s♣❧❛♥ts t♦ ❖✈❡r❝♦♠❡✬❈❤r♦♥♦❧♦❣✐❝❛❧ ■♥❝♦♠♣❛t✐❜✐❧✐t②✬❇❡t✇❡❡♥ ▲✐✈✐♥❣ ❉♦♥♦rs ❛♥❞ ❘❡❝✐♣✐❡♥ts✳✱✑ ❚r❛♥s♣❧❛♥t❛t✐♦♥✱ ✷✵✶✼✳

❲❛❧❧✱ ❆♥❥✐ ❊✱ ❏❡✛r❡② ▲ ❱❡❛❧❡✱ ❛♥❞ ▼❛r❝ ▲ ▼❡❧❝❤❡r✱ ✏❆❞✈❛♥❝❡❞ ❉♦♥❛t✐♦♥ Pr♦❣r❛♠s ❛♥❞ ❉❡❝❡❛s❡❞ ❉♦♥♦r ■♥✐t✐❛t❡❞ ❈❤❛✐♥s✲✷ ■♥♥♦✈❛t✐♦♥s ✐♥ ❑✐❞♥❡② P❛✐r❡❞ ❉♦♥❛t✐♦♥✳✱✑ ❚r❛♥s✲ ♣❧❛♥t❛t✐♦♥✱ ✷✵✶✼✳

❲❡②❧✱ ❊ ●❧❡♥✱ ✏❆ ♣r✐❝❡ t❤❡♦r② ♦❢ ♠✉❧t✐✲s✐❞❡❞ ♣❧❛t❢♦r♠s✱✑ ❚❤❡ ❆♠❡r✐❝❛♥ ❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇✱ ✷✵✶✵✱ ✶✵✵ ✭✹✮✱ ✶✻✹✷✕✶✻✼✷✳

✺✺

❲♦❧❢❡✱ ❘♦❜❡rt ❆✱ ❱❛❧❛r✐❡ ❇ ❆s❤❜②✱ ❊❞❣❛r ▲ ▼✐❧❢♦r❞✱ ❆❦✐♥❧♦❧✉ ❖ ❖❥♦✱ ❘♦❜❡rt ❊ ❊tt❡♥❣❡r✱ ▲❛✇r❡♥❝❡ ❨❈ ❆❣♦❞♦❛✱ P❤✐❧✐♣ ❏ ❍❡❧❞✱ ❛♥❞ ❋r✐❡❞r✐❝❤ ❑ P♦rt✱ ✏❈♦♠♣❛r✲

✐s♦♥ ♦❢ ♠♦rt❛❧✐t② ✐♥ ❛❧❧ ♣❛t✐❡♥ts ♦♥ ❞✐❛❧②s✐s✱ ♣❛t✐❡♥ts ♦♥ ❞✐❛❧②s✐s ❛✇❛✐t✐♥❣ tr❛♥s♣❧❛♥t❛t✐♦♥✱ ❛♥❞ r❡❝✐♣✐❡♥ts ♦❢ ❛ ✜rst ❝❛❞❛✈❡r✐❝ tr❛♥s♣❧❛♥t✱✑

✸✹✶

✭✷✸✮✱ ✶✼✷✺✕✶✼✸✵✳

◆❡✇ ❊♥❣❧❛♥❞ ❏♦✉r♥❛❧ ♦❢ ▼❡❞✐❝✐♥❡✱

✶✾✾✾✱

❆♣♣❡♥❞✐① t♦ ✏▼❛r❦❡t ❋❛✐❧✉r❡ ✐♥ ❑✐❞♥❡② ❊①❝❤❛♥❣❡✑ ◆✐❦❤✐❧ ❆❣❛r✇❛❧✱ ■t❛✐ ❆s❤❧❛❣✐✱ ❊❞✉❛r❞♦ ❆③❡✈❡❞♦✱ ❈❧❛②t♦♥ ❋❡❛t❤❡rst♦♥❡✱ ❛♥❞ Ö♠❡r ❑❛r❛❞✉♠❛♥

❆♣r✐❧ ✶✽✱ ✷✵✶✽

✷

❆ Pr♦♦❢s ❆✳✶

Pr♦♦❢ ♦❢ t❤❡ ▼❛✐♥ ❚❤❡♦r❡♠

▲❡t R+ ❜❡ t❤❡ s❡t ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧ ♥✉♠❜❡rs ❛❣✐♥❞ R++ ❜❡ t❤❡ s❡t ♦❢ str✐❝t❧② ♣♦s✐t✐✈❡ r❡❛❧ ♥✉♠❜❡rs✳ ▲❡t t❤❡ s✉♣♣❧② ❝♦rr❡s♣♦♥❞❡♥❝❡ ♦❢ ❤♦s♣✐t❛❧ h ❜❡ S h (p) = arg max p · q − C h (q)✳ q∈RI+

❚❤❛t ✐s S h (p) ✐s t❤❡ s❡t ♦❢ q✉❛♥t✐t✐❡s t❤❛t ❛r❡ ♦♣t✐♠❛❧ ❢♦r ❤♦s♣✐t❛❧ h ❣✐✈❡♥ t❤❡ ✈❡❝t♦r ♦❢ r❡✇❛r❞s p✳ ▲❡t t❤❡ ❛❣❣r❡❣❛t❡ s✉♣♣❧② ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡ S(p) = arg max p · q − C(q)✳ q∈RI+

❲❡ ❜❡❣✐♥ ✇✐t❤ ❛ ❧❡♠♠❛ s❤♦✇✐♥❣ t❤❛t ❤♦s♣✐t❛❧ s✉♣♣❧② ❝❛♥ ❜❡ ❛❣❣r❡❣❛t❡❞✱ s♦ t❤❛t ❤♦s♣✐t❛❧s ❜❡❤❛✈❡ ❛s ❛ s✐♥❣❧❡ ❤♦s♣✐t❛❧ t❤❛t t❛❦❡s t❤❡ ❛❣❣r❡❣❛t❡ ❝♦st ❝✉r✈❡ ✐♥t♦ ❛❝❝♦✉♥t✳ ❚❤✐s ✐s s✐♠✐❧❛r t♦ st❛♥❞❛r❞ ❛❣❣r❡❣❛t✐♦♥ r❡s✉❧ts ✐♥ ♥❡♦❝❧❛ss✐❝❛❧ ✜r♠ t❤❡♦r② ✭▼❛s✲❈♦❧❡❧❧ ❡t ❛❧✳✱ ✶✾✾✺ ♣✳ ✶✹✽✮✳ ▲❡♠♠❛ ✶✳ ❋✐① ❛ ✈❡❝t♦r ♦❢ r❡✇❛r❞s ✐♥❞✐✈✐❞✉❛❧ s✉♣♣❧②

h

S (p)

p✳

❆❣❣r❡❣❛t❡ s✉♣♣❧②

❢♦r ❛❧❧ ❤♦s♣✐t❛❧s✳

h H ✈❡❝t♦rs (q )h=1 ✇✐t❤ ❡❛❝❤

h

h

q ∈ S (p)✱ H X

✐s t❤❡ ▼✐♥❦♦✇s❦✐ s✉♠ ♦❢

▼♦r❡♦✈❡r✱ ✐❢ t❤❡r❡ ✐s ❛ s❡t ♦❢ ✐♥❞✐✈✐❞✉❛❧ s✉♣♣❧②

t❤❡♥

C h (q h ) = C

h=1

Pr♦♦❢✳

S(p)

H X

qh

h=1

!

✭❆✶✮

✳

◆♦t❡ t❤❛t

max p ·

(q h )H h=1

H X

q

h=1

h

!

−

H X h=1

h

h

max P

C (q ) = max

q∈RI+ (q h )H h=1 :

H h=1

q h =q

p·

= max p · q − C(q)✳

H X h=1

q∈RI+

❈♦♥s✐❞❡r q 0 =

PH

h=1

q h0 ✇✐t❤ ❡❛❝❤ q h0 ✐s ✐♥ S h (p)✳ ❲❡ ❤❛✈❡ t❤❛t ! H H X X h p · q 0 − C(q 0 ) ≥ p · q0 − C h (q h0 )✳ h=1

q

h

!

−

H X

C h (q h )

h=1

✭❆✷✮

✭❆✸✮

h=1

❖♣t✐♠❛❧✐t② ♦❢ t❤❡ q h0 ✐♠♣❧✐❡s t❤❛t t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤✐s ✐♥❡q✉❛❧✐t② ❛tt❛✐♥s t❤❡ ♠❛①✐♠✉♠ ✐♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ ❡q✉❛t✐♦♥ ✭❆✷✮✳ ❍❡♥❝❡✱ p · q 0 − C(q 0 ) ≥ max p · q − C(q)✱ q∈RI+

✸ s♦ t❤❛t q 0 ✐s ✐♥ S(p)✳ ■♥❡q✉❛❧✐t② ✭❆✸✮ ❤♦❧❞s ❛s ❛♥ ❡q✉❛❧✐t②✱ ✇❤✐❝❤ ✐♠♣❧✐❡s ✭❆✶✮ ❛s ❞❡s✐r❡❞✳ ❈♦♥✈❡rs❡❧②✱ ❝♦♥s✐❞❡r q 0 ✐♥ S(p)✳ ❊q✉❛t✐♦♥ ✭❆✷✮ ✐♠♣❧✐❡s t❤❛t p · q 0 − C(q 0 ) = max p · (q h )H h=1

H X h=1

q

h

!

−

H X

C h (q h )✳

h=1

▲❡t (q h0 ) ❜❡ s✉♣♣❧② ✈❡❝t♦rs t❤❛t ♠✐♥✐♠✐③❡ t❤❡ s✉♠ ♦❢ ❝♦sts ❝♦♥❞✐t✐♦♥❛❧ ♦♥ t♦t❛❧ s✉♣♣❧② ❜❡✐♥❣ q 0 ✳ ❚❤❡♥ ! p · q 0 − C(q 0 ) = p ·

❚❤❡r❡❢♦r❡✱ ❡❛❝❤

q h0

✐s ✐♥ S (q )✳ h

h

H X

q h0

h=1

−

H X

C h (q h0 )✳

h=1

▲❡t (p∗ , (q h∗ )h=1,...,H ) ♠❛①✐♠✐③❡ ❤♦s♣✐t❛❧ ✇❡❧❢❛r❡ s✉❜❥❡❝t t♦ ❛❧❧ ❤♦s♣✐t❛❧s ❝❤♦♦s✐♥❣ s✉♣♣❧② ♦♣t✐♠❛❧❧② ❣✐✈❡♥ p ❛♥❞ s✉❜❥❡❝t t♦ ♥♦t ♣r♦♠✐s✐♥❣ ♠♦r❡ tr❛♥s♣❧❛♥ts t❤❛♥ ❛r❡ ♣r♦❞✉❝❡❞✳ ▼❛t❤❡♠❛t✐❝❛❧❧②✱ t❤❡ t✉♣❧❡ (p∗ , q ∗ , (q h∗ )h=1,...,H ) ✐♥ RI × RI+ × RIH + ♠❛①✐♠✐③❡s Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳

f (q) −

s✉❜❥❡❝t t♦

h X

C h (q h )

h=1

H X

qh✱

✭❆✹✮

q h ∈ S h (p),

✭❆✺✮

p · q = f (q)✳

✭❆✻✮

q=

h=1

t♦ ❡❛❝❤ ❛♥❞ t♦

▲❡♠♠❛ ✶ ✐♠♣❧✐❡s t❤❛t t❤✐s ♠❛①✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐s ❡q✉✐✈❛❧❡♥t t♦ ✜♥❞✐♥❣ ❛ ♣❛✐r (p∗ , q ∗ ) ✐♥ RI × RI+ t❤❛t ♠❛①✐♠✐③❡s f (q) − C(q) ✭❆✼✮

s✉❜❥❡❝t t♦

❛♥❞ t♦ ✭❆✻✮✳

q ∈ S(p)

✭❆✽✮

❆❣❣r❡❣❛t❡ ❝♦st ✐s s♠♦♦t❤ ❛♥❞ str✐❝t❧② ❝♦♥✈❡①✱ ❛♥❞ t❤❡ s♦❧✉t✐♦♥ ✐s ✐♥t❡r✐♦r✳ ❙♦ ❝♦♥str❛✐♥t ✭❆✽✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ p = ∇C(q)✳ ❚❤✉s✱ q ∗ ♠❛①✐♠✐③❡s ✭❆✼✮ ✐♥ RI++ s✉❜❥❡❝t t♦ ∇C(q) · q = f (q)✳

✹ ❚❤❡ ♣r♦❞✉❝t✐♦♥ ❢✉♥❝t✐♦♥ ❛♥❞ ❛❣❣r❡❣❛t❡ ❝♦st ❛r❡ s♠♦♦t❤✱ ❛♥❞ RI++ ✐s ❛♥ ♦♣❡♥ s❡t✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r t❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ❡①✐sts λ s✉❝❤ t❤❛t q ∗ ♠❛①✐♠✐③❡s f (q) − C(q) + λ · {f (q) − ∇C(q) · q} ✳

❙❡tt✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡ ❡q✉❛❧ t♦ ③❡r♦✱ ✇❡ ❤❛✈❡ ∇f − ∇C =

λ q′ D2 C ✳ 1+λ

❚♦ s♦❧✈❡ ❢♦r t❤❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r✱ ✇❡ ♠✉❧t✐♣❧② ❜② q ♦♥ t❤❡ r✐❣❤t✱ ❛♥❞ ✉s❡ t❤❡ ❡q✉❛❧✐t② ∇C · q = f ✳ ❲❡ ❤❛✈❡ ∇f · q − f =

λ q ′ D 2 Cq. 1+λ

❚❤❡r❡❢♦r❡✱ (∇f · q − f )(q ′ D 2 Cq)−1 =

λ ✳ 1+λ

❙✉❜st✐t✉t✐♥❣ λ ✇❡ ❣❡t t❤❡ ✜♥❛❧ ❢♦r♠✉❧❛✱ ∇C = ∇f − (∇f · q − f ) · (q ′ D 2 Cq)−1 · (q ′ D 2 C)✳

P❛rt ✶ ♦❢ t❤❡ t❤❡♦r❡♠ ❢♦❧❧♦✇s ❜② s✉❜st✐t✉t✐♥❣ p = ∇C ✳ P❛rt ✷ ❢♦❧❧♦✇s ❜❡❝❛✉s❡✱ ✇✐t❤ ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡✱ t❤❡ ❝♦♥str❛✐♥t ✭❆✻✮ ✐s ❛✉t♦♠❛t✐❝❛❧❧② s❛t✐s✜❡❞✳ P❛rt ✸ ❢♦❧❧♦✇s ❜❡❝❛✉s❡ ✇❡❧❢❛r❡ ❛♥❞ s♦❝✐❛❧ ✇❡❧❢❛r❡ ❝♦✐♥❝✐❞❡✳

❆✳✷

❆❞❞✐t✐♦♥❛❧ Pr♦♦❢s

◆♦t❡ t❤❛t✱ ❜② ▲❡♠♠❛ ✭✶✮ ❛♥❞ t❤❡ ❢❛❝t t❤❛t q 0 ❛♥❞ q ∗ ❛r❡ ✐♥t❡r✐♦r✱ p0 = ∇C(q 0 )✱ ❛♥❞ t❤❡ ♦♣t✐♠❛❧ r❡✇❛r❞ ✈❡❝t♦r p∗ = ∇C(q ∗ )✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ t❛❦❡ q 0 s✉✣❝✐❡♥t❧② ❝❧♦s❡ t♦ q ∗ s✉❝❤ t❤❛t t❤❡ ♠❛tr✐① D2 C(q 0 ) − D2 f (q 0 ) ✐s ♥♦♥✲s✐♥❣✉❧❛r✳

Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✶✳

P❛rt ✶✿ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ q ∗ − q 0 ✳

❚❤❡♦r❡♠ ✭✶✮ ✐♠♣❧✐❡s t❤❛t

❚❛❦✐♥❣ ❞❡r✐✈❛t✐✈❡s✱ ✇❡ ❤❛✈❡

∇C(q ∗ ) − ∇f (q ∗ ) = −A✳

∇C(q 0 ) − ∇f (q 0 ) + (q ∗ − q 0 )′ (D 2 C(q 0 ) − D 2 f (q 0 )) + ǫ1 (q 0 ) = −A✱

✇❤❡r❡ ǫ1 (q 0 ) ✐s ✐♥ o(kq ∗ − q 0 k)✳ ❚❤❡r❡❢♦r❡✱ q ∗ − q 0 = (D 2 C(q 0 ) − D 2 f (q 0 ))−1 · (∇f (q 0 ) − p0 )′ + ǫ2 (q 0 )✱

✇❤❡r❡ t❤❡ ❡rr♦r t❡r♠ ǫ2 (q 0 ) = −(D 2 C(q 0 ) − D 2 f (q 0 ))−1 (ǫ1 (q 0 ) + A)′ ✳

✭❆✾✮

✺ P❛rt ✷✿ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss✳ ❉❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ g(t) = f (q ∗ ) − C(q ∗ ) − [f (q ∗ + t · (q 0 − q ∗ )) − C(q ∗ + t · (q 0 − q ∗ ))]✳

❲❡ ❤❛✈❡ t❤❛t g(0) = 0✱ ❛♥❞ g(1) ✐s t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ g ❡q✉❛❧s g ′ (t) = −[∇f (q ∗ + t · (q 0 − q ∗ )) − ∇C(q ∗ + t · (q 0 − q ∗ ))] · (q 0 − q ∗ )✳

❚❛❦✐♥❣ ❞❡r✐✈❛t✐✈❡s ✇❡ ❤❛✈❡ g ′ (t) = −[∇f (q ∗ ) − ∇C(q ∗ ) + t(q 0 − q ∗ )′ (D 2 f (q 0 ) − D 2 C(q 0 )) + ǫ3 (q 0 )] · (q 0 − q ∗ )✱

✇❤❡r❡ t❤❡ ❡rr♦r ǫ3 (q 0 ) ✐s ✐♥ o(kq ∗ − q 0 k)✳ ❇② ❚❤❡♦r❡♠ ✭✶✮✱ g ′ (t) =t(q ∗ − q 0 )′ (D 2 C(q 0 ) − D 2 f (q 0 )(q ∗ − q 0 ) + [A + ǫ3 (q 0 )] · (q ∗ − q 0 )✳

❲❡ ❝❛♥ s✉❜st✐t✉t❡ (D2 C(q 0 ) − D2 f (q 0 )(q ∗ − q 0 ) ✉s✐♥❣ ❡q✉❛t✐♦♥ ✭❆✾✮✳ ❚❤✐s ②✐❡❧❞s g ′ (t) =t(q ∗ − q 0 )′ · (∇f (q 0 ) − p0 )′ − t(q ∗ − q 0 )′ · (ǫ1 (q 0 ) + A)′ + [A + ǫ3 (q 0 )] · (q ∗ − q 0 )✳

■♥t❡❣r❛t✐♥❣ ✐♥ t ❢r♦♠ 0 t♦ 1✱ ❛♥❞ ♥♦t✐♥❣ t❤❛t t❤❡ ❞♦t ♣r♦❞✉❝ts ❛r❡ t❤❡ s❛♠❡ ✐❢ ✇❡ t❛❦❡ tr❛♥s♣♦s❡s✱ t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ❡q✉❛❧s 1 g(1) = (∇f (q 0 ) − p0 ) · (q ∗ − q 0 ) 2 1 + A · (q ∗ − q 0 ) 2   1 + ǫ3 (q 0 ) − ǫ1 (q 0 ) · (q ∗ − q 0 )✳ 2

✭❆✶✵✮

❚❤✐s ❡st❛❜❧✐s❤❡s t❤❡ ✜rst ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛✳ ❚♦ ❡st❛❜❧✐s❤ t❤❡ s❡❝♦♥❞ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛ ✇❡ s✉❜st✐t✉t❡ q ∗ − q 0 ❢r♦♠ ❡q✉❛t✐♦♥ ✭❆✾✮✱ ②✐❡❧❞✐♥❣ 1 g(1) = (∇f (q 0 ) − p0 )(D 2 C(q 0 ) − D 2 f (q 0 ))−1 · (∇f (q 0 ) − p0 )′ 2 1 − (∇f (q 0 ) − p0 )(D 2 C(q 0 ) − D 2 f (q 0 ))−1 A′ + A · (q ∗ − q 0 ) 2 + ǫ4 (q 0 )✱

✇❤❡r❡ ǫ4 (q 0 ) ✐s ✐♥ o(kq ∗ − q 0 k2 )✳

✭❆✶✶✮

✻ ❚❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t C h ❛♥❞ C ❛r❡ str✐❝t❧② ❝♦♥✈❡① ✐♠♣❧✐❡s t❤❛t t❤❡ s✉♣♣❧② ❛♥❞ ❛❣❣r❡❣❛t❡ s✉♣♣❧② ❝♦rr❡s♣♦♥❞❡♥❝❡s ❛r❡ s✐♥❣❧❡✲✈❛❧✉❡❞✳ ❉❡✜♥❡ t❤❡ ❛❣❣r❡❣❛t❡ ❡①t❡r♥❛❧✐t② ❣✐✈❡♥ q✉❛♥t✐t② q ❛s ˜ E(q) = E(S 1 (∇C(q)), . . . , S 1 (∇C(q)))✳ ✭❆✶✷✮ Pr♦♣♦s✐t✐♦♥ ✷✳

▲❡t (p∗ , (q h∗ )h=1,...,H ) ♠❛①✐♠✐③❡ s♦❝✐❛❧ ✇❡❧❢❛r❡ s✉❜❥❡❝t t♦ ❛❧❧ ❤♦s♣✐t❛❧s ❝❤♦♦s✐♥❣ s✉♣♣❧② ♦♣t✐✲ ♠❛❧❧② ❣✐✈❡♥ p ❛♥❞ s✉❜❥❡❝t t♦ ♥♦t ♣r♦♠✐s✐♥❣ ♠♦r❡ tr❛♥s♣❧❛♥ts t❤❛♥ ❛r❡ ♣r♦❞✉❝❡❞✳ ▼❛t❤❡♠❛t✲ ✐❝❛❧❧②✱ t❤❡ ♣❛✐r (p∗ , (q h∗ )h=1,...,H )✱ ✐♥ RI × RIH + ♠❛①✐♠✐③❡s f (q) −

H X

h

h

C (q ) +

h=1

H X

E h (q h )

h=1

s✉❜❥❡❝t t♦ ❝♦♥str❛✐♥ts ✭❆✹✮✱ ✭❆✺✮✱ ❛♥❞ ✭❆✻✮✳ ❇② ▲❡♠♠❛ ✶✱ t❤❡ ♠❛①✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐s ❡q✉✐✈❛❧❡♥t t♦ ✜♥❞✐♥❣ (p∗ , q ∗ ) ✐♥ RI × RI+ t❤❛t ♠❛①✐♠✐③❡ f (q) − C(q) + E(S 1 (p), . . . , S h (p))

s✉❜❥❡❝t t♦ ❝♦♥str❛✐♥ts ✭❆✽✮ ❛♥❞ ✭❆✻✮✳ ❇❡❝❛✉s❡ t❤❡ ♠❛①✐♠✉♠ ✐s ❛ss✉♠❡❞ t♦ ❜❡ str✐❝t❧② ♣♦s✐t✐✈❡✱ ✇❡ ❝❛♥ r❡str✐❝t ❛tt❡♥t✐♦♥ t♦ q ✐♥ RI++ ✳ ■♥ t❤✐s ❝❛s❡✱ ❜❡❝❛✉s❡ C ✐s s♠♦♦t❤ ❛♥❞ ❝♦♥✈❡①✱ ❝♦♥str❛✐♥t ✭❆✽✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ p = ∇C(q)✳ ❍❡♥❝❡✱ t❤❡ ♠❛①✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐s ❡q✉✐✈❛❧❡♥t t♦ ✜♥❞✐♥❣ q ∗ ✐♥ RI++ t❤❛t ♠❛①✐♠✐③❡s ˜ f (q) − C(q) + E(q)

s✉❜❥❡❝t t♦

∇C(q) · q = f (q)✳

❇② t❤❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r t❤❡♦r❡♠✱ t❤❡r❡ ❡①✐sts ❛ ♠✉❧t✐♣❧✐❡r λSW s✉❝❤ t❤❛t✱ ❛t t❤❡ ♦♣t✐♠✉♠✱ ˜ + λSW (∇f − ∇C − q ′ D 2 C) = 0✳ ∇f − ∇C + ∇E

❙✉❜st✐t✉t✐♥❣ p∗ = ∇C(q ∗ ) ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞❥✉st♠❡♥t t❡r♠✳ ❚♦ ♦❜t❛✐♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r ✇❡ ♠✉❧t✐♣❧② ♦♥ t❤❡ r✐❣❤t ❜② q ∗ ❛♥❞ ✉s❡ t❤❡ ❢❛❝t t❤❛t ∇f (q ∗ ) · q ∗ = p∗ · q ∗ = f (q ∗ )✳ ❚❤❡ s❡❝♦♥❞ ❛♥❞ t❤✐r❞ ♣❛rts ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥ ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r✳

Pr♦♣♦s✐t✐♦♥ ✸✳

❚❤❡ ♣❧❛t❢♦r♠ ❝❤♦♦s❡s q ✐♥ RI+ t♦ ♠❛①✐♠✐③❡ f (q)

s✉❜❥❡❝t t♦

f (q) = PS (q) · q ✳

❇❡❝❛✉s❡ t❤❡ s♦❧✉t✐♦♥ ✐s ✐♥t❡r✐♦r✱ t❤❡r❡ ❡①✐sts ❛ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r λ s✉❝❤ t❤❛t ∇f + λ(∇f − PS − q ′ DPS ) = 0✳

✼ ❙✉❜st✐t✉t✐♥❣ t❤❛t t❤❡ ♦♣t✐♠❛❧ r❡✇❛r❞s p = PS (q)✱ ✇❡ ♦❜t❛✐♥ 1 p = ∇f + ∇f − q ′ DPS ✳ λ

❚♦ ❝❛❧❝✉❧❛t❡ t❤❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r✱ ✇❡ r✐❣❤t ♠✉❧t✐♣❧② ❜② q ′ ❛♥❞ ✉s❡ pq ′ = ∇f q ′ = f (q) t♦ ♦❜t❛✐♥ f (q) ✳ λ= ′ q DPS (q)q

❚❤❡s❡ t✇♦ ❢♦r♠✉❧❛s ✐♠♣❧② ♣❛rt ✶ ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥ st❛t❡♠❡♥t✳ ❚❤❡ ♦❜s❡r✈❛t✐♦♥ ✐♥ ♣❛rt ✷ ❢♦❧❧♦✇s ❞✐r❡❝t❧② ❢r♦♠ t❤❡ ❢♦r♠✉❧❛ ✐♥ ♣❛rt ✶✳

❇ ❉❛t❛ ❆♣♣❡♥❞✐① ❚❤✐s st✉❞② ✉s❡❞ ✜✈❡ ♠❛✐♥ ❛♥♦♥②♠✐③❡❞ ❞❛t❛ s❡ts✿ ❛ ❞❛t❛❜❛s❡ ♦❢ ❛❧❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s✲ ♣❧❛♥ts ❞♦♥❡ ✐♥ t❤❡ ❯❙ ❢r♦♠ ❏❛♥✉❛r② ✶✱ ✷✵✵✽ t❤r♦✉❣❤ ❉❡❝❡♠❜❡r ✹✱ ✷✵✶✹ ✭t❤❡ ❖P❚◆ tr❛♥s♣❧❛♥t ❞❛t❛✮✱ ❞❛t❛❜❛s❡s ♦❢ ❛❧❧ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ♦r❣❛♥✐③❡❞ ❜② ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ❧❛r❣❡st ♠✉❧t✐✲❤♦s♣✐t❛❧ ♣❧❛t❢♦r♠s ✐♥ t❤❡ ❯❙ ✭t❤❡ ◆❑❘✱ ❆P❉✱ ❛♥❞ ❯◆❖❙ tr❛♥s♣❧❛♥t ❞❛t❛✮✱ ❛♥❞ ❛ ❞❛t❛❜❛s❡ ♦❢ ❛❧❧ ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r r❡❣✐str❛t✐♦♥s t♦ t❤❡ ❧❛r❣❡st ♦❢ t❤♦s❡ ♣❧❛t❢♦r♠s ✭t❤❡ ◆❑❘ r❡❣✐str❛t✐♦♥ ❞❛t❛✮✳ ❇✳✶

❚r❛♥s♣❧❛♥t ❞❛t❛

■♥ ♦r❞❡r t♦ ❞♦❝✉♠❡♥t t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♠❛r❦❡t✱ ✇❡ ♠❡r❣❡❞ t❤❡ ❖P❚◆ tr❛♥s♣❧❛♥t ❞❛t❛ ✇✐t❤ t❤❡ tr❛♥s♣❧❛♥t ❞❛t❛ ❢r♦♠ ◆❑❘✱ ❆P❉✱ ❛♥❞ ❯◆❖❙✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ ✇✐❧❧ ❞❡s❝r✐❜❡ t❤❡s❡ ❞❛t❛ ❛♥❞ t❤❡ ♠❡r❣❡ ♣r♦❝❡❞✉r❡ ✇❡ ✉s❡❞✳

❖❜t❛✐♥✐♥❣ t❤❡ ❞❛t❛s❡ts

❚❤❡ ❖P❚◆ ♣r♦✈✐❞❡❞ ✉s ✇✐t❤ ❛ ❞❛t❛s❡t ♦♥ ❛❧❧ tr❛♥s♣❧❛♥ts ❝♦♥✲ ❞✉❝t❡❞ ✐♥ t❤❡ ❯❙✱ ❦♥♦✇♥ ❛s t❤❡ ❙t❛♥❞❛r❞ ❚r❛♥s♣❧❛♥t ❆♥❛❧②s✐s ❛♥❞ ❘❡s❡❛r❝❤ ✭❙❚❆❘✮ ❞❛t❛s❡t✳ ❚❤❡ ❙❚❆❘ ❞❛t❛s❡t ❜② ✐ts❡❧❢ ❧❛❝❦s t✇♦ ❦❡② ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥✿ t❤❡ tr❛♥s♣❧❛♥t ❤♦s♣✐t❛❧s ✇❤❡r❡ t❤❡ ❦✐❞♥❡② ✇❛s ♣✉t ✐♥t♦ t❤❡ ♣❛t✐❡♥t ❛♥❞ r❡♠♦✈❡❞ ❢r♦♠ t❤❡ ❞♦♥♦r ✭✇❤✐❝❤ ✇❡ ✉s❡ t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❛ tr❛♥s♣❧❛♥t ✐s ✐♥t❡r♥❛❧ ♦r ❡①t❡r♥❛❧✮ ❛♥❞ t❤❡ ✉♥❛❝❝❡♣t❛❜❧❡ ❛♥t✐❣❡♥s ❢♦r t❤❡ ♣❛t✐❡♥t ✭✇❤✐❝❤ ✇❡ ♥❡❡❞ t♦ ♠❡❛s✉r❡ s❡♥s✐t✐③❛t✐♦♥✮✳ ❚❤❡s❡ s✉♣♣❧❡♠❡♥t❛❧ ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❛r❡ ❛❧s♦ ❛✈❛✐❧❛❜❧❡ ❢r♦♠ t❤❡ ❖P❚◆ ♦♥ r❡q✉❡st✳ ▼❡r❣✐♥❣ ✐s ❞♦♥❡ ❜② ✉s✐♥❣ ❖P❚◆ ✐❞❡♥t✐✜❡rs✳ ❚❤❡ ❖P❚◆ ❞❛t❛❜❛s❡ ❝♦♥t❛✐♥s r❡❝♦r❞s ♦♥ ✹✸✼✼ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts✳ ❲❡ ♦❜t❛✐♥❡❞ ❡❛❝❤ ♦❢ t❤❡ ♣❧❛t❢♦r♠ ❞❛t❛s❡ts ❞✐r❡❝t❧② ❢r♦♠ t❤❡ ♣❧❛t❢♦r♠✳ ❚❤❡ ♣❧❛t❢♦r♠ ❞❛t❛s❡ts ❝♦♥t❛✐♥ r❡❝♦r❞s ♦♥ ✶✹✵✵ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ tr❛♥s♣❧❛♥ts ✐♥ t♦t❛❧✿ ✶✶✾✸ ❢r♦♠ ◆❑❘✱ ✶✵✵ ❢r♦♠ ❆P❉✱ ❛♥❞ ✶✵✼ ❢r♦♠ ❯◆❖❙✳

✽

❉❛t❛s❡t ♠❡r❣❡ ❛❧❣♦r✐t❤♠

■♥ ♦r❞❡r t♦ ✐❞❡♥t✐❢② ✇❤✐❝❤ tr❛♥s♣❧❛♥ts ✐♥ t❤❡ ❝♦♠♣r❡❤❡♥s✐✈❡ ❖P❚◆ ❞❛t❛❜❛s❡ ✇❡r❡ ♦r❣❛♥✐③❡❞ ❜② ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ♣❧❛t❢♦r♠s✱ ✇❡ ♠❛t❝❤❡❞ r❡❝♦r❞s ✐♥ t❤❡ ♣❧❛t❢♦r♠ ✜❧❡s t♦ r❡❝♦r❞s ✐♥ t❤❡ ❖P❚◆ ✜❧❡✳ ❍♦✇❡✈❡r✱ ❜❡❝❛✉s❡ ❛❧❧ ❞❛t❛s❡ts ❛r❡ ❛♥♦♥②♠✐③❡❞✱ t❤❡ ♠❡r❣❡ ♠✉st ❜❡ ❞♦♥❡ ♦♥ t❤❡ ❜✐♦❧♦❣✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❡❛❝❤ tr❛♥s♣❧❛♥t✬s r❡❝✐♣✐❡♥t ❛♥❞ ❞♦♥♦r ❛♥s ❧♦❣✐st✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ tr❛♥s♣❧❛♥t ✐ts❡❧❢✳ ❋♦rt✉♥❛t❡❧②✱ ✇❡ ❝❛♥ ❛❧s♦ ✉s❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ❖P❚◆ ❞❛t❛❜❛s❡ ✐s ❝♦♠♣r❡❤❡♥s✐✈❡ ✇❤✐❧❡ t❤❡ ♣❧❛t❢♦r♠ ❞❛t❛❜❛s❡ ✐s ♥♦t✳✶ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ tr❛♥s♣❧❛♥ts ✐♥ ♣❧❛t❢♦r♠ ❞❛t❛ ❛r❡ ❛ s✉❜s❡t ♦❢ tr❛♥s♣❧❛♥ts ✐♥ t❤❡ ❖P❚◆ ❞❛t❛ ✐❢ r❡❝♦r❞✲❦❡❡♣✐♥❣ ✐s ♣❡r❢❡❝t✳ ❍♦✇❡✈❡r✱ ✇✐t❤♦✉t ♣❡r❢❡❝t r❡❝♦r❞✲ ❦❡❡♣✐♥❣✱ ✇❡ s❤♦✉❧❞ ❜❡ ✇❛r② ♦❢ ❢❛❧s❡ ♠❛t❝❤❡s✳ ❖✉r ♠❛t❝❤✐♥❣ ♣r♦❝❡❞✉r❡ ✐s ❞❡s✐❣♥❡❞ t♦ ❧✐♠✐t ❢❛❧s❡ ♠❛t❝❤❡s ❜② ❦❡❡♣✐♥❣ ♦♥❧② t❤❡ ❤✐❣❤❡st q✉❛❧✐t② ♠❛t❝❤ t❤❛t ♠❡❡ts ❛ ♠✐♥✐♠✉♠ t❤r❡s❤♦❧❞✳ ❲❡ ❢♦r♠❛❧✐③❡ t❤✐s ✐❞❡❛ ❜❡❧♦✇✳ ▲❡t t❤❡ s❡t ♦❢ r❡❝♦r❞s ✐♥ t❤❡ ♣❧❛t❢♦r♠ ❞❛t❛ ❜❡ Dp ❛♥❞ t❤❡ s❡t ✐♥ t❤❡ ❖P❚◆ ❞❛t❛ ❜❡ Do ✳ ▲❡t t❤❡ ✉♥✐✈❡rs❡ ♦❢ ❛❝❝❡♣t❛❜❧❡ ♠❛t❝❤❡s ♦❢ ♦♥❡ ♣❧❛t❢♦r♠ r❡❝♦r❞ rp ∈ Dp t♦ ♦♥❡ ❖P❚◆ r❡❝♦r❞ ro ∈ Do ❜❡ r❡♣r❡s❡♥t❡❞ ❜② U ⊆ Dp × Do ✳ ❉❡✜♥❡ (rp , ro ) ❤❛s  ′ ′ t❤❡
s❡t ♦❢′ ❝♦❧❧✐s✐♦♥s′ ❛ ♠❛t❝❤ r❡❧❛t✐✈❡ t♦ s♦♠❡ s❡t X t♦ ❜❡ κ (rp , ro , X) ≡ rp , ro ∈ X | rp = rp ♦r ro = ro \ {(rp , ro )}✳ ❆ s❡t ♦❢ ♠❛t❝❤❡s M ⊆ U ✐s ❝❛❧❧❡❞ ❛ ♠❡r❣❡ ✐❢ ♥♦♥❡ ✐ts ♠❛t❝❤❡s ❤❛✈❡ ❛♥② ❝♦❧❧✐s✐♦♥s r❡❧❛t✐✈❡ t♦ t❤❡ ♠❡r❣❡✱ t❤❛t ✐s✱ ❢♦r ❡❛❝❤ (rp , ro ) ∈ M ✱ ✇❡ ❤❛✈❡ κ (rp , ro , M ) = ∅✳ ❆♥② r❡❝♦r❞ t❤❛t ✐s ♥♦t ♣❛rt ♦❢ ❛ ♠❛t❝❤ ✐♥ ♠❡r❣❡ M s❤♦✉❧❞ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ✉♥♠❛t❝❤❡❞✳ ◆♦✇✱ ✇❡ ❝❛♥ ❞✐s❝✉ss ✇❤❛t ♠❛❦❡s ❛♥ ✐♥❞✐✈✐❞✉❛❧ ♠❛t❝❤ ❣♦♦❞✳ ❉❡✜♥❡ ❛ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ t♦ n n+1 ❜❡ ❛ s❡r✐❡s ♦❢ N s✉❜s❡ts ♦❢ t❤❡ ✉♥✐✈❡rs❡ C = (C n )N ✱ C 0 = ∅✱ ❛♥❞ n=0 s✉❝❤ t❤❛t C ⊆ C C N = U ✳ ❚❤❡ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ ❝♦❞✐✜❡s ❛ ❤✐❡r❛r❝❤② ♦❢ ♠❛t❝❤ q✉❛❧✐t② ❧❡✈❡❧s✳ ❋✉rt❤❡r✱ ❞❡✜♥❡ t❤❡ r❛♥❦ ♦❢ ❛ ♠❛t❝❤ (rp , ro ) t♦ ❜❡ ρ (rp , ro ) ≡ min {n | (rp , ro ) ∈ C n }✳✷

❋✐♥❛❧❧②✱ ✇❡ ❛r❡ r❡❛❞② t♦ st❛t❡ ♦✉r ♥♦t✐♦♥ ♦❢ ♠❡r❣❡ q✉❛❧✐t②✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t❤❛t ❛ ♠❛t❝❤ ♦❢ r❛♥❦ n s❤♦✉❧❞ ❜❡ ✐♥ t❤❡ ♠❡r❣❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s t❤❡ ✉♥✐q✉❡ ♠❛t❝❤ ♦❢ r❛♥❦ n ♦r ❜❡tt❡r ✇❤♦s❡ ❝♦♠♣♦♥❡♥t r❡❝♦r❞s ❛r❡♥✬t ♣❛rt ♦❢ ❛ ❜❡tt❡r ♠❛t❝❤ ✐♥ t❤❡ ♠❡r❣❡✳ ❋♦r ❛ ❣✐✈❡♥ s❡t X ❛♥❞ r❛♥❦ ❧❡✈❡❧ n✱ ❞❡✜♥❡ t❤❡ ✉♥✐✈❡rs❡ ❡①❝❧✉❞✐♥❣ r❡❝♦r❞s t❤❛t ♠❛t❝❤❡❞ ❜❡tt❡r t❤❛♥ n ✐♥ X t♦ ❜❡   U n (X) ≡ U \ 

(rp ,ro )∈X∩C n−1

❚❤❡♥✱ ❛ s❡t ♦❢ ♠❛t❝❤❡s M ⊆ U ✐s st❛❜❧❡ (rp , ro ) ∈ M ⇔

  

[

κ (rp , ro , U ) .

r❡❧❛t✐✈❡ t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ C ✐❢ ✐t s❛t✐s✜❡s (rp , ro ) ∈ U ρ(rp ,ro ) (M )

κ rp , ro , C

ρ(rp ,ro )

❛♥❞

∩U

ρ(rp ,ro )




(M ) = ∅.

 

.



❚❤❡ ✜rst ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❡♥s✉r❡s t❤❛t (rp , ro ) ❞♦❡s♥✬t ❝♦❧❧✐❞❡ ✇✐t❤ ❛ str✐❝t❧② ❜❡tt❡r ♠❛t❝❤ ✐♥ M ✱ ✇❤✐❧❡ t❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ❡♥s✉r❡s t❤❛t (rp , ro ) ✐s t❤❡ ✉♥✐q✉❡ s✉❝❤ ♠❛t❝❤✳ ✶ ❚❤❡ ❖P❚◆ ✐s r❡q✉✐r❡❞ ❜② ❢❡❞❡r❛❧ ❛❞♠✐♥✐str❛t✐✈❡ ❧❛✇ t♦ ✏ ❬♠❪❛✐♥t❛✐♥ r❡❝♦r❞s ♦❢ ❛❧❧ tr❛♥s♣❧❛♥t ❝❛♥❞✐❞❛t❡s✱ ❛❧❧ ♦r❣❛♥ ❞♦♥♦rs ❛♥❞ ❛❧❧ tr❛♥s♣❧❛♥t r❡❝✐♣✐❡♥ts✑ ✭✹✷ ❈✳❋✳❘✳ ➓ ✶✷✶✳✶✶✭❛✮✭✶✮✭✐✐✮✮✳

✷ ◆♦t❡ t❤❛t ✇❡ ❝♦✉❧❞ ❡q✉✐✈❛❧❡♥t❧② st❛rt ✇✐t❤ ❛ r❛♥❦ ❢✉♥❝t✐♦♥ ❛♥❞ ❝♦♥str✉❝t

♦❢ t❤❛t ❢✉♥❝t✐♦♥✳

C

t♦ ❜❡ t❤❡ ❧♦✇❡r ❝♦♥t♦✉r s❡ts

✾ ◆♦t❡ t❤❛t ❛♥② s❡t M t❤❛t ✐s st❛❜❧❡ r❡❧❛t✐✈❡ t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ C ✐s ♥❡❝❡ss❛r✐❧② ❛ ♠❡r❣❡ ✭s❡❡ ▲❡♠♠❛ ✺ ❜❡❧♦✇✮✳ ❋✐♥❞✐♥❣ ❛ st❛❜❧❡ ♠❡r❣❡ r❡❧❛t✐✈❡ t♦ s♦♠❡ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ C ✐s ❛ s✐♠♣❧❡ ♠❛tt❡r ♦❢ ❢♦❧❧♦✇✐♥❣ ❆❧❣♦r✐t❤♠ ✶✳ ■t ✐s ❛❧s♦ tr✉❡ t❤❛t ❢♦r ❛ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ C ✱ t❤❡ st❛❜❧❡ ♠❡r❣❡ ✐s ✉♥✐q✉❡✱ s♦ ✇❡ ♥❡❡❞ ♥♦t ✇♦rr② t❤❛t t❤❡r❡ ✐s s♦♠❡ ♦t❤❡r st❛❜❧❡ ♠❡r❣❡ t❤❛t ✇❡ ♠✐❣❤t ♣r❡❢❡r ♦♥ ❞✐✛❡r❡♥t ❣r♦✉♥❞s✳ Pr♦♦❢s ♦❢ t❤❡s❡ ❝❧❛✐♠s ✭✇❤✐❝❤ ❛r❡ s✉♠♠❛r✐③❡❞ ✐♥ Pr♦♣♦s✐t✐♦♥ ✶✮ ❝❛♥ ❜❡ ❢♦✉♥❞ ❛t t❤❡ ❜♦tt♦♠ ♦❢ t❤✐s s✉❜s❡❝t✐♦♥✳ M 0 = ∅✳ ❚❤❡♥ ❢♦r n ∈ {1, . . . , N }✱
X n = C n ∩ U n M n−1 M n = M n−1 ∪ {(rp , ro ) ∈ X n | κ (rp , ro , X n ) = ∅} .

❆❧❣♦r✐t❤♠ ✶✳ ■♥✐t✐❛❧✐③❡ ❜② s❡tt✐♥❣

❚❤❡ ♦✉t♣✉t ♠❡r❣❡ ✐s

MN✳

❇❛s✐❝❛❧❧②✱ ✐♥ ❙t❡♣ n✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛❞❞s ❛♥② ♠❛t❝❤ t♦ t❤❡ ♦✉t♣✉t s❡t t❤❛t ✐s t❤❡ ✉♥✐q✉❡ r❛♥❦✲n ♦r ❜❡tt❡r ♠❛t❝❤ ✇❤♦s❡ r❡❝♦r❞s ❛r❡ ♥♦t ✐♥✈♦❧✈❡❞ ✐♥ ❛ r❛♥❦ n − 1 ♦r ❜❡tt❡r ♠❛t❝❤ t❤❛t ✐s ❛❧r❡❛❞② ✐♥ t❤❡ ♦✉t♣✉t s❡t✳ Pr♦♣♦s✐t✐♦♥ ✶✳ ❆❧❣♦r✐t❤♠ ✶ ②✐❡❧❞s t❤❡ ✉♥✐q✉❡ s❡t t❤❛t ✐s st❛❜❧❡ r❡❧❛t✐✈❡ t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥

C✳

❋✉rt❤❡r♠♦r❡✱ t❤✐s s❡t ✐s ❛ ♠❡r❣❡✳

❲❡ ♠❡r❣❡❞ t❤❡ ❖P❚◆ ❛♥❞ ♣❧❛t❢♦r♠ tr❛♥s♣❧❛♥t ❞❛t❛❜❛s❡s ✉s✐♥❣ ❆❧❣♦r✐t❤♠ ✶✳ ❚❤❡ s❡t ♦❢ ❛❝❝❡♣t❛❜❧❡ ♠❛t❝❤❡s✱ U ✱ ✇❛s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦t❡♥t✐❛❧ ♠❛t❝❤❡s ✇❤❡r❡ t❤❡ tr❛♥s♣❧❛♥t ❞❛t❡s ❛r❡ ✇✐t❤✐♥ ✸✶ ❞❛②s ♦❢ ❡❛❝❤ ♦t❤❡r ❛♥❞ t❤❡ ❛❣❡s ♦❢ t❤❡ ❞♦♥♦r ❛♥❞ t❤❡ r❡❝✐♣✐❡♥t ❛r❡ ❡❛❝❤ ✇✐t❤✐♥ ✶✵ ②❡❛rs ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ ✇❡ ✉s❡❞ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✳ ◗✉❛❧✐t② ♦❢ t❤❡ ♠❡r❣❡

❼ C 1 ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❛❝❝❡♣t❛❜❧❡ ♠❛t❝❤❡s ✇❤❡r❡ t❤❡ ❞♦♥♦r ❛♥❞ r❡❝✐♣✐❡♥t ❡❛❝❤ ♠❛t❝❤ ❡①❛❝t❧②

♦♥ ❜❧♦♦❞ t②♣❡✱ s❡①✱ t❤❡ ❤♦s♣✐t❛❧ ✇❤❡r❡ t❤❡ tr❛♥s♣❧❛♥t ✇❛s ❝♦♥❞✉❝t❡❞✱ ❛♥❞ ❛❧❧ s✐① ♠❛❥♦r ❍▲❆ ❛❧❧❡❧❡s ✭t✇♦ ❡❛❝❤ ♦♥ t❤❡ ❍▲❆✲❆✱ ❍▲❆✲❇✱ ❛♥❞ ❍▲❆✲❉❘ ❧♦❝✐✮✳

❼ ❋♦r n ∈ {2, . . . , 5}✱ C n ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❛❝❝❡♣t❛❜❧❡ ♠❛t❝❤❡s ✇❤❡r❡ t❤❡ ❞♦♥♦r ❛♥❞ r❡❝✐♣✐❡♥t ❡❛❝❤ ♠❛t❝❤ ♦♥ ❡✐t❤❡r ❜❧♦♦❞ t②♣❡ ♦r s❡① ❛♥❞ ❛❧s♦ ❡❛❝❤ ♠❛t❝❤ ♦♥ ❛t ❧❡❛st 7 − n ♦✉t ♦❢

t❤❡ s✐① ♠❛❥♦r ❍▲❆ ❛❧❧❡❧❡s✳

●✐✈❡♥ t❤❡ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ t❤❡ ♠❡r❣❡ ❛❧❣♦r✐t❤♠ ♣❡r❢♦r♠❡❞ ✇❡❧❧✳ ❚❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ ♣❧❛t❢♦r♠ r❡❝♦r❞s ♠❛t❝❤❡❞ t♦ ❛♥ ❖P❚◆ r❡❝♦r❞ ✇❛s ✾✹✪ ♦✈❡r❛❧❧ ✭✾✹✪ ❢♦r ◆❑❘✱ ✾✼✪ ❢♦r ❆P❉✱ ❛♥❞ ✾✹✪ ❢♦r ❯◆❖❙✮✳ ▼♦r❡♦✈❡r✱ t❤❡ ♠❛t❝❤❡s s❡❡♠ t♦ ❜❡ ❤✐❣❤ q✉❛❧✐t②✿ ❚❛❜❧❡ ❇✶ r❡♣♦rts t❤❡ ♣❡r❝❡♥t❛❣❡ ♦❢ ♠❛t❝❤❡s t❤❛t ♠❡❡t ✈❛r✐♦✉s ❝r✐t❡r✐❛✳

✶✵ ❚❛❜❧❡ ❇✶✿ ❆❣r❡❡♠❡♥t ❇❡t✇❡❡♥ ▼❛t❝❤❡❞ ❘❡❝♦r❞s ✐♥ t❤❡ ❚r❛♥s♣❧❛♥t ▼❡r❣❡ At most 1 No Transplant 5 or more Blood type Transplant Age within criterion to criterion to date within HLA alleles and gender center 5 years the left is the left is 1 day match match matches violated violated

Platform NKR APD UNOS All of the above

97.8% 92.8% 87.1%

97.4% 95.9% 99.0%

94.9% 93.8% 96.0%

95.1% 96.9% 99.0%

97.2% 95.9% 99.0%

97.6% 96.9% 99.0%

87.6% 90.7% 83.2%

96.6%

97.4%

94.9%

95.5%

97.3%

97.6%

89.9%

❈♦♥✜r♠✐♥❣ t❤❡ ♠❡r❣❡ ✐s ❝♦rr❡❝t ■t ✇❛s ♥❡❝❡ss❛r② t♦ ✉s❡ t❤❡ ♠❡r❣❡ ❞❡s❝r✐❜❡❞ ❛❜♦✈❡ t♦

♠❛t❝❤ tr❛♥s♣❧❛♥ts ✐♥ t❤❡ ❆P❉ ❛♥❞ ◆❑❘ ❞❛t❛❜❛s❡s t♦ t❤❡ ❖P❚◆ tr❛♥s♣❧❛♥t ❞❛t❛❜❛s❡✱ s✐♥❝❡ ❛❧❧ ♦❢ t❤❡s❡ ❞❛t❛s❡ts ❛r❡ ❛♥♦♥②♠✐③❡❞ t♦ ❞✐✛❡r❡♥t s❡ts ♦❢ ✐❞❡♥t✐✜❡rs✳ ❍♦✇❡✈❡r✱ s✐♥❝❡ ❯◆❖❙ ✐s ❛ ❝♦♥tr❛❝t♦r ❢♦r t❤❡ ❖P❚◆✱ t❤❡ ❯◆❖❙ ❛♥❞ ❖P❚◆ ❞❛t❛❜❛s❡s s❤❛r❡ ❝♦♠♠♦♥ ✐❞❡♥t✐✜❡rs t❤❛t ❛❧❧♦✇ ✉s t♦ s❡❡ t❤❡ ❛❝t✉❛❧ tr✉❡ ♠❡r❣❡ ♦❢ t❤❡ ❯◆❖❙ tr❛♥s♣❧❛♥t ❞❛t❛s❡t t♦ t❤❡ ❖P❚◆ ❞❛t❛s❡t✳ ❈♦♠♣❛r✐♥❣ tr✉❡ ♠❛t❝❤❡s t♦ t❤❡ ♠❛t❝❤❡s s❡❧❡❝t❡❞ ❜② ♦✉r ❛❧❣♦r✐t❤♠✱ ✇❡ ✜♥❞ t❤❛t ♦♥❧② ♦♥❡ ❯◆❖❙ r❡❝♦r❞ ✇❛s ✐♥❝♦rr❡❝t❧② ♠❛t❝❤❡❞✳ ❚❤❛t ✐s✱ ♦✉r ❛❧❣♦r✐t❤♠ ❝❤♦s❡ t❤❡ ❝♦rr❡❝t ♠❛t❝❤ ✐♥ t❤❡ ❖P❚◆ ❞❛t❛s❡t ❢♦r ✾✾✪ ♦❢ t❤❡ ❯◆❖❙ r❡❝♦r❞s ✐t ♠❛t❝❤❡❞✳ ❚❤✐s ❣✐✈❡s ✉s ❛❞❞❡❞ ❝♦♥✜❞❡♥❝❡ t❤❛t ♦✉t ♠❡r❣❡ ❛❧❣♦r✐t❤♠ ✐s ✇♦r❦✐♥❣ ✇❡❧❧✳

Pr♦♦❢s ❝♦♥❝❡r♥✐♥❣ ❆❧❣♦r✐t❤♠ ✶ ❛♥❞ st❛❜❧❡ ♠❡r❣❡s ❚♦ ♣r♦✈❡ Pr♦♣♦s✐t✐♦♥ ✶✱ t❤r❡❡

❧❡♠♠❛s ❛r❡ ❤❡❧♣❢✉❧✳ ❋✐rst✱ ✇❡ s❤♦✇ t❤❛t ✐♥ ❆❧❣♦r✐t❤♠ ✶✱ ♦♥❝❡ ❛ ♠❛t❝❤ ❝♦❧❧✐❞❡s ✐♥ s♦♠❡ st❡♣ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✱ t❤❡r❡ ✐s ♥♦ ❝❤❛♥❝❡ t❤❛t t❤❡ ♠❛t❝❤ ✇✐❧❧ ❡✈❡r ❜❡ ❛❞❞❡❞ t♦ t❤❡ ♠❡r❣❡✳

▲❡♠♠❛ ✷✳ ♦❢

M

N

✳

■♥ ❆❧❣♦r✐t❤♠ ✶✱ ✐❢

(rp , ro ) ∈ X n ✱

t❤❡♥ ❛♥②


rp′ , ro′ ∈ κ (rp , ro , X n )

❝❛♥♥♦t ❜❡ ♣❛rt

❈♦♥s✐❞❡r t❤❡ ❜❛s❡ ❝❛s❡ n = N
✳ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦❧❧✐s✐♦♥✱ rp′ , ro′ ∈ X n ✱ ❛♥❞ ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✱ rp′ , ro′ ∈/ M N ✳


Pr♦♦❢✳

◆♦✇✱ ✇❡ ♣r♦✈❡ t❤❡ ✐♥❞✉❝t✐♦♥ st❡♣✿
❢♦r n < N ✱ ✐❢ (rp , ro ) ∈ X n ❛♥❞ rp′ , ro′ ∈ κ (rp , ro , X n )✱ n+1 t❤❡♥ (r ❛♥❞ rp′ , ro′ ∈ κ (rp
, ro , X n+1 )✳ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦❧❧✐s✐♦♥✱ ❜♦t❤ p , ro ) ∈ X
rp′ , ro′ ∈ X n ❛♥❞ (rp , ro ) ∈ κ rp′ , ro′ , X n ♠✉st ❤♦❧❞✱ ❛♥❞ ❤❡♥❝❡ ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡

  ⊆ C n+1 s✐♥❝❡ ❝❧❡❛r❧② (rp , ro ) , rp′ , ro′ ❛❧❣♦r✐t❤♠✱ (rp , ro ) , rp′ , ro′ ∩M n = ∅✳ ◆♦✇✱
⊆ X n+1 ✱ ✇❡ ❥✉st ♥❡❡❞ t♦ ❡st❛❜❧✐s❤ t❤❛t C n ⊆ C n+1 ✱ s♦ t♦ s❤♦✇ t❤❛t (rp , ro ) , rp′ , ro′ 
(rp , ro ) , rp′ , ro′ ⊆ U n+1 (M n )✳


❇② ✇❛② ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❛ss✉♠❡ ♦t❤❡r✇✐s❡✳ ❚❤❡♥✱ t❤❡r❡ ♠✉st ❡①✐st s♦♠❡ rp′′ , ro′′ ∈ M n ∩ C n


❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ s✉❝❤ t❤❛t ❡✐t❤❡r (rp , ro ) ∈
κ rp′′ , ro′′ , U ♦r rp′ , ro′ ∈ κ rp′′ , ro′′ , U ✳ ❲✐t❤♦✉t
′′ ′′ n ′′ ′′ ❧❡t (rp
, ro ) ∈ κ rp , ro , U ✳ ❙✐♥❝❡ (rp , ro ) ∈ X
✱ ✇❡ ❦♥♦✇ t❤❛t rp , ro ∈/ M n−1 ✳ ❚❤❡♥✱ ❢♦r
rp′′ , ro′′ ∈ M n t♦ ❤♦❧❞✱ ✐t ♠✉st ❜❡ t❤❛t rp′′ , ro′′ ∈ X n ✳ ❇✉t t❤❡♥✱ (rp , ro ) ∈ κ rp′′ , ro′′ , X n ✱


✶✶ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts r
p′′ , ro′′ ∈ M n ✳ ❍❡♥❝❡✱ ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t (rp , ro ) , rp′ , ro′ ⊆ X n+1 ✳ ❈❧❡❛r❧② t❤❡♥✱ rp′ , ro′ ∈ κ (rp , ro , X n+1 )✳ ❚❤✉s ✇❡ ❤❛✈❡ ♣r♦✈❡❞ t❤❡ r❡s✉❧t ✈✐❛ ✐♥❞✉❝t✐♦♥✳







◆❡①t✱ ✇❡ s❤♦✇ t❤❛t ✐♥ ❆❧❣♦r✐t❤♠ ✶✱ ❛♥② ♠❛t❝❤ ✐s ❡✐t❤❡r ❛❞❞❡❞ t♦ t❤❡ ♠❡r❣❡ ✐♥ t❤❡ st❡♣ ❡q✉❛❧ t♦ ✐ts r❛♥❦✱ ♦r ✐t ✐s ♥❡✈❡r ❛❞❞❡❞ ❛t ❛❧❧✳ ▲❡♠♠❛ ✸✳ ■♥ ❆❧❣♦r✐t❤♠ ✶✱ ✐❢

(rp , ro )

✐s♥✬t ✐♥

M ρ(rp ,ro ) ✱

t❤❡♥ ✐t ✐s ♥♦t ✐♥

MN✳

Pr♦♦❢✳ ❚❤❡ st❛t❡♠❡♥t ✐s tr✐✈✐❛❧❧② tr✉❡ ✐❢ ρ (rp , ro ) = N ✱ s♦ ✇❡ ❝♦♥s✐❞❡r ρ (rp , ro ) = n < N ✳ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✱ ❙t❡♣ n ✐s t❤❡ ✜rst st❡♣ ❛t ✇❤✐❝❤ (rp , ro ) ❝♦✉❧❞ ❥♦✐♥ t❤❡ ♠❡r❣❡✳

◆♦✇✱ ✐❢ (rp , ro ) ∈ / M n ❜❡❝❛✉s❡ (rp , ro ) ∈ / X n ✱ t❤❡♥ ❢♦r ❛♥② n′ > n✱ (rp , ro ) ∈ / M n ✳ ❚♦ s❡❡ t❤✐s✱ ♥♦t❡ t❤❛t s✐♥❝❡ (rp , ro ) ∈ C n ✱ ✐❢ (rp , ro ) ∈ / X n ✱ t❤❡♥ ✐t ♠✉st ❜❡ t❤❛t (rp , ro ) ∈ / U n (M n−1 )✳ n′ ′ n ′ ❇② ❞❡✜♥✐t✐♦♥✱ ✇❡ ❦♥♦✇ t❤❛t U (X ) ⊆ U (X)✱ ❢♦r ❛♥② n > n ❛♥❞ X ⊆ X ′ ✳ ❍❡♥❝❡✱
′ ′ ′ U n M n −1 ⊆ U n (M n−1 ) ❢♦r ❛♥② n′ > n ❛♥❞ ❤❡♥❝❡ (rp , ro ) ∈ / X n ✳ ❋r♦♠ t❤✐s✱ ✐t ❢♦❧❧♦✇s t❤❛t ′ ❢♦r ❛♥② n′ > n✱ (rp , ro ) ∈ / M n ❛♥❞ ❤❡♥❝❡ (rp , ro ) ∈ / MN✳

◆♦✇✱ ✐❢ (rp , ro ) ❢❛✐❧s
t♦ ❥♦✐♥ t❤❡ ♠❡r❣❡ ❛t ❙t❡♣ n ❛♥❞ (rp , ro ) ∈ X n ✱ t❤❡♥ ✐t ♠✉st ❜❡ t❤❛t t❤❡r❡
❡①✐sts s♦♠❡ rp′ , ro′ ∈ κ rp , ro , X N ✳ ❇② ▲❡♠♠❛ ✷✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t {(rp , ro ) , (rp , ro )} ∩ M N = ∅✳ ❚❤❡♥✱ ✇❡ ❡st❛❜❧✐s❤ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ s❡t t❤❛t ✐s st❛❜❧❡ r❡❧❛t✐✈❡ t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ C ❜② s❤♦✇✐♥❣ t❤❛t ❆❧❣♦r✐t❤♠ ✶ ♣r♦❞✉❝❡s ✐t✳ ▲❡♠♠❛ ✹✳ ❆❧❣♦r✐t❤♠ ✶ ♦✉t♣✉ts ❛ s❡t t❤❛t ✐s st❛❜❧❡ r❡❧❛t✐✈❡ t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥

C✳

Pr♦♦❢✳ ❚♦ ♣r♦✈❡ t❤❛t t❤❡ ❛❧❣♦r✐t❤♠✬s ♦✉t♣✉t✱ M ✱ ✐s st❛❜❧❡ r❡❧❛t✐✈❡ t♦ C ✱ ✇❡ ♠✉st ❝♦♥s✐❞❡r ❜♦t❤ t❤❡ ❢♦r✇❛r❞ ❛♥❞ ❜❛❝❦✇❛r❞ ✐♠♣❧✐❝❛t✐♦♥ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥✳ ❋✐rst✱ ❝♦♥s✐❞❡r t❤❡ ❢♦r✇❛r❞ ✐♠♣❧✐❝❛t✐♦♥✱ ✇❤✐❝❤ r❡q✉✐r❡s t❤❛t ❢♦r ❛♥② (rp , ro ) ∈ M ❜♦t❤ κ (rp , ro , C n ∩ U n (M )) = ∅ ❛♥❞ (rp , ro ) ∈ U n (M )✱ ✇❤❡r❡ n = ρ (rp , ro )✳ ❲❡ ✇✐❧❧ ♣r♦✈❡ ❜♦t❤ ✐♠♣❧✐❝❛t✐♦♥s ❜② ❝♦♥tr❛❞✐❝t✐♦♥✳

❇② ✇❛② ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❛ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐sts rp′ , ro′ ∈ κ (rp , ro , C n ∩ U n (M ))✳ ❇② ▲❡♠♠❛ ✸✱ (rp , ro ) ♠✉st ❥♦✐♥ t❤❡ n✳ ❋♦r t❤✐s t♦ ❜❡ tr✉❡✱ ✐t ♠✉st ❜❡ t❤❛t
♠❡r❣❡ ✐♥ ❙t❡♣ n n ′ ′ (rp , ro ) ∈ X ✳ ❇✉t s✐♥❝❡ rp , ro ∈ κ (rp , ro , C ∩ U n (M )) ❛♥❞ U n (M ) ⊆ U n (M n−1 )✱ ✐t ♠✉st
❛❧s♦ ❜❡ t❤❛t rp′ , ro′ ∈ κ (rp , ro , X n )✱ ✇❤✐❝❤ ❜② ▲❡♠♠❛ ✷ ♠❡❛♥s t❤❛t (rp , ro ) ∈ / M ✱ ♣r♦✈✐❞✐♥❣ t❤❡ r❡q✉✐r❡❞ ❝♦♥tr❛❞✐❝t✐♦♥✳


◆♦✇✱ ❜② ✇❛② ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❛ss✉♠❡ t❤❛t (rp , ro ) ∈ / U n (M )✳ ❙✐♥❝❡ ▲❡♠♠❛ ✸ t❡❧❧s ✉s t❤❛t M n−1 = M ∩ C n−1 ✱ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ t❤❛t U n (M ) = U n (M n−1 )✳ ❚❤✐s t❤❡♥ t❡❧❧s ✉s t❤❛t (rp , ro ) ∈ / X n ✱ ❛♥❞ ❤❡♥❝❡ t❤❛t (rp , ro ) ∈ / M n ✱ ♣r♦✈✐❞✐♥❣ t❤❡ r❡q✉✐r❡❞ ❝♦♥tr❛❞✐❝t✐♦♥✳

◆♦✇✱ ❝♦♥s✐❞❡r t❤❡ ❜❛❝❦✇❛r❞ ✐♠♣❧✐❝❛t✐♦♥✳ ❇② ✇❛② ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❛ss✉♠❡ t❤❛t κ (rp , ro , C n ∩ U n (M )) = ∅ ❛♥❞ (rp , ro ) ∈ U n (M )✱ ❜✉t (rp , ro ) ∈ / M ✱ ✇❤❡r❡ n = ρ (rp , ro )✳ n n n n−1 ■t ♠✉st ❜❡ t❤❛t (rp , ro ) ∈ X
✱ s✐♥❝❡ U (M ) ⊆ U (M )✳ ❍❡♥❝❡✱ (rp , ro ) ∈ / M r❡✲ n n n−1 ′ ′ ▲❡♠♠❛ ✸ t❡❧❧s ✉s t❤❛t q✉✐r❡s t❤❛t t❤❡r❡ ❡①✐sts rp , ro ∈ κ (rp , ro , C ∩ U (M ))✳ ❇✉t✱
n−1 n−1 n n−1 n ′ ′ M = M ∩ C ✱ s♦ U (M ) = U (M )✳ ❍❡♥❝❡✱ rp , ro ∈ κ (rp , ro , C n ∩ U n (M ))✱ ✇❤✐❝❤ ♣r♦✈✐❞❡s t❤❡ r❡q✉✐r❡❞ ❝♦♥tr❛❞✐❝t✐♦♥✳

✶✷ ❲❡ ❛❧s♦ ♥❡❡❞ t♦ s❤♦✇ t❤❛t ❛♥② s❡t t❤❛t ✐s st❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ C ✐s ❛ ♠❡r❣❡✳ ▲❡♠♠❛ ✺✳ ❆♥② s❡t

M ⊆U

t❤❛t ✐s st❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ ❈ ♠✉st ❜❡ ❛ ♠❡r❣❡✳

Pr♦♦❢✳ ❇② ✇❛② ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❛ss✉♠❡ t❤❛t M ✐s st❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ C ✱ ❜✉t
✐s ♥♦t ❛ ♠❡r❣❡✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts (rp , ro ) ∈
M s✉❝❤ t❤❛t κ (rp , ro , M ) 6= ∅✳ ▲❡t rp′ , ro′ ∈ κ (rp , ro , M )✳ ◆♦✇✱ ✐❢ ρ (rp , ro ) 6= ρ rp′ , ro′ ✱ t❤❡♥ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ❧❡t
/ U ρ(rp ,ro ) (M )✱ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts t❤❡ ρ (rp , ro ) > ρ rp′ , ro′ ✳ ❚❤❡♥✱ ✐t ♠✉st ❜❡ t❤❛t (rp , ro ) ∈ ❞❡✜♥✐t✐♦♥ ♦❢ st❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ C ✳

❙♦✱ ❛ss✉♠❡ t❤❛t ρ (rp , ro ) = ρ rp′ , ro′ ✳ ❋✉rt❤❡r✱ ❛ss✉♠❡ t❤❛t (rp , ro ) , rp′ ,ro′ ⊆ U ρ(rp ,ro )
(M )❀ ⊆ ♦t❤❡r✇✐s❡ ✇❡ ❤❛✈❡ ❛❧r❡❛❞② ❝♦♥tr❛❞✐❝t❡❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ st❛❜❧❡✳ ❚❤❡♥✱ (rp , ro ) , rp′ , ro′
ρ(rp ,ro ) ρ(rp ,ro ) ρ(rp ,ro ) ρ(rp ,ro ) C ∩U (M )✱ ❛♥❞ ❤❡♥❝❡ κ rp , ro , C ∩U (M ) 6= ∅✱ ❝♦♥tr❛❞✐❝t✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ st❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ r❛♥❦❡❞ ❝r✐t❡r✐♦♥ C ✳

❋✐♥❛❧❧②✱ ✇❡ ❛r❡ r❡❛❞② t♦ ♣r♦✈❡ t❤❛t t❤❡ st❛❜❧❡ ♠❛t❝❤ r❡❧❛t✐✈❡ t♦ C ♣r♦❞✉❝❡❞ ❜② ❆❧❣♦r✐t❤♠ ✶ ✐s ✐♥ ❢❛❝t t❤❡ ✉♥✐q✉❡ st❛❜❧❡ ♠❡r❣❡ r❡❧❛t✐✈❡ t♦ C ✳ ▲❡♠♠❛ ✹ s❤♦✇s t❤❛t ❆❧❣♦r✐t❤♠ ✶ ♦✉t♣✉ts ❛ st❛❜❧❡ ♠❛t❝❤✳ ❯♥✐q✉❡✲ ♥❡ss ❝❛♥ ❜❡ ❡st❛❜❧✐s❤❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ r❛♥❦✳ ❋♦r t❤❡ ❜❛s❡ ❝❛s❡✱ ❝♦♥s✐❞❡r s♦♠❡ (rp , ro ) ∈ M ∩ C 1 ✳ ❙✐♥❝❡ M ✐s st❛❜❧❡✱ ✐t ♠✉st ❜❡ t❤❛t κ (rp , ro , C 1 ) = ∅✳ ❇✉t t❤✐s ✇♦✉❧❞ r❡q✉✐r❡ t❤❛t (rp , ro ) ❜❡ ✐♥❝❧✉❞❡❞ ✐♥ M ′ ❛s ✇❡❧❧✳ ❙♦✱ M ❛♥❞ M ′ ♠✉st ❛❣r❡❡ ♦♥ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥ ✇✐t❤ C 1 ✳ ◆♦✇✱ ❢♦r t❤❡ ✐♥❞✉❝t✐✈❡ st❡♣✱ ❛ss✉♠❡ t❤❛t M ∩ C n = M ′ ∩ C n ✳ ❚❛❦❡ ❛ ♠❛t❝❤ (rp , ro ) ∈ (M \ C n ) ∩ C n+1 ✳ ❙✐♥❝❡ M ✐s st❛❜❧❡✱ ✐t ♠✉st ❜❡ t❤❛t κ (rp , ro , C n+1 ∩ U n+1 (M ∩ C n )) = ∅. ❇✉t s✐♥❝❡ M ∩ C n = M ′ ∩ C n ✱ ✐t ✐s ❛❧s♦ tr✉❡ t❤❛t κ (rp , ro , C n+1 ∩ U n+1 (M ′ ∩ C n )) = ∅✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t (rp , ro ) ∈ M ′ ✳ ❙♦✱ ❜② ✐♥❞✉❝t✐♦♥✱ ❛♥② t✇♦ ♠❡r❣❡s ♠✉st ❝♦♥t❛✐♥ t❤❡ s❛♠❡ ♠❛t❝❤❡s✱ t❤❛t ✐s✱ t❤❡ st❛❜❧❡ ♠❛t❝❤ ✐s ✉♥✐q✉❡✳ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✶✳

❇✳✷

❘❡❣✐str❛t✐♦♥ ❞❛t❛

■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ ✇❡ ❞❡s❝r✐❜❡ ❤♦✇ t❤❡ ❧✐st ♦❢ r❡❣✐str❛t✐♦♥s t♦ t❤❡ ◆❑❘ ✇❛s ❛ss❡♠❜❧❡❞✳ ❚❤❡ ◆❑❘ ♣r♦✈✐❞❡❞ ✉s ✇✐t❤ s♥❛♣s❤♦t ✜❧❡s ♦❢ t❤❡ ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r ♣♦♦❧ ❜❡t✇❡❡♥ ❆♣r✐❧ ✶✱ ✷✵✶✷ ❛♥❞ ❉❡❝❡♠❜❡r ✹✱ ✷✵✶✹✳ ❚❤❡s❡ ✜❧❡s ❛r❡ t②♣✐❝❛❧❧② ❞❛✐❧② s♥❛♣s❤♦ts ❡①❝❡♣t ❢♦r s♦♠❡ ♠✐ss✐♥❣ ♣❡r✐♦❞s✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s ✉♣ t♦ ❛ ♠♦♥t❤ ✐♥ ❧❡♥❣t❤✳ ❊❛❝❤ s♥❛♣s❤♦t ❝♦rr❡s♣♦♥❞s t♦ ❛ ❞✐✛❡r❡♥t ❞❛t❡ ❛♥❞ ✐♥❝❧✉❞❡s ❜❛s✐❝ ♠❡❞✐❝❛❧ r❡❝♦r❞s ❢♦r ❡❛❝❤ ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r ✐♥ t❤❡ ♣♦♦❧✱ t❤❡✐r ❧✐st✐♥❣ ❞❛t❡s✱ t❤❡ r❡❧❛t❡❞ ♣❛t✐❡♥t ❢♦r ❡❛❝❤ ❞♦♥♦r ✭✐❢ ❛♥②✮✱ ❛♥❞ ✇❤❡t❤❡r ❛ ♣❛t✐❡♥t ✐s ✉♥♣❛✐r❡❞✳ ❋r♦♠ t❤❡s❡ s♥❛♣s❤♦ts ✇❡ r❡❝♦✈❡r ♣❛t✐❡♥t ❛♥❞ ❞♦♥♦rs ❞❡♣❛rt✉r❡s✱ ✇❤✐❝❤ ♠❛② ❜❡ ❞✉❡ t♦ ❜❡✐♥❣ tr❛♥s♣❧❛♥t❡❞ ♦r ♦t❤❡r ♥♦♥✲❞♦❝✉♠❡♥t❡❞ r❡❛s♦♥s✳ ❆ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❞❡♣❛rt ✇✐t❤♦✉t ❛ tr❛♥s♣❧❛♥t ❞✉r✐♥❣ t❤❡ ♣❡r✐♦❞ ♦❢ ❛ ♠✐ss✐♥❣ s♥❛♣s❤♦t❀ ❢♦r t❤❡s❡✱ ✇❡ ✉s❡ ❜♦✉♥❞s ♦♥ t❤❡ ❞❡♣❛rt✉r❡ t✐♠❡ ✉s✐♥❣ t❤❡ t✇♦ ❝❧♦s❡st ❛✈❛✐❧❛❜❧❡ s♥❛♣s❤♦ts✱ ❜❡❢♦r❡ ❛♥❞ ❛❢t❡r t❤❡ r❡❛❧ ❞❡♣❛rt✉r❡ ❞❛t❡✳ ❚❤❡s❡ s♥❛♣s❤♦ts ❛❧s♦ ✐♥❝❧✉❞❡ ❡❛❝❤ ♣❛t✐❡♥t✬s s❡t ♦❢ ❞♦♥♦rs t❤❛t ♠❛② ♥♦t ❜❡ ♠❛t❝❤❡❞ ✭✐✳❡✳ ❛r❡ ❜❧♦❝❦❡❞✮ ❞❡s♣✐t❡ ❜❡✐♥❣ ✈✐rt✉❛❧❧② ❝♦♠♣❛t✐❜❧❡✳ ❙♦♠❡ ♦❢ t❤❡s❡ ❜❧♦❝❦❡❞ ❞♦♥♦rs ❛r❡ ❞✉❡ t♦ ♣❛t✐❡♥t ♣r❡❢❡r❡♥❝❡ ✭♥♦t t♦ ♠❛t❝❤ ✇✐t❤ t❤❡s❡ ❞♦♥♦rs✮ ❛♥❞ ♦t❤❡rs ❛r❡ ❞✉❡ t♦ ♠❛t❝❤ ❢❛✐❧✉r❡s✳

✶✸

❈ ❙✐♠✉❧❛t✐♦♥ ❉❡t❛✐❧s ❲❡ ♥♦✇ ♣r♦✈✐❞❡ ❞❡t❛✐❧s ♦♥ t❤❡ ♣r♦❝❡❞✉r❡ ✉s❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✶✳

❈✳✶ ▼❛t❝❤✐♥❣ ❖✛❡rs ❈✳✶✳✶

❚❤❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❡①❛♠♣❧❡s

❲❡ s♦❧✈❡ t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠ ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✶ ❛s ❢♦❧❧♦✇s✳ ❇❡❝❛✉s❡ ✐t ✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❜✉r❞❡♥s♦♠❡ t♦ ❝♦♠♣✉t❡ ❛❧❧ ❝②❝❧❡s ✐♥ ❛ ♣♦♦❧ ✇✐t❤ ♠❛♥② ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✱ ✇❡ ✜rst s♦❧✈❡ t❤❡ r❡❧❛①❡❞ ♣r♦❜❧❡♠ ❜② ✐❣♥♦r✐♥❣ t❤❡ ❝♦♥str❛✐♥t t❤❛t ❝②❝❧❡s ❝❛♥♥♦t ✐♥✈♦❧✈❡ ♠♦r❡ t❤❛♥ t❤r❡❡ tr❛♥s♣❧❛♥ts✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ s♦❧✈❡✿

max

xjk ∈{0,1} s✳t✳xjk

X j

−

X

cjk wjk xjk

jk

X

xkl = 0

❢♦r ❛❧❧

l

xjk ≤ 1,

X k

k∈ P

xjk ≤ 1

xjk = 1 ❞❡♥♦t❡s ❛ ♣r♦♣♦s❡❞ tr❛♥s♣❧❛♥t ❢r♦♠ t❤❡ ❞♦♥♦r ✐♥ k ∈ A ∪ P t♦ t❤❡ ♣❛t✐❡♥t ✐♥ j ∈ P ∪ U ✱ wjk ✐s t❤❡ ✇❡✐❣❤ts ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ❈✳✶✳✷ ❜❡❧♦✇✱ ❛♥❞ cjk = 1 ✐❢ ❛ tr❛♥s♣❧❛♥t ❢r♦♠ k t♦ j ✐s ❛❧❧♦✇❡❞ ❛♥❞ 0 ♦t❤❡r✇✐s❡✳ ❚❤❡ ✜rst ❝♦♥str❛✐♥t ❡♥s✉r❡s t❤❛t ❛ ❞♦♥♦r ✇❤♦ ✐s ✇❤❡r❡

♣❛rt ♦❢ ❛ ♣❛✐r ✐s ♦♥❧② ❛s❦❡❞ t♦ ❞♦♥❛t❡ ❛♥ ♦r❣❛♥ ✐❢ t❤❡ ✐♥t❡♥❞❡❞ r❡❝✐♣✐❡♥t ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ❛ tr❛♥s♣❧❛♥t✳ ❚❤❡ s❡❝♦♥❞ ❝♦♥str❛✐♥t ❡♥s✉r❡s t❤❛t ♥♦ ❞♦♥♦r ♦r r❡❝✐♣✐❡♥t ✐s ✐♥✈♦❧✈❡❞ ✐♥ ♠♦r❡ t❤❛♥ ♦♥❡ tr❛♥s♣❧❛♥t✳ ■❢ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠ ❞♦❡s ♥♦t ✐♥✈♦❧✈❡ ❛♥② ❧♦♥❣ ❝②❝❧❡s✱ ✐✳❡✳ t❤❡r❡ ❞♦ ♥♦t ❡①✐st j1 , . . . , j4 ∈ P s✉❝❤ t❤❛t x∗jk+1 jk = 1 ❢♦r k ∈ {1, . . . , 3}✱ t❤❡♥ ✐t ♠✉st ❜❡ t❤❛t x∗ ✐s ♦♣t✐♠❛❧✱ ❣✐✈❡♥ t❤❡ ♥♦ ❧♦♥❣✲❝②❝❧❡ ❝♦♥str❛✐♥ts✱ ❛♥❞ ✐s ♦✉r ❞❡s✐r❡❞ s♦❧✉t✐♦♥✳ ■♥ ♦✉r s✐♠✉❧❛t✐♦♥s ✽✼✳✷✪ ♦❢ ✏s✐♠✉❧❛t✐♦♥ ❞❛②s✑ t❤❡ s♦❧✉t✐♦♥ t♦ t❤✐s r❡❧❛①❡❞ ♣r♦❜❧❡♠ ②✐❡❧❞s ❛ ❢❡❛s✐❜❧❡ ♠❛t❝❤ ✇✐t❤♦✉t ❛♥② ❢✉rt❤❡r ❝②❝❧❡ r❡str✐❝t✐♦♥s✳ ■❢ t❤❡ s♦❧✉t✐♦♥ t♦ t❤✐s ♣r♦❜❧❡♠ ❝♦♥t❛✐♥s ❛t ❧❡❛st ♦♥❡ ❧♦♥❣ ❝②❝❧❡✱ t❤❡♥ ✇❡ ♣r♦❝❡❡❞ ❛s ❢♦❧❧♦✇s✳ ❲❡ ❜❡❣✐♥ ❜② ❢♦❧❧♦✇✐♥❣ t❤❡ ❛❧❣♦r✐t❤♠ ✐♥ ❆♥❞❡rs♦♥ ❡t ❛❧✳ ✭✷✵✶✺✮✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✐♥❝❧✉❞❡s ❛ ∗ ❝♦♥str❛✐♥t t❤❛t ❡①♣❧✐❝✐t❧② ♣r♦❤✐❜✐ts ❛❧❧ ❧♦♥❣ ❝②❝❧❡s ✐♥ x ✱ ✐✳❡✳ ❢♦r ❡❛❝❤ s❡q✉❡♥❝❡ j1 , . . . , jK ∈ P ∗ s✉❝❤ t❤❛t xj = 1 ❢♦r k ∈ {1, . . . , K − 1}✱ ✇❡ ✐♥❝❧✉❞❡ ❛ ❝♦♥str❛✐♥t ✐♥ t❤❡ ♣r♦❜❧❡♠ ❛❜♦✈❡ k+1 jk QK−1 t♦ ❡♥s✉r❡ t❤❛t k=1 xjk+1 jk = 0. ■❢ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♠♦❞✐✜❡❞ ♣r♦❜❧❡♠ ❛❧s♦ ❝♦♥t❛✐♥s ❧♦♥❣ ❝②❝❧❡s✱ ✇❡ ♠♦❞✐❢② t❤❡ ♣r♦❜❧❡♠ ❛❣❛✐♥ t♦ ♣r♦❤✐❜✐t t❤♦s❡ ❝②❝❧❡s✳ ❲❡ ✐t❡r❛t❡ t❤✐s ♣r♦❝❡❞✉r❡ ✉♣

t♦ t❡♥ t✐♠❡s✳ ❚❤✐s ♣r♦❝❡❞✉r❡ ②✐❡❧❞s ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ ✐♥ ❛❜♦✉t ✺✵✪ ♦❢ t❤❡ r❡♠❛✐♥✐♥❣ ❝❛s❡s ✭❛❜♦✉t ✼✳✽✪ ♦❢ ❛❧❧ ❝❛s❡s✮ ✇✐t❤ ❛♥ ❛✈❡r❛❣❡ ♦❢ ❛♣♣r♦①✐♠❛t❡❧② ✻✳✸ ✐t❡r❛t✐♦♥s✳ ■❢ t❤❡ ❛❧❣♦r✐t❤♠ ❛❜♦✈❡ ❞♦❡s ♥♦t ②✐❡❧❞ ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ ❡✈❡♥ ❛❢t❡r ✶✵ r❡♣❡t✐t✐♦♥s✱ ✇❡ ♣r♦❝❡❡❞ t♦ t❤❡ ♥❡①t ♣❤❛s❡ ✐♥ ✇❤✐❝❤ ✇❡ ✉s❡ ❏♦❤♥s♦♥ ✭✶✾✼✼✮✬s ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ ❝②❝❧❡s ❛♥❞ ❡①♣❧✐❝✐t❧②

❛❞❞ ❝♦♥str❛✐♥ts t❤❛t ♣r♦❤✐❜✐t ❧♦♥❣ ❝②❝❧❡s✳ ❚❤✐s ❛❧❣♦r✐t❤♠ s❡❛r❝❤❡s t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ❣r❛♣❤

✶✹

✐♥❞✉❝❡❞ ❜②

c

t♦ ❝❛❧❝✉❧❛t❡ ❝②❝❧❡s✳

❲❡ ❡♥✉♠❡r❛t❡ ❛♥❞ ❛❞❞ ❛ ❝♦♥str❛✐♥t t♦ ♦✉r ♣r♦❣r❛♠ t♦

♣r♦❤✐❜✐t ❛♥② ❧♦♥❣ ❝②❝❧❡s ✇❡ ❤❛✈❡ ❢♦✉♥❞✳ ❙✐♥❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ❢❡❛s✐❜❧❡ ❧♦♥❣ ❝②❝❧❡s ✐s ❡①tr❡♠❡❧② ❧❛r❣❡✱ ✇❡ s❡❛r❝❤ ❢♦r ❝②❝❧❡s ✇✐t❤ ❛ t✐♠❡✲♦✉t ♦❢ ♦♥❡ s❡❝♦♥❞✳ ❲❡ ✜♥❞ ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠ ✇✐t❤ t❤❡s❡ ❛❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥ts ❛♥❞ t❡r♠✐♥❛t❡ ♦✉r ❛❧❣♦r✐t❤♠ ✐❢ t❤❡ s♦❧✉t✐♦♥ ✐s ❢❡❛s✐❜❧❡✳ ❚❤✐s ♣r♦❝❡❞✉r❡ ✐s r❡♣❡❛t❡❞ ♦♥❝❡ ♠♦r❡✱ ✐❢ ♥❡❝❡ss❛r②✳ ❆t t❤❡ ❡♥❞ ♦❢ t❤✐s ♣❤❛s❡✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ✜♥❞ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ t♦ t❤❡ ❢✉❧❧ ♣r♦❜❧❡♠ ✐♥ ❛❜♦✉t ✾✾✳✵✪ ♦❢ t❤❡ ✏s✐♠✉❧❛t✐♦♥ ❞❛②s✳✑ ❋♦r t❤❡ r❡♠❛✐♥✐♥❣ ✶✳✵✪ ♦❢ ❞❛②s ✐♥ t❤❡ s✐♠✉❧❛t✐♦♥✱ ♦✉r ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠ st✐❧❧ ❡♥❞s ✉♣ ✇✐t❤ ❧♦♥❣ ❝②❝❧❡s✳ ❲❤❡♥❡✈❡r t❤✐s ✐s t❤❡ ❝❛s❡✱ ✇❡ ❛❜❛♥❞♦♥ t❤❡ ♠❛t❝❤ ♦♥ t❤❛t ✏s✐♠✉❧❛t✐♦♥ ❞❛②✑ t❤❛t ♥♦ ♠❛t❝❤ ✐s ♦✛❡r❡❞✱ ❛♥❞ ❛❧❧ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ❛r❡ r❡t✉r♥❡❞ t♦ t❤❡ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣♦♦❧ t♦ ✇❛✐t ❢♦r t❤❡ ♥❡①t ❞❛② ❢♦r t❤❡ tr❛♥s♣❧❛♥t ♦✛❡r✳ ❚❤❡ t②♣✐❝❛❧ ❛✈❡r❛❣❡ ❞✉r❛t✐♦♥ ❢♦r t❤❡ ❤❡✉r✐st✐❝ ♦❢ ✧♥♦ ♠❛t❝❤✧ ✐s ❛r♦✉♥❞ ✷✳✾ ❞❛②s✳ ❋✐❣✉r❡ ❈✶ ✐❧❧✉str❛t❡s ❛ ❢❡✇ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣♦♦❧s✳ ❝❛♣t✉r❡❞ ❜②

c✱

❚❤❡ ❧❡❢t ♣❛♥❡❧ s❤♦✇s ❝♦♠♣❛t✐❜✐❧✐t② ❛s

❢❡❛s✐❜❧❡ tr❛♥s♣❧❛♥ts✱ ❛♥❞ t❤❡ ♦♣t✐♠❛❧ ♠❛t❝❤✳ ❇❧✉❡ ❞♦ts ❞❡♥♦t❡ ♣❛t✐❡♥t✲❞♦♥♦r

♣❛✐rs✱ ❛♥❞ ♠❛❣❡♥t❛ ❞♦ts ❞❡♥♦t❡ ❛❧tr✉✐st✐❝ ❞♦♥♦rs✳

❲❡ ✐❣♥♦r❡ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥ts ✐♥ t❤✐s ✐❧✲

❧✉str❛t✐♦♥ ❢♦r s✐♠♣❧✐❝✐t②✳ ❆ ❜❧✉❡ ❛rr♦✇s ❞❡♣✐❝ts ❛ ❢❡❛s✐❜❧❡ tr❛♥s♣❧❛♥t ✇✐t❤ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ❛rr♦✇ ❞❡♥♦t✐♥❣ t❤❡ ❞♦♥♦r✳ ❘❡❞ ❛♥❞ ❣r❡❡♥ ❛rr♦✇s ❞❡♣✐❝t ❝②❝❧❡s ❛♥❞ ❝❤❛✐♥s✱ r❡s♣❡❝t✐✈❡❧②✱ ✐♥ t❤❡ ♦♣t✐♠❛❧ ♠❛t❝❤✳ ●✐✈❡♥ ❢❡❛s✐❜❧❡ tr❛♥s♣❧❛♥ts ♦♥ ❧❡❢t✱ ♦✉r ♠❛t❝❤ ❛❧❣♦r✐t❤♠ ♦✛❡rs t❤❡ ♦♥❡ ♦♥ t❤❡ r✐❣❤t✳ ❚❤❡ ✜❣✉r❡ s❤♦✇s t❤❡r❡ ♠❛② ❜❡ s❡✈❡r❛❧ ❢❡❛s✐❜❧❡ tr❛♥s♣❧❛♥ts✱ ❛♥❞ ✐♥ t❤❡s❡ ❝❛s❡s✱ t❤❡ ♦♣t✐♠❛❧ ♠❛t❝❤ ♠❛② ❜❡ r❡❧❛t✐✈❡❧② ❡❛s② t♦ ❞❡t❡r♠✐♥❡✳ ❋✐❣✉r❡ ❈✷✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐❧❧✉str❛t❡s ❛ r❡❧❛t✐✈❡❧② ❤❛r❞✲t♦✲♠❛t❝❤ ♣r♦❜❧❡♠ ✇❤❡r❡ t❤❡ ♦♣t✐♠❛❧ ♠❛t❝❤ ✐s r❡❧❛t✐✈❡❧② ♠♦r❡ ❞✐✣❝✉❧t t♦ ❞❡t❡r♠✐♥❡✳

❋✐❣✉r❡ ❈✶✿ ❆♥ ♦♣t✐♠❛❧ ♠❛t❝❤ ✐♥ ❛ s✐♠♣❧❡ ♣♦♦❧

❈✳✶✳✷

❲❡✐❣❤ts

❲❡ ❛tt❡♠♣t t♦ ❝❧♦s❡❧② ♠❛t❝❤ t❤❡ ✇❡✐❣❤ts✱ ❞♦♥♦r

j✳

wij ✱

♦♥ ❛ ◆❑❘ tr❛♥s♣❧❛♥t ❜❡t✇❡❡♥ ♣❛t✐❡♥t

i

❛♥❞

❚❤❡s❡ ✇❡✐❣❤ts ❛r❡ ❞❡s✐❣♥❡❞ t♦ ❢❛✈♦r ♣❛t✐❡♥ts ✇❤♦ ❛r❡ ❤✐❣❤❧② s❡♥s✐t✐③❡❞✱ ✐♥ ♦t❤❡r

✇♦r❞s✱ ✇❤♦ ❛r❡ ❤❛r❞❡r t♦ tr❛♥s♣❧❛♥t✳ ♣♦✇❡r ❢♦r ❡❛❝❤ s✉❜♠✐ss✐♦♥✳

❚♦ ❞❡✜♥❡ ✇❡✐❣❤ts✱

w✱

◆❑❘ ✜rst ❞❡✜♥❡s ❛ ♠❛t❝❤✐♥❣

❊❛❝❤ ♣❛t✐❡♥t ❤❛s ❛ P❛t✐❡♥t ▼❛t❝❤ P♦✇❡r ✭P▼P✮✱ ❛ ♥✉♠❜❡r

✶✺

❋✐❣✉r❡ ❈✷✿ ❆ ❝♦♠♣❧✐❝❛t❡❞ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ♣♦♦❧ ❜❡t✇❡❡♥ ✵ ❛♥❞ ✶✱ t❤❛t ✐s ❛ ❢r❛❝t✐♦♥ ♦❢ ❝♦♠♣❛t✐❜❧❡ ❞♦♥♦rs ✐♥ t❤❡ ◆❑❘ ♣♦♦❧ ❢♦r t❤❛t ♣❛t✐❡♥t✳ ❆ ❧♦✇ P▼P ❢♦r ♣❛t✐❡♥t i ✐♠♣❧✐❡s t❤❛t ❢❡✇ ❞♦♥♦rs ❛r❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ♣❛t✐❡♥t i✳ ❙✐♠✐❧❛r❧②✱ t❤❡ ❉♦♥♦r ▼❛t❝❤ P♦✇❡r ✭❉▼P✮ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❢r❛❝t✐♦♥ ♦❢ ♣❛t✐❡♥ts ✐♥ t❤❡ ◆❑❘ ♣♦♦❧ ✇✐t❤ ✇❤♦♠ t❤❛t ❞♦♥♦r ✐s ❝♦♠♣❛t✐❜❧❡✳ ❇❡❝❛✉s❡ t❤❡s❡ q✉❛♥t✐t✐❡s ❛♥❞ t❤❡ ♣♦♦❧ ✉s❡❞ ❜② t❤❡ ◆❑❘ t♦ ❝♦♠♣✉t❡ t❤❡s❡ ♠❛t❝❤ ♣♦✇❡rs ❛r❡ ♥♦t ❞✐r❡❝t❧② ♦❜s❡r✈❡❞ ✐♥ ♦✉r ❞❛t❛s❡t✱ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡♠ ✉s✐♥❣ ♦✉r s❛♠♣❧❡✳ ●✐✈❡♥ t❤❡s❡ ❝❤❛r❛❝t❡r✐st✐❝s✱ ◆❑❘ ❝❛❧❝✉❧❛t❡s ❛ s❝❛❧❡❞ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ❧✐❦❡❧② ❛ ❢❡❛s✐❜❧❡ tr❛♥s✲ ♣❧❛♥t ❝❛♥ ♦❝❝✉r ❜❡t✇❡❡♥ i ❛♥❞ j ✱ W N KRij ✳ ❙♣❡❝✐✜❝❛❧❧②✱ W N KRij = P M P ∗ DM P ∗ 10200✳

❆ ❧♦✇ W N KRij ❝♦rr❡❧❛t❡s ✇✐t❤ ❛ tr❛♥s♣❧❛♥t ❜❡t✇❡❡♥ i ❛♥❞ j ❜❡✐♥❣ ✉♥❧✐❦❡❧②✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ W N KRij ✐s ♥♦t r❡❧❛t❡❞ t♦ t❤❡ s✉❝❝❡ss ♦❢ ❛ tr❛♥s♣❧❛♥t ✐❢ ✐t t✉r♥s ♦✉t t♦ ❜❡ ❢❡❛s✐❜❧❡✳ ❚❤❡s❡ ✇❡✐❣❤ts t❤❡r❡❢♦r❡ ❛❝❝♦r❞ ❤✐❣❤❡r ♣r✐♦r✐t② t♦ ❤❛r❞✲t♦✲♠❛t❝❤ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs✳ ❯s✐♥❣ W N KRij ✱ ◆❑❘ ❛ss✐❣♥s t❤❡ ✇❡✐❣❤ts wij ❛s ❢♦❧❧♦✇s✿ W N KRij interval W N KRij > 70 25 < W N KRij ≤ 70 5 < W N KRij ≤ 25 W N KRij ≤ 5

wij

✶ ✶✳✵✶ ✶✳✷ ✶✳✺

❇❡❝❛✉s❡ t❤❡s❡ ✇❡✐❣❤ts ❛r❡ ❧❡ss t❤❛♥ ✷✱ t❤❡② t②♣✐❝❛❧❧② ♠❛①✐♠✐③❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s✲ ♣❧❛♥ts✳ ❍♦✇❡✈❡r✱ t❤❡s❡ ✇❡✐❣❤ts ♠❛② s♦♠❡t✐♠❡s r❡s✉❧t ✐♥ t✇♦ tr❛♥s♣❧❛♥ts✱ ❡❛❝❤ ✇✐t❤ ✇❡✐❣❤t ✶✳✺✱ ✐♥st❡❛❞ ♦❢ t❤r❡❡ tr❛♥s♣❧❛♥ts ✇✐t❤ ✇❡✐❣❤t ✶ ❡❛❝❤✳

✶✻ ❈✳✷

❆rr✐✈❛❧ ❛♥❞ ❉❡♣❛rt✉r❡

❈✳✷✳✶

❆rr✐✈❛❧ ♣r♦❝❡ss

❲❡ ❛ss✉♠❡ t❤❡ ❞❛✐❧② ♥✉♠❜❡r ♦❢ s✉❜♠✐ss✐♦♥s ✐♥ t❤❡ ◆❑❘ ✐s ❣✐✈❡♥ ❜② t❤❡ ❛ ♣♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡r λ. ❲❡ ❡st✐♠❛t❡ t❤❛t ♣❛r❛♠❡t❡r λ = 1.975✱ t❤❡ ♠❡❛♥ ❛rr✐✈❛❧ r❛t❡ ❢♦r ◆❑❘✳ ■♥ ❡❛❝❤ ♣❡r✐♦❞✱ ♦✉r s✐♠✉❧❛t✐♦♥s ❞r❛✇ ❛ ♥✉♠❜❡r✱ s❛② nt ,❢r♦♠ t❤✐s ❞✐str✐❜✉t✐♦♥✳ ❚❤❡♥ ✇❡ ❞r❛✇ nt s✉❜♠✐ss✐♦♥s ✇✐t❤ r❡♣❧❛❝❡♠❡♥t ❢r♦♠ t❤❡ ❡♥t✐r❡ ♣♦♦❧ t❤❛t ❡✈❡r r❡❣✐st❡r❡❞ ✐♥ t❤❡ ◆❑❘ ❞✉r✐♥❣ t❤❡ ❆♣r✐❧ ✷✵✶✷ t♦ ❏✉♥❡ ✷✵✶✹ s❛♠♣❧❡ ♣❡r✐♦❞✳

❋✐❣✉r❡ ❈✸✿ ❉✐str✐❜✉t✐♦♥ ♦❢ ◆❑❘ ❛♥❞ P♦✐ss♦♥ ◆✉♠❜❡r ♦❢ ❙✉❜♠✐ss✐♦♥s P❡r ❉❛② ❋✐❣✉r❡ C3 s❤♦✇s t❤❡ ✜t ♦❢ t❤❡ ❛rr✐✈❛❧ ♣❡r ❞❛② ❞✐str✐❜✉t✐♦♥s ♦❢ ◆❑❘ ❛♥❞ P♦✐ss♦♥✳ ◆♦t✐❝❡ t❤❛t ◆❑❘✬s ❞✐str✐❜✉t✐♦♥ ❤❛s ♠♦r❡ ✵ ❛rr✐✈❛❧s ♣❡r ❞❛②s t❤❛♥ t❤❡ ♣♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥✳ ❚❤✐s ♠❛ss ♣♦✐♥t ✐s ❡①♣❧❛✐♥❡❞ ❜② ✇❡❡❦❡♥❞s✱ ✇❤✐❝❤ ❛♣♣❡❛r t♦ ❤❛✈❡ ❛ ♠✉❝❤ ❧♦✇❡r ❛rr✐✈❛❧ r❛t❡✳✸ ❋✐❣✉r❡ C4 s❤♦✇s t❤❡ ❛rr✐✈❛❧ ♣❡r ❞❛② ❞✐str✐❜✉t✐♦♥ ♦❢ ◆❑❘ ❢♦r ✇❡❡❦❞❛②s ❛♥❞ ♦✉r ❡st✐♠❛t❡❞ ♣♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥✱ ✇❤✐❝❤ s❤♦✇s ❛ ❜❡tt❡r ✜t✳ ❈✳✷✳✷

❉❡♣❛rt✉r❡ ♣r♦❝❡ss

❚♦ ♠♦❞❡❧ ❞❡♣❛rt✉r❡s✱ ✇❡ ❡st✐♠❛t❡ ❛♥ ✐♥t❡r✈❛❧ ❝❡♥s♦r❡❞ ❤❛③❛r❞ ♠♦❞❡❧ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ r❛t❡ ❛t ✇❤✐❝❤ ♣❛t✐❡♥ts ❛♥❞✴♦r ❞♦♥♦rs ❞❡♣❛rt t❤❡ ◆❑❘ ✇✐t❤♦✉t ❛ tr❛♥s♣❧❛♥t✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❧❡t tai ✱ ✸ ❖♥❧②

✹✵ ❛rr✐✈❛❧s ✐♥ ✇❡❡❦❡♥❞s ♦✈❡r t❤❡ ❝♦✉rs❡ ♦❢ ✶✹✵ ✇❡❡❦s✳

✶✼

❋✐❣✉r❡ ❈✹✿ ❉✐str✐❜✉t✐♦♥ ♦❢ ◆❑❘ ❛♥❞ P♦✐ss♦♥ ◆✉♠❜❡r ♦❢ ❙✉❜♠✐ss✐♦♥s P❡r ❉❛② ❢♦r ❲❡❡❦❞❛②s t∗i ✱ ❛♥❞ tdi ❜❡ t❤❡ ✭❧❛t❡♥t✮ ❛rr✐✈❛❧✱ tr❛♥s♣❧❛♥t✱ ❛♥❞ ❞❡♣❛rt✉r❡ ❞❛t❡s ❢♦r ❛♥ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥t✱ ❞♦♥♦r✱ ♦r ♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐r i✳ ❖✉r ❞❛t❛s❡t r❡❝♦r❞s t0i ❛♥❞ t∗i ✐❢ i ✇❛s tr❛♥s♣❧❛♥t❡❞✳ ❋✉rt❤❡r✱ ✐❢ i ✐s tr❛♥s♣❧❛♥t❡❞✱ t❤❡♥ ✇❡ ❦♥♦✇ t❤❛t tdi > t∗i . ■❢ i ✐s ♥♦t tr❛♥s♣❧❛♥t❡❞✱ t❤❡♥  ❝❛s❡s ✇❡  a−✐♥ ♠♦st a+ a a ♦❜s❡r✈❡ ti ✱ ❜✉t ✐♥ s♦♠❡ ❝❛s❡s✱ ✇❡ ♦♥❧② ❦♥♦✇ t❤❛t ti ❜❡❧♦♥❣s t♦ ❛♥ ✐♥t❡r✈❛❧ ti , ti ✭t②♣✐❝❛❧❧② ✇✐t❤✐♥ ❛ ✇❡❡❦✮✳ ■❢ i ❞❡♣❛rt❡❞ ✇✐t❤♦✉t ❛ tr❛♥s♣❧❛♥t✱ ✇❡ ♦❜s❡r✈❡ tdi ❡✐t❤❡r ❡①❛❝t❧② ♦r ✉♣ t♦ ❛ s♠❛❧❧ ✐♥t❡r✈❛❧✳ ■❢ i r❡♠❛✐♥s ✐♥ t❤❡ ◆❑❘ ❛t t❤❡ ❡♥❞ ♦❢ ♦✉r s❛♠♣❧❡✱ t❤❡♥ ✇❡ ❦♥♦✇ t❤❛t tdi > T. ❯s✐♥❣ t❤❡s❡ ♦❜s❡r✈❛t✐♦♥s✱ ✇❡ ❝❛♥ ❝♦♥str✉❝t ❜♦✉♥❞s ♦♥ t❤❡ ❞✉r❛t✐♦♥ τi t❤❛t ❡❛❝❤ ✉♥✐t i r❡♠❛✐♥s

✐♥ t❤❡ ◆❑❘ ✇✐t❤♦✉t ❛ tr❛♥s♣❧❛♥t✳ ❲✐t❤ t❤❡s❡ ♦❜s❡r✈❛t✐♦♥s✱ ✇❡ ❡st✐♠❛t❡ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❤❛③❛r❞s ♠♦❞❡❧✳ ❚❤❡ ♠♦❞❡❧ ✐s ❝❤❛r❛❝✲ t❡r✐③❡❞ ❜② ❛ s✉r✈✐✈❛❧ ❢✉♥❝t✐♦♥ Si (τ ) = exp (−λi t) ,

✇❤❡r❡ ✇❡ ✉s❡ t❤❡ ♣❛r❛♠❡tr✐❝ ❢♦r♠ λi = α + zi β.

❚❤❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ t❤❡ ♠♦❞❡❧ ❢♦r t❤❡ ✐♥t❡r✈❛❧ ❝❡♥s♦r❡❞ s✉r✈✐✈❛❧ ❞❛t❛ ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ ❞❡r✐✈❡✱ ❛♥❞ ❡st✐♠❛t✐♦♥ ✈✐❛ ✐♥t❝❡♥s ✐♥ ❙❚❆❚❆ ✐s str❛✐❣❤t❢♦r✇❛r❞✳ ❚❛❜❧❡ ❈✷ ♣r❡s❡♥ts t❤❡ ❡st✐♠❛t❡s✳ ◆♦t❡ t❤❛t t❤❡ ❤②♣♦t❤❡s✐s t❡sts ✐♥ ❛ ❤❛③❛r❞ ♠♦❞❡❧ ❛r❡ r❡♣♦rt❡❞ r❡❧❛t✐✈❡ t♦ ✶✱ ✇❤✐❝❤ ✐♠♣❧✐❡s ♥♦ ❡✛❡❝t✳ ❆s ❝❛♥ ❜❡ s❡❡♥✱ ♣❛t✐❡♥ts ✇✐t❤ ❜❧♦♦❞ t②♣❡s t❤❛t ❛r❡ ❡❛s✐❡r t♦ ♠❛t❝❤ ♦r ✇❤♦ ❛r❡ ♣❛✐r❡❞ ✇✐t❤ ❡❛s✐❡r✲t♦✲♠❛t❝❤ ❞♦♥♦rs ❤❛✈❡ ❛ ❤✐❣❤❡r ❞❡♣❛rt✉r❡ r❛t❡✳ ❚❤✐s ✜♥❞✐♥❣ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ♣❛t✐❡♥ts ❛♥❞ ♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐rs ❞❡♣❛rt✐♥❣ ✐♥ r❡s♣♦♥s❡

✶✽

❚❛❜❧❡ ❈✷✿ ❉❡♣❛rt✉r❡ ❍❛③❛r❞ ❘❛t❡ ❊st✐♠❛t❡s

P❛t✐❡♥t ▼❛t❝❤✐♥❣ P♦✇❡r ❉♦♥♦r ▼❛t❝❤✐♥❣ P♦✇❡r P❛t✐❡♥t ❆❣❡ ❉♦♥♦r ❆❣❡ ❆❇ ❇❧♦♦❞✲t②♣❡ P❛t✐❡♥t ❆ ❇❧♦♦❞✲t②♣❡ P❛t✐❡♥t ❇ ❇❧♦♦❞✲t②♣❡ P❛t✐❡♥t ❆❇ ❇❧♦♦❞✲t②♣❡ ❉♦♥♦r

✭✶✮

✭✷✮

✭✸✮

P❛t✐❡♥t✲❉♦♥♦r P❛✐rs

❯♥♣❛✐r❡❞ P❛t✐❡♥ts

❆❧tr✉✐st✐❝ ❉♦♥♦rs

✶✳✽✶✾✯✯✯

✶✻✳✷✹✯✯✯

✭✵✳✷✹✶✮

✭✷✳✹✷✻✮

✵✳✵✽✸✺

✹✳✸✹❡✲✵✺

✭✵✳✶✻✹✮

✭✵✳✵✵✵✸✸✽✮

✵✳✾✾✸✯

✶✳✵✵✷

✭✵✳✵✵✸✹✾✮

✭✵✳✵✵✸✽✸✮

✶✳✵✵✼

✶✳✵✶✶

✭✵✳✵✵✹✹✹✮

✭✵✳✵✶✹✵✮

✷✳✹✺✷✯✯✯

✶✳✺✻✻✯

✭✵✳✻✾✹✮

✭✵✳✸✾✷✮

✶✳✶✼✷

✶✳✷✾✾

✭✵✳✶✺✽✮

✭✵✳✷✻✻✮

✶✳✵✼✾

✵✳✻✹✵✯

✭✵✳✶✼✸✮

✭✵✳✶✺✾✮

✵✳✺✽✺✯✯

✶✳✷✽✻

✭✵✳✶✺✵✮

✭✶✳✸✾✻✮

❆ ❇❧♦♦❞✲t②♣❡ ❉♦♥♦r

✵✳✻✻✵✯✯✯

✶✳✺✽✶

✭✵✳✵✽✷✶✮

✭✵✳✻✻✷✮

❇ ❇❧♦♦❞✲t②♣❡ ❉♦♥♦r

✵✳✻✷✹✯✯✯

✵✳✼✻✸

✭✵✳✵✾✽✼✮ ❈♦♥st❛♥t

❖❜s❡r✈❛t✐♦♥s

✭✵✳✺✶✾✮

✵✳✵✵✺✺✸✯✯✯

✵✳✵✵✵✽✾✹✯✯✯

✵✳✶✶✽

✭✵✳✵✵✺✸✺✮

✭✵✳✵✵✵✷✵✹✮

✭✵✳✹✷✼✮

✶✱✷✻✹

✹✾✽

✶✻✹

◆♦t❡✿ ■♥t❡r✈❛❧ ❝❡♥s♦r❡❞ ❡①♣♦♥❡♥t✐❛❧ ❤❛③❛r❞ ♠♦❞❡❧✳ P❛t✐❡♥t ✭❉♦♥♦r✮ ▼❛t❝❤ P♦✇❡r ✐s t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❞♦♥♦rs ✭♣❛t✐❡♥t✮ ✐♥ t❤❡ ◆❑❘ ♣♦♦❧ ♦✈❡r t❤❡ ❝♦✉rs❡ ♦❢ ❛ s❛♠♣❧❡ ❛ ❣✐✈❡♥ ♣❛t✐❡♥t ✭❞♦♥♦r✮ ✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤✳ ❙❛♠♣❧❡ r❡str✐❝t❡❞ t♦ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs t❤❛t r❡❣✐st❡r❡❞ ❛❢t❡r ❆♣r✐❧ ✷✵✶✷✳

✶✾

t♦ tr❛♥s♣❧❛♥t❛t✐♦♥ ♦♣♣♦rt✉♥✐t✐❡s ❡❧s❡✇❤❡r❡✱ ❡✐t❤❡r t❤r♦✉❣❤ ❞✐r❡❝t ❞♦♥❛t✐♦♥✱ ❞❡❝❡❛s❡❞ ❞♦♥♦r tr❛♥s♣❧❛♥ts✱ ♦r ❧✐✈❡✲❞♦♥♦r ❡①❝❤❛♥❣❡s ♦✉ts✐❞❡ t❤❡ ◆❑❘✳

❈✳✸

❈♦♠♣❛t✐❜✐❧✐t② ❛♥❞ ❛❝❝❡♣t❛♥❝❡

❚♦ ❝❛❧❝✉❧❛t❡ ✇❤❡t❤❡r ❞♦♥♦r

j

✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ♣❛t✐❡♥t

i✱

✇❡ ✉s❡ t❤❡ ❜❧♦♦❞ t②♣❡s ♦❢ t❤❡

♣❛t✐❡♥t ❛♥❞ ❞♦♥♦r✱ t❤❡ t✐ss✉❡ t②♣❡ ♦❢ t❤❡ ❞♦♥♦r✱ ❛♥❞ t❤❡ ❧✐st ♦❢ ✉♥❛❝❝❡♣t❛❜❧❡ ❛♥t✐❣❡♥s ❧✐st❡❞ ❜② t❤❡ ♣❛t✐❡♥t✳ ❚❤❡r❡ ❛r❡ t❤r❡❡ ❛❞❞✐t✐♦♥❛❧ ✇❛②s ✐♥ ✇❤✐❝❤ ❛ tr❛♥s♣❧❛♥t ❜❡t✇❡❡♥ ❛ ♣❛t✐❡♥t ❛♥❞ ❛ ❞♦♥♦r ❝❛♥ ❜❡ ♣r♦❤✐❜✐t❡❞✳ ❋✐rst✱ ✉♣♦♥ r❡❣✐str❛t✐♦♥✱ ❡❛❝❤ ♣❛t✐❡♥t ❝❛♥ ❞❡❝❧❛r❡ ❝r✐t❡r✐❛ ❢♦r ❡①❝❧✉❞✐♥❣ ❞♦♥♦rs ❜❛s❡❞ ♦♥ ❛ ✈❛r✐❡t② ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❚❤❡s❡ ✐♥❝❧✉❞❡ t❤r❡s❤♦❧❞s ❢♦r t❤❡ ♠❛①✐♠✉♠ ❞♦♥♦r ❛❣❡ ❛♥❞ ♠✐♥✐♠✉♠ ❞♦♥♦r ✇❡✐❣❤t t❤❛t ❛r❡ ❛❝❝❡♣t❛❜❧❡✳ ❚❤❡s❡ ❝r✐t❡r✐❛ ❛r❡ r❡❝♦r❞❡❞ ✐♥ ♦✉r ❞❛t❛s❡t✳ ❙❡❝♦♥❞✱ ✉♣♦♥ ❛rr✐✈❛❧ ❛ ♣❛t✐❡♥t ❝❛♥ ❧✐st ❛s ✉♥❛❝❝❡♣t❛❜❧❡ ❛♥② ♥✉♠❜❡r ♦❢ s♣❡❝✐✜❝ ❞♦♥♦rs ✇❤♦ ✇❡r❡ ✐♥ t❤❡ ◆❑❘ ♣♦♦❧ ❛t t❤❡ t✐♠❡✳ ❚❤✐s r❡❥❡❝t✐♦♥ ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r ❛♥② r❡❛s♦♥✱ ✐♥❝❧✉❞✐♥❣ ❦♥♦✇♥ ♣❛t❤♦❧♦❣✐❡s✳ P❛t✐❡♥ts ❝❛♥ ❛❧s♦ ❡①❝❧✉❞❡ ❞♦♥♦rs ❧❛t❡r✱ ❜✉t ❛❝❝♦r❞✐♥❣ t♦ ♦✉r ✉♥❞❡rst❛♥❞✐♥❣✱ t❤❡ ♣r❛❝t✐❝❡ ✐s ♠♦st ❝♦♠♠♦♥ ❞✉r✐♥❣ r❡❣✐str❛t✐♦♥✳ ❖✉r ❞❛t❛s❡t ✐♥❝❧✉❞❡s t❤❡ ❛♥♦♥②♠✐③❡❞ ✐❞❡♥t✐✜❡rs ❢♦r t❤❡s❡ ❡①❝❧✉❞❡❞ ❞♦♥♦rs✳ ❚❤✐r❞✱ ✇❤❡♥ ❛ tr❛♥s♣❧❛♥t ✐s ♣r♦♣♦s❡❞✱ ❛ ♣❛t✐❡♥t ♠❛② r❡❢✉s❡ t❤❡ s♣❡❝✐✜❝ ❞♦♥♦r✳ ❆ ✜rst ♣❤❛s❡ ♦❢ r❡❢✉s❛❧s ✐s ❛t t❤❡ ♣❛t✐❡♥t✬s ❞✐s❝r❡t✐♦♥ ✭✇✐t❤ ❛❞✈✐s❡ ❢r♦♠ ❤✐s✴❤❡r s✉r❣❡♦♥✮✳ ■❢ ❛ ♣❛t✐❡♥t ❝❤♦♦s❡s t♦ ♣r♦❝❡❡❞ ❛❢t❡r t❤❡ ✜rst ♣❤❛s❡✱ ❛ ✜♥❛❧ t✐ss✉❡✲t②♣❡ ❝♦♠♣❛t✐❜✐❧✐t② t❡st ✐s ❝♦♥❞✉❝t❡❞✳ ❲❡ r❡❢❡r t♦ t❤✐s ❛s t❤❡ s❡❝♦♥❞ ♣❤❛s❡✳ ■♥ ♦✉r s✐♠✉❧❛t✐♦♥s✱ ✇❡ ✐♥✐t✐❛❧✐③❡ ❖t❤❡r✇✐s❡✱ ✇❡ s❡t

cij = 0✳

■❢

j

cij = 1 ✐❢ j

✐s ♦✛❡r❡❞ t♦

✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤

i

i ❛♥❞ ✐❢ j

✐s ♥♦t ❡①❝❧✉❞❡❞ ❜② i✳

❞✉r✐♥❣ t❤❡ s✐♠✉❧❛t✐♦♥ ❛♥❞ ❛ tr❛♥s♣❧❛♥t ✐s r✉❧❡❞

♦✉t ❞✉r✐♥❣ ❡✐t❤❡r t❤❡ ✜rst ♦r s❡❝♦♥❞ ♣❤❛s❡s ✐♥ t❤❡ t❤✐r❞ t②♣❡ ♦❢ ❡①❝❧✉s✐♦♥✱ t❤❡♥ ✇❡ s❡t

cij = 0

❢♦r ❢✉t✉r❡ ✏s✐♠✉❧❛t✐♦♥ ❞❛②s✳✑

❈✳✹

❇✉r♥✲✐♥ ❛♥❞ ❝❛❧❝✉❧❛t✐♥❣ st❛♥❞❛r❞ ❡rr♦rs

❲❡ ♠❛② st❛rt ♦✉r s✐♠✉❧❛t✐♦♥s ❢r♦♠ ❛♥② ✐♥✐t✐❛❧ st❛t❡ ❢♦r t❤❡ ◆❑❘ ❜❡❝❛✉s❡ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❝❤♦s❡♥ ✐♥✐t✐❛❧ st❛t❡ ❢❛❞❡s ♦✈❡r t✐♠❡✳ ❆ ❝♦♥✈❡♥✐❡♥t ❝❤♦✐❝❡ ✐s t♦ ♣✐❝❦ ❛♥ ✐♥✐t✐❛❧ ♣♦♦❧ ✇✐t❤ ♥♦ ✉♥♣❛✐r❡❞ ♣❛t✐❡♥ts✱ ❛❧tr✉✐st✐❝ ❞♦♥♦rs✱ ♦r ♣❛t✐❡♥t✲❞♦♥♦r ♣❛✐rs✳ ❆❧t❤♦✉❣❤ t❤❡ ✐♥✐t✐❛❧ ♣♦♦❧ ❞♦❡s ♥♦t ❛✛❡❝t ❧♦♥❣✲r✉♥ ❛✈❡r❛❣❡s ✇✐t❤ ❡♥♦✉❣❤ s✐♠✉❧❛t✐♦♥s✱ ✐t ✐s ❛❞✈✐s❛❜❧❡ t♦ ❞✐s❝❛r❞ ♦r ❜✉r♥✲✐♥ ❛ ♣♦rt✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝❤❛✐♥ ✐♥ ♦r❞❡r t♦ ✐♠♣r♦✈❡ t❤❡ ❡st✐♠❛t❡s✬ ♣r❡❝✐s✐♦♥✳ ❆ ❜✉r♥✲✐♥ ♦❢ ❛❜♦✉t ✷✵✵✵ ❞❛②s ❛♣♣❡❛rs t♦ ②✐❡❧❞ ♣♦t❡♥t✐❛❧ s❝❛❧❡ r❡❞✉❝t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ♣❡r ❞❛② t❤❛t ✐s ❝❧♦s❡ t♦ ✶✱ s✉❣❣❡st✐♥❣ t❤❛t t❤❡ ❝❤❛✐♥ ✐s ❧✐❦❡❧② t♦ ❤❛✈❡ ❝♦♥✈❡r❣❡❞ ❛t t❤❛t ♣♦✐♥t✳

y1 , . . . , yT ♦❢ t❤❡ tr❛♥s♣❧❛♥ts t❤❛t ♦❝❝✉r ♦♥ ❡❛❝❤ ❞❛② ❛❢t❡r ❛♥ ❲❡ ❡st✐♠❛t❡ f (q) ❛s t❤❡ s❛♠♣❧❡ ♠❡❛♥ ♦❢ t❤❡ yt ❛♥❞ ❝❛❧❝✉❧❛t❡ t❤❡ st❛♥❞❛r❞

❖✉r s✐♠✉❧❛t✐♦♥s ♣r♦❞✉❝❡ ❛ s❡r✐❡s ✐♥✐t✐❛❧ ❜✉r♥ ♣❡r✐♦❞✳

❡rr♦rs ♦❢ t❤✐s ❡st✐♠❛t❡ ✉s✐♥❣ t❤❡ ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❜❛t❝❤ ♠❡❛♥s ❡st✐♠❛t♦r ❜② ❢♦❧❧♦✇✐♥❣ ❈❤❛♣t❡r ✶✷ ✐♥ ❘♦❜❡rt ❛♥❞ ❈❛s❡❧❧❛ ✭✷✵✵✹✮✳

❚❤❡ ♠❡t❤♦❞ ❞✐✈✐❞❡s t❤❡ t✐♠❡ s❡r✐❡s ♦❢

yt

✐♥t♦ ❜❛t❝❤❡s✱

❝❛❧❝✉❧❛t❡s t❤❡ s❛♠♣❧❡ ♠❡❛♥ ✐♥ ❡❛❝❤ ♦❢ t❤♦s❡ ❜❛t❝❤❡s✱ ❛♥❞ ✉s❡s t❤❡ ✈❛r✐❛❜✐❧✐t② ✐♥ s❛♠♣❧❡ ♠❡❛♥s t♦ ❡st✐♠❛t❡ t❤❡ st❛♥❞❛r❞ ❡rr♦r ♦❢

f (q)✳

❲❡ ✉s❡ t❤❡ ❝♦♠♠♦♥❧② r❡❝♦♠♠❡♥❞❡❞ ❜❛t❝❤

✷✵ √

s✐③❡ ♦❢ ❛♣♣r♦①✐♠❛t❡❧② T ✳ ❚❤❡ ♣r♦❝❡❞✉r❡ ✐s ❛ s✐♠♣❧❡ ❛♥❞ ♣♦♣✉❧❛r ♠❡t❤♦❞ t❤❛t ❛❝❝♦✉♥ts ❢♦r ❛✉t♦❝♦rr❡❧❛t✐♦♥ ♦❢ t❤❡ yt ✳ ❈✳✺ ❈✳✺✳✶

❈❛❧✐❜r❛t✐♦♥ ❈❛❧✐❜r❛t✐♦♥ Pr♦❝❡❞✉r❡

❖✉r s✐♠✉❧❛t✐♦♥ ♣r♦❝❡❞✉r❡ ✐s t❛✐❧♦r❡❞ t♦ ♠❛t❝❤ t❤❡ ♣r♦❝❡❞✉r❡s ❛♥❞ ♣r❛❝t✐❝❡s ✉s❡❞ ❜② t❤❡ ◆❑❘✳ ■♥ ♠♦st ❝❛s❡s✱ t❤❡ ❞❛t❛ ♦r ✐♥st✐t✉t✐♦♥❛❧ ❦♥♦✇❧❡❞❣❡ ❞✐r❡❝t❧② t❡❧❧ ✉s t❤❡ ♣❛r❛♠❡t❡rs❀ ❡✳❣✳✱ t❤❡ ✇❡✐❣❤ts wij ❛r❡ ❝❤♦s❡♥ t♦ ♠❛t❝❤ ◆❑❘✬s ♣r❛❝t✐❝❡s✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ❛ ❢❡✇ ❛s♣❡❝ts ♦❢ t❤❡ r❡❛❧✲✇♦r❧❞ ♣r♦❝❡❞✉r❡s ❛♥❞ ♦✉t❝♦♠❡s ✐♥ t❤❡ ◆❑❘ ♦♥ ✇❤✐❝❤ ✇❡ ❞♦♥✬t ❞✐r❡❝t❧② ❤❛✈❡ ✐♥❢♦r♠❛t✐♦♥✳ ❲❡ ♠♦❞❡❧ ❛♥❞ ♣❛r❛♠❡tr✐③❡ t❤❡s❡ ❛s♣❡❝ts ✐♥ ♦✉r s✐♠✉❧❛t✐♦♥ ♠♦❞❡❧ ❛♥❞ ❝❛❧✐❜r❛t❡ t❤❡♠ t♦ ♠❛t❝❤ t❤❡ r❡❛❧✐③❡❞ ♥✉♠❜❡r ♦❢ ♠❛t❝❤❡s ✐♥ t❤❡ ◆❑❘✳ ❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ s❡ts ♦❢ ♣❛r❛♠❡t❡rs ✇❡ ♥❡❡❞ t♦ ❝❛❧✐❜r❛t❡✳ ❋✐rst✱ ✇❡ ❞♦ ♥♦t ❤❛✈❡ ❞✐r❡❝t ❞❛t❛ ♦♥ t❤❡ ❢r✐❝t✐♦♥s ♦❢ tr❛♥s❧❛t✐♥❣ ♣r♦♣♦s❡❞ tr❛♥s♣❧❛♥ts ✐♥t♦ s✉r❣❡r✐❡s✳ ❆s ♠❡♥t✐♦♥❡❞ ✐♥ ❙❡❝t✐♦♥ ❈✳✸✱ t❤❡ ✈❛r✐♦✉s ❛❝❝❡♣t❛♥❝❡ ♣❤❛s❡s ♠❛② r❡s✉❧t ✐♥ s♦♠❡ tr❛♥s♣❧❛♥ts ♥♦t ❜❡✐♥❣ ❝♦♥s✉♠♠❛t❡❞✳ ❊❛❝❤ ♣❤❛s❡ ✐♥tr♦❞✉❝❡s ❛ t✐♠❡✲❧❛❣ ❜❡t✇❡❡♥ tr❛♥s♣❧❛♥ts ❜❡✐♥❣ ♣r♦♣♦s❡❞ ❛♥❞ ✜♥❛❧✐③❡❞ ❛s ✇❡❧❧ ❛s t❤❡ ❝❤❛♥❝❡ ♦❢ ❛ ♠❛t❝❤ ❜❡✐♥❣ ❛❜♦rt❡❞✳ ❘♦✉❣❤❧② s♣❡❛❦✐♥❣✱ t❤❡s❡ ❢r✐❝t✐♦♥s r❡❞✉❝❡ t❤❡ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ❢❛❝✐❧✐t❛t❡❞ ❜② t❤❡ ◆❑❘✳ ❲❡ ♣❛r❛♠❡tr✐③❡ t❤❡s❡ ♣❤❛s❡s ❛♥❞ ❝❛❧✐❜r❛t❡ t❤❡ ♣❛r❛♠❡t❡rs t♦ ❜❡st ✜t t❤❡ ♦❜s❡r✈❡❞ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥ts ❜② ♣❛t✐❡♥t t②♣❡✳ ❚♦ ❞♦ s♦✱ ✇❡ s✐♠✉❧❛t❡ ♦✉t❝♦♠❡s ♣r❡❞✐❝t❡❞ ❜② ♦✉r ♠♦❞❡❧ ❢♦r ✈❛r✐♦✉s ❧❡♥❣t❤s ✭♥✉♠❜❡r ♦❢ ❞❛②s✮ ❛♥❞ ✈❛r✐♦✉s ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t✐❡s ❢♦r ❜♦t❤ ♣❤❛s❡s✳ ❚❤❡ ✜rst ♣❤❛s❡ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❝♦♥tr♦❧❧✐♥❣ t❤❡ ❢r✐❝t✐♦♥s ✐♥ t❤❡ s②st❡♠ ❜❡❝❛✉s❡ ♣r♦♣♦s❡❞ ♠❛t❝❤❡s ❛r❡ r❡❢✉s❡❞✱ ✇❤❡r❡❛s t❤❡ s❡❝♦♥❞ ♣❤❛s❡ ♣❛r❛♠❡t❡rs ❣♦✈❡r♥ t❤❡ ❢r✐❝t✐♦♥s ❞✉❡ t♦ ❜✐♦❧♦❣✐❝❛❧ ❝♦♠♣❛t✐❜✐❧✐t② t❡sts✳ ❙❡❝♦♥❞✱ ❛s ♠❡♥t✐♦♥❡❞ ❡❛r❧✐❡r✱ ✇❤❡♥ ❝❤❛✐♥s ❛r❡ ❛❜♦rt❡❞ ❜❡❝❛✉s❡ ♦❢ ❛ r❡❢✉s❛❧✱ ◆❑❘ ✉s✉❛❧❧② tr✐❡s t♦ ✉s❡ t❤❡ ❞♦♥♦r✱ ❝❛❧❧❡❞ t❤❡ ❜r✐❞❣❡ ❞♦♥♦r✱ ♦❢ t❤❡ ❧❛st tr❛♥s♣❧❛♥t❡❞ ♣❛t✐❡♥t ❢♦r ❛ ♥❡✇ ❝❤❛✐♥✳ ❍♦✇❡✈❡r✱ t❤❡ ❡①❝❤❛♥❣❡ ♣r❡❢❡rs ♥♦t t♦ ✇❛✐t t♦♦ ❧♦♥❣ t♦ st❛rt ❛ ❝❤❛✐♥ ✇✐t❤ t❤✐s ❞♦♥♦r✳ ■❢ ❛ ♥❡✇ ❝❤❛✐♥ ❝❛♥♥♦t ❜❡ ❢♦✉♥❞✱ t❤❡ ❞♦♥♦r ✐s ♦✛❡r❡❞ t♦ ❛ ♣❛t✐❡♥t ✇✐t❤♦✉t ❛ r❡❧❛t❡❞ ❞♦♥♦r✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ✇❡ ❞♦ ♥♦t ❦♥♦✇ ♦❢ ❛ ❝♦♥s✐st❡♥t ♣♦❧✐❝② r✉❧❡ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ◆❑❘✳ ❲❡ t❤❡r❡❢♦r❡ ❛❧s♦ ❡①♣❡r✐♠❡♥t❡❞ ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s t❤❡ ◆❑❘ tr✐❡s t♦ ♠❛t❝❤ ❛ ❜r✐❞❣❡ ❞♦♥♦r✳ ■♥ s✉♠♠❛r②✱ ✇❡ ❝❛❧✐❜r❛t❡❞ ✜✈❡ ♣❛r❛♠❡t❡rs✿ ✭✐✮ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s ❛ ❜r✐❞❣❡ ❞♦♥♦r ❝❛♥ ✐♥✐t✐❛t❡ ❛ ♥❡✇ ❝❤❛✐♥✱ ✭✐✐✮ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s ❢♦r ❝♦♥s❡♥t ❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❝♦♥s❡♥t ❢♦r t❤❡ t✇♦ ♣❤❛s❡s ♦❢ ♠❛t❝❤ ❛❝❝❡♣t❛♥❝❡✱ ❛♥❞ ✭✐✐✐✮ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s t❛❦❡♥ ❢♦r t❡st✐♥❣ ❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❢❛✐❧✐♥❣ t❤❡ t✐ss✉❡ t②♣❡ ❜✐♦❧♦❣✐❝❛❧ t❡st✳ ❚❤✐s ❛♣♣r♦❛❝❤ ♣❛rs✐♠♦♥✐♦✉s❧② ♣❛r❛♠❡tr✐③❡s t❤❡ ♥❛t✉r❡ ♦❢ ❢r✐❝t✐♦♥s ✐♥ t❤❡ ◆❑❘✳ ❍♦✇❡✈❡r✱ t❤❡s❡ ♠♦❞❡❧s✬ s✐♥❣❧❡ ♣❛r❛♠❡t❡r ✈❡rs✐♦♥s ❛r❡ ❝❧❡❛r❧② ❛ s✐♠♣❧✐✜❝❛t✐♦♥✳ ▼♦st ❧✐❦❡❧②✱ ◆❑❘ ♣♦❧✐❝✐❡s ❛♥❞ t❤❡✐r ❛❜✐❧✐t② t♦ tr❛♥s❧❛t❡ ♣r♦♣♦s❡❞ ♠❛t❝❤❡s ✐♥t♦ tr❛♥s♣❧❛♥ts ❡✈♦❧✈❡s ♦✈❡r t✐♠❡ ❛♥❞ ✐♥❝❧✉❞❡s s♦♠❡ ❛❞ ❤♦❝ ♠♦❞✐✜❝❛t✐♦♥s t♦ t❤❡✐r ❜❛s✐❝ ♣r♦❝❡❞✉r❡✳ ❋♦r ♦✉r ❝❛❧✐❜r❛t✐♦♥s✱ ✇❡ ❝♦♥❞✉❝t ♦✉r s✐♠✉❧❛t✐♦♥s ❜② s❡tt✐♥❣ ❛♥ ✐♥✐t✐❛❧ ♠❛r❦❡t ✇✐t❤ t❤❡ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✇❤♦ ✇❡r❡ ♣r❡s❡♥t ✐♥ t❤❡ ◆❑❘ ♦♥ ✶ ❆♣r✐❧✱ ✷✵✶✷✳ ❚❤✐s ✐s t❤❡ ❞❛t❡ ❢r♦♠ ✇❤✐❝❤ ✇❡

✷✶ ❤❛✈❡ ❝❧❡❛r r❡❣✐str❛t✐♦♥ ❞❛t❛✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♣❛r❛♠❡t❡r s❡t✱ ✇❡ r✉♥ ✶✵✵ s✐♠✉❧❛t✐♦♥s ✉♥t✐❧ ❉❡❝❡♠❜❡r ✷✵✶✹✱ t❤❡ ❧❛st ❞❛t❡ ♦❢ t❤❡ ❛✈❛✐❧❛❜❧❡ ❞❛t❛✳ ❲❡ ❝❛❧✐❜r❛t❡❞ ♦✉r ♣❛r❛♠❡t❡rs t♦ ♠❛t❝❤ t❤❡ tr❛♥s♣❧❛♥t❛t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s✱ ❞❛②s ✐♥ t❤❡ ❡①❝❤❛♥❣❡ ❜r♦❦❡♥ ❜② ✇❤❡t❤❡r ♦r ♥♦t t❤❡ ♣❛t✐❡♥t✴❞♦♥♦r ✇❛s tr❛♥s♣❧❛♥t❡❞✱ t❤❡ t♦t❛❧ st♦❝❦ ♦❢ ♣❛t✐❡♥ts ❛♥❞ ❞♦♥♦rs ✐♥ t❤❡ ❡①❝❤❛♥❣❡✱ ❛♥❞ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♣❧❛♥t❛t✐♦♥s✳ ❚❤❡ ❜❡st ✜tt❡❞ ♣❛r❛♠❡t❡rs ✇❡r❡ ✶✹ ❞❛②s ♦❢ ✇❛✐t✐♥❣ ❛♥❞ ✽✵✪ s✉❝❝❡ss r❛t❡ ❢♦r ❡❛❝❤ ♦❢ t❤❡ t✇♦ ♣❤❛s❡s✳ ❈✳✺✳✷

❈❛❧✐❜r❛t✐♦♥ ❘❡s✉❧ts

❋✐❣✉r❡s C5 ✲ C8 ❝♦♠♣❛r❡ t❤❡ tr❡♥❞s ❢♦r t❤❡s❡ st❛t✐st✐❝s ❛❝r♦ss t❤❡ ❞❛t❛ ❛♥❞ t❤❡ s✐♠✉❧❛t✐♦♥s ❢♦r ♦✉r ❝❤♦s❡♥ ♣❛r❛♠❡t❡rs✳ ❚❤❡ ❞❛s❤❡❞ ❧✐♥❡s ❞❡♣✐❝t ❞❛t❛✱ ❞❛r❦❡r ❧✐♥❡s ❛r❡ t❤❡ ♠❡❛♥ ♦❢ ✶✵✵ s✐♠✉❧❛t✐♦♥s✱ ❛♥❞ ❞❛s❤❡❞ ❧✐♥❡s ❛r❡ ✾✺✪ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧ ❢r♦♠ t❤❡ s✐♠✉❧❛t✐♦♥s✳ ●r❡❡♥ ❧✐♥❡s s❤♦✇ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜♠✐ss✐♦♥s ♦❢ ❛ ❣✐✈❡♥ t②♣❡ ✭❛❧tr✉✐st✐❝✱ ♣❛✐r✱ ♦r ✉♥♣❛✐r❡❞✮✱ r❡❞ ❧✐♥❡s s❤♦✇ t❤❡ ❝✉♠✉❧❛t✐✈❡ ♥✉♠❜❡r ♦❢ s✉❜♠✐ss✐♦♥s t❤❛t ❞❡♣❛rt❡❞ t❤❡ ♠❛r❦❡t ✇✐t❤♦✉t ❛ tr❛♥s♣❧❛♥t✱ ❛♥❞ ❜❧✉❡ ❧✐♥❡s s❤♦✇ ❝✉♠✉❧❛t✐✈❡ ♥✉♠❜❡r ♦❢ s✉❜♠✐ss✐♦♥s t❤❛t ✇❡r❡ tr❛♥s♣❧❛♥t❡❞✳ ❋✐❣✉r❡s C5 ❛♥❞ C6 s❤♦✇ t❤❛t✱ ❛t t❤❡ ❝❛❧✐❜r❛t❡❞ ♣❛r❛♠❡t❡rs✱ t❤❡ ♠♦❞❡❧ ✜ts t❤❡ ❞❛t❛ ❡①tr❡♠❡❧② ✇❡❧❧✳ ❆❧t❤♦✉❣❤ t❤❡ ❛✈❡r❛❣❡s ❛r❡ ♥♦t ❛s ✇❡❧❧ ♠❛t❝❤❡❞ ✐♥ ❋✐❣✉r❡s C8 ❛♥❞ C7✱ t❤❡ ♦❜s❡r✈❡❞ q✉❛♥t✐t✐❡s ❛r❡ ✇✐t❤✐♥ t❤❡ ✾✺✪ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ❢♦r t❤❡ ♠♦❞❡❧✳ ❚❛❜❧❡ C3 ♣r❡s❡♥ts t❤❡ s✉♠♠❛r② st❛t✐st✐❝s ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ s✐♠✉❧❛t✐♦♥s ❛♥❞ ◆❑❘ ❞❛t❛✳ ❆❧t❤♦✉❣❤ ✇❡ ❝❛❧✐❜r❛t❡❞ ♦♥❧② ✜✈❡ ♣❛r❛♠❡t❡rs✱ t❤❡ t❛❜❧❡ s❤♦✇s t❤❛t t❤❡ ♠♦❞❡❧ ♠❛t❝❤❡s s❡✈❡r❛❧ ❞❛t❛ ♠♦♠❡♥ts ❡①tr❡♠❡❧② ✇❡❧❧✳

✷✷ 1200 Number of Transplants Number of Registrations Left without Transplant Stock

Number of Registrations

1000

800

600

400

200

0 0

100

200

300

400

500

600

700

800

900

1000

900

1000

Calendar/Simulation Days

❋✐❣✉r❡ ❈✺✿ ❈❛❧✐❜r❛t✐♦♥ ❢♦r ❆❧❧ ❙✉❜♠✐ss✐♦♥s 800 Number of Transplants Number of Registrations Left without Transplant Stock

700

Number of Pairs

600

500

400

300

200

100

0 0

100

200

300

400

500

600

700

800

Calendar/Simulation Days

❋✐❣✉r❡ ❈✻✿ ❈❛❧✐❜r❛t✐♦♥ ❢♦r P❛✐rs

◆♦t❡✿ ❖❜s❡r✈❡❞ ◗✉❛♥t✐t✐❡s ❛♥❞ ❙✐♠✉❧❛t✐♦♥s ❛r❡ s❤♦✇♥ ✇✐t❤ s♦❧✐❞ ❛♥❞ ❞❛s❤❡❞ ❧✐♥❡s r❡s♣❡❝t✐✈❡❧②✳ ❙♦❧✐❞ ❣r❡② ❧✐♥❡s r❡♣r❡s❡♥t ✾✺✪ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s✳

✷✸ 180 Number of Transplants Number of Registrations Left without Transplant Stock

160

Number of Altruistic Donors

140

120

100

80

60

40

20

0 0

100

200

300

400

500

600

700

800

900

1000

900

1000

Calendar/Simulation Days

❋✐❣✉r❡ ❈✼✿ ❈❛❧✐❜r❛t✐♦♥ ❢♦r ❆❧tr✉✐st✐❝ ❉♦♥♦rs 450 Number of Transplants Number of Registrations Left without Transplant Stock

400

Number of Unpaired Patients

350

300

250

200

150

100

50

0 0

100

200

300

400

500

600

700

800

Calendar/Simulation Days

❋✐❣✉r❡ ❈✽✿ ❈❛❧✐❜r❛t✐♦♥ ❢♦r ❯♥♣❛✐r❡❞ P❛t✐❡♥ts

◆♦t❡✿ ❖❜s❡r✈❡❞ ◗✉❛♥t✐t✐❡s ❛♥❞ ❙✐♠✉❧❛t✐♦♥s ❛r❡ s❤♦✇♥ ✇✐t❤ s♦❧✐❞ ❛♥❞ ❞❛s❤❡❞ ❧✐♥❡s r❡s♣❡❝t✐✈❡❧②✳ ❙♦❧✐❞ ❣r❡② ❧✐♥❡s r❡♣r❡s❡♥t ✾✺✪ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s✳

❚❛❜❧❡ ❈✸✿ ▼♦❞❡❧ ❋✐t

Patient Match Probability

Donor Match Probability

Mean Days in the NKR Mean Days in the NKR Mean Days in the NKR Mean Days in the NKR for Untransplanted for Transplanted for Untransplanted for Transplanted Donors Donors Patients Patients

Panel A: Observed Quantities from the National Kidney Registry (NKR) Pair

0.56

0.55

146.67

197.58

Chip Patients

0.30

-

63.85

135.40

-

-

-

0.79

-

-

105.98

74.91

Under-Demanded

0.41

0.40

195.19

223.27

187.97

215.60

Over-Demanded

0.65

0.66

117.55

129.45

115.37

128.52

Self-Demanded

0.64

0.62

133.09

188.89

126.22

178.06

179.42

Altruistic Donors

140.53

189.92

Panel B: Simulated Quantities Pair

0.60

0.57

99.85

183.35

120.00

Chip Patients

0.27

-

87.87

105.40

-

-

-

0.89

-

-

50.00

47.72

Altruistic Donors Under-Demanded

0.38

0.36

164.01

211.06

190.64

209.50

Over-Demanded

0.71

0.69

69.62

135.61

78.82

132.20

Self-Demanded

0.73

0.69

84.87

153.87

106.51

148.88

✷✹

✷✺

❋✐❣✉r❡ ❈✾✿ ❈❤❛✐♥ ▲❡♥❣t❤ ❉✐str✐❜✉t✐♦♥

✷✻

1 0.9 0.8

Frequency

0.7 0.6

Chains Chains

0.5 0.4 0.3 0.2 3-Cycle 3-Cycle

0.1 2-Cycle

2-Cycle

Data

Simulation

0

Cycle - Chain Ratio ❋✐❣✉r❡ ❈✶✵✿ ❈②❝❧❡ ✲ ❈❤❛✐♥ ❘❛t✐♦

❚❛❜❧❡ ❈✹✿ ❘❡❣r❡ss✐♦♥ ❚r❡❡ ❙✉♠♠❛r② ❙t❛t✐st✐❝s Match Probability N

Mean

S.E.

Marginal Product

Within Category Standard Deviation

Mean

S.E.

Within Category Standard Deviation

Panel A: Altruistic Donors Non-O Donor

102

0.86

(0.01)

0.04

0.84

(0.05)

0.21

O Donor

62

0.94

(0.01)

0.00

1.86

(0.06)

0.17

Panel B: Patient-Donor Pairs O Patient, Non-O Donor

493

0.28

(0.01)

0.16

0.08

(0.02)

0.19

O Patient, O Donor, PRA >= 91%

100

0.30

(0.01)

0.23

0.13

(0.05)

0.23

O Patient, O Donor, PRA < 91%

148

0.81

(0.01)

0.03

0.72

(0.04)

0.22

Non-O Patient, O Donor, PRA >= 96%

124

0.28

(0.01)

0.21

0.09

(0.05)

0.21

Non-O Patient, Non-O Donor, PRA < 96%

292

0.85

(0.01)

0.03

0.69

(0.03)

0.26

Non-O Patient, O Donor, PRA < 96%

108

0.84

(0.01)

0.01

1.44

(0.05)

0.30

Unpaired Patients

501

0.24

(0.01)

0.13

(0.02)

0.20

Panel C: Unpaired Patients 0.07

◆♦t❡s ✿

❈❛t❡❣♦r✐❡s ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② r❡❣r❡ss✐♦♥ tr❡❡ ❛♥❛❧②s✐s t♦ ♣r❡❞✐❝t ♠❛r❣✐♥❛❧ ♣r♦❞✉❝ts ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✇❤❡t❤❡r ❛ s✉❜♠✐ss✐♦♥ ✐s ❛ ♣❛✐r ♦r ❛❧tr✉✐st✱ ❜❧♦♦❞ t②♣❡s✱ ❛♥❞ t❤❡ ♣❛t✐❡♥t✬s P❘❆✳ ❖✉r ♣r♦❝❡❞✉r❡ ❢♦❧❧♦✇❡❞ st❛♥❞❛r❞ r❡❝♦♠♠❡♥❞❛t✐♦♥s ✐♥ ❋r✐❡❞♠❛♥ ❡t ❛❧✳ ✭✷✵✵✶✮✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ✉s❡❞ ✶✵✲❢♦❧❞ ❝r♦ss✲✈❛❧✐❞❛t✐♦♥ t♦ ♣✐❝❦ t❤❡ ♣❡♥❛❧t② ♣❛r❛♠❡t❡r ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ♥♦❞❡s✱ r❡q✉✐r❡❞ ❡❛❝❤ ❧❡❛❢ t♦ ❤❛✈❡ ❛t ❧❡❛st ✷✵ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ ♣r✉♥❡❞ ❛ ❧❡❛❢ ✐❢ ✐t ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ t❤❡ ♦✈❡r❛❧❧ ✜t ❜② ❛t ❧❡❛st ✷✪✳ ❚❤❡ r❡s✉❧t✐♥❣ tr❡❡ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✉r❡ ✽✳ ❙t❛♥❞❛r❞ ❡rr♦rs ❢♦r t❤❡ s✐♠✉❧❛t✐♦♥s ❛r❡ ❝❛❧❝✉❧❛t❡❞ ❜② ❢♦❧❧♦✇✐♥❣ ❈❤❛♣t❡r ✶✷ ♦❢ ❘♦❜❡rt ❛♥❞ ❈❛s❡❧❧❛ ✭✷✵✵✹✮✳ ❚❤❡ ✇✐t❤✐♥ ❝❛t❡❣♦r② st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ✐s ❡st✐♠❛t❡❞ ✉s✐♥❣ s❤r✐♥❦❛❣❡ ♠❡t❤♦❞s r❡❝♦♠♠❡♥❞❡❞ ✐♥ ▼♦rr✐s ✭✶✾✽✸✮✳

✷✼

Average Number of Transplantation Per Registration

✷✽

0.7

0.6

0.5

0.4

Baseline Lower Waittime, Higher Friction Higher Waittime, Lower Friction

0.3

0.2

0.1

0 0

100

200

300

400

500

600

700

800

900

1000

Number of Registrations Per Year ❋✐❣✉r❡ ❉✶✶✿ ❘♦❜✉st♥❡ss✿ Pr♦❞✉❝t✐♦♥ ❋✉♥❝t✐♦♥ ✈❡rs✉s ❙❝❛❧❡

◆♦t❡s ✿ ❈♦♥str✉❝t❡❞ ❛s ✐♥ ✜❣✉r❡ ✻✳

❉ ❘♦❜✉st♥❡ss ❛♥❛❧②s❡s ❚❤✐s s❡❝t✐♦♥ ❛ss❡ss❡s t❤❡ r♦❜✉st♥❡ss ♦❢ ♦✉r r❡s✉❧ts t♦ ❝❛❧✐❜r❛t❡❞ ♣❛r❛♠❡t❡rs✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❢r✐❝✲ t✐♦♥s ✐♥ ❝♦♥s✉♠♠❛t✐♥❣ ♣r♦♣♦s❡❞ tr❛♥s♣❧❛♥ts ❞✉❡ t♦ ❧♦♥❣❡r ✇❛✐t✐♥❣ t✐♠❡s ❜✉t ❤✐❣❤❡r ❛♣♣r♦✈❛❧ r❛t❡s ♣r♦❞✉❝❡ s✐♠✐❧❛r ♠♦♠❡♥ts ❛s ❧♦✇❡r ✇❛✐t✐♥❣ t✐♠❡s ❛♥❞ ❧♦✇❡r ❛❝❝❡♣t❛♥❝❡ r❛t❡s✳

❈❤❛✐♥

❧❡♥❣t❤s✱ ❤♦✇❡✈❡r✱ ❛r❡ ✐♥❝r❡❛s✐♥❣ ✐♥ ❛❝❝❡♣t❛♥❝❡ r❛t❡s ❛♥❞ ❛r❡ ❜❡st ♠❛t❝❤❡❞ ❜② ♦✉r ❜❛s❡❧✐♥❡ ♣❛r❛♠❡t❡rs✳

❲❡ ❝♦♠♣❛r❡ ♦✉r ❜❛s❡❧✐♥❡ r❡s✉❧ts ✇✐t❤ t✇♦ s✉❜st❛♥t✐❛❧❧② ❞✐✛❡r❡♥t ♣❛r❛♠❡t❡rs✳

❚❤❡ ✜rst✱ ❧❛❜❡❧❧❡❞ ✏❍✐❣❤❡r ❲❛✐t✲t✐♠❡ ❛♥❞ ▲♦✇❡r ❋r✐❝t✐♦♥s✱✑ ❤❛s t✇♦ ✇❡❡❦s ❛♥❞ t❤r❡❡ ✇❡❡❦s ❢♦r ❡❛❝❤ ♦❢ t❤❡ t✇♦ ❛♣♣r♦✈❛❧ ♣❡r✐♦❞s ✭❛♣♣r♦✈❛❧ ❛♥❞ ❜✐♦❧♦❣✐❝❛❧ t❡st✐♥❣✮ ❜✉t ✐♥❝r❡❛s❡s t❤❡ ❛❝✲ ❝❡♣t❛♥❝❡ r❛t❡s ✐♥ ❡❛❝❤ ♣❤❛s❡ ❢r♦♠ ✽✵✪ t♦ ✽✺✪✳ ❚❤❡ ♦t❤❡r✱ ❧❛❜❡❧❧❡❞ ✏▲♦✇❡r ❲❛✐t✲t✐♠❡ ❛♥❞ ❍✐❣❤❡r ❋r✐❝t✐♦♥✱✑ ✉s❡s t❤r❡❡ ❞❛②s ❛♥❞ t❤r❡❡ ✇❡❡❦s ❢♦r ❡❛❝❤ ♣❤❛s❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❜✉t ❞❡❝r❡❛s❡s t❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡s ✐♥ ❡❛❝❤ ♣❤❛s❡ ❢r♦♠ ✽✵✪ t♦ ✼✺✪✳ ❚❤❡ q✉❛❧✐t❛t✐✈❡ ❛♥❞ q✉❛♥t✐t❛t✐✈❡ ✜♥❞✐♥❣s ❛r❡ r♦❜✉st t♦ t❤❡s❡ ❛❧t❡r♥❛t✐✈❡ ♣❛r❛♠❡t❡rs✳ ❋✐❣✉r❡ ❉✶✶ ♣❧♦ts ❛✈❡r❛❣❡ ♣r♦❞✉❝ts✱ ❛s ✐♥ ❋✐❣✉r❡ ✻✳

❚❤❡s❡ ❛❧t❡r♥❛t✐✈❡ ♣❛r❛♠❡t❡rs ②✐❡❧❞ ❛✈❡r❛❣❡

♣r♦❞✉❝t ❢✉♥❝t✐♦♥s t❤❛t ❝❧♦s❡❧② ❢♦❧❧♦✇ t❤❡ ❜❛s❡❧✐♥❡✳ ❚❛❜❧❡ ❉✺ s❤♦✇s t❤❡ ✐♥❡✣❝✐❡♥❝② ❡st✐♠❛t❡s ❛s ✐♥ ❚❛❜❧❡ ✸✳ ❚❤❡ ❡st✐♠❛t❡❞ ✐♥❡✣❝✐❡♥❝② ✐s ✇✐t❤✐♥ ✺✲✶✵✪ ♦❢ t❤❡ ❜❛s❡❧✐♥❡✳ ❋✐❣✉r❡ ❉✶✷ s❤♦✇s ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t ✈❡rs✉s ♠❛t❝❤✐♥❣ ♣r♦❜❛❜✐❧✐t② ♦❢ r❡❣✐str❛t✐♦♥s ❛❣❣r❡❣❛t❡❞ ❜② ❝❛t❡❣♦r②✱ ❛s ✐♥ ✼❜✳ ❚❤❡s❡ r❡s✉❧ts ❛r❡ ❛❧s♦ q✉❛❧✐t❛t✐✈❡❧② s✐♠✐❧❛r✳ ❚❛❜❧❡ ❉✻ s❤♦✇s ♠❛r❣✐♥❛❧ ♣r♦❞✉❝t✱ ♠❛t❝❤✐♥❣ ♣r♦❜❛❜✐❧✐t②✱ ❛♥❞ ♣♦✐♥t s②st❡♠ s✉♠♠❛r② st❛t✐st✐❝s✱ ❛s ✐♥ ❚❛❜❧❡ ❈✹✳ ❆❣❛✐♥✱ t❤❡ ♣♦✐♥ts s②st❡♠ ✉♥❞❡r t❤❡ ❛❧t❡r♥❛t✐✈❡ ♣❛r❛♠❡t❡rs ❛r❡ s✐♠✐❧❛r ✐♥ ♠❛❣♥✐t✉❞❡✳

✷✾ ❚❛❜❧❡ ❉✺✿ ❘♦❜✉st♥❡ss✿ ❚♦t❛❧ ❊✣❝✐❡♥❝② ▲♦ss Efficiency Loss Additional Kidney Exchange Transplants

Number of Hospitals Base

Higher Waittime Lower Friction

Lower Waittime Higher Friction

Panel A: All Hospitals All Hospitals

164

447.7

454.2

357.1

Panel B: By hospital size (number of PKEs per year) Top Quartile

42

237.5

229.7

238.0

2nd Quartile

48

132.7

126.8

77.4

3rd Quartile

40

57.9

72.1

22.0

Bottom Quartile

34

19.7

25.7

19.7

NKR

68

234.8

231.2

215.6

Only UNOS and APD

45

106.0

107.2

73.4

None

51

106.9

115.8

68.0

Panel C: By Platform Membership

Panel D: By NKR Participation Rate (Fraction of PKEs facilitated through the NKR) Top Quartile

17

14.5

15.8

9.0

2nd Quartile

17

44.2

44.2

28.7

3rd Quartile

17

81.5

80.9

78.3

Bottom Quartile

17

94.7

90.3

99.6

◆♦t❡s

✿ ❈♦♥str✉❝t❡❞ ❛s ✐♥ ❚❛❜❧❡

✸✳

✸✵

Matching Probability

1.00

PRA

0.75

(10,90]

● (90,100] [0,10] Altruist

0.50

●

Category ● Altruist non−O ● Altruist O ● Overdemanded ● Selfdemanded ● Underdemanded

● 0.25 ●

0.00 0.0

0.5

1.0

1.5

2.0

Marginal Product

✭❛✮ ❇❛s❡❧✐♥❡

Matching Probability

1.00

PRA

0.75

(10,90]

● (90,100] [0,10] Altruist

0.50

●

Category

●

● Altruist non−O ● Altruist O ● Overdemanded ● Selfdemanded ● Underdemanded

0.25

● 0.00 0.0

0.5

1.0

1.5

2.0

Marginal Product

✭❜✮ ❍✐❣❤❡r ❲❛✐t✲t✐♠❡ ▲♦✇❡r ❋r✐❝t✐♦♥

Matching Probability

1.00

PRA

0.75

(10,90]

● (90,100] [0,10] Altruist

0.50

●

Category ● Altruist non−O ● Altruist O ● Overdemanded ● Selfdemanded ● Underdemanded

● 0.25

● 0.00 0.0

0.5

1.0

1.5

2.0

Marginal Product

✭❝✮ ▲♦✇❡r ❲❛✐t✲t✐♠❡ ❍✐❣❤❡r ❋r✐❝t✐♦♥ ❋✐❣✉r❡ ❉✶✷✿ ❘♦❜✉st♥❡ss✿ Pr✐✈❛t❡ ✈❡rs✉s ❙♦❝✐❛❧❧② ❖♣t✐♠❛❧ ❘❡✇❛r❞s ❢♦r ❙✉❜♠✐ss✐♦♥ ❚②♣❡s

◆♦t❡s ✿ ❈♦♥str✉❝t❡❞ ❛s ✐♥ ❋✐❣✉r❡ ✼✳

❚❛❜❧❡ ❉✻✿ ❘♦❜✉st♥❡ss✿ P♦✐♥ts ❙②st❡♠

Match Probability Baseline

Marginal Product

Higher Waittime Lower Waittime Lower Friction Higher Friction

Baseline

Points per Transplantation

Higher Waittime Lower Waittime Lower Friction Higher Friction

Baseline

Higher Waittime Lower Waittime Lower Friction Higher Friction

Panel A: Altruistic Donors Non-O Donor

0.86

0.85

0.87

0.84

0.62

0.82

0.97

0.74

0.95

O Donor

0.94

0.93

0.94

1.86

1.83

1.76

1.99

1.98

1.87

Panel B: Patient-Donor Pairs O Patient, Non-O Donor

0.27

0.29

0.28

0.08

0.03

0.00

-0.72

-0.89

-1.00

O Patient, O Donor, PRA >= 91%

0.29

0.30

0.28

0.13

0.10

0.06

-0.55

-0.67

-0.78

O Patient, O Donor, PRA < 91%

0.81

0.81

0.82

0.72

0.72

0.79

-0.11

-0.12

-0.04

Non-O Patient, O Donor, PRA >= 96%

0.28

0.29

0.27

0.09

0.08

0.01

-0.66

-0.73

-0.96

Non-O Patient, Non-O Donor, PRA < 96%

0.85

0.86

0.87

0.69

0.68

0.62

-0.19

-0.21

-0.28

Non-O Patient, O Donor, PRA < 96%

0.84

0.84

0.86

1.44

1.40

1.44

0.72

0.66

0.68

Unpaired Patients

0.24

0.23

0.24

0.07

-0.71

-1.12

-0.92

Panel C: Unpaired Patients

✿ ❈♦♥str✉❝t❡❞ ❛s ✐♥ ❚❛❜❧❡ ❈✹✳

0.02

✸✶

◆♦t❡s

-0.03

✸✷

❘❡❢❡r❡♥❝❡s ❆♥❞❡rs♦♥✱ ❘♦ss✱ ■t❛✐ ❆s❤❧❛❣✐✱ ❉❛✈✐❞ ●❛♠❛r♥✐❦✱ ❛♥❞ ❆❧✈✐♥ ❊ ❘♦t❤✱ ✏ ❋✐♥❞✐♥❣ ❧♦♥❣

❝❤❛✐♥s ✐♥ ❦✐❞♥❡② ❡①❝❤❛♥❣❡ ✉s✐♥❣ t❤❡ tr❛✈❡❧✐♥❣ s❛❧❡s♠❛♥ ♣r♦❜❧❡♠✳✱✑

Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆❛✲

t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ♦❢ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ♦❢ ❆♠❡r✐❝❛✱ ❥❛♥ ✷✵✶✺✱ ✶✶✷ ✭✸✮✱ ✻✻✸✕✽✳ ❋r✐❡❞♠❛♥✱ ❏❡r♦♠❡✱ ❚r❡✈♦r ❍❛st✐❡✱ ❛♥❞ ❘♦❜❡rt ❚✐❜s❤✐r❛♥✐✱

❚❤❡ ❡❧❡♠❡♥ts ♦❢ st❛t✐st✐❝❛❧

❧❡❛r♥✐♥❣✱ ❱♦❧✳ ✶✱ ❙♣r✐♥❣❡r s❡r✐❡s ✐♥ st❛t✐st✐❝s ◆❡✇ ❨♦r❦✱ ✷✵✵✶✳ ❏♦❤♥s♦♥✱ ❉♦♥❛❧❞ ❇✱ ✏❊✣❝✐❡♥t ❛❧❣♦r✐t❤♠s ❢♦r s❤♦rt❡st ♣❛t❤s ✐♥ s♣❛rs❡ ♥❡t✇♦r❦s✱✑

❏♦✉r♥❛❧

♦❢ t❤❡ ❆❈▼ ✭❏❆❈▼✮✱ ✶✾✼✼✱ ✷✹ ✭✶✮✱ ✶✕✶✸✳ ▼❛s✲❈♦❧❡❧❧✱ ❆♥❞r❡✉✱ ▼✐❝❤❛❡❧ ❉❡♥♥✐s ❲❤✐♥st♦♥✱ ❛♥❞ ❏❡rr② ❘ ●r❡❡♥✱

▼✐❝r♦❡❝♦♥♦♠✐❝

❚❤❡♦r②✱ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✺✳ ▼♦rr✐s✱ ❈❛r❧ ◆✱ ✏P❛r❛♠❡tr✐❝ ❡♠♣✐r✐❝❛❧ ❇❛②❡s ✐♥❢❡r❡♥❝❡✿ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱✑

❏♦✉r♥❛❧

♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❙t❛t✐st✐❝❛❧ ❆ss♦❝✐❛t✐♦♥✱ ✶✾✽✸✱ ✼✽ ✭✸✽✶✮✱ ✹✼✕✺✺✳ ❘♦❜❡rt✱ ❈❤r✐st✐❛♥ ❛♥❞ ●❡♦r❣❡ ❈❛s❡❧❧❛✱ ✏▼♦♥t❡ ❈❛r❧♦ ❙t❛t✐st✐❝❛❧ ▼❡t❤♦❞s ❙♣r✐♥❣❡r✲

❱❡r❧❛❣✱✑

◆❡✇ ❨♦r❦✱ ✷✵✵✹✳