630:. In a quantal-response-equilibrium, the best-response curves are not sharp as in a standard Nash equilibrium. Rather, they change smoothly from the action whose probability is 0 to the action whose probability 1 (in other words, while in a Nash-equilibrium, a player chooses the best response with probability 1 and the worst response with probability 0, in a quantal-response-equilibrium the player chooses the best response with high probability that is smaller than 1 and the worst response with smaller probability that is higher than 0). The equilibrium point is the intersection point of the smoothed curves of the two players, which is different from the Nash-equilibrium point.
675:, the kicker has two options – kick left or kick right – and the goalie has two options – jump left or jump right. The kicker's probability of scoring a goal is higher when the choices do not match, and lower when the choices match. In general, the payoffs are asymmetric because each kicker has a stronger leg (usually the right leg) and his chances are better when kicking to the opposite direction (left). In a close examination of the actions of kickers and goalies, it was found that their actions do not deviate significantly from the prediction of a Nash equilibrium.
578:, the change in Even's payoff affects Odd's equilibrium strategy and not Even's own equilibrium strategy. This may be unintuitive at first. The reasoning is that in equilibrium, the choices must be equally appealing. The +7 possibility for Even is very appealing relative to +1, so to maintain equilibrium, Odd's play must lower the probability of that outcome to compensate and equalize the expected values of the two choices, meaning in equilibrium Odd will play Heads less often and Tails more often.
143:
609:. When the Odd player is named "the misleader" and the Even player is named "the guesser", the former focuses on trying to randomize and the latter focuses on trying to detect a pattern, and this increases the chances of success of the guesser. Additionally, the fact that Even wins when there is a match gives him an advantage, since people are better at matching than at mismatching (due to the
206:
Varying the payoffs in the matrix can change the equilibrium point. For example, in the table shown on the right, Even has a chance to win 7 if both he and Odd play Heads. To calculate the equilibrium point in this game, note that a player playing a mixed strategy must be indifferent between his two
150:
game. The leftmost mapping is for the Even player, the middle shows the mapping for the Odd player. The sole Nash equilibrium is shown in the right hand graph. x is a probability of playing heads by Odd player, y is a probability of playing heads by Even. The unique intersection is the only point
621:
Players tend to increase the probability of playing an action which gives them a higher payoff, e.g. in the payoff matrix above, Even will tend to play more Heads. This is intuitively understandable, but it is not a Nash equilibrium: as explained above, the mixing probability of a player should
637:. Players tend to underestimate high gains and overestimate high losses; this moves the quantal-response curves and changes the quantal-response-equilibrium point. This apparently contradicts theoretical results regarding the irrelevance of risk-aversion in finitely-repeated zero-sum games.
88:
and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match (both heads or both tails), then Even wins and keeps both pennies. If the pennies do not match (one heads and one tails), then Odd wins and keeps both pennies.
139:: each player chooses heads or tails with equal probability. In this way, each player makes the other indifferent between choosing heads or tails, so neither player has an incentive to try another strategy. The best-response functions for mixed strategies are depicted in Figure 1 below:
663:
To overcome these difficulties, several authors have done statistical analyses of professional sports games. These are zero-sum games with very high payoffs, and the players have devoted their lives to become experts. Often such games are strategically similar to matching pennies:
682:
serve-and-return plays, the situation is similar. It was found that the win rates are consistent with the minimax hypothesis, but the players' choices are not random: even professional tennis players are not good at randomizing, and switch their actions too
101:
because each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the participants' total gains are added up and their total losses subtracted, the sum will be zero.
586:
Human players do not always play the equilibrium strategy. Laboratory experiments reveal several factors that make players deviate from the equilibrium strategy, especially if matching pennies is played repeatedly:
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to a best response. In other words, there is no pair of pure strategies such that neither player would want to switch if told what the other would do. Instead, the unique Nash equilibrium of this game is in
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The payoffs in lab experiments are small, so subjects do not have much incentive to play optimally. In real life, the market may punish such irrationality and cause players to behave more rationally.
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Humans are not good at randomizing. They may try to produce "random" sequences by switching their actions from Heads to Tails and vice versa, but they switch their actions too often (due to a
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Humans are trained to detect patterns. They try to detect patterns in the opponent's sequence, even when such patterns do not exist, and adjust their strategy accordingly.
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595:). This makes it possible for expert players to predict their next actions with more than 50% chance of success. In this way, a positive
109:(pictured right - from Even's point of view). Each cell of the matrix shows the two players' payoffs, with Even's payoffs listed first.
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Subjects have other considerations besides maximizing monetary payoffs, such as to avoid looking foolish or to please the experimenter.
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Moreover, when the payoff matrix is asymmetric, other factors influence human behavior even when the game is not repeated:
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Eliaz, Kfir; Rubinstein, Ariel (2011). "Edgar Allan Poe's riddle: Framing effects in repeated matching pennies games".
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Wooders, John; Shachat, Jason M. (2001). "On the
Irrelevance of Risk Attitudes in Repeated Two-Outcome Games".
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973:"Testing Mixed-Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer"
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Mookherjee, Dilip; Sopher, Barry (1994). "Learning
Behavior in an Experimental Matching Pennies Game".
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Lab experiments are short and subjects do not have sufficient time to learn the optimal strategy.
20:
19:
This article is about the two-person game studied in game-theory. For the confidence trick, see
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Games in lab experiments are artificial and simplistic and do not mimic real-life behavior.
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Ochs, Jack (1995). "Games with Unique, Mixed
Strategy Equilibria: An Experimental Study".
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696:- a game with the same strategic structure, that is played with fingers instead of coins.
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There is also the option of kicking/standing in the middle, but it is less often used.
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where the strategy of Even is the best response to the strategy of Odd and vice versa.
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actions (otherwise he would switch to a pure strategy). This gives us two equations:
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The conclusions of laboratory experiments have been criticized on several grounds.
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When either player plays the equilibrium, everyone's expected payoff is zero.
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708:- a much more complicated two-player logic game, played on a colored graph.
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702:- a similar game in which each player has three strategies instead of two.
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player's payoff, not his own payoff. This deviation can be explained as a
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29:
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84:. It is played between two players, Even and Odd. Each player has a
1603:
141:
85:
1076:
Walker, Mark; Wooders, John (2001). "Minimax Play at
Wimbledon".
112:
Matching pennies is used primarily to illustrate the concept of
1120:
895:
Goeree, Jacob K.; Holt, Charles A.; Palfrey, Thomas R. (2003).
211:
For the Even player, the expected payoff when playing Heads is
370:
For the Odd player, the expected payoff when playing Heads is
858:(1995). "Quantal Response Equilibria for Normal Form Games".
897:"Risk averse behavior in generalized matching pennies games"
500:
probability of playing Heads), and these must be equal, so
341:
probability of playing Heads), and these must be equal, so
146:
Figure 1. Best response correspondences for players in the
130:
since there is no pure strategy (heads or tails) that is a
1028:
Palacios-Huerta, I. (2003). "Professionals Play
Minimax".
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731:. Princeton University Press. pp. 29–33.
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633:The own-payoff effects are mitigated by
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1021:
469:{\displaystyle -1\cdot y+1\cdot (1-y)}
416:{\displaystyle +1\cdot y-1\cdot (1-y)}
310:{\displaystyle -1\cdot x+1\cdot (1-x)}
257:{\displaystyle +7\cdot x-1\cdot (1-x)}
7:
1188:First-player and second-player win
728:Game Theory for Applied Economists
16:Simple game studied in game theory
14:
1295:Coalition-proof Nash equilibrium
755:. GameTheory.net. Archived from
605:Humans' behavior is affected by
611:Stimulus-Response compatibility
1305:Evolutionarily stable strategy
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304:
292:
251:
239:
1:
1233:Simultaneous action selection
916:10.1016/s0899-8256(03)00052-6
105:The game can be written in a
2165:List of games in game theory
1345:Quantal response equilibrium
1335:Perfect Bayesian equilibrium
1270:Bayes correlated equilibrium
1078:The American Economic Review
628:quantal response equilibrium
574:is the Heads-probability of
550:is the Heads-probability of
1634:Optional prisoner's dilemma
1365:Self-confirming equilibrium
931:Games and Economic Behavior
904:Games and Economic Behavior
860:Games and Economic Behavior
826:Games and Economic Behavior
799:Games and Economic Behavior
772:Games and Economic Behavior
2217:
2099:Principal variation search
1815:Aumann's agreement theorem
1478:Strategy-stealing argument
1390:Trembling hand equilibrium
1320:Markov perfect equilibrium
1315:Mertens-stable equilibrium
1031:Review of Economic Studies
23:. For other variants, see
18:
2135:Combinatorial game theory
1794:Princess and monster game
1350:Quasi-perfect equilibrium
1275:Bayesian Nash equilibrium
1004:10.1257/00028280260344678
971:; Groseclose, T. (2002).
811:10.1016/j.geb.2009.05.010
197:
64:
2150:Evolutionary game theory
1883:Antoine Augustin Cournot
1769:Guess 2/3 of the average
1566:Strictly determined game
1360:Satisfaction equilibrium
1178:Escalation of commitment
981:American Economic Review
725:Gibbons, Robert (1992).
2155:Glossary of game theory
1754:Stackelberg competition
1380:Strong Nash equilibrium
1054:10.1111/1467-937X.00249
423:and when playing Tails
264:and when playing Tails
2180:Tragedy of the commons
2160:List of game theorists
2140:Confrontation analysis
1850:Sprague–Grundy theorem
1370:Sequential equilibrium
1290:Correlated equilibrium
943:10.1006/game.2000.0808
882:10.1006/game.1995.1023
838:10.1006/game.1995.1030
784:10.1006/game.1994.1037
582:Laboratory experiments
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97:Matching Pennies is a
2201:Non-cooperative games
1953:Jean-François Mertens
1100:10.1257/aer.91.5.1521
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519:{\displaystyle y=0.5}
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362:
360:{\displaystyle x=0.2}
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116:and a mixed strategy
2082:Search optimizations
1958:Jennifer Tour Chayes
1845:Revelation principle
1840:Purification theorem
1779:Nash bargaining game
1744:Bertrand competition
1729:El Farol Bar problem
1694:Electronic mail game
1659:Lewis signaling game
1203:Hierarchy of beliefs
599:might be attainable.
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78:non-cooperative game
2130:Bounded rationality
1749:Cournot competition
1699:Rock paper scissors
1674:Battle of the sexes
1664:Volunteer's dilemma
1536:Perfect information
1463:Dominant strategies
1300:Epsilon-equilibrium
1183:Extensive-form game
700:Rock paper scissors
622:depend only on the
2109:Paranoid algorithm
2089:Alpha–beta pruning
1968:John Maynard Smith
1799:Rendezvous problem
1639:Traveler's dilemma
1629:Gift-exchange game
1624:Prisoner's dilemma
1541:Large Poisson game
1508:Bargaining problem
1413:Backward induction
1385:Subgame perfection
1340:Proper equilibrium
753:"Matching Pennies"
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21:coin-matching game
2188:
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2094:Aspiration window
2063:Suzanne Scotchmer
2018:Oskar Morgenstern
1913:Donald B. Gillies
1855:Zermelo's theorem
1784:Induction puzzles
1739:Fair cake-cutting
1714:Public goods game
1644:Coordination game
1518:Intransitive game
1448:Forward induction
1330:Pareto efficiency
1310:Gibbs equilibrium
1280:Berge equilibrium
1228:Simultaneous game
852:McKelvey, Richard
738:978-0-691-00395-5
593:gambler's fallacy
567:{\displaystyle y}
543:{\displaystyle x}
489:{\displaystyle y}
330:{\displaystyle x}
204:
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123:This game has no
71:
70:
2208:
2175:Topological game
2170:No-win situation
2068:Thomas Schelling
2048:Robert B. Wilson
2008:Merrill M. Flood
1978:John von Neumann
1888:Ariel Rubinstein
1873:Albert W. Tucker
1724:War of attrition
1684:Matching pennies
1325:Nash equilibrium
1248:Mechanism design
1213:Normal-form game
1168:Cooperative game
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1084:(5): 1521–1538.
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988:(4): 1138–1151.
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530:Note that since
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137:mixed strategies
128:Nash equilibrium
118:Nash equilibrium
114:mixed strategies
74:Matching pennies
66:Matching pennies
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2073:William Vickrey
2033:Reinhard Selten
1988:Kenneth Binmore
1903:David K. Levine
1898:Daniel Kahneman
1865:
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1835:Negamax theorem
1825:Minimax theorem
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1764:Diner's dilemma
1619:All-pay auction
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1571:Stochastic game
1523:Mean-field game
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1458:Markov strategy
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1223:Sequential game
1208:Information set
1193:Game complexity
1163:Congestion game
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1091:10.1.1.614.5372
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1609:Infinite chess
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1576:Symmetric game
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1038:(2): 395–415.
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873:10.1.1.30.5152
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759:on 2006-10-01.
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694:Odds and evens
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642:Real-life data
640:
639:
638:
631:
615:
614:
603:
600:
583:
580:
563:
539:
528:
527:
515:
512:
509:
485:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
412:
409:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
368:
356:
353:
350:
326:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
253:
250:
247:
244:
241:
238:
235:
232:
229:
226:
223:
220:
202:
201:
195:
194:
191:
188:
184:
183:
180:
177:
173:
172:
169:
166:
160:
157:
94:
91:
69:
68:
62:
61:
58:
55:
51:
50:
47:
44:
40:
39:
36:
33:
15:
13:
10:
9:
6:
4:
3:
2:
2213:
2202:
2199:
2198:
2196:
2181:
2178:
2176:
2173:
2171:
2168:
2166:
2163:
2161:
2158:
2156:
2153:
2151:
2148:
2146:
2143:
2141:
2138:
2136:
2133:
2131:
2128:
2127:
2125:
2123:Miscellaneous
2121:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2086:
2084:
2080:
2074:
2071:
2069:
2066:
2064:
2061:
2059:
2058:Samuel Bowles
2056:
2054:
2053:Roger Myerson
2051:
2049:
2046:
2044:
2043:Robert Aumann
2041:
2039:
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2019:
2016:
2014:
2011:
2009:
2006:
2004:
2001:
1999:
1998:Lloyd Shapley
1996:
1994:
1991:
1989:
1986:
1984:
1983:Kenneth Arrow
1981:
1979:
1976:
1974:
1971:
1969:
1966:
1964:
1963:John Harsanyi
1961:
1959:
1956:
1954:
1951:
1949:
1946:
1944:
1941:
1939:
1936:
1934:
1933:Herbert Simon
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1870:
1868:
1862:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1812:
1810:
1806:
1800:
1797:
1795:
1792:
1790:
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1734:Fair division
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1709:Dictator game
1707:
1705:
1702:
1700:
1697:
1695:
1692:
1690:
1687:
1685:
1682:
1680:
1677:
1675:
1672:
1670:
1667:
1665:
1662:
1660:
1657:
1655:
1652:
1650:
1647:
1645:
1642:
1640:
1637:
1635:
1632:
1630:
1627:
1625:
1622:
1620:
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1596:
1594:
1592:
1588:
1582:
1581:Zero-sum game
1579:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1551:Repeated game
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1530:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1500:
1498:
1496:
1490:
1484:
1481:
1479:
1476:
1474:
1471:
1469:
1468:Pure strategy
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1433:De-escalation
1431:
1429:
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1409:
1406:
1405:
1403:
1401:
1397:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1375:Shapley value
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1346:
1343:
1341:
1338:
1336:
1333:
1331:
1328:
1326:
1323:
1321:
1318:
1316:
1313:
1311:
1308:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1286:
1283:
1281:
1278:
1276:
1273:
1271:
1268:
1267:
1265:
1263:
1259:
1255:
1249:
1246:
1244:
1243:Succinct game
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1201:
1199:
1196:
1194:
1191:
1189:
1186:
1184:
1181:
1179:
1176:
1174:
1171:
1169:
1166:
1164:
1161:
1160:
1158:
1154:
1150:
1142:
1137:
1135:
1130:
1128:
1123:
1122:
1119:
1109:
1105:
1101:
1097:
1092:
1087:
1083:
1079:
1072:
1069:
1063:
1060:
1055:
1051:
1046:
1041:
1037:
1033:
1032:
1024:
1022:
1018:
1013:
1009:
1005:
1001:
996:
991:
987:
983:
982:
974:
970:
963:
961:
957:
952:
948:
944:
940:
936:
932:
925:
922:
917:
913:
909:
905:
898:
891:
888:
883:
879:
874:
869:
865:
861:
857:
853:
847:
844:
839:
835:
831:
827:
820:
817:
812:
808:
804:
800:
793:
790:
785:
781:
777:
773:
766:
763:
758:
754:
748:
745:
740:
734:
730:
729:
721:
718:
712:
707:
704:
701:
698:
695:
692:
691:
687:
681:
677:
674:
673:penalty kicks
671:
667:
666:
665:
658:
655:
652:
649:
648:
647:
641:
636:
635:risk aversion
632:
629:
625:
620:
619:
618:
612:
608:
604:
601:
598:
594:
590:
589:
588:
581:
579:
577:
561:
553:
537:
513:
510:
507:
499:
483:
460:
457:
454:
448:
445:
442:
439:
436:
433:
430:
407:
404:
401:
395:
392:
389:
386:
383:
380:
377:
369:
354:
351:
348:
340:
324:
301:
298:
295:
289:
286:
283:
280:
277:
274:
271:
248:
245:
242:
236:
233:
230:
227:
224:
221:
218:
210:
209:
208:
200:
196:
192:
189:
186:
185:
181:
178:
175:
174:
170:
167:
165:
164:
158:
156:
149:
144:
140:
138:
133:
132:best response
129:
126:
125:pure strategy
121:
119:
115:
110:
108:
107:payoff matrix
103:
100:
99:zero-sum game
92:
90:
87:
83:
79:
75:
67:
63:
59:
56:
53:
52:
48:
45:
42:
41:
37:
34:
32:
31:
26:
22:
2028:Peyton Young
2023:Paul Milgrom
1938:Hervé Moulin
1878:Amos Tversky
1820:Folk theorem
1683:
1531:-player game
1528:
1453:Grim trigger
1081:
1077:
1071:
1062:
1035:
1029:
985:
979:
934:
930:
924:
907:
903:
890:
863:
859:
846:
829:
825:
819:
802:
798:
792:
775:
771:
765:
757:the original
747:
727:
720:
662:
645:
623:
616:
585:
575:
551:
529:
497:
338:
205:
198:
154:
147:
122:
111:
104:
96:
73:
72:
65:
25:morra (game)
2145:Coopetition
1948:Jean Tirole
1943:John Conway
1923:Eric Maskin
1719:Blotto game
1704:Pirate game
1513:Global game
1483:Tit for tat
1418:Bid shading
1408:Appeasement
1258:Equilibrium
1238:Solved game
1173:Determinacy
1156:Definitions
1149:game theory
832:: 202–217.
706:Parity game
82:game theory
80:studied in
1789:Trust game
1774:Kuhn poker
1443:Escalation
1438:Deterrence
1428:Cheap talk
1400:Strategies
1218:Preference
1147:Topics of
969:Levitt, S.
937:(2): 342.
910:: 97–113.
713:References
1973:John Nash
1679:Stag hunt
1423:Collusion
1086:CiteSeerX
1040:CiteSeerX
990:CiteSeerX
868:CiteSeerX
805:: 88–99.
778:: 62–91.
458:−
449:⋅
437:⋅
431:−
405:−
396:⋅
390:−
384:⋅
299:−
290:⋅
278:⋅
272:−
246:−
237:⋅
231:−
225:⋅
2195:Category
2114:Lazy SMP
1808:Theorems
1759:Deadlock
1614:Checkers
1495:of games
1262:concepts
866:: 6–38.
688:See also
613:effect).
159:Variants
1866:figures
1649:Chicken
1503:Auction
1493:Classes
1108:2677937
1012:3083302
951:2401322
476:(where
317:(where
193:+1, -1
190:-1, +1
182:-1, +1
179:+7, -1
60:+1, −1
57:−1, +1
49:−1, +1
46:+1, −1
1106:
1088:
1042:
1010:
992:
949:
870:
735:
683:often.
680:tennis
670:soccer
498:Even's
187:Tails
176:Heads
171:Tails
168:Heads
93:Theory
54:Tails
43:Heads
38:Tails
35:Heads
1604:Chess
1591:Games
1104:JSTOR
1008:JSTOR
976:(PDF)
947:S2CID
900:(PDF)
624:other
339:Odd's
86:penny
76:is a
1285:Core
733:ISBN
576:Even
554:and
1864:Key
1096:doi
1050:doi
1000:doi
939:doi
912:doi
878:doi
834:doi
807:doi
780:doi
678:In
668:In
552:Odd
514:0.5
496:is
355:0.2
337:is
2197::
1599:Go
1102:.
1094:.
1082:91
1080:.
1048:.
1036:70
1034:.
1020:^
1006:.
998:.
986:92
984:.
978:.
959:^
945:.
935:34
933:.
908:45
906:.
902:.
876:.
864:10
862:.
854:;
830:10
828:.
803:71
801:.
774:.
120:.
1529:n
1140:e
1133:t
1126:v
1110:.
1098::
1056:.
1052::
1014:.
1002::
953:.
941::
918:.
914::
884:.
880::
840:.
836::
813:.
809::
786:.
782::
776:7
741:.
562:y
538:x
526:.
511:=
508:y
484:y
464:)
461:y
455:1
452:(
446:1
443:+
440:y
434:1
411:)
408:y
402:1
399:(
393:1
387:y
381:1
378:+
367:.
352:=
349:x
325:x
305:)
302:x
296:1
293:(
287:1
284:+
281:x
275:1
252:)
249:x
243:1
240:(
234:1
228:x
222:7
219:+
27:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
