of assumptions, for this see Kreps (1988) or Fishburn (1970). Definition A.1 A ....
We will stick to the vN0M theory, the interested reader should consult. Kreps ...
D D14
H{shfwhg xwlolw| wkhru| Wkh sodq
Zh vwduw e| frqvlghulqj dq lqglylgxdov suhihuhqfhv lq d zruog zlwkrxw xqfhuwdlqw|1 Zh qhhg wr eh deoh wr pdnh fkrlfhv ehwzhhq frqvxpswlrq exqgohv1 H{dpsohv ri frqvxps0 wlrq exqgohv duh= 5 dssohv/ 5 rudqjhv/ d '4133 wrgd| soxv '5133 lq rqh |hdu1 Iurp khuh rq lq zloo dvvxph wkdw doo frqvxpswlrq lwhpv duh ydoxhg lq prqh| +grooduv,1 Li wkhuh lv qr lq dwlrq wkhq wklv dvvxpswlrq lv uhodwlyho| kdupohvv1 Wklv uhgxfhv frqvxpswlrq exqgohv wr prqh| frqvxphg dw ydulrxv wlphv1 Lq rwkhu zruogv d frqvxpswlrq exqgoh kdv ehhq vlpsolhg wr d yhfwru ri groodu dprxqwv1 H{dpsoh D14 Vxssrvh zh frqvlghu frqvxpswlrq exqgohv lq U2n +50glphqvlrqdo srv0 lwlyh uhdo qxpehuv, dqg ohw wlph 3 eh qrz dqg wlph 4 d |hdu khqfh1 Li S ' Emffc mDf dqg S2 ' Em2Dc m2D wkhq suhihuulqj S wr S2 zrxog eh d suhihuhqfh iru mff wrgd| dqg mDf lq d |hdu wr m2D lq hdfk |hdu1 Li zh duh jrlqj wr prgho fkrlfhv lq dq xqfhuwdlq zruog/ wkhq zh qhhg d zd| wr prgho xqfhuwdlqw|1 Wkh wkh vwdwh ri qdwxuh lv ghqrwhg e| / Wkh froohfwlrq ri vwdwhv ri qdwxuh lv fdoohg l Wkh vwdwh ri qdwxuh / whoov xv hyhu|wklqj zh qhhg wr nqrz derxw wkh zruog1 Iru h{dpsoh / whoov xv/ lqwhuhvw udwhv lq wkh Xqlwhg Vwdwhv/ xqhpsor|phqw ohyhov lq Hxursh/ zruog srsxodwlrq/ zklfk frxqwulhv duh dw zdu/ dqg hyhu|wklqj hovh |rx fdq wklqn ri1 Wr exlog xvhixo prghov / uhdoo| qhhgv wr eh d ixqfwlrq ri wlph/ glvfuhwh ru frqwlqxrxv1 Dowkrxjk wklv h{whqvlrq wr pxwlshulrgv lv vwudljkw iruzdug lw lv fxpehuvrph/ khqfh zh zloo uhvwulfw rxu glvfxvvlrq wr wkh vwdwh ri wkh zruog dw d vlqjoh srlqw lq wlph1 Reylrxvo|/ l lv d odujh vhw/ lq pdq| fdvhv lw lv lqqlwh +h1j1 li frqvxpswlrq jrrgv duh lqqlwho| glylvleoh,1 Zh zloo prgho d frqvxpswlrq vfkhgxoh dv d udqgrp yduldeoh rq l Lw lv hdvlhvw wr looxvwudwh zlwk d vlpsoh h{dpsoh1 Vxssrvh wkdw wkhuh duh wkuhh srvvleoh vwdwhv ri qdwxuh l ' i/ c / 2 c / j wkhq d frqvxpswlrq vfkhgxoh ru udqgrp yduldeoh %E/ pljkw eh / / 2 / %E/ 0'8 '3 '53 Vxssrvh zh kdg d ixqfwlrq L E% E/ wkdw uhsuhvhqwhg suhihuhqfhv/ lq rwkhu zrugv % E/ lv suhihuuhg wr + E/ li dqg rqo| l L E% : L E+ +L*oo zulwh % iru % E/ wr vlpsoli| qrwdwlrq/ exw uhphpehu wkdw % lv d ixqfwlrq ri /,1 Wklv zrxog doorz xv wr frpsduh udqgrp yduldeohv e| frpsdulqj qxpehuv/ wklv lv d jrrg wklqj1 L E% E/ lv d ixqfwlrq ri d ixqfwlrq/ d frpsoh{ remhfw1 Vlqfh zh kdyh wdnhq % wr eh grooduv/ lw lv qrw doo wkdw gl!fxow wr vshfli| L E% Krzhyhu/ wr vshfli| L E% E/ lv doprvw lpsrvvleoh1 L zrxog eh pxfk hdvlhu wr zrun zlwk li +dqg vshfli|, li lw frxog eh uhsuhvhqwhg dv dq
4
h{shfwdwlrq ru dyhudjh ri vrph rwkhu ixqfwlrq zkhuh lv d ixqfwlrq ri d srvlwlyh uhdo qxpehu 1 Zh duh orrnlqj iru d ixqfwlrq vxfk wkdw L E% ' , dE%o ru li l lv qlwh