461:. These functions are similar to the best response correspondence, except that the function does not "jump" from one pure strategy to another. The difference is illustrated in Figure 8, where black represents the best response correspondence and the other colors each represent different smoothed best response functions. In standard best response correspondences, even the slightest benefit to one action will result in the individual playing that action with probability 1. In smoothed best response as the difference between two actions decreases the individual's play approaches 50:50.
317:, the "Cooperate" move is not optimal for any probability of opponent Cooperation. Figure 5 shows the reaction correspondence for such a game, where the dimensions are "Probability play Cooperate", the Nash equilibrium is in the lower left corner where neither player plays Cooperate. If the dimensions were defined as "Probability play Defect", then both players best response curves would be 1 for all opponent strategy probabilities and the reaction correspondences would cross (and form a Nash equilibrium) at the top right corner.
372:
450:
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67:
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368:-axis) wins if the players discoordinate. Player Y's reaction correspondence is that of a coordination game, while that of player X is a discoordination game. The only Nash equilibrium is the combination of mixed strategies where both players independently choose heads and tails with probability 0.5 each.
601:
There are several advantages to using smoothed best response, both theoretical and empirical. First, it is consistent with psychological experiments; when individuals are roughly indifferent between two actions they appear to choose more or less at random. Second, the play of individuals is uniquely
341:
game. The axes are assumed to show the probability that the player plays their strategy 1. From left to right: A) Always play 2, strategy 1 is dominated, B) Always play 1, strategy 2 is dominated, C) Strategy 1 best when opponent plays his strategy 1 and 2 best when opponent plays his 2, D) Strategy
261:
in which players score highest when they choose opposite strategies, i.e., discoordinate, are called anti-coordination games. They have reaction correspondences (Figure 4) that cross in the opposite direction to coordination games, with three Nash equilibria, one in each of the top left and bottom
350:
games (of which one is trivial), the five different best response curves per player allow for a larger number of payoff asymmetric game types. Many of these are not truly different from each other. The dimensions may be redefined (exchange names of strategies 1 and 2) to produce symmetrical games
329:
games with payoff asymmetries. For each player there are five possible best response shapes, shown in Figure 6. From left to right these are: dominated strategy (always play 2), dominated strategy (always play 1), rising (play strategy 2 if probability that the other player plays 2 is above
237:. These games have reaction correspondences of the same shape as Figure 3, where there is one Nash equilibrium in the bottom left corner, another in the top right, and a mixing Nash somewhere along the diagonal between the other two.
212:
games: coordination games, discoordination games, and games with dominated strategies (the trivial fourth case in which payoffs are always equal for both moves is not really a game theoretical problem). Any payoff symmetric
571:
363:
game. In this game one player, the row player (graphed on the y dimension) wins if the players coordinate (both choose heads or both choose tails) while the other player, the column player (shown in the
396:
represents a class of strategy updating rules, where players strategies in the next round are determined by their best responses to some subset of the population. Some examples include:
411:. Players do not consider the effect that choosing a strategy on the next round would have on future play in the game. This constraint results in the dynamical rule often being called
88:
must only have one value per argument, and many reaction correspondences will be undefined, i.e., a vertical line, for some opponent strategy choice. One constructs a correspondence
379:
game. The leftmost mapping is for the coordinating player, the middle shows the mapping for the discoordinating player. The sole Nash equilibrium is shown in the right hand graph.
330:
threshold), falling (play strategy 1 if probability that the other player plays 2 is above threshold), and indifferent (both strategies play equally well under all conditions).
288:
Figure 4. Reaction correspondence for both players in the hawk-dove game. Nash equilibria shown with points, where the two player's correspondences agree, i.e. cross
249:
Figure 3. Reaction correspondence for both players in the Stag Hunt game. Nash equilibria shown with points, where the two player's correspondences agree, i.e. cross
469:
464:
There are many functions that represent smoothed best response functions. The functions illustrated here are several variations on the following function:
400:
In a large population model, players choose their next action probabilistically based on which strategies are best responses to the population as a whole.
266:
which lies along the diagonal from the bottom left to top right corners. If the players do not know which one of them is which, then the mixed Nash is an
309:
strategies have reaction correspondences which only cross at one point, which will be in either the bottom left, or top right corner in payoff symmetric
342:
1 best when opponent plays his strategy 2 and 2 best when opponent plays his 1, E) Both strategies play equally well no matter what the opponent plays.
95:, for each player from the set of opponent strategy profiles into the set of the player's strategies. So, for any given set of opponent's strategies
58:, the point at which each player in a game has selected the best response (or one of the best responses) to the other players' strategies.
262:
right corners, where one player chooses one strategy, the other player chooses the opposite strategy. The third Nash equilibrium is a
171:
964:
407:
Importantly, in these models players only choose the best response on the next round that would give them the highest payoff
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198:
variables in the opposite axes to those normally used, so that it may be superimposed onto the previous graph, to show the
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In a spatial model, players choose (in the next round) the action that is the best response to all of their neighbors.
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is a parameter that determines the degree to which the function deviates from the true best response (a larger
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182:-axis). In Figure 2 the dotted line shows the optimal probability that player X plays 'Stag' (shown in the
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for a player, taking other players' strategies as given. The concept of a best response is central to
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There are three distinctive reaction correspondence shapes, one for each of the three types of
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Games in which players score highest when both players choose the same strategy, such as the
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In any finite potential game, best response dynamics always converge to a Nash equilibrium.
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186:-axis), as a function of the probability that player Y plays Stag (shown in the
178:-axis), as a function of the probability that player X plays Stag (shown in the
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Proceedings of the
National Academy of Sciences of the United States of America
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2020:
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566:{\displaystyle {\frac {e^{E(1)/\gamma }}{e^{E(1)/\gamma }+e^{E(2)/\gamma }}}}
84:
Nash equilibria. Reaction correspondences are not "reaction functions" since
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610:. Finally, using smoothed best response with some learning rules (as in
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Figure 8. A BR correspondence (black) and smoothed BR functions (colors)
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Figure 6 - The five possible reaction correspondences for a player in a
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Figure 5. Reaction correspondence for a game with a dominated strategy.
202:
at the points where the two player's best responses agree in Figure 3.
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786:
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Figure 2. Reaction correspondence for player X in the Stag Hunt game.
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Figure 1. Reaction correspondence for player Y in the Stag Hunt game.
762:
1939:
1439:
163:. Figures 1 to 3 graphs the best response correspondences for the
27:"Reaction function" redirects here. For the economic concept, see
1660:
1650:
1328:
898:
Nisan, N.; Roughgarden, T.; Tardos, É.; Vazirani, V. V. (2007),
449:
325:
A wider range of reaction correspondences shapes is possible in
897:
713:
1429:
346:
While there are only four possible types of payoff symmetric
598:
implies that the player is more likely to make 'mistakes').
719:
457:
Instead of best response correspondences, some models use
278:
375:
Figure 7. Reaction correspondences for players in the
472:
277:
is said to exist, and the corner Nash equilibria are
359:One well-known game with payoff asymmetries is the
695:
565:
320:
292:
46:(or strategies) which produces the most favorable
914:
689:
430:by computing the best response for every player:
2018:
792:
776:
744:"Learning, Local Interaction, and Coordination"
725:
677:
659:
649:
958:
167:game. The dotted line in Figure 1 shows the
965:
951:
240:
972:
881:
871:
614:) can result in players learning to play
586:represents the expected payoff of action
217:game will take one of these three forms.
448:
370:
332:
313:games. For instance, in the single-play
296:
283:
244:
135:
65:
826:
741:
701:
653:
602:determined in all cases, since it is a
14:
2019:
946:
930:
220:
190:-axis). Note that Figure 2 plots the
842:
665:
354:
174:that player Y plays 'Stag' (in the
24:
1014:First-player and second-player win
25:
2038:
933:Strategic Learning and Its Limits
144:Response correspondences for all
61:
29:monetary policy reaction function
1121:Coalition-proof Nash equilibrium
459:smoothed best response functions
846:(1950), "Equilibrium points in
783:The Theory of Learning in Games
735:
690:Osborne & Rubinstein (1994)
351:which are logically identical.
321:Other (payoff asymmetric) games
293:Games with dominated strategies
1131:Evolutionarily stable strategy
547:
541:
517:
511:
489:
483:
269:evolutionarily stable strategy
13:
1:
1059:Simultaneous action selection
726:Fudenberg & Levine (1998)
678:Fudenberg & Tirole (1991)
650:Fudenberg & Tirole (1991)
636:
426:refers to a way of finding a
54:best-known contribution, the
1991:List of games in game theory
1171:Quantal response equilibrium
1161:Perfect Bayesian equilibrium
1096:Bayes correlated equilibrium
923:, Cambridge, Massachusetts:
806:, Cambridge, Massachusetts:
785:, Cambridge, Massachusetts:
7:
1460:Optional prisoner's dilemma
1191:Self-confirming equilibrium
624:
444:
383:
10:
2043:
1925:Principal variation search
1641:Aumann's agreement theorem
1304:Strategy-stealing argument
1216:Trembling hand equilibrium
1146:Markov perfect equilibrium
1141:Mertens-stable equilibrium
909:Cambridge University Press
26:
1961:Combinatorial game theory
1948:
1907:
1689:
1633:
1620:Princess and monster game
1415:
1317:
1224:
1176:Quasi-perfect equilibrium
1101:Bayesian Nash equilibrium
1082:
981:
1976:Evolutionary game theory
1709:Antoine Augustin Cournot
1595:Guess 2/3 of the average
1392:Strictly determined game
1186:Satisfaction equilibrium
1004:Escalation of commitment
831:, Harvester-Wheatsheaf,
390:evolutionary game theory
1981:Glossary of game theory
1580:Stackelberg competition
1206:Strong Nash equilibrium
937:Oxford University Press
921:A Course in Game Theory
901:Algorithmic Game Theory
829:A Primer in Game Theory
241:Anti-coordination games
2006:Tragedy of the commons
1986:List of game theorists
1966:Confrontation analysis
1676:Sprague–Grundy theorem
1196:Sequential equilibrium
1116:Correlated equilibrium
567:
454:
424:best response dynamics
394:best response dynamics
380:
343:
302:
289:
275:uncorrelated asymmetry
250:
141:
71:
1779:Jean-François Mertens
931:Young, H. P. (2005),
568:
452:
374:
336:
300:
287:
248:
155:for each player in a
139:
69:
1908:Search optimizations
1784:Jennifer Tour Chayes
1671:Revelation principle
1666:Purification theorem
1605:Nash bargaining game
1570:Bertrand competition
1555:El Farol Bar problem
1520:Electronic mail game
1485:Lewis signaling game
1029:Hierarchy of beliefs
873:10.1073/pnas.36.1.48
827:Gibbons, R. (1992),
742:Ellison, G. (1993),
470:
413:myopic best response
151:can be drawn with a
126:s best responses to
1956:Bounded rationality
1575:Cournot competition
1525:Rock paper scissors
1500:Battle of the sexes
1490:Volunteer's dilemma
1362:Perfect information
1289:Dominant strategies
1126:Epsilon-equilibrium
1009:Extensive-form game
864:1950PNAS...36...48N
714:Nisan et al. (2007)
438: —
231:battle of the sexes
1935:Paranoid algorithm
1915:Alpha–beta pruning
1794:John Maynard Smith
1625:Rendezvous problem
1465:Traveler's dilemma
1455:Gift-exchange game
1450:Prisoner's dilemma
1367:Large Poisson game
1334:Bargaining problem
1239:Backward induction
1211:Subgame perfection
1166:Proper equilibrium
563:
455:
436:
381:
344:
315:prisoner's dilemma
303:
290:
253:Games such as the
251:
235:coordination games
221:Coordination games
142:
119:represents player
72:
2014:
2013:
1920:Aspiration window
1889:Suzanne Scotchmer
1844:Oskar Morgenstern
1739:Donald B. Gillies
1681:Zermelo's theorem
1610:Induction puzzles
1565:Fair cake-cutting
1540:Public goods game
1470:Coordination game
1344:Intransitive game
1274:Forward induction
1156:Pareto efficiency
1136:Gibbs equilibrium
1106:Berge equilibrium
1054:Simultaneous game
917:Rubinstein, Ariel
716:, Section 19.3.2.
656:, pp. 33–49.
561:
434:
418:In the theory of
409:on the next round
149:normal form games
16:(Redirected from
2034:
2001:Topological game
1996:No-win situation
1894:Thomas Schelling
1874:Robert B. Wilson
1834:Merrill M. Flood
1804:John von Neumann
1714:Ariel Rubinstein
1699:Albert W. Tucker
1550:War of attrition
1510:Matching pennies
1151:Nash equilibrium
1074:Mechanism design
1039:Normal-form game
994:Cooperative game
967:
960:
953:
944:
943:
939:
927:
915:Osborne, M. J.;
911:
906:
894:
885:
875:
850:-person games",
839:
820:
789:
779:Levine, David K.
773:
757:(5): 1047–1071,
748:
729:
723:
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711:
705:
699:
693:
687:
681:
680:, Section 1.3.B.
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428:Nash equilibrium
377:matching pennies
367:
361:matching pennies
355:Matching pennies
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328:
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280:
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56:Nash equilibrium
21:
2042:
2041:
2037:
2036:
2035:
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2031:
2017:
2016:
2015:
2010:
1944:
1930:max^n algorithm
1903:
1899:William Vickrey
1859:Reinhard Selten
1814:Kenneth Binmore
1729:David K. Levine
1724:Daniel Kahneman
1691:
1685:
1661:Negamax theorem
1651:Minimax theorem
1629:
1590:Diner's dilemma
1445:All-pay auction
1411:
1397:Stochastic game
1349:Mean-field game
1320:
1313:
1284:Markov strategy
1220:
1086:
1078:
1049:Sequential game
1034:Information set
1019:Game complexity
989:Congestion game
977:
971:
904:
818:
794:Fudenberg, Drew
777:Fudenberg, D.;
763:10.2307/2951493
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619:Nash equilibria
612:Fictitious play
606:that is also a
595:
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420:potential games
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255:game of chicken
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1864:Robert Axelrod
1861:
1856:
1851:
1846:
1841:
1839:Olga Bondareva
1836:
1831:
1829:Melvin Dresher
1826:
1821:
1819:Leonid Hurwicz
1816:
1811:
1806:
1801:
1796:
1791:
1786:
1781:
1776:
1771:
1766:
1761:
1756:
1754:Harold W. Kuhn
1751:
1746:
1744:Drew Fudenberg
1741:
1736:
1734:David M. Kreps
1731:
1726:
1721:
1719:Claude Shannon
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1701:
1695:
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1656:Nash's theorem
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1527:
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1515:Ultimatum game
1512:
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1495:Dollar auction
1492:
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1480:Centipede game
1477:
1472:
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1437:
1435:Infinite chess
1432:
1427:
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1412:
1410:
1409:
1404:
1402:Symmetric game
1399:
1394:
1389:
1387:Signaling game
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1382:Screening game
1379:
1374:
1372:Potential game
1369:
1364:
1359:
1351:
1346:
1341:
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1299:Mixed strategy
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1181:Risk dominance
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1024:Graphical game
1021:
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912:
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824:
816:
790:
774:
737:
734:
731:
730:
718:
706:
702:Ellison (1993)
694:
692:, Section 2.2.
682:
670:
658:
654:Gibbons (1992)
652:, p. 29;
641:
640:
638:
635:
634:
633:
626:
623:
616:mixed strategy
604:correspondence
557:
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294:
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267:
264:mixed strategy
259:hawk-dove game
242:
239:
222:
219:
129:
113:
106:
98:
82:mixed strategy
63:
62:Correspondence
60:
9:
6:
4:
3:
2:
2039:
2028:
2025:
2024:
2022:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1953:
1951:
1949:Miscellaneous
1947:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1913:
1912:
1910:
1906:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1884:Samuel Bowles
1882:
1880:
1879:Roger Myerson
1877:
1875:
1872:
1870:
1869:Robert Aumann
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1837:
1835:
1832:
1830:
1827:
1825:
1824:Lloyd Shapley
1822:
1820:
1817:
1815:
1812:
1810:
1809:Kenneth Arrow
1807:
1805:
1802:
1800:
1797:
1795:
1792:
1790:
1789:John Harsanyi
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1760:
1759:Herbert Simon
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1696:
1694:
1688:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1638:
1636:
1632:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1561:
1560:Fair division
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1535:Dictator game
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1456:
1453:
1451:
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1422:
1420:
1418:
1414:
1408:
1407:Zero-sum game
1405:
1403:
1400:
1398:
1395:
1393:
1390:
1388:
1385:
1383:
1380:
1378:
1377:Repeated game
1375:
1373:
1370:
1368:
1365:
1363:
1360:
1358:
1356:
1352:
1350:
1347:
1345:
1342:
1340:
1337:
1335:
1332:
1330:
1327:
1326:
1324:
1322:
1316:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1294:Pure strategy
1292:
1290:
1287:
1285:
1282:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1262:
1260:
1259:De-escalation
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1237:
1235:
1232:
1231:
1229:
1227:
1223:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1201:Shapley value
1199:
1197:
1194:
1192:
1189:
1187:
1184:
1182:
1179:
1177:
1174:
1172:
1169:
1167:
1164:
1162:
1159:
1157:
1154:
1152:
1149:
1147:
1144:
1142:
1139:
1137:
1134:
1132:
1129:
1127:
1124:
1122:
1119:
1117:
1114:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1093:
1091:
1089:
1085:
1081:
1075:
1072:
1070:
1069:Succinct game
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
986:
984:
980:
976:
968:
963:
961:
956:
954:
949:
948:
945:
938:
934:
929:
926:
922:
918:
913:
910:
903:
902:
896:
893:
889:
884:
879:
874:
869:
865:
861:
857:
853:
849:
845:
844:Nash, John F.
841:
838:
834:
830:
825:
823:
822:Book preview.
819:
817:9780262061414
813:
809:
805:
804:
799:
795:
791:
788:
784:
780:
775:
772:
768:
764:
760:
756:
752:
745:
740:
739:
727:
722:
715:
710:
703:
698:
691:
686:
679:
674:
667:
662:
655:
651:
646:
642:
632:
629:
628:
622:
620:
617:
613:
609:
605:
599:
583:
579:
573:
555:
551:
544:
538:
534:
530:
525:
521:
514:
508:
504:
497:
493:
486:
480:
476:
465:
462:
460:
451:
441:
431:
429:
425:
421:
416:
414:
408:
402:
399:
398:
397:
395:
391:
378:
373:
369:
362:
352:
335:
331:
318:
316:
308:
299:
286:
282:
276:
270:
265:
260:
256:
247:
238:
236:
233:, are called
232:
228:
218:
208:
203:
201:
197:
193:
173:
170:
166:
162:
158:
154:
150:
138:
134:
116:
109:
92:
87:
83:
79:
78:
68:
59:
57:
53:
49:
45:
41:
40:best response
37:
30:
19:
1854:Peyton Young
1849:Paul Milgrom
1764:Hervé Moulin
1704:Amos Tversky
1646:Folk theorem
1357:-player game
1354:
1279:Grim trigger
932:
920:
907:, New York:
900:
858:(1): 48–49,
855:
851:
847:
828:
802:
798:Tirole, Jean
782:
754:
751:Econometrica
750:
736:Bibliography
721:
709:
697:
685:
673:
661:
645:
600:
581:
577:
574:
466:
463:
458:
456:
433:
423:
417:
412:
406:
393:
387:
358:
345:
324:
304:
252:
224:
204:
143:
111:
104:
90:
74:
73:
39:
33:
18:Optimal play
2027:Game theory
1971:Coopetition
1774:Jean Tirole
1769:John Conway
1749:Eric Maskin
1545:Blotto game
1530:Pirate game
1339:Global game
1309:Tit for tat
1244:Bid shading
1234:Appeasement
1084:Equilibrium
1064:Solved game
999:Determinacy
982:Definitions
975:game theory
803:Game Theory
666:Nash (1950)
631:Solved game
305:Games with
192:independent
172:probability
157:unit square
52:John Nash's
36:game theory
1615:Trust game
1600:Kuhn poker
1269:Escalation
1264:Deterrence
1254:Cheap talk
1226:Strategies
1044:Preference
973:Topics of
637:References
1799:John Nash
1505:Stag hunt
1249:Collusion
925:MIT Press
808:MIT Press
787:MIT Press
556:γ
526:γ
498:γ
307:dominated
227:stag hunt
207:symmetric
165:stag hunt
159:strategy
86:functions
75:Reaction
2021:Category
1940:Lazy SMP
1634:Theorems
1585:Deadlock
1440:Checkers
1321:of games
1088:concepts
919:(1994),
892:16588946
837:10248389
800:(1991),
781:(1998),
625:See also
608:function
445:Smoothed
384:Dynamics
196:response
44:strategy
1692:figures
1475:Chicken
1329:Auction
1319:Classes
883:1063129
860:Bibcode
771:2951493
435:Theorem
169:optimal
48:outcome
42:is the
890:
880:
835:
814:
769:
590:, and
575:where
38:, the
1430:Chess
1417:Games
905:(PDF)
833:S2CID
767:JSTOR
747:(PDF)
348:2 × 2
339:2 × 2
327:2 × 2
311:2 × 2
279:ESSes
271:(ESS)
215:2 × 2
210:2 × 2
161:space
146:2 × 2
124:'
1111:Core
888:PMID
812:ISBN
257:and
229:and
194:and
153:line
1690:Key
878:PMC
868:doi
759:doi
388:In
93:(·)
34:In
2023::
1425:Go
935:,
886:,
876:,
866:,
856:36
854:,
810:,
796:;
765:,
755:61
753:,
749:,
621:.
422:,
415:.
392:,
281:.
133:.
130:−i
114:−i
102:,
99:−i
1355:n
966:e
959:t
952:v
870::
862::
848:n
761::
728:.
704:.
668:.
596:γ
592:γ
588:x
584:)
582:x
580:(
578:E
552:/
548:)
545:2
542:(
539:E
535:e
531:+
522:/
518:)
515:1
512:(
509:E
505:e
494:/
490:)
487:1
484:(
481:E
477:e
366:x
188:y
184:x
180:x
176:y
128:σ
121:i
117:)
112:σ
110:(
107:i
105:b
97:σ
91:b
31:.
20:)
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