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RAPERSpecialSectionon
2717
Theory and itsApptications Nonlinear
Multi-Population Replicator DynamicswithChanges of of Strategies Interpretations [[hkafUmiKANAZAWA'a),
SUMMARY
Ifsome differences of perceptionsarise betweenpopulations,then strategies which are regarded as thesame straLegy in a populationmay be perceived distinguishahly intheether populations, Tb disreplicator dynamies formulti-population hypergarnes cuss such u situation, hms been preposed,However. itjsas$umed thatplayers' perceptionsare givenand fixed.In thispuper,we consider thateach popu]ationhasyarious interpretation fun¢ tions and choose one of them dependingon payoffs, and we propose a hybridsystem representation ofreplicator dynamics
StudentMember
and
[[bshimitsu USHIOtb),Mbmber
and methods of analysis taking players' perceptions intoaccount have been proposedforthe hypergames concepts
The hypergame theoryinvestigates theeffbcts of [13]-[15],
about players' perceptions
strategies andpayoffs
on
theirde-
clslons.
:[bdiscussdynamicalbehaviorsof the selection prowhen taking intoconsideration players' perceptioils, with changes of jnterpretation functions. Moreover, we apply our proposed has replicator dynamics for multi-population hypergames model to a we]1-known example of a hypergame Hooliganism" been in A rigorous definition of [16]. player's perand show that behaviors converging to hetereclinic orbits cari appear by the proposed ception isgivenby an interpretation functionwhich specchanges ot'theinterpretation functions, kay words: ei,ott{tionat),game, replicator clynamics, 1rybridsystem, interifies therelationship betweeneach player's real strategies pretationjunction and strategiesperceived by the other players. Replicator dyintroducing namics for hypergames isa model the interpre1. Intreduction tationfunction intomulti-population evolutionary garnes, In n-population models where each population hasdiferent T:hegarnetheoryhas made consiiderable achievements in bothan ESS and replicator dynamicsare fbrmuperception, including economics, and somany scientific fie]ds politics, lated[161, However,ittsassumed thatplayers' perceptions cial science. Itsextensions havebeen investigated inseveral are given and fixed. Among them, we fbcuson an evoludifferent directions. Each playercan have varieus interpretation functions. tionarygame and a hypergame [1], [2], So,we can consider thateach player's current interpretation In evolutionary game theory, the distributien of stratefi:Lnction isan element of a set of interpretation functions, in the is changed according to Interpretation functions are groupingef strategies based on gies population payoffswhich L41. policies, institutions, or perceptions, Sincepolicies or instipiayersearn depending on theirselected strategies [3], In thisprocess,a strategy iscalled an evoiutionarily stabte tutions are changed dependingon payoffs, states, population if,whenever all membe!s of the population strategy (ESS) andlor various costs, interpretation functions are changed by adopt it,no dissident strategy could invadethe population them. Each player's strategy and payoffare affected by poli[5][6],On the other hand,a dynamicalbehaviorsof the secies as well as the other players'strategies, So, changes of lection are modeled by repticator ct})namics [7]-[9],interpretation fOnctiens rrrustbe taken intoco"sideration, process Replicatordynarnics describesevolutions of the distribution In this paper,we consider multi-population replicator of strategies inthepQpulation itself. Though no players' radynamics with changes of players'interpretations, Each tionalities are assumed inevolutionary games, an ESS and has various interpretation functions and choose population an equilibrium pointof replicator dy]amicsare closely inone of them dependingon payoffs. We proposea model, repvolved with equilibrium concepts of noncooperative resenting changes of the interpretation game functions and switchtheory[10], ing of them based on payoffsas thesimplest case,･ Soccer On theother hand,inmany conflict situations, play- Hooliganism[12] isa well-known example of a hypergame, ers' perceptions are djfferent other and their bchaviors .each and its replicator dynamicshas been derivedin [16], We depend on their perceptions to the game. Such a situation apply our proposedmodel tothe example and show thatbeismodeled by a hypergame [2], Varioussolution [1ll,L12]. haviors converging to heteroclinic orbits can appear by the switching of the interpretation functions. Manuscriptreceived January19.2006. cess
"Soccer
,
Manuscriptrevised May 16.2006. Finalrnanuscript received June 15,2006. i'The authors are wnh the Divisienof MathematicalSciences for SocialSystems, Department of Systems Innovation, Graduate Schoolef Engineering Science, Osaka University, Tbyonaka-shi, 560-8531Japan. a)E-mail:kanazawa@hopf,sys.es.osaka-u.ac.jp
b)E-mail:
[email protected],osaka-u.ac.jp DOI:1O.1093!ietfec/e89-alO.2717 Copyright
@
2. ReplicatorDynamicsforHypergames A
in a population may not be always perception jn other populations, For example, ifsome difierences of perceptions arise betweenpopu}ations, then strategieswhich are regarded as the same strategy ina population may be perceived distinguishably in theother populations. common
acceptable
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In order to discusssuch situations, replieator dynamicsfor multi-population hypergarnes has been proposed[16], Suppose that thereexist n Iargepopulations Pi,P2,･･･, = P.. For each population1'i (iE N let {1,･･･,n}), Oi be a set of pure strategies of Pi,and {1,2,･･･,mi} Si be a set of populationstates of Pi, where a population state isa distribution of strategies ina population ofplayers with a purestrategy. In our proposedmodel, the players' interpretations of the set of pure strategies and population Let ¢ ij ar]d SiJ･ states can be different from one another, be Pi's sets of pure strategies and population states perceived by Pj, respectively. The first subscript i stands for a popthe strategy ulation to which (orthe population state) beiongswhile thesecond one jstands fora population which = the strategy, Denoted by si E Si (sl,･･･,sl") perceives is a populationstate of Pi,where sts,is the proportion of E Oi in Pi, We define sets playerswith a pure strategy k of population state combinations S by Cartesianproduct S XiENSi, For s E S, denotedby s-i isthepopulation state combination which results from s by remeving si E S i: i･e., s7i = (si,･･･,si-i,si+i,･･･,sn). Let Ri : ×.iE,vSji R be the payofffunctionforpopu]ation Pi,where Sii= S i. The mapping FiJ･: ¢ i . Oij is called P,i'sinterfunctionto Pi. Depending on the interpretation pretation functionFiJ･of purestrategies, an interpretation function nj : Si -> Sijof populationstates is defined as fo11ows: foreach population state si E Si, =
=
,flt,i(si)=
Z
sft･,
isthekth element of ijand i,istheidentity map,fli = si. We define Pj's interpretation of a populaping:,fii(si) ''' = tienstate combination s by (,fi ,i(si),
L･(s)
forall i E N vector
such
-R, (,fi(s))] (2) [R, (.fi(ets.,,s-,)) and that
kE
¢ i, where
thekthelement
,
,
Pi's interpretation of s, Denoted by aik'
a set of
isthe event that Pi changes itsinteTpretation function from to Let Z be a set of all such events, We asjlk(s)i･k'(s), sume thatthe event E Z occurs when Pi uses Lk(s) and the difference betweenthe average payoffof Pi with fle'(s) and thatwith flic(s) becomesequal or greater than a specified thresho]d 15,k' > O. Thus,Pi changes itsinterpretation func･ ･ ･ tionfrom to ifthere exist k and k'(k, k' = 1, mfi)
ofk'
ff ff'
,
satisfying o-Ik'
EE
R,
and
-R, (Ylk'(s)) (jik(s))(3) ) ISk'.
Note that Pi must change itsinterpretation functions ifthe holds,For a pepulation state combination s condition (3) and an interpretation combination f,ifthereexist i(iE N),
k,and k' (k, k' = 1,･･･,m,fi)satisfying the condition (3),sis said to be infeasihle forf,otherwise feasible forf, interpretation funcAccoTdingto theabove definition, tions of each population change discretely, and evolutions ofpopulation
states
viously,
a system
sueh
depend on replicator dynamics(2), Obisa hybridsystem,
3.2 AHybridAutomatonRepresentation We
describeour the fo]lowing tuple: can
RD
as ahybrid
inodel
[17]by
automaton
S init:' Z,G), (F,S,IIInit
=
,
,
(4)
,
functioncombinations: of interpretation i.e., itistheset of discrete states; S isthe set ofpopulation state combinations: i,e., itis
S' isthe set
.
.
,A.i(sn>)･
In each population, suppose that the rate of increase of playerswith a strategy k is expressed as the difference between the payoffsof a playerwith a strategy k and the Replicatordynamics for average payoffof the population. hypergamesisgiven by sf
of
･･･ ,frif"} isan e]f･E {Ji! f12,
where
where
where
=
ement
a)
kE ¢ ij,
hE{hEO,IF,v(h}=k}
SS
function combination,
isthe t-dimensional unit equals 1. ef
the set of
tiens.
states;
×
.
the continuous
Z g
.
states;
E N, k,k' {ofk'Li
1,･-･,mpt.] isthe set of
=
events;
and
G:Z
.
-) 2S istheguardforeach element 'f
Equation (2) impliesthat,foreach finesthe population state combination
ofZ.
SF,fiowsef E :.de-
E
s
E
S
as
Si =
gt･i(s),
where
[R, (i(eX,,(fi(s))
s5･ gtf(s)=
3. AModelwithChangesoflnterpretations considering Equation(2)isan evolutionary game model [16].However, itisassu[ned thatthe players' perceptions interpretation functionsare gtvenand fixed.In this section, with changes ofthe interpretation funcwe proposea model
continuous
states; g 9:' × S isthe set of initial SFInitSinit :' = {4f: S -->xiENRM'},f.f- istheset of flowsdefining
.
sLi))-Ri
(5)
,
By Eq.(3), the gf (Cf,,･･･,gf.). gf, (gfl,･･･,gfln'), and
=
=
guardG isdefinbdas follows: - Ri Ul･k(s)) AS,"'}, (6) G(a'f") {sE SIRi(flk'(s)) =
where
the
rnust
satisfy
guard isthe condition that the when
continuous
state
each event occurs.
we make the fo11owing two assamptions Additionally, assumption isthat population states in our model. The first is has various iiiterpretationdo not jumpwhen events occur. [Ehesecond assumption We censider thateach population the guard fortheeventis satdependingon paythatan event occurs whenever functions and changes itsinterpretations = be an interpretation isfied. When an event occurs, ifthe guard fbran another offs. Let f(s) Ul(s),.fi(s),･･･,f,(s))
3.1
Changesoflnterpretations
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REPI.ICafOR DYNAMTCS
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WITH CHANGES
OF INTERPRETrmON'S
QF STRATEGIES
2719
Ri(f3(s))- Ri
Itable 1
) Al2 Ut(s))
Perceived on SoccerHooliganism L payeffs Authorities' Game
Fans' Game Tou.Tol. Beh.1,53,6
't'nt..Non-Int. Acc.1,23,4 Unace.2,34,1
P.H,2,36,4 R,H.4,2S,1
sSi
:a
s;i
:
a
proponionof playerswith the strategy of players with proportion
the
"Tbu.;'
`'Ibl."
strategy
In authorities' game, population states are described as follows: .
Fans' population state
si2
s?2]T: [sl2,
=
proponionof playerswith thestrategy s?2 : a proportion of players with thestrategy Authorities' state s2 = [sl, si]T: popu]ation sS : a proportion of players with the strategy s; : a proportion of players with thestrategy sl2 : a
Ri(fl(s)}-Ri(fi(s))zAii Fig.1
A hybrid automaton
model
of
.
dynamics
repl{cator
with
changes
ofinterpretations,
is satisfied
at
thatinstant in the
the event occurs without changes
However,an infinite chain
of
of
events
discrete state,
new
the continuous
without
state.
changes
of
the
Af.k' > O in By changing of the interpretation functions, theavEq.(6), erage payoffof the population must increase. So itisimto visit thesame discrete state more than once by a possible state cannot
continuous
sequence
"UnaccP
"Int,;'
"Nen-Int:'
We
event
`EAcc,;'
of events
exist since we
without
changes
of
assume
the continuous
states.
two types of jnterpretation functjon comIn both suppose binations. cases, thatfansperceive consider
"Int."
and
"Non-Int;'
,fi(s) =
as
"Tbu:'
and
Moreover, (sl,s2).
"[Ibl.;'
respectively.
in the type 1, suppose
thorities both perceive
"Beh7'
and
t`P,H."
as
"Acc,;'
Hence,
that
au-
and only
Hence, f!(s) ([sl + si,s?]T,s2). In thetype 2, however,suppose that authorities onty perceive as and perceive the rest as Hence, "R.HJ'
as
"Behl'
`GUnaccY
=:
"Acc.;'
"Unacci'
s? + fl;(s)([sl,
si]T,
=
s2),
Figure1 shows an example of a hybridautomaton of The interpretation functionfl and fh2 means thatau"accept two populations.Each population has two interpretatiorithorities' are P,H:'and "don't accept RH,;' policies functions so thatfourdiscrete states exist, respectively. In thecase of SoccerHooliganism, changes of interpretation functions are switching betweenthesepolicies 4. SoccerHooliganism dependingon costs and population states.
(1) Simulationl SoccerHooliganisrn which is a noncooperative [12], game We consider perceivedpayoffson Soccer Hooliganism as betweenfansand authorities, isa well-known example of a hypergame anct itsreplicator dynamicshas been de;ivedin shown in Tlable 1 where each cell indicates perceivedpaychanges of interpretations offs, and the 1stand the 2nd number in each cell are payoffs [16].In thissection, we introduce intothereplicator dynamics. of fansand authorities, respectively. Note thatthe payoffs in [Ibble 1 are givento show a preference In thegame perceivedby fans,which will be called order among the "fans' the fans have three strategies, behavcombinationsofstrategies. game;' ior(Beh.);' hooligan(RH.);' and hooliganism Note that Zsi ZsS 1. From Eq.(2),replicator "Tbugh Authorities have two strategies, response dynamics of Pi strategies isfbrmulated as follows: (R.H.)7' and response ([Ibl,)i' On theother hand,in ([[bu,)" -3sS) s?), sl sl (7) the game perceived by theauthorities, which will be called game;' the fans have two strategies, s'? s? sD sl +(1 -3s6)(1 - s?)). (8) ,
"Orderly
"Play
"Real
=
=
's
"Tblerant
=
"authorities'
able
"Accept-
and (Acc.)"
itieshave two
and the (Unacc,);'
"Unacceptable
strategies,
and (Int.)"
"Intervention
Intervention(Non-Int,):'
In fans' game, populationstates lows:. Fans' population state
si
=
are
author"Non-
describedas fo1-
((2+sD (sl-l)-(1 ((2+
For each interpretation function of P2, dynarnicsof P2's strategies isgivenby '2s?)
s?,si]T: [sl,
"Beh,;' proponionof playerswith thestrategy s? : a proportion of players with the strategy si : a proportion of players with the strategy Authorities' state s2i s;i]T: [sSi, population "RH.;'
"R.H;'
=
::;ii･ (g)
si-(iii,ElI;llElIi;l,
sl : a
-
=
Im the case P2's interpretation function,fiisfixedto jll there exist two asymptoticalIy stahle equilibrium points, (sis2) = ,
,
1]T)and ([O,O,l]T,[1,O]T), and in the ([O,1,O]T,[O,
thatfiisfixedto f;, thereexists the unique
case
asymptotically
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stableequilibrium point, s2) = (LO, O,1]T, (si, [1,O]T). In thispaper, suppose thatitispossible forP2 to change
ECs))
theinterpretation functions from fll(s) (resp. to .f (s) Hence, Z From condition (3), (resp, ,fll(s)). {al2,a;i}. the guardisgivenas fbllows: =
-R - 4s3) (s)) -si (3 (f;(s)) (.XII R2 (,ftl (s))-R (f:'(s)) . (3
R2
=
(10)(11) (3
,
) A S2
sZ
4sS)
sl
=
Thus,we havethefoliowing hybridautoTnaton:
f2],s (fl,
f.
=S,
×
S2,
(fi (f2
× Sznit Y:lntt =
×
G(crS2)) u
×
G(u-gi)) ,
Mg,2
(4fi,gfi),
Z･ -
A hybridautomaton
gfil=efzl=sl((2+sS)(sl-1)-(I-3sS)s :)･ s? + s5) sl + 3sS) si gfi l 4fii gfiJ 2se - ,5)(1 - 2,l- 2,i), =
O.8
(l &2J (1 2 (crl2, l - 4sS) Ai2] Gi
Suppose that
£
=
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2721
O,O.O.O, ) A;2
(3
s{
o
ff
1 Fig.5
A hybriclauternaton
Simulation ll
representing
Li100ff
o 20406080
1o
O,O,O,O.
G'
o 20
O
40
o Fig.8 cording
Time
60
80'
100
Example ot'periodic orbits on fi inthecase h can and its transientbehax'ior. payoffs
change
ac-
to the earned
Fig.6 The phaseportrait ef thecase j}isfixed to.nj ,
:--' :--':-:": :ll;1
oj1O.8O,6O,4O.2 at::--1'
O,O,O,O.
:'`"'7""i
I l f
1 l
1 t I
1 l l
1
t t
t
l
1 1 1
%
-6-''-Si'`'i}-" ,
ii
1 Mg.7Tbe
and point,
of the phaseportrait
atl orbits
intheinterior of S
Figures 6 and case thisfixedto ]21 and odic
orbits.
case
7
fiis fixed
te
are noniselated
peri-
change
o
respectively. fl;,
9
show
examples
of
such
orbits.
io
2 ,,
ff
Li100'4z
204060se
1
P2 does
Oo
2o
not
itsinitial interpretation function ,f!(resp, .f;)on the
inFig.8 (resp. Fig,9).On the other hand,many in interior of S are accompanied by the swjtching of
orbit shown orbits
and
iil l
o
We consider h can change according to the expeeted earned function. As shown in payoffsby each interpretation Fi'g,3,near the plainsl = O, the guardfbrthe events isnot satisfied and fodoesnot change. Se, thereexist nonisolated switching of ,fibetween ,4 and periodicorbits without ,fZ. Figures 8
--' -'--- ""Z"Z
. ,f3
the phaseportraits of the
show
1 1 1
Fig9 cording
4o
Time
6o
so
Exan]ple of periodic orbits on f2 in the case h can to the eamed payoffsa]d itstransient behavior.
llski chafige
ac-
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[i,o]T)
([1,O
sl
Interpretation"""ff
1
o
150
o501OO
200 s?
[o,llT)
,4 300
250 sl""'"
ff
1
([1,o,o]T,[o,11T) Illustrationot'thestructuTeoFtheheteroclinicorbitofSimulation
Fig.10
ff. O s5--'-t-----------.------------d-p-p---L--------U-1 ----1 : '-t'
Example
100 ofa
150 Time
200
300
2SO
transientbehavioTef Eqs. (9). and (12), (13),
P2's
interpretation(i).
:1 ::
6 4
:
2
ooo.--O.4O.6
Exttmple uf
,thbetween
orbits
in the
G and E, of
and
orbit$, By changes
,8,6-Pp--.4sl2
O.81
payoffs(i),
case
fican
they are
O.O.O.O.
o
....--.-----..---.-
s7 Fig.11
fig,12
SO
::
O.8O.6O,402 8
earned
Li
o
not
theorbits are jli,
change
according
{o the
periodic
nonisolated
betweenthe
switched
on J'L and those on f2, and they converge to heteroclinicorbit on the boundary of S, Figure IO shows an il]ustration ot' the structure of the heteroclinic orbit of
periodicorbits
o
Fig. 13
Example
of orbits
in thecase
ltcan
change
accerding
to the
earnedpayoffs(li).
a
SimulationI. Sinceeach
of saddle
pointsA
two-dimensional unstable manifold as shown s"'ucture of the heteroclinic orbit which orbits
D has a inFig.10,the and
with
converge
Interpretation"""'
1
,g
changes
isnot simple heteroclinic cycle but the fi branchednetwork of heteroclinic oi`bits. Thus, each orbit to a orbit consisting of the with changes of ,ficonverges heteroclinic orbits CD, FA, orbits from D to F on theplain DEF, and fromA to C on the plainABC. Note thatorbits on ABC (resp. DEF) does not converge to ABC nor AC (resp, DEF nor DF), of such orbits, and Figures11 and 13 show examples theirtransienlbehaviorsare shown in Figs.12 and 14,respectively. The orbits shown in Figs.11 and 13 converges heteroclinic orbit on theboundaryof S similarly, butthe heteroclinic orbits they converge to are different on theplains ABC and DEF, This result alsoshows that erbits do not eonverge to a simple heteroclmic cyele such as ABCDEFA oT ACDFA inFig.1O. In the case of Sifnulation a there exjsts no asymptotiequilibrium So, cally stable point. in contrast to Simulations L the populationstate is ever-changing, and P2 switches itsinterpretation functionfrom .4 to fE2and from .E!to many times. Moreover,Figs.12and 14 show 6i,infinite of
sl
to
4i
o o
50
100s?
150
200 sS''""'
1
/'
/t1:11:!liIi/hl :/::::/ i"'til 11"'1 :il f""/::::L1/'
/t
't-----.'::/::://
Ji]
:::/::'
i''' 'i
i
/IL. o O
' 50
100 Time
150
4i200
Fig.14 ExanipleefatransientbehaviorofEqs,(9), (12), (13),and P2's interpretation (ii).
thattheorbit converges to thehetcroclinic orbit while visiting tbe saddle points([O, O,1IT,[O,1]T),([o, o, 1]',[1,o]T), and O]T,LO,1]T)repeatedly O]T,rl,O]T), ([1,O, ([1,O,
with
duration the duration, the switching interval of .ti alsokeepsincreasing. ever-growing
of
stays.
With the growthof the
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USHTO: MUIJTI-POPULATION
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IMPLICafOR
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OF INTERPRETATIONS
OF SIRATEGIES 2723
Sucha phenomenon haye been observed in replicator In this simulation, however,rep]icator dydynarnics [18].
or jlcloes not have such .Eiisfixed to fll fbr itself, Changes of interpretatiQn funcphenomenon switching of multiple dynamics. tions in thispapermeans Itisimportantthatcomplex phenomena a$ shown in Fig,12 can be caused by switching betweenvery simple dynamics such as al] orbits of them are nonisolated periodic orbits. An existence of such a phenomenon impliesthat¢ hangesof interpretation functions must be taken intoconsideTation.
nannics
in the
case
a
Society, vol,31, 19SO, pp.621-635, [13]M, Wang, K.W. Hipel,and N.M. Fraser,
'tSolutjon
concepts
inhy-
pergames;' Appl. Math. Comput., vol.34, pp.147-171, 1989. `"Adaptive ]eaniingof hy[14]U,S, Putre,K, Kis'ima,and S, Thkahasbi, situations using a algorithm;' IEEE Trans, Syst. pergame genetic Man Cybern.A, Syst.Humans, voL30. no.5, pp.562-572, 20UO. hypergamesto inerease [L5]R. Vlineand P, hehner, p!annedpayoff and reduce risk;' Autonomous Agents and MuLti-Agent Systerns, "Using
vol,5,pp.365-380,2002,
[16]T.Kanazawa,T.Ushio,and T,Yan]asald,
tLReplicator
esrolutionai}, ence on
dynamicsof
hypergarnes;'Proc. 2003 IEEE InternationalCon'fer-
Systoms, Man, and Cybernetics, 2003. pp.3828-3833, theory of hybridautomata," Proc.] tthAn-
[17]T,A.Henzinger, "The
5. Conclusions
IEEE Symposiurn on Logic in ComputeT Seience,pp,278-292, 1996. [18]T, Chawanya, new type of irregulurmotion in a c]ass of game dynamics systems;' Prog. Theor. Phys.,vol.94, pp,163-179, 1995. nua]
In replicator dynamics of hypergames
LlbL
where
each
player
has several interpretation functions, itis natural that each his interpretation functions dependingon switches player hisearned payoffs.So, thispaperhas proposeda hybrid replicator dynamicsto model such a switching. Moreover, we have applied our proposedmodel to a well-known ex"Soccer ample of a hypergame Hooliganism;' and we have shown thattheinterpretation changes dependingon the evolutions of distributions of strategies. In thecase of SimulationL especially, we have shown thattheorbits converging to the heteroclinic orbits appear hy switching betweentwo discrete states. Thus, various complex･phenomena can be
Tbkafumi Kanazawa
by switching
student memher
investigate of the replicator dyglobalbifurcation properties namics. Moreover,itisalso future work to discuss stability conditions for intexpretation functions and population states to converge to an equilibrium pointby using stability theory of hybridsystems.
and [1]I.Maynard SrnideEvolutien
the Theoryof Games,Cambridge UniversityPress,Cannbridge,UK, 1982. a theoryof hypergames;' Omega, vol.5, [2]RG. Bennett, `;Tbward
pp.749-7Sl,1977.
sor
Press,Carnbridge,
of
ISCIEand IEEE,
in 1994, and is currently a Professor. His interestsjnclude nonlinear oscMation
research
MA, 1995,
[4] J. Hofoauer and K. Sigmund, EvolutionaryGames and Population Dynamics,CambridgeUniversity Press, Cambridge,UK, 1998, logieof animal conflicts;' Nar5]J.Maynard Smith aiid G. Price,
and
Tbshimitsu Ushio received B,S. M,S., and Ph.D,degreesin1980,19g2,]985,respective]y, from Kobe University,He was a Research Assistant at the University of California, Berkeley in 1985, From 19g6 to 1990,he was a Research Associate in Kobe Universjty, and becamea LectureT at Kobe College in 1990,He Osaka University as an Assoeiate Prufesjoined
References
EvolutionaryGarneTheory, MIT [3]J.W.Weibull,
B,E,
the Ph.D. degree. Hjs research interests include evolutionary games and hypergames,He isa
the interpretation functions basedon theearned payoffsforthefunctions. Itisour futurework to caused
received
M.E.degreesin2003,2oo4,respectively, Osaka University, where he iscurrently studying for
and member
of
control
of
discrete evellt systems, He isa
SICE,ISCIE,and MEE.
"Thc
ture,vol,246, pp.15-18, 1973.
[6]R, Cressrnan,J,Garay,andJ. Hefoauer,
"Evolutionay
stabiljty con-
ceptsforn-species'Frequency-dependentinteractions;:J.Theoretical
Bjology,voL211, pp.1-le, 2001. "Evolutionarily stable strategies [7]RD, Tbylor andL. Jonker,
and game dynamics;' Mathematical Bioscience, vol.40, pp,]45-156, 1978. [81 C, Hauert. S.D. Monte, J. Hodbauer, und K. Sigmund, dynamics foroptienal pablic l.Theoretieal Biology, good garnes;' "Replicator
vol.2!8,pp.lg7-]94,2002.
[9] K.M. Page
Newak, evolutionary dynamics;' J, TheoreticalBlology, vol.219, pp.93-98, 2002. [1OlE. van Dame, Stabilityamd Perfection of Nash Equilibria, 2nd ed,, Sprtnger Nk)rlag, Berlin, 1991. Developing a modet of confiict;' FuTl1]RG. Bermett, and
M,A,
C'Unifying
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tures,vol,12,
1980,
pp.489-507, L`Using hypergames to model [12]P.G.Bennett,
diMcult socialissues/
An
approachtothecaseofsoccerhooliganism;'J.OperationalResearch
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