399:. K stands for the number of times a cycle of reasoning is repeated. A Level-k model usually assumes that k-level 0 agents would approach the game naively and make choices distributed uniformly over the range . In accordance with cognitive hierarchy theory, level 1 players select the best responses to level 0 choices, while level 2 players select the best responses to level 1 choices. Level 1 players would assume that everyone else was playing at level 0, responding to an assumed average of 50 in relation to naive play, and thus their guess would be 33 (2/3 of 50). At k-level 2, a player would play more sophisticatedly and assume that all other players are playing at k-level 1, so they would choose 22 (2/3 of 33). Players are presumptively aware of the probability distributions of selections at each higher level. It would take approximately 21 k-levels to reach 0, the Nash equilibrium of the game.
437:
403:
play at k-levels 0 to 3, so you would just have to think one step ahead of that to have a higher chance at winning the game. Therefore, being aware of this logic allow players to adjust their strategy. This means that perfectly rational players playing in such a game should not guess 0 unless they know that the other players are rational as well, and that all players' rationality is common knowledge. If a rational player reasonably believes that other players will not follow the chain of elimination described above, it would be rational for him/her to guess a number above 0 as their best response.
75:
343:. This process will continue as this logic is continually applied, If the same group of people play the game consistently, with each step, the highest possible logical answer keeps getting smaller, the average will move close to 0, all other numbers above 0 have been eliminated. If all players understand this logic and select 0, the game reaches its Nash equilibrium, which also happens to be the
133:. The competitors had to pick out the six prettiest faces from 100 photos, and the winner is the competitor whose choices best matches the average preferences of all the competitors. Keynes observed that "It is not a case of choosing those that, to the best of oneâs judgment, are really the prettiest, nor even those that average opinion genuinely thinks the prettiest. We have reached the
466:
have been successful previously. This demonstrates the importance of social learning in arriving at the equilibrium of any decision-making. Empirical studies show, shrewd traders like hedge fund managers frequently benefit from the cognitive biases of ordinary investors, "Second level thinking" is essential for active investors to achieve superior returns.
473:
Similarly, during penalty kicks in soccer, both the shooter and goalie simultaneously decide whether to go left or right depending on what they expect the other person to do. Goalies tend to memorise the behavioural patterns of their opponents, but penalty shooters know that and will act accordingly.
465:
Another example of K-level reasoning is when stock traders evaluate stocks based on the value that others place on those stocks. Their goal is to foresee changes in valuation ahead of the general public. Their choice is also likely influenced by other individualsâ choices, especially if those choices
461:
K-level reasoning can be useful in several social and competitive interactions. For example, deciding when to sell or buy stock in the stock market before too many others do it and decrease your profitability. Philosophers and psychologists observe this as an ability to consider otherâs mental states
452:
Sbrigliaâs investigation also revealed that non-winners often try to imitate winnersâ understanding of the gameâs structure. Accordingly, other players adopt strategies which are best responses to the imitatorsâ behaviour instead of to the average level of rationality. This accelerates the attainment
422:
Experiments demonstrate that many people make mistakes and do not assume common knowledge of rationality. It has been demonstrated that even economics graduate students do not guess 0. When performed among ordinary people it is usually found that the winner's guess is much higher than 0: the winning
444:
Grosskopf and Nagelâs investigation also revealed that most players do not choose 0 the first time they play this game. Instead, they realise that 0 is the Nash
Equilibrium after some repetitions. A study by Nagel reported an average initial choice of around 36. This corresponds to approximately two
391:
of all players. To achieve its Nash equilibrium of 0, this game requires all players to be perfectly rational, rationality to be common knowledge, and all players to expect everyone else to behave accordingly. Common knowledge means that every player has the same information, and they also know that
402:
The guessing game depends on three elements: (1) the subject's perceptions of the level 0 would play; (2) the subject's expectations about the cognitive level of other players; and (3) the number of in-game reasoning steps that the subject is capable of completing. Evidence suggest that most people
418:
This game is a common demonstration in game theory classes. It reveals the significant heterogeneity of behaviour. It is unlikely that many people will play rationally according to the Nash equilibrium. This is because the game has no strictly dominant strategy, so it requires players to consider
448:
Kocher and Sutter compared the behaviours between individual and groups in playing this type of game. They observed that while both subjects applied roughly the same level of reasoning, groups learned faster. This demonstrated that repetition enabled a group of individuals to observe othersâ
469:
Howard Marks, co-founder of one of the largest hedge funds in distressed securities, has given the example that when a company reports good news about future profits, first-level retail investors will buy its shares based on that good news alone. However, a second-level thinker with more
193:
Intuitively, guessing any number higher than 2/3 of what you expect others to guess on average cannot be part of a Nash equilibrium. The highest possible average that would occur if everyone guessed 100 is 66+2/3. Therefore, choosing a number that lies above
354:
However, this degeneration does not occur in quite the same way if choices are restricted to, for example, the integers between 0 and 100. In this case, all integers except 0 and 1 vanish; it becomes advantageous to select 0 if you expect that at least
1333:
of 33.3), indicating a second iteration of this theory based on an assumption that players would guess 33.3. The final number of 33 was slightly below this peak, implying that on average each player iterated their assumption 1.07
419:
what others will do. For Nash equilibrium to be played, players would need to assume both that everyone else is rational and that there is common knowledge of rationality. However, this is a strong assumption.
392:
everyone else knows that, and that everyone else knows that everyone else knows that, and so on, infinitely. Common knowledge of rationality of all players is the reason why the winning guess is 0.
470:
sophistication would argue that if everyone only buys in response to good news, then the good news actually becomes bad news because it overvalues the stock's price, making it a bad choice.
636:
Ledoux, Alain (1981). "Concours résultats complets. Les victimes se sont plu à jouer le 14 d'atout" [Competition results complete. The victims were pleased to play the trump 14].
406:
In reality, we can assume that most players are not perfectly rational, and do not have common knowledge of each other's rationality. As a result, they will also expect others to have a
106:. He asked about 4,000 readers, who reached the same number of points in previous puzzles, to state an integer between 1 and 1,000,000,000. The winner was the one who guessed closest to
474:
In each example, individuals will weigh their own understanding of the best response against how well they think others understand the situation (i.e., how rational they are).
371:
of all players will do so, and select 1 otherwise. (In this way, it is a lopsided version of the so-called "consensus game", where one wins by being in the majority.)
122:
of the average guess. Rosemarie Nagel (1995) revealed the potential of guessing games of that kind: They are able to disclose participants' "depth of reasoning."
137:
where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practice the
919:
722:
148:
Due to the analogy to Keynes's comparison of newspaper beauty contests and stock market investments the guessing game is also known as the
1278:
879:
156:. The forgotten inventor of this game was unearthed in 2009 during an online beauty contest experiment with chess players provided by the
213:
is strictly dominated for every player. These guesses can thus be eliminated. Once these strategies are eliminated for every player,
1317:
of 50), indicating an assumption that players would guess randomly. A smaller but significant number of players guessed 22.2 (i.e.
43:" is a game that explores how a playerâs strategic reasoning process takes into account the mental process of others in the game.
1681:
1418:
698:
395:
Economic game theorists have modelled this relationship between rationality and the common knowledge of rationality through
160:: Alain Ledoux, together with over 6,000 other chess players, participated in that experiment which looked familiar to him.
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1932:
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behaviour in previous games and correspondingly choose a number that increases their chances of winning the game.
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of the guesses. Note that some of the players guessed close to 100. A large number of players guessed 33.3 (i.e.
801:
2743:
2176:
1847:
1298:
1220:
Alba-Fernåndez, Virtudes; Brañas-Garza, Pablo; Jiménez-Jiménez, Francisca; Rodero-Cosano, Javier (2010-08-07).
2015:
802:"Chess Players Performance Beyond 64 Squares: A Case Study on the Limitations of Cognitive Abilities Transfer"
2357:
1775:
1750:
2707:
2133:
1887:
1877:
1812:
878:
Nagel, Bosch-DomÚnech, Satorra, and Garcia-Montalvo, Rosemarie, Antoni, Albert and José (5 December 2002).
1927:
1907:
2641:
2392:
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2020:
1862:
1857:
1611:"Testing Mixed-Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer"
493:
396:
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2677:
2600:
2336:
1892:
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50:
between 0 and 100, inclusive. The winner of the game is the player(s) who select a number closest to
1449:"The Decision Maker Matters: Individual Versus Group Behaviour in Experimental BeautyâContest Games"
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660:
483:
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149:
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1496:"Other minds in the brain: a functional imaging study of "theory of mind" in story comprehension"
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103:
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2010:
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The mean number chosen when playing the "guess 2/3 of the average" game four consecutive rounds
153:
2181:
2166:
1740:
1015:"HOW PORTABLE IS LEVEL-0 BEHAVIOR? A TEST OF LEVEL-k THEORY IN GAMES WITH NON-NEUTRAL FRAMES"
987:
Advances in
Economics and Econometrics: Theory and Applications: Seventh World Congress Vol I
913:
102:
of the average" game. In 1981, Ledoux used this game as a tie breaker in his French magazine
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8:
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2005:
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value was found to be 33 in a large online competition organized by the Danish newspaper
407:
380:
2331:
1346:"Rational Reasoning or Adaptive Behavior? Evidence from Two-Person Beauty Contest Games"
534:
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2171:
2050:
1955:
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1259:
1222:"Teaching Nash Equilibrium and Dominance: A Classroom Experiment on the Beauty Contest"
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Agranov, Marina; Potamites, Elizabeth; Schotter, Andrew; Tergiman, Chloe (July 2012).
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846:"Inspired and inspiring: Hervé Moulin and the discovery of the beauty contest game"
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183:
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2000:
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1705:
1163:"Bounded rationality in Keynesian beauty contests: a lesson for central bankers?"
2480:
982:
880:"One, Two, (Three), Infinity, ...: Newspaper and Lab Beauty-Contest Experiments"
658:
Nagel, Rosemarie (1995). "Unraveling in
Guessing Games: An Experimental Study".
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895:
519:"Neural correlates of depth of strategic reasoning in medial prefrontal cortex"
130:
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348:
180:
74:
543:
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2420:
1995:
570:
430:
126:
1537:
1399:"Unraveling in guessing games: An experimental study (by Rosemarie Nagel)"
125:
In his influential book, Keynes compared the determination of prices in a
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2055:
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1960:
1950:
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47:
20:
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1357:
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951:
152:. Rosemarie Nagel's experimental beauty contest became a famous game in
2316:
1970:
1189:
1161:
Mauersberger, Felix; Nagel, Rosemarie; BĂŒhren, Christoph (2020-06-04).
778:
753:"On the Robustness of Behaviour in Experimental 'Beauty Contest' Games"
673:
589:"On the Robustness of Behaviour in Experimental 'Beauty Contest' Games"
232:
becomes the new highest possible average (that is, if everyone chooses
1246:
1039:
1030:
688:
270:
is weakly dominated for every player since no player will guess above
2221:
2141:
1965:
1366:
1283:
904:
425:
374:
2656:
2156:
1013:
Heap, Shaun
Hargreaves; Arjona, David Rojo; Sugden, Robert (2014).
172:
2377:
2367:
2045:
1121:"Beliefs and endogenous cognitive levels: An experimental study"
1118:
1654:
1070:
Agranov, Marina; Caplin, Andrew; Tergiman, Chloe (2015-05-19).
351:
strategy for themselves, given what everyone else is choosing.
2146:
347:
solution. At this state, every player has chosen to play the
940:"Revealing the Depth of Reasoning in p-Beauty Contest Games"
989:, Cambridge: Cambridge University Press, pp. 87â172,
844:
Nagel, Rosemarie; BĂŒhren, Christoph; Frank, Björn (2016).
822:
BĂŒhren, Christoph; Frank, Björn; Nagel, Rosemarie (2012).
456:
1608:
1609:
Chiappori, P.-A; Levitt, S; Groseclose, T (2002-08-01).
1072:"Naive play and the process of choice in guessing games"
1279:"GĂŠt-et-tal konkurrence afslĂžrer at vi er irrationelle"
1160:
1167:
Economics: The Open-Access, Open-Assessment e-Journal
1069:
690:
The
General Theory of Employment, Interest, and Money
78:
Distribution of the 2898 answers to 1983 tie breaker
738:
The
General Theory of Interest, Employment and Money
429:. 19,196 people participated and the prize was 5000
517:Coricelli, Giorgio; Nagel, Rosemarie (2009-06-09).
188:
iterated elimination of weakly dominated strategies
1447:Kocher, Martin G.; Sutter, Matthias (2004-12-22).
375:Rationality versus common knowledge of rationality
86:Alain Ledoux is the founding father of the "guess
379:This game illustrates the difference between the
66:of the average of numbers chosen by all players.
2735:
1012:
843:
831:MAGKS Joint Discussion Paper Series in Economics
821:
809:MAGKS Joint Discussion Paper Series in Economics
523:Proceedings of the National Academy of Sciences
1343:
516:
46:In this game, players simultaneously select a
1675:
1564:"Informed speculation with k-level reasoning"
1397:Kagel, John H.; Penta, Antonio (2021-07-12),
1446:
918:: CS1 maint: multiple names: authors list (
721:: CS1 maint: multiple names: authors list (
587:Duffy, John; Nagel, Rosemarie (1997-11-01).
1076:Journal of the Economic Science Association
799:
1682:
1668:
1396:
1344:Grosskopf, Brit; Nagel, Rosemarie (2001).
983:"Rationality and knowledge in game theory"
750:
586:
1689:
1519:
1365:
1270:
1245:
1188:
1178:
1087:
1038:
903:
824:"A Historical Note on the Beauty Contest"
768:
604:
560:
542:
1493:
937:
800:BĂŒhren, Christoph; Frank, Björn (2010).
686:
435:
73:
1405:, London: Routledge, pp. 109â118,
653:
651:
457:Real-life examples of K-level reasoning
413:
410:and thus guess a number higher than 0.
163:
2736:
751:Duffy, John; Nagel, Rosemarie (1997).
735:
635:
1663:
1557:
1555:
1276:
1180:10.5018/economics-ejournal.ja.2020-16
1156:
1154:
933:
931:
929:
837:
693:. Springer International Publishing.
657:
1561:
648:
582:
580:
512:
510:
508:
1651:Short video explanation of the game
1277:Schou, Astrid (22 September 2005).
815:
186:. This equilibrium can be found by
13:
1731:First-player and second-player win
1552:
1151:
926:
770:10.1111/j.1468-0297.1997.tb00075.x
606:10.1111/j.1468-0297.1997.tb00075.x
14:
2755:
1644:
1403:The Art of Experimental Economics
1226:The Journal of Economic Education
980:
793:
740:. London: Macmillan. p. 156.
577:
505:
1838:Coalition-proof Nash equilibrium
1465:10.1111/j.1468-0297.2004.00966.x
865:10.1016/j.mathsocsci.2016.09.001
453:of the gameâs Nash equilibrium.
1602:
1487:
1440:
1390:
1337:
1213:
1112:
1063:
1006:
974:
871:
687:Maynard., Keynes, John (2018).
1848:Evolutionarily stable strategy
744:
729:
680:
629:
251:). Therefore, any guess above
1:
1776:Simultaneous action selection
1494:Fletcher, P (November 1995).
445:levels of k-level reasoning.
2708:List of games in game theory
1888:Quantal response equilibrium
1878:Perfect Bayesian equilibrium
1813:Bayes correlated equilibrium
1512:10.1016/0010-0277(95)00692-r
853:Mathematical Social Sciences
7:
2177:Optional prisoner's dilemma
1908:Self-confirming equilibrium
1125:Games and Economic Behavior
938:Sbriglia, Patrizia (2004).
477:
10:
2760:
2642:Principal variation search
2358:Aumann's agreement theorem
2021:Strategy-stealing argument
1933:Trembling hand equilibrium
1863:Markov perfect equilibrium
1858:Mertens-stable equilibrium
1568:Journal of Economic Theory
995:10.1017/ccol0521580110.005
896:10.1257/000282802762024737
494:Unexpected hanging paradox
462:to predict their actions.
168:In this game, there is no
69:
2678:Combinatorial game theory
2665:
2624:
2406:
2350:
2337:Princess and monster game
2132:
2034:
1941:
1893:Quasi-perfect equilibrium
1818:Bayesian Nash equilibrium
1799:
1698:
1627:10.1257/00028280260344678
1580:10.1016/j.jet.2021.105384
1562:Zhou, Hang (2022-03-01).
1521:21.11116/0000-0001-A1FA-F
1238:10.3200/jece.37.3.305-322
1137:10.1016/j.geb.2012.02.002
1089:10.1007/s40881-015-0003-5
175:, but there are strongly
2693:Evolutionary game theory
2426:Antoine Augustin Cournot
2312:Guess 2/3 of the average
2109:Strictly determined game
1903:Satisfaction equilibrium
1721:Escalation of commitment
1615:American Economic Review
1411:10.4324/9781003019121-10
884:American Economic Review
736:Keynes, John M. (1936).
661:American Economic Review
499:
484:Keynesian beauty contest
150:Keynesian beauty contest
2698:Glossary of game theory
2297:Stackelberg competition
1923:Strong Nash equilibrium
1350:SSRN Electronic Journal
944:SSRN Electronic Journal
544:10.1073/pnas.0807721106
2723:Tragedy of the commons
2703:List of game theorists
2683:Confrontation analysis
2393:SpragueâGrundy theorem
1913:Sequential equilibrium
1833:Correlated equilibrium
441:
154:experimental economics
83:
2744:Non-cooperative games
2496:Jean-François Mertens
439:
77:
2625:Search optimizations
2501:Jennifer Tour Chayes
2388:Revelation principle
2383:Purification theorem
2322:Nash bargaining game
2287:Bertrand competition
2272:El Farol Bar problem
2237:Electronic mail game
2202:Lewis signaling game
1746:Hierarchy of beliefs
1453:The Economic Journal
757:The Economic Journal
638:Jeux & Stratégie
593:The Economic Journal
414:Experimental results
383:of an actor and the
179:. There is a unique
177:dominated strategies
164:Equilibrium analysis
158:University of Kassel
2673:Bounded rationality
2292:Cournot competition
2242:Rock paper scissors
2217:Battle of the sexes
2207:Volunteer's dilemma
2079:Perfect information
2006:Dominant strategies
1843:Epsilon-equilibrium
1726:Extensive-form game
1358:10.2139/ssrn.286573
952:10.2139/ssrn.656586
535:2009PNAS..106.9163C
408:bounded rationality
381:perfect rationality
2652:Paranoid algorithm
2632:Alphaâbeta pruning
2511:John Maynard Smith
2342:Rendezvous problem
2182:Traveler's dilemma
2172:Gift-exchange game
2167:Prisoner's dilemma
2084:Large Poisson game
2051:Bargaining problem
1956:Backward induction
1928:Subgame perfection
1883:Proper equilibrium
908:– via JSTOR.
599:(445): 1684â1700.
489:Unique bid auction
442:
84:
2731:
2730:
2637:Aspiration window
2606:Suzanne Scotchmer
2561:Oskar Morgenstern
2456:Donald B. Gillies
2398:Zermelo's theorem
2327:Induction puzzles
2282:Fair cake-cutting
2257:Public goods game
2187:Coordination game
2061:Intransitive game
1991:Forward induction
1873:Pareto efficiency
1853:Gibbs equilibrium
1823:Berge equilibrium
1771:Simultaneous game
1420:978-1-003-01912-1
1031:10.3982/ECTA11132
700:978-3-319-70344-2
529:(23): 9163â9168.
397:K-level reasoning
170:strictly dominant
104:Jeux et Stratégie
80:Jeux et Stratégie
16:Mathematical game
2751:
2718:Topological game
2713:No-win situation
2611:Thomas Schelling
2591:Robert B. Wilson
2551:Merrill M. Flood
2521:John von Neumann
2431:Ariel Rubinstein
2416:Albert W. Tucker
2267:War of attrition
2227:Matching pennies
1868:Nash equilibrium
1791:Mechanism design
1756:Normal-form game
1711:Cooperative game
1684:
1677:
1670:
1661:
1660:
1639:
1638:
1621:(4): 1138â1151.
1606:
1600:
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1549:
1523:
1491:
1485:
1484:
1459:(500): 200â223.
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1025:(3): 1133â1151.
1010:
1004:
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1002:
1001:
978:
972:
971:
935:
924:
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917:
909:
907:
890:(5): 1687â1702.
875:
869:
868:
850:
841:
835:
834:
828:
819:
813:
812:
806:
797:
791:
790:
772:
748:
742:
741:
733:
727:
726:
720:
712:
684:
678:
677:
655:
646:
645:
633:
627:
626:
608:
584:
575:
574:
564:
546:
514:
385:common knowledge
370:
368:
367:
364:
361:
342:
340:
339:
336:
333:
329:
323:
321:
320:
317:
314:
310:
304:
302:
301:
298:
295:
288:
286:
285:
282:
279:
275:
269:
267:
266:
263:
260:
256:
250:
248:
247:
244:
241:
237:
231:
229:
228:
225:
222:
218:
212:
210:
209:
206:
203:
199:
184:Nash equilibrium
121:
119:
118:
115:
112:
101:
99:
98:
95:
92:
65:
63:
62:
59:
56:
40:
38:
37:
34:
31:
2759:
2758:
2754:
2753:
2752:
2750:
2749:
2748:
2734:
2733:
2732:
2727:
2661:
2647:max^n algorithm
2620:
2616:William Vickrey
2576:Reinhard Selten
2531:Kenneth Binmore
2446:David K. Levine
2441:Daniel Kahneman
2408:
2402:
2378:Negamax theorem
2368:Minimax theorem
2346:
2307:Diner's dilemma
2162:All-pay auction
2128:
2114:Stochastic game
2066:Mean-field game
2037:
2030:
2001:Markov strategy
1937:
1803:
1795:
1766:Sequential game
1751:Information set
1736:Game complexity
1706:Congestion game
1694:
1688:
1647:
1642:
1607:
1603:
1560:
1553:
1492:
1488:
1445:
1441:
1433:
1431:
1421:
1395:
1391:
1342:
1338:
1327:
1324:
1321:
1320:
1318:
1311:
1308:
1305:
1304:
1302:
1290:
1288:
1275:
1271:
1218:
1214:
1159:
1152:
1117:
1113:
1068:
1064:
1011:
1007:
999:
997:
979:
975:
936:
927:
911:
910:
876:
872:
848:
842:
838:
826:
820:
816:
804:
798:
794:
749:
745:
734:
730:
714:
713:
701:
685:
681:
656:
649:
634:
630:
585:
578:
515:
506:
502:
480:
459:
416:
377:
365:
362:
359:
358:
356:
337:
334:
331:
330:
327:
325:
318:
315:
312:
311:
308:
306:
299:
296:
293:
292:
290:
283:
280:
277:
276:
273:
271:
264:
261:
258:
257:
254:
252:
245:
242:
239:
238:
235:
233:
226:
223:
220:
219:
216:
214:
207:
204:
201:
200:
197:
195:
166:
116:
113:
110:
109:
107:
96:
93:
90:
89:
87:
72:
60:
57:
54:
53:
51:
35:
32:
29:
28:
26:
17:
12:
11:
5:
2757:
2747:
2746:
2729:
2728:
2726:
2725:
2720:
2715:
2710:
2705:
2700:
2695:
2690:
2685:
2680:
2675:
2669:
2667:
2663:
2662:
2660:
2659:
2654:
2649:
2644:
2639:
2634:
2628:
2626:
2622:
2621:
2619:
2618:
2613:
2608:
2603:
2598:
2593:
2588:
2583:
2581:Robert Axelrod
2578:
2573:
2568:
2563:
2558:
2556:Olga Bondareva
2553:
2548:
2546:Melvin Dresher
2543:
2538:
2536:Leonid Hurwicz
2533:
2528:
2523:
2518:
2513:
2508:
2503:
2498:
2493:
2488:
2483:
2478:
2473:
2471:Harold W. Kuhn
2468:
2463:
2461:Drew Fudenberg
2458:
2453:
2451:David M. Kreps
2448:
2443:
2438:
2436:Claude Shannon
2433:
2428:
2423:
2418:
2412:
2410:
2404:
2403:
2401:
2400:
2395:
2390:
2385:
2380:
2375:
2373:Nash's theorem
2370:
2365:
2360:
2354:
2352:
2348:
2347:
2345:
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2259:
2254:
2249:
2244:
2239:
2234:
2232:Ultimatum game
2229:
2224:
2219:
2214:
2212:Dollar auction
2209:
2204:
2199:
2197:Centipede game
2194:
2189:
2184:
2179:
2174:
2169:
2164:
2159:
2154:
2152:Infinite chess
2149:
2144:
2138:
2136:
2130:
2129:
2127:
2126:
2121:
2119:Symmetric game
2116:
2111:
2106:
2104:Signaling game
2101:
2099:Screening game
2096:
2091:
2089:Potential game
2086:
2081:
2076:
2068:
2063:
2058:
2053:
2048:
2042:
2040:
2032:
2031:
2029:
2028:
2023:
2018:
2016:Mixed strategy
2013:
2008:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1947:
1945:
1939:
1938:
1936:
1935:
1930:
1925:
1920:
1915:
1910:
1905:
1900:
1898:Risk dominance
1895:
1890:
1885:
1880:
1875:
1870:
1865:
1860:
1855:
1850:
1845:
1840:
1835:
1830:
1825:
1820:
1815:
1809:
1807:
1797:
1796:
1794:
1793:
1788:
1783:
1778:
1773:
1768:
1763:
1758:
1753:
1748:
1743:
1741:Graphical game
1738:
1733:
1728:
1723:
1718:
1713:
1708:
1702:
1700:
1696:
1695:
1687:
1686:
1679:
1672:
1664:
1658:
1657:
1646:
1645:External links
1643:
1641:
1640:
1601:
1551:
1506:(2): 109â128.
1486:
1439:
1419:
1389:
1336:
1269:
1232:(3): 305â322.
1212:
1150:
1131:(2): 449â463.
1111:
1082:(2): 146â157.
1062:
1005:
981:Dekel, Eddie,
973:
925:
870:
836:
814:
792:
743:
728:
699:
679:
668:(5): 1313â26.
647:
628:
576:
503:
501:
498:
497:
496:
491:
486:
479:
476:
458:
455:
415:
412:
376:
373:
345:Pareto optimal
165:
162:
143:higher degrees
131:beauty contest
71:
68:
41:of the average
15:
9:
6:
4:
3:
2:
2756:
2745:
2742:
2741:
2739:
2724:
2721:
2719:
2716:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2670:
2668:
2666:Miscellaneous
2664:
2658:
2655:
2653:
2650:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2630:
2629:
2627:
2623:
2617:
2614:
2612:
2609:
2607:
2604:
2602:
2601:Samuel Bowles
2599:
2597:
2596:Roger Myerson
2594:
2592:
2589:
2587:
2586:Robert Aumann
2584:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2542:
2541:Lloyd Shapley
2539:
2537:
2534:
2532:
2529:
2527:
2526:Kenneth Arrow
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2506:John Harsanyi
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2476:Herbert Simon
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2413:
2411:
2405:
2399:
2396:
2394:
2391:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2355:
2353:
2349:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2277:Fair division
2275:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2255:
2253:
2252:Dictator game
2250:
2248:
2245:
2243:
2240:
2238:
2235:
2233:
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2139:
2137:
2135:
2131:
2125:
2124:Zero-sum game
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2094:Repeated game
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2073:
2069:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2043:
2041:
2039:
2033:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2011:Pure strategy
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1976:De-escalation
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1948:
1946:
1944:
1940:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1918:Shapley value
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1866:
1864:
1861:
1859:
1856:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1810:
1808:
1806:
1802:
1798:
1792:
1789:
1787:
1786:Succinct game
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1703:
1701:
1697:
1693:
1685:
1680:
1678:
1673:
1671:
1666:
1665:
1662:
1656:
1652:
1649:
1648:
1636:
1632:
1628:
1624:
1620:
1616:
1612:
1605:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1558:
1556:
1547:
1543:
1539:
1535:
1531:
1527:
1522:
1517:
1513:
1509:
1505:
1501:
1497:
1490:
1482:
1478:
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1443:
1430:
1426:
1422:
1416:
1412:
1408:
1404:
1400:
1393:
1385:
1381:
1377:
1373:
1368:
1363:
1359:
1355:
1351:
1347:
1340:
1300:
1286:
1285:
1280:
1273:
1265:
1261:
1257:
1253:
1248:
1243:
1239:
1235:
1231:
1227:
1223:
1216:
1208:
1204:
1200:
1196:
1191:
1186:
1181:
1176:
1172:
1168:
1164:
1157:
1155:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1115:
1107:
1103:
1099:
1095:
1090:
1085:
1081:
1077:
1073:
1066:
1058:
1054:
1050:
1046:
1041:
1036:
1032:
1028:
1024:
1020:
1016:
1009:
996:
992:
988:
984:
977:
969:
965:
961:
957:
953:
949:
945:
941:
934:
932:
930:
921:
915:
906:
901:
897:
893:
889:
885:
881:
874:
866:
862:
858:
854:
847:
840:
832:
825:
818:
810:
803:
796:
788:
784:
780:
776:
771:
766:
763:(445): 1684.
762:
758:
754:
747:
739:
732:
724:
718:
710:
706:
702:
696:
692:
691:
683:
675:
671:
667:
663:
662:
654:
652:
643:
640:(in French).
639:
632:
624:
620:
616:
612:
607:
602:
598:
594:
590:
583:
581:
572:
568:
563:
558:
554:
550:
545:
540:
536:
532:
528:
524:
520:
513:
511:
509:
504:
495:
492:
490:
487:
485:
482:
481:
475:
471:
467:
463:
454:
450:
446:
438:
434:
432:
431:Danish kroner
428:
427:
420:
411:
409:
404:
400:
398:
393:
390:
386:
382:
372:
352:
350:
349:best response
346:
191:
189:
185:
182:
181:pure strategy
178:
174:
171:
161:
159:
155:
151:
146:
144:
140:
139:fourth, fifth
136:
132:
129:to that of a
128:
123:
105:
81:
76:
67:
49:
44:
42:
22:
2571:Peyton Young
2566:Paul Milgrom
2481:Hervé Moulin
2421:Amos Tversky
2363:Folk theorem
2311:
2074:-player game
2071:
1996:Grim trigger
1618:
1614:
1604:
1571:
1567:
1503:
1499:
1489:
1456:
1452:
1442:
1432:, retrieved
1402:
1392:
1349:
1339:
1289:. Retrieved
1282:
1272:
1229:
1225:
1215:
1170:
1166:
1128:
1124:
1114:
1079:
1075:
1065:
1022:
1019:Econometrica
1018:
1008:
998:, retrieved
986:
976:
943:
914:cite journal
887:
883:
873:
856:
852:
839:
830:
817:
808:
795:
760:
756:
746:
737:
731:
689:
682:
665:
659:
644:(10): 10â11.
641:
637:
631:
596:
592:
526:
522:
472:
468:
464:
460:
451:
447:
443:
424:
421:
417:
405:
401:
394:
378:
353:
192:
167:
147:
142:
138:
135:third degree
134:
127:stock market
124:
85:
79:
45:
24:
18:
2688:Coopetition
2491:Jean Tirole
2486:John Conway
2466:Eric Maskin
2262:Blotto game
2247:Pirate game
2056:Global game
2026:Tit for tat
1961:Bid shading
1951:Appeasement
1801:Equilibrium
1781:Solved game
1716:Determinacy
1699:Definitions
1692:game theory
1297:Includes a
1287:(in Danish)
1190:10230/45169
859:: 191â207.
389:rationality
48:real number
21:game theory
2332:Trust game
2317:Kuhn poker
1986:Escalation
1981:Deterrence
1971:Cheap talk
1943:Strategies
1761:Preference
1690:Topics of
1574:: 105384.
1434:2022-04-26
1247:10261/2097
1040:2381/44091
1000:2022-04-26
833:. 11â2012.
811:. 19â2010.
709:1055269540
2516:John Nash
2222:Stag hunt
1966:Collusion
1635:0002-8282
1596:244095022
1588:0022-0531
1530:0010-0277
1500:Cognition
1473:0013-0133
1429:237752741
1376:1556-5068
1367:10230/686
1299:histogram
1291:29 August
1284:Politiken
1256:0022-0485
1207:212631702
1199:1864-6042
1098:2199-6776
1049:0012-9682
968:197657612
960:1556-5068
905:10230/573
787:153447786
717:cite book
623:153447786
615:0013-0133
553:0027-8424
426:Politiken
326:44
307:66
272:66
253:44
234:66
215:66
196:66
2738:Category
2657:Lazy SMP
2351:Theorems
2302:Deadlock
2157:Checkers
2038:of games
1805:concepts
1546:16321133
1384:14073840
1264:49574187
1057:24029309
571:19470476
478:See also
173:strategy
82:contest.
2409:figures
2192:Chicken
2046:Auction
2036:Classes
1538:8556839
1481:7339369
1331:
1319:
1315:
1303:
1145:1632208
1106:7593331
779:2957901
674:2950991
562:2685737
531:Bibcode
387:of the
369:
357:
341:
322:
303:
291:
287:
268:
249:
230:
211:
120:
108:
100:
88:
70:History
64:
52:
39:
27:
1655:TED-Ed
1633:
1594:
1586:
1544:
1536:
1528:
1479:
1471:
1427:
1417:
1382:
1374:
1334:times.
1262:
1254:
1205:
1197:
1143:
1104:
1096:
1055:
1047:
966:
958:
785:
777:
707:
697:
672:
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