92:
probability vector (i.e. S â„ 0 and S_1 + S_2... + S_n = 1) this is called the state vector of the population. Using this the function F(i|s) can be made, F(i|s) refers to the fitness of I in state S. The state vector of the population (S) is not static. The idea behind it is that the more fit a strategy at the moment the more likely it is to be employed in the future, thus the state vector (S) will change. Using game theory we can look how (S) changes over time and try to figure out in what state it has reached an equilibrium. Let K be the set of all probability vectors of length N, this is the state space of the population. Thus element P in K represents a possible strategy mix. A state P in K is called an equilibrium state if F(i|p) is equal for all pure strategies i for which P_i > 0, That is, supp(p) = {i :p,â 0}. If Q is in K: F(q|p) + (ÎŁQ_1 x F(i|p). We can see F(q|p) as the expected fitness of an individual using mixed strategy Q against the population in state P. If P is an equilibrium state and the supp(q) is contained in supp(p) then F(q|p) = F(q|p).(supp(p) are the I's for which P_i > 0). Thus a state p is called an ESS (evolutionary stable state) if for every state Q â P, if we let pÌ
=(1-Δ)p + Δq (the perturbed state), then F(q|p) < F(p|pÌ
) for sufficiently small Δ>0
67:
While
Maynard Smith had originally defined an ESS as being a single "uninvadable strategy," Thomas generalized this to include a set of multiple strategies employed by individuals. In other words, a collection of simultaneously present strategies could be considered uninvadable as a group. Thomas noted that evolutionary stability can exist in either model, allowing for an evolutionarily stable state to exist even when multiple strategies are used within the population.
41:
describes a population that returns as a whole to its previous composition even after being disturbed. In short: the ESS refers to the strategy itself, uninterrupted and supported through natural selection, while the evolutionarily stable state refers more broadly to a population-wide balance of one or more strategies that may be subjected to temporary change.
75:
The strategy employed by individuals (or ESS) is thought to depend on fitness: the better the strategy is at supporting fitness, the more likely the strategy is to be used. When it comes to an evolutionarily stable state, all of the strategies used within the population must have equal fitness. While
66:
There has been variation in how the term is used and exploration of under what conditions an evolutionarily stable state might exist. In 1984, Benhard Thomas compared "discrete" models in which all individuals use only one strategy to "continuous" models in which individuals employ mixed strategies.
40:
An ESS is a strategy that, if adopted by all individuals within a population, cannot be invaded by alternative or mutant strategies. This strategy becomes fixed in the population because alternatives provide no fitness benefit that would be selected for. In comparison, an evolutionarily stable state
120:
as a whole provides a theoretical framework examining interactions of organisms in a system where individuals have repeated interactions within a population that persists on an evolutionarily relevant timescale. This framework can be used to better understand the evolution of interaction strategies
91:
In a game of individuals in competition with each other there are (N) possible strategies available. Thus each individual is using one of these (N) strategies. If we denote each strategy as I we let S_i be the proportion of individuals who are currently using strategy I. Then S=(S_1 -> S_n) is a
104:
It has been suggested by Ross
Cressman that criteria for evolutionary stability include strong stability, as it would describe evolution of both frequency and density (whereas Maynard Smith's model focused on frequency). Cressman further demonstrated that in habitat selection games modeling only a
241:
620:
20:
when that population's "genetic composition is restored by selection after a disturbance, provided the disturbance is not too large" (Maynard Smith, 1982). This population as a whole can be either monomorphic or
96:
In summary, a state P is evolutionarily stable whenever a small change from P to state pÌ
the expected fitness in the perturbed state is less than the expected fitness of the remaining population.
79:
One of the base mathematical models for identifying an evolutionarily stable state was outlined by Taylor & Jonker in 1978. Their base equilibrium model for ES states stipulates that
401:
83:
A state p is called an ESS (evolutionary stable state) if for every state q â p, if we let pÌ
=(1-Δ)p + Δq (the perturbed state), then F(q|p) < F(p|pÌ
) for sufficiently small Δ>0.
76:
the equilibrium may be disturbed by external factors, the population is considered to be in an evolutionarily stable state if it returns to the equilibrium state after the disturbance.
132:
For the purpose of predicting evolutionary outcomes, the replicator equation is also a frequently utilized tool. Evolutionarily stable states are often taken as solutions to the
308:
337:
273:
129:
are closely related to the evolutionarily stable state. There are various potential refinements proposed to account for different theory games and behavioral models.
52:
Maynard Smith developed the ESS drawing in part from game theory and
Hamilton's work on the evolution of sex ratio. The ESS was later expanded upon in his book
142:
702:
603:
Cressman, R., & KĆivan, V. (2010). The ideal free distribution as an evolutionarily stable state in densityâdependent population games.
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447:
1423:
953:
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54:
858:
1332:
868:
34:
1202:
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908:
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833:
948:
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913:
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345:
1718:
1451:
1337:
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923:
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1723:
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943:
731:
106:
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1327:
1232:
1091:
1041:
1036:
963:
933:
853:
781:
22:
761:
37:(ESS), evolutionarily stable states are not identical and the two terms cannot be used interchangeably.
1207:
1192:
278:
1541:
1526:
1413:
1408:
1312:
1297:
1262:
1227:
821:
766:
688:
1698:
1317:
1267:
1104:
1031:
1006:
863:
746:
133:
121:
and stable states, though many different specific models have been used under this framework. The
1677:
1536:
1367:
1347:
1197:
1076:
976:
903:
848:
45:
313:
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1662:
1631:
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1352:
1307:
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1212:
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1001:
893:
843:
791:
653:
587:
Cressman, R. (1990). Strong stability and density-dependent evolutionarily stable strategies.
509:
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888:
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811:
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1601:
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1187:
1139:
1021:
786:
756:
726:
1506:
429:
Maynard Smith, J.. (1982) Evolution and the Theory of Games. Cambridge
University Press.
621:
Game Theory, Evolutionary Stable
Strategies and the Evolution of Biological Interactions
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1302:
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1119:
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996:
938:
873:
806:
668:
Cressman, R., & Tao, Y. (2014). The replicator equation and other game dynamics.
544:
Maynard Smith, J. (1974). The theory of games and the evolution of animal conflicts.
571:
Taylor, P. D, Jonker, L. B. (1978). Evolutionarily stable states and Game
Dynamics.
1596:
1591:
1446:
1016:
652:
Cressman, R. (2003) Evolutionary
Dynamics and Extensive Form Games. The MIT Press.
608:
638:
Computing Nash
Equilibria and Evolutionarily Stable States of Evolutionary Games.
1713:
1516:
1511:
1491:
1287:
1272:
1081:
1051:
981:
971:
801:
736:
712:
592:
637:
576:
549:
493:
1342:
991:
236:{\displaystyle {\dot {x_{i}}}=x_{i}\left(\left(Ax\right)_{i}-x^{T}Ax\right),}
1247:
1167:
986:
109:(IFD) is itself an evolutionarily stable state containing mixed strategies.
87:
In greater detail, the Taylor & Jonker model can be understood this way
673:
1682:
1182:
680:
1403:
1393:
1071:
528:
Maynard Smith, J., Price, G. R. (1973). The logic of animal conflict.
504:
Maynard Smith, J. (1972). Game Theory and the
Evolution of Fighting.
533:
1172:
488:
Thomas, B. (1984). Evolutionary stability: States and strategies.
684:
448:"Evolutionary game theory: ESS, convergence stability, and NIS"
58:
in 1982, which also discussed the evolutionarily stable state.
29:
History & connection to evolutionary stable strategy
71:
Mathematical formulation & evolutionary game theory
348:
316:
281:
252:
145:
670:
Proceedings of the
National Academy of Sciences, 111
25:. This is now referred to as convergent stability.
1691:
1650:
1432:
1376:
1158:
1060:
962:
820:
719:
395:
331:
302:
267:
235:
641:IEEE Transactions on Evolutionary Computation, 20
446:Apaloo, J.; Brown, J. S.; Vincent, T. L. (2009).
609:https://doi.org/10.1111/j.1600-0706.2010.17845.x
275:is said to be evolutionarily stable if for all
696:
593:https://doi.org/10.1016/S0022-5193(05)80112-2
16:A population can be described as being in an
8:
577:https://doi.org/10.1016/0025-5564(78)90077-9
550:https://doi.org/10.1016/0022-5193(74)90110-6
494:https://doi.org/10.1016/0040-5809(84)90023-6
396:{\displaystyle x^{T}Ax<{\hat {x}}^{T}Ax}
703:
689:
681:
636:Li, J., Kendall, G., and John, R. (2015).
381:
370:
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213:
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674:https://doi.org/10.1073/pnas.1400823111
411:
7:
567:
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524:
522:
484:
482:
480:
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476:
425:
423:
421:
419:
417:
415:
589:Journal of Theoretical Biology, 145
33:While related to the concept of an
752:First-player and second-player win
490:Theoretical Population Biology, 26
14:
55:Evolution and the Theory of Games
859:Coalition-proof Nash equilibrium
534:https://doi.org/10.1038/246015a0
303:{\displaystyle x\neq {\hat {x}}}
48:in an essay from the 1972 book
44:The term ESS was first used by
869:Evolutionarily stable strategy
508:. Edinburgh University Press.
375:
323:
294:
259:
136:, here in linear payoff form:
35:evolutionarily stable strategy
1:
797:Simultaneous action selection
672:(Supplement 3), 10810-10817.
452:Evolutionary Ecology Research
1734:List of games in game theory
909:Quantal response equilibrium
899:Perfect Bayesian equilibrium
834:Bayes correlated equilibrium
625:Nature Education Knowledge 3
1203:Optional prisoner's dilemma
929:Self-confirming equilibrium
573:Mathematical Biosciences 40
113:In evolutionary game theory
18:evolutionarily stable state
1786:
1668:Principal variation search
1384:Aumann's agreement theorem
1047:Strategy-stealing argument
954:Trembling hand equilibrium
884:Markov perfect equilibrium
879:Mertens-stable equilibrium
332:{\displaystyle {\hat {x}}}
268:{\displaystyle {\hat {x}}}
62:Mixed v. single strategies
1704:Combinatorial game theory
1363:Princess and monster game
914:Quasi-perfect equilibrium
839:Bayesian Nash equilibrium
458:: 489â515. Archived from
1770:Evolutionary game theory
1719:Evolutionary game theory
1452:Antoine Augustin Cournot
1338:Guess 2/3 of the average
1135:Strictly determined game
924:Satisfaction equilibrium
742:Escalation of commitment
310:in some neighborhood of
118:Evolutionary game theory
1724:Glossary of game theory
1323:Stackelberg competition
944:Strong Nash equilibrium
107:ideal free distribution
1749:Tragedy of the commons
1729:List of game theorists
1709:Confrontation analysis
1419:SpragueâGrundy theorem
934:Sequential equilibrium
854:Correlated equilibrium
397:
333:
304:
269:
237:
94:
85:
1522:Jean-François Mertens
619:Cowden, C. C. (2012)
398:
334:
305:
270:
238:
89:
81:
1651:Search optimizations
1527:Jennifer Tour Chayes
1414:Revelation principle
1409:Purification theorem
1348:Nash bargaining game
1313:Bertrand competition
1298:El Farol Bar problem
1263:Electronic mail game
1228:Lewis signaling game
767:Hierarchy of beliefs
346:
314:
279:
250:
143:
105:single species, the
100:Additional proposals
1699:Bounded rationality
1318:Cournot competition
1268:Rock paper scissors
1243:Battle of the sexes
1233:Volunteer's dilemma
1105:Perfect information
1032:Dominant strategies
864:Epsilon-equilibrium
747:Extensive-form game
134:replicator equation
1678:Paranoid algorithm
1658:Alphaâbeta pruning
1537:John Maynard Smith
1368:Rendezvous problem
1208:Traveler's dilemma
1198:Gift-exchange game
1193:Prisoner's dilemma
1110:Large Poisson game
1077:Bargaining problem
977:Backward induction
949:Subgame perfection
904:Proper equilibrium
393:
329:
300:
265:
233:
46:John Maynard Smith
1757:
1756:
1663:Aspiration window
1632:Suzanne Scotchmer
1587:Oskar Morgenstern
1482:Donald B. Gillies
1424:Zermelo's theorem
1353:Induction puzzles
1308:Fair cake-cutting
1283:Public goods game
1213:Coordination game
1087:Intransitive game
1012:Forward induction
894:Pareto efficiency
874:Gibbs equilibrium
844:Berge equilibrium
792:Simultaneous game
378:
326:
297:
262:
162:
1777:
1744:Topological game
1739:No-win situation
1637:Thomas Schelling
1617:Robert B. Wilson
1577:Merrill M. Flood
1547:John von Neumann
1457:Ariel Rubinstein
1442:Albert W. Tucker
1293:War of attrition
1253:Matching pennies
1027:Pairing strategy
889:Nash equilibrium
812:Mechanism design
777:Normal-form game
732:Cooperative game
705:
698:
691:
682:
676:
666:
660:
650:
644:
634:
628:
617:
611:
607:(8), 1231-1242.
601:
595:
585:
579:
569:
552:
546:J Theor Biol. 47
542:
536:
526:
517:
502:
496:
486:
471:
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402:
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394:
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371:
358:
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338:
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319:
309:
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205:
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177:
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164:
163:
158:
157:
148:
123:Nash Equilibrium
1785:
1784:
1780:
1779:
1778:
1776:
1775:
1774:
1760:
1759:
1758:
1753:
1687:
1673:max^n algorithm
1646:
1642:William Vickrey
1602:Reinhard Selten
1557:Kenneth Binmore
1472:David K. Levine
1467:Daniel Kahneman
1434:
1428:
1404:Negamax theorem
1394:Minimax theorem
1372:
1333:Diner's dilemma
1188:All-pay auction
1154:
1140:Stochastic game
1092:Mean-field game
1063:
1056:
1022:Markov strategy
958:
824:
816:
787:Sequential game
772:Information set
757:Game complexity
727:Congestion game
715:
709:
679:
667:
663:
651:
647:
635:
631:
618:
614:
602:
598:
586:
582:
570:
555:
543:
539:
532:(5427), 15-18.
527:
520:
503:
499:
487:
474:
465:
463:
445:
444:
440:
428:
413:
409:
368:
349:
344:
343:
312:
311:
277:
276:
248:
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209:
188:
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149:
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115:
102:
73:
64:
31:
12:
11:
5:
1783:
1781:
1773:
1772:
1762:
1761:
1755:
1754:
1752:
1751:
1746:
1741:
1736:
1731:
1726:
1721:
1716:
1711:
1706:
1701:
1695:
1693:
1689:
1688:
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1685:
1680:
1675:
1670:
1665:
1660:
1654:
1652:
1648:
1647:
1645:
1644:
1639:
1634:
1629:
1624:
1619:
1614:
1609:
1607:Robert Axelrod
1604:
1599:
1594:
1589:
1584:
1582:Olga Bondareva
1579:
1574:
1572:Melvin Dresher
1569:
1564:
1562:Leonid Hurwicz
1559:
1554:
1549:
1544:
1539:
1534:
1529:
1524:
1519:
1514:
1509:
1504:
1499:
1497:Harold W. Kuhn
1494:
1489:
1487:Drew Fudenberg
1484:
1479:
1477:David M. Kreps
1474:
1469:
1464:
1462:Claude Shannon
1459:
1454:
1449:
1444:
1438:
1436:
1430:
1429:
1427:
1426:
1421:
1416:
1411:
1406:
1401:
1399:Nash's theorem
1396:
1391:
1386:
1380:
1378:
1374:
1373:
1371:
1370:
1365:
1360:
1355:
1350:
1345:
1340:
1335:
1330:
1325:
1320:
1315:
1310:
1305:
1300:
1295:
1290:
1285:
1280:
1275:
1270:
1265:
1260:
1258:Ultimatum game
1255:
1250:
1245:
1240:
1238:Dollar auction
1235:
1230:
1225:
1223:Centipede game
1220:
1215:
1210:
1205:
1200:
1195:
1190:
1185:
1180:
1178:Infinite chess
1175:
1170:
1164:
1162:
1156:
1155:
1153:
1152:
1147:
1145:Symmetric game
1142:
1137:
1132:
1130:Signaling game
1127:
1125:Screening game
1122:
1117:
1115:Potential game
1112:
1107:
1102:
1094:
1089:
1084:
1079:
1074:
1068:
1066:
1058:
1057:
1055:
1054:
1049:
1044:
1042:Mixed strategy
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
999:
994:
989:
984:
979:
974:
968:
966:
960:
959:
957:
956:
951:
946:
941:
936:
931:
926:
921:
919:Risk dominance
916:
911:
906:
901:
896:
891:
886:
881:
876:
871:
866:
861:
856:
851:
846:
841:
836:
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828:
818:
817:
815:
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809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
762:Graphical game
759:
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749:
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734:
729:
723:
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710:
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693:
685:
678:
677:
661:
645:
629:
612:
596:
591:(3), 319-330.
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553:
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518:
497:
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13:
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6:
4:
3:
2:
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1697:
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1692:Miscellaneous
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1684:
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1659:
1656:
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1643:
1640:
1638:
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1633:
1630:
1628:
1627:Samuel Bowles
1625:
1623:
1622:Roger Myerson
1620:
1618:
1615:
1613:
1612:Robert Aumann
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1605:
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1567:Lloyd Shapley
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1560:
1558:
1555:
1553:
1552:Kenneth Arrow
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1533:
1532:John Harsanyi
1530:
1528:
1525:
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1508:
1505:
1503:
1502:Herbert Simon
1500:
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1405:
1402:
1400:
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1385:
1382:
1381:
1379:
1375:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1324:
1321:
1319:
1316:
1314:
1311:
1309:
1306:
1304:
1303:Fair division
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1279:
1278:Dictator game
1276:
1274:
1271:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
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1234:
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1211:
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1199:
1196:
1194:
1191:
1189:
1186:
1184:
1181:
1179:
1176:
1174:
1171:
1169:
1166:
1165:
1163:
1161:
1157:
1151:
1150:Zero-sum game
1148:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1120:Repeated game
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1099:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1069:
1067:
1065:
1059:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1037:Pure strategy
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
997:De-escalation
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
969:
967:
965:
961:
955:
952:
950:
947:
945:
942:
940:
939:Shapley value
937:
935:
932:
930:
927:
925:
922:
920:
917:
915:
912:
910:
907:
905:
902:
900:
897:
895:
892:
890:
887:
885:
882:
880:
877:
875:
872:
870:
867:
865:
862:
860:
857:
855:
852:
850:
847:
845:
842:
840:
837:
835:
832:
831:
829:
827:
823:
819:
813:
810:
808:
807:Succinct game
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
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750:
748:
745:
743:
740:
738:
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733:
730:
728:
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724:
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714:
706:
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687:
686:
683:
675:
671:
665:
662:
659:
658:9780262033053
655:
649:
646:
643:(3), 460-469.
642:
639:
633:
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626:
622:
616:
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594:
590:
584:
581:
578:
574:
568:
566:
564:
562:
560:
558:
554:
551:
548:(1). 209-221.
547:
541:
538:
535:
531:
525:
523:
519:
515:
514:0-85224-223-9
511:
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462:on 2017-08-09
461:
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435:0-521-28884-3
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139:
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128:
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119:
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99:
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88:
84:
80:
77:
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61:
59:
57:
56:
51:
50:On Evolution.
47:
42:
38:
36:
28:
26:
24:
19:
1597:Peyton Young
1592:Paul Milgrom
1507:Hervé Moulin
1447:Amos Tversky
1389:Folk theorem
1100:-player game
1097:
1017:Grim trigger
669:
664:
648:
640:
632:
624:
615:
604:
599:
588:
583:
572:
545:
540:
529:
506:On Evolution
505:
500:
492:(1), 49-67.
489:
464:. Retrieved
460:the original
455:
451:
441:
245:
131:
127:folk theorem
116:
103:
95:
90:
86:
82:
78:
74:
65:
53:
49:
43:
39:
32:
17:
15:
1714:Coopetition
1517:Jean Tirole
1512:John Conway
1492:Eric Maskin
1288:Blotto game
1273:Pirate game
1082:Global game
1052:Tit for tat
982:Bid shading
972:Appeasement
822:Equilibrium
802:Solved game
737:Determinacy
720:Definitions
713:game theory
575:, 145-156.
23:polymorphic
1358:Trust game
1343:Kuhn poker
1007:Escalation
1002:Deterrence
992:Cheap talk
964:Strategies
782:Preference
711:Topics of
605:Oikos, 119
530:Nature 246
466:2018-01-10
407:References
246:The state
1542:John Nash
1248:Stag hunt
987:Collusion
376:^
324:^
295:^
286:≠
260:^
207:−
160:˙
125:(NE) and
1764:Category
1683:Lazy SMP
1377:Theorems
1328:Deadlock
1183:Checkers
1064:of games
826:concepts
1435:figures
1218:Chicken
1072:Auction
1062:Classes
627:(10):6.
656:
512:
433:
1173:Chess
1160:Games
849:Core
654:ISBN
510:ISBN
431:ISBN
366:<
1433:Key
1766::
1168:Go
623:.
556:^
521:^
475:^
456:11
454:.
450:.
414:^
339:.
1098:n
704:e
697:t
690:v
516:.
469:.
391:x
388:A
383:T
373:x
363:x
360:A
355:T
351:x
321:x
292:x
283:x
257:x
231:,
227:)
223:x
220:A
215:T
211:x
202:i
197:)
193:x
190:A
186:(
180:(
174:i
170:x
166:=
155:i
151:x
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