108:. Perfect Bayesian equilibria are used to solve the outcome of games where players take turns but are unsure of the "type" of their opponent, which occurs when players don't know their opponent's preference between individual moves. A classic example of a dynamic game with types is a war game where the player is unsure whether their opponent is a risk-taking "
1184:
Now, suppose that this game is repeated two times. The two plays are independent, i.e., each day the players decide simultaneously whether to build a public good in that day, get a payoff of 1 if the good is built in that day, and pay their cost if they built in that day. The only connection between
905:
Note that option 3 is a Nash equilibrium. If we ignore beliefs, then rejecting can be considered a best-response for the receiver, since it does not affect their payoff (since there is no gift anyway). Moreover, option 3 is even a SPE, since the only subgame here is the entire game. Such implausible
918:
To summarize: in this variant of the gift game, there are two SPEs: either the sender always gives and the receiver always accepts, or the sender always does not give and the receiver always rejects. From these, only the first one is a PBE; the other is not a PBE since it cannot be supported by any
914:
must contain complete information sets, sometimes there is only one subgame—the entire game—and so every Nash equilibrium is trivially subgame perfect. Even if a game does have more than one subgame, the inability of subgame perfection to cut through information sets can result in implausible
1181:, then there is a symmetric equilibrium in which the threshold cost of both players is 2/3. This means that a player whose cost is between 2/3 and 1 will not contribute, even though their cost is below the benefit, because of the possibility that the other player will contribute.
1582:
willing to build than in the one-stage game. Intuitively, the reason is that, when a player does not contribute in the first day, they make the other player believe their cost is high, and this makes the other player more willing to contribute in the second day.
626:
so the sender is very likely a friend, but the receiver still would refuse any gift because he thinks enemies are much more likely than friends to give gifts. This shows how pessimistic beliefs can result in an equilibrium bad for both players, one that is not
1064:
In the one-stage game, each player builds if-and-only-if their cost is smaller than their expected gain from building. The expected gain from building is exactly 1 times the probability that the other player does NOT build. In equilibrium, for every player
1599:- some bidders raise the current price much more than the minimal increment. One explanation to this is that it serves as a signal to the other bidders. There is a PBE in which each bidder jumps if-and-only-if their value is above a certain threshold. See
674:
if an enemy. The receiver's beliefs are updated accordingly: if he receives a gift, he believes the sender is an enemy; otherwise, he believes the sender is a friend. The receiver's best-response strategy is
1308:. They update their belief accordingly, and conclude that there is a smaller chance that their opponent will build in day 2. Therefore, they increase their threshold cost, and the threshold cost in day 2 is
1382:. They update their belief accordingly, and conclude that there is a larger chance that their opponent will build in day 2. Therefore, they decrease their threshold cost, and the threshold cost in day 2 is
650:
if an enemy. The receiver's beliefs are updated accordingly: if he receives a gift, he believes the sender is a friend; otherwise, he believes the sender is an enemy. Thus, the receiver will respond with
1699:
561:, since both types take that action and it is uninformative about the sender's type in this equilibrium. The out-of-equilibrium belief does not matter, since the sender would not want to deviate to
557:
The sender prefers the payoff of 1 from giving to the payoff of 0 from not giving, expecting that his gift will be accepted. In equilibrium, Bayes's Rule requires the receiver to have the belief
139:
of a player in a given information set specifies his choice of action in that information set, which may depend on the history (on actions taken previously in the game). This is similar to a
1057:- each player knows their own cost but not the other's cost. It is only known that each cost is drawn independently at random from some probability distribution. This makes this game a
703:
In the following example, the set of PBEs is strictly smaller than the set of SPEs and BNEs. It is a variant of the above gift-game, with the following change to the receiver's utility:
1422:
1348:
436:
1518:) in each of these situations. Since the threshold player should be indifferent between contributing and not contributing, it is possible to calculate the day-1 threshold cost
733:
The sender's strategy is: always give. The receiver's beliefs are not updated: they still believe in the a-priori probability, that the sender is a friend with probability
158:
of the other players. Formally, a belief system is an assignment of probabilities to every node in the game such that the sum of probabilities in any information set is 1.
1545:
1516:
1483:
1454:
1380:
1306:
1273:
1244:
1215:
1185:
the games is that, by playing in the first day, the players may reveal some information about their costs, and this information might affect the play in the second day.
1147:
1115:
595:
506:
348:
465:
624:
1576:
1051:
777:
248:
552:
292:
1083:
1024:
892:
840:
820:
797:
751:
222:
1179:
872:
1275:, we work backwards and analyze the players' actions in day 2. Their actions depend on the history (= the two actions in day 1), and there are three options:
1149:. This threshold cost can be calculated based on the probability distribution of the players' costs. For example, if the costs are distributed uniformly on
1485:. There is an equilibrium in which the actions in day 2 are identical to the actions in day 1 - player 1 builds and player 2 does not build.
802:
The sender's strategy is: never give. Suppose the receiver's beliefs when receiving a gift is that the sender is a friend with probability
361:
at all. That belief must be pessimistic enough that the receiver prefers the payoff of 0 from rejecting a gift to the expected payoff of
631:. These beliefs seem unrealistic, though, and game theorists are often willing to reject some perfect Bayesian equilibria as implausible.
150:
of a player in a given information set determines what node in that information set he believes the game has reached. The belief may be a
894:, the receiver's optimal strategy is: accept. This is NOT a PBE, since the sender can improve their payoff from 0 to 1 by giving a gift.
2841:
438:
from accepting, so the requirement that the receiver's strategy maximize his expected payoff given his beliefs necessitates that
1615:- a refinement of PBE, that restricts the beliefs that can be assigned to off-equilibrium information sets to "reasonable" ones.
1774:
96:(PBE) is a solution with Bayesian probability to a turn-based game with incomplete information. More specifically, it is an
2678:
638:, in which the two types of sender choose different actions, and see why they do not exist as perfect Bayesian equilibria:
2495:
2025:
1823:
729:
Similarly to example 1, there is no separating equilibrium. Let's look at the following potential pooling equilibria:
1674:
2314:
2133:
1737:
931:. See also for more examples. There is a recent application of this concept in Poker, by Loriente and Diez (2023).
1930:
1427:
In day 1, exactly one player built; suppose it is player 1. So now, it is known that the cost of player 1 is below
907:
1217:
the threshold cost of both players in day 1 (so in day 1, each player builds if-and-only-if their cost is at most
353:
The sender prefers the payoff of 0 from not giving to the payoff of -1 from sending and not being accepted. Thus,
256:
The sender's utility is 1 if his gift is accepted, -1 if his gift is rejected, and 0 if he does not give any gift.
2404:
471:
is required by Bayes's Rule, since both types take that action and it is uninformative about the sender's type.
2846:
2274:
1940:
779:. Their payoff from accepting is always higher than from rejecting, so they accept (regardless of the value of
2113:
2455:
1868:
1843:
2805:
2231:
1980:
1905:
897:
The sender's strategy is: never give, and the receiver's strategy is: reject. This is NOT a PBE, since for
1595:, the bidders can raise the current price in small steps (e.g. in $ 1 each time). However, often there is
1385:
1311:
364:
2020:
2000:
634:
Equilibria 1 and 2 are the only equilibria that might exist, but we can also check for the two potential
2739:
2490:
2460:
2118:
1955:
1950:
906:
equilibria might arise also in games with complete information, but they may be eliminated by applying
179:
on every path reached in equilibrium with positive probability. On paths of zero probability, known as
1489:
It is possible to calculate the expected payoff of the "threshold player" (a player with cost exactly
2775:
2698:
2434:
1985:
1910:
1767:
154:
over the nodes in the information set, and is typically a probability distribution over the possible
117:
46:
943:
is a sequence of simultaneous games played one after the other. These games may be identical (as in
2790:
2523:
2409:
2206:
1995:
1813:
1006:
or not build. Each player gains 1 if the public good is built and 0 if not; in addition, if player
707:
If the sender is a friend, then the receiver's utility is 1 (if they accept) or 0 (if they reject).
175:: each belief should be updated according to the equilibrium strategies, the observed actions, and
151:
2593:
265:
If the sender is an enemy, then the receiver's utility is -1 (if he accepts) or 0 (if he rejects).
2795:
2394:
2364:
2015:
1803:
1648:
1003:
262:
If the sender is a friend, then the receiver's utility is 1 (if he accepts) or 0 (if he rejects).
1521:
1492:
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1088:
635:
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477:
324:
105:
444:
2279:
2264:
1833:
1744:
1578:- the threshold in the one-stage game. This means that, in a two-stage game, the players are
954:
508:, a second pooling equilibrium exists as well as Equilibrium 1, based on different beliefs:
1723:
600:
2613:
2598:
2485:
2480:
2384:
2369:
2334:
2299:
1893:
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In day 1, both players built. So now both players know that their opponent's cost is below
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8:
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1935:
1818:
1618:
756:
295:
253:
The receiver has only one type, and two strategies: either accept the gift, or reject it.
227:
97:
2429:
1279:
In day 1, no player built. So now both players know that their opponent's cost is above
928:
534:
274:
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2608:
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1975:
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877:
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805:
782:
736:
207:
120:(BNE), which is a solution concept with Bayesian probability for non-turn-based games.
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27:
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has zero probability in equilibrium and Bayes's Rule does not restrict the belief
2713:
2673:
2628:
2543:
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2259:
2211:
2093:
1858:
1828:
1798:
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910:. However, Bayesian games often contain non-singleton information sets and since
140:
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1117:, such that the player contributes if-and-only-if their cost is less than
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2153:
2123:
2053:
2043:
1873:
1808:
1784:
1719:
89:
31:
1752:
687:, to increase his payoff from -1 to 0, so this cannot be an equilibrium.
2414:
2063:
799:). This is a PBE - it is a best-response for both sender and receiver.
663:, to increase his payoff from 0 to 1, so this cannot be an equilibrium.
204:
The sender has two possible types: either a "friend" (with probability
162:
The strategies and beliefs also must satisfy the following conditions:
2319:
2239:
2058:
1729:
113:
109:
169:: each strategy should be optimal in expectation, given the beliefs.
2754:
2254:
2475:
2465:
2143:
911:
250:). Each type has two strategies: either give a gift, or not give.
519:, whether they are the friend type or the enemy type. Receiver:
309:, whether they are the friend type or the enemy type. Receiver:
2244:
187:
A perfect
Bayesian equilibrium is always a Nash equilibrium.
100:
that uses
Bayesian updating to describe player behavior in
710:
If the sender is an enemy, then the receiver's utility is
190:
1002:. There are two players, each of whom can either build a
901:
belief of the receiver, rejecting is not a best-response.
116:" type. Perfect Bayesian Equilibria are a refinement of
298:
in which both types of sender choose the same action:
123:
Any perfect
Bayesian equilibrium has two components --
259:
The receiver's utility depends on who gives the gift:
1698:
Loriente, MartĂn Iñaki & Diez, Juan Cruz (2023).
1557:
1524:
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1462:
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1314:
1285:
1252:
1223:
1194:
1155:
1123:
1091:
1071:
1032:
1012:
998:
The following game is a simple representation of the
880:
848:
828:
808:
785:
759:
739:
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537:
480:
447:
367:
327:
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183:, the beliefs must be specified but can be arbitrary.
1026:builds the public good, they have to pay a cost of
565:no matter what response the receiver would have.
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1448:
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866:
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589:
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500:
459:
430:
342:
286:
242:
216:
722:Note that in this variant, accepting is a weakly
683:, though, the enemy sender will deviate to
659:, though, the enemy sender will deviate to
2833:
1714:
1188:We are looking for a symmetric PBE. Denote by
1768:
1700:"Perfect Bayesian Equilibrium in Kuhn Poker"
950:
934:
1775:
1761:
1782:
1672:
1625:- other refinements of PBE, specific to
691:We conclude that in this game, there is
191:Examples of perfect Bayesian equilibria
2834:
1710:
1708:
1646:
1547:. It turns out that this threshold is
1756:
1417:{\displaystyle c^{11}<{\hat {c}}}
1343:{\displaystyle c^{00}>{\hat {c}}}
431:{\displaystyle x(1)+(1-x)(-1)=2x-1,}
1705:
13:
1824:First-player and second-player win
1456:and the cost of player 2 is above
666:Suppose the sender's strategy is:
642:Suppose the sender's strategy is:
224:) or an "enemy" (with probability
14:
2858:
915:equilibria not being eliminated.
2842:Game theory equilibrium concepts
1931:Coalition-proof Nash equilibrium
922:
908:subgame perfect Nash equilibrium
1586:
16:Solution concept in game theory
1941:Evolutionarily stable strategy
1692:
1675:"Perfect Bayesian Equilibrium"
1666:
1649:"Perfect Bayesian Equilibrium"
1640:
1531:
1502:
1469:
1440:
1408:
1366:
1334:
1292:
1259:
1230:
1201:
1168:
1156:
861:
849:
753:and an enemy with probability
698:
407:
398:
395:
383:
377:
371:
195:
1:
1869:Simultaneous action selection
1702:. Universidad de San Andres.
1633:
568:Equilibrium 1 is perverse if
200:Consider the following game:
2806:List of games in game theory
1981:Quantal response equilibrium
1971:Perfect Bayesian equilibrium
1906:Bayes correlated equilibrium
1728:. Cambridge, Massachusetts:
1085:, there is a threshold cost
94:Perfect Bayesian Equilibrium
22:Perfect Bayesian Equilibrium
7:
2275:Optional prisoner's dilemma
2001:Self-confirming equilibrium
1606:
655:. If the receiver chooses
10:
2863:
2740:Principal variation search
2456:Aumann's agreement theorem
2119:Strategy-stealing argument
2026:Trembling hand equilibrium
1956:Markov perfect equilibrium
1951:Mertens-stable equilibrium
1680:. University of California
1540:{\displaystyle {\hat {c}}}
1511:{\displaystyle {\hat {c}}}
1478:{\displaystyle {\hat {c}}}
1449:{\displaystyle {\hat {c}}}
1375:{\displaystyle {\hat {c}}}
1301:{\displaystyle {\hat {c}}}
1268:{\displaystyle {\hat {c}}}
1239:{\displaystyle {\hat {c}}}
1210:{\displaystyle {\hat {c}}}
927:For further examples, see
313:, with the beliefs that
2776:Combinatorial game theory
2763:
2722:
2504:
2448:
2435:Princess and monster game
2230:
2132:
2034:
1986:Quasi-perfect equilibrium
1911:Bayesian Nash equilibrium
1892:
1791:
1142:{\displaystyle C_{i}^{*}}
1110:{\displaystyle C_{i}^{*}}
989:
951:Repeated public-good game
590:{\displaystyle p\geq .5.}
531:, choosing any value for
529:Prob(Friend|Not give) = x
501:{\displaystyle p\geq 1/2}
469:Prob(Friend|Not give) = p
343:{\displaystyle x\leq .5.}
315:Prob(Friend|Not Give) = p
118:Bayesian Nash equilibrium
76:
65:
57:
52:
47:Bayesian Nash equilibrium
42:
37:
26:
21:
2791:Evolutionary game theory
2524:Antoine Augustin Cournot
2410:Guess 2/3 of the average
2207:Strictly determined game
1996:Satisfaction equilibrium
1814:Escalation of commitment
935:PBE in multi-stage games
695:separating equilibrium.
679:If the receiver chooses
460:{\displaystyle \leq .5.}
294:Equilibrium 1 exists, a
152:probability distribution
112:" type or a pacifistic "
2796:Glossary of game theory
2395:Stackelberg competition
2016:Strong Nash equilibrium
1654:. Ohio State University
929:signaling game#Examples
2821:Tragedy of the commons
2801:List of game theorists
2781:Confrontation analysis
2491:Sprague–Grundy theorem
2006:Sequential equilibrium
1926:Correlated equilibrium
1613:Sequential equilibrium
1601:Jump bidding#signaling
1572:
1541:
1512:
1479:
1450:
1418:
1376:
1344:
1302:
1269:
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1211:
1175:
1143:
1111:
1079:
1047:
1020:
888:
868:
836:
816:
793:
773:
747:
620:
619:{\displaystyle p=.99,}
591:
548:
523:with the beliefs that
502:
461:
432:
344:
319:Prob(Friend|Give) = x,
288:
244:
218:
167:Sequential rationality
106:incomplete information
2847:Non-cooperative games
2594:Jean-François Mertens
1573:
1571:{\displaystyle c^{*}}
1542:
1513:
1480:
1451:
1419:
1377:
1345:
1303:
1270:
1241:
1212:
1176:
1144:
1112:
1080:
1048:
1046:{\displaystyle C_{i}}
1021:
889:
869:
837:
817:
794:
774:
748:
636:separating equilibria
621:
592:
559:Prob(Friend|Give) = p
549:
525:Prob(Friend|Give) = p
503:
462:
433:
345:
289:
245:
219:
181:off-equilibrium paths
2723:Search optimizations
2599:Jennifer Tour Chayes
2486:Revelation principle
2481:Purification theorem
2420:Nash bargaining game
2385:Bertrand competition
2370:El Farol Bar problem
2335:Electronic mail game
2300:Lewis signaling game
1839:Hierarchy of beliefs
1555:
1522:
1493:
1460:
1431:
1386:
1357:
1312:
1283:
1250:
1221:
1192:
1153:
1121:
1089:
1069:
1030:
1010:
878:
846:
826:
806:
783:
757:
737:
714:(if they accept) or
601:
597:The game could have
572:
535:
478:
445:
365:
325:
275:
228:
208:
2771:Bounded rationality
2390:Cournot competition
2340:Rock paper scissors
2315:Battle of the sexes
2305:Volunteer's dilemma
2177:Perfect information
2104:Dominant strategies
1936:Epsilon-equilibrium
1819:Extensive-form game
1619:Intuitive criterion
1138:
1106:
1055:private information
772:{\displaystyle 1-p}
467:On the other hand,
296:pooling equilibrium
243:{\displaystyle 1-p}
98:equilibrium concept
2750:Paranoid algorithm
2730:Alpha–beta pruning
2609:John Maynard Smith
2440:Rendezvous problem
2280:Traveler's dilemma
2270:Gift-exchange game
2265:Prisoner's dilemma
2182:Large Poisson game
2149:Bargaining problem
2049:Backward induction
2021:Subgame perfection
1976:Proper equilibrium
1623:Divine equilibrium
1591:In an open-outcry
1568:
1537:
1508:
1475:
1446:
1414:
1372:
1340:
1298:
1265:
1236:
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1075:
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1016:
1000:free-rider problem
884:
864:
832:
812:
789:
769:
743:
726:for the receiver.
616:
587:
547:{\displaystyle x.}
544:
498:
457:
428:
340:
287:{\displaystyle p,}
284:
240:
214:
2829:
2828:
2735:Aspiration window
2704:Suzanne Scotchmer
2659:Oskar Morgenstern
2554:Donald B. Gillies
2496:Zermelo's theorem
2425:Induction puzzles
2380:Fair cake-cutting
2355:Public goods game
2285:Coordination game
2159:Intransitive game
2084:Forward induction
1966:Pareto efficiency
1946:Gibbs equilibrium
1916:Berge equilibrium
1864:Simultaneous game
1534:
1505:
1472:
1443:
1411:
1369:
1337:
1295:
1262:
1233:
1204:
1078:{\displaystyle i}
1019:{\displaystyle i}
996:
995:
887:{\displaystyle q}
842:is any number in
835:{\displaystyle q}
815:{\displaystyle q}
792:{\displaystyle p}
746:{\displaystyle p}
724:dominant strategy
718:(if they reject).
440:Prob(Friend|Give)
359:Prob(Friend|Give)
321:choosing a value
271:For any value of
217:{\displaystyle p}
86:
85:
2854:
2816:Topological game
2811:No-win situation
2709:Thomas Schelling
2689:Robert B. Wilson
2649:Merrill M. Flood
2619:John von Neumann
2529:Ariel Rubinstein
2514:Albert W. Tucker
2365:War of attrition
2325:Matching pennies
2099:Pairing strategy
1961:Nash equilibrium
1884:Mechanism design
1849:Normal-form game
1804:Cooperative game
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1305:
1304:
1299:
1297:
1296:
1288:
1274:
1272:
1271:
1266:
1264:
1263:
1255:
1246:). To calculate
1245:
1243:
1242:
1237:
1235:
1234:
1226:
1216:
1214:
1213:
1208:
1206:
1205:
1197:
1180:
1178:
1177:
1174:{\displaystyle }
1172:
1148:
1146:
1145:
1140:
1137:
1132:
1116:
1114:
1113:
1108:
1105:
1100:
1084:
1082:
1081:
1076:
1053:. The costs are
1052:
1050:
1049:
1044:
1042:
1041:
1025:
1023:
1022:
1017:
991:Public good game
955:
947:) or different.
941:multi-stage game
893:
891:
890:
885:
874:. Regardless of
873:
871:
870:
867:{\displaystyle }
865:
841:
839:
838:
833:
821:
819:
818:
813:
798:
796:
795:
790:
778:
776:
775:
770:
752:
750:
749:
744:
629:Pareto efficient
625:
623:
622:
617:
596:
594:
593:
588:
553:
551:
550:
545:
507:
505:
504:
499:
494:
466:
464:
463:
458:
437:
435:
434:
429:
349:
347:
346:
341:
293:
291:
290:
285:
249:
247:
246:
241:
223:
221:
220:
215:
28:Solution concept
19:
18:
2862:
2861:
2857:
2856:
2855:
2853:
2852:
2851:
2832:
2831:
2830:
2825:
2759:
2745:max^n algorithm
2718:
2714:William Vickrey
2674:Reinhard Selten
2629:Kenneth Binmore
2544:David K. Levine
2539:Daniel Kahneman
2506:
2500:
2476:Negamax theorem
2466:Minimax theorem
2444:
2405:Diner's dilemma
2260:All-pay auction
2226:
2212:Stochastic game
2164:Mean-field game
2135:
2128:
2094:Markov strategy
2030:
1896:
1888:
1859:Sequential game
1844:Information set
1829:Game complexity
1799:Congestion game
1787:
1781:
1751:
1750:
1740:
1716:Fudenberg, Drew
1713:
1706:
1697:
1693:
1683:
1681:
1677:
1673:Zack Grossman.
1671:
1667:
1657:
1655:
1651:
1645:
1641:
1636:
1627:signaling games
1609:
1593:English auction
1589:
1562:
1558:
1556:
1553:
1552:
1526:
1525:
1523:
1520:
1519:
1497:
1496:
1494:
1491:
1490:
1464:
1463:
1461:
1458:
1457:
1435:
1434:
1432:
1429:
1428:
1403:
1402:
1393:
1389:
1387:
1384:
1383:
1361:
1360:
1358:
1355:
1354:
1329:
1328:
1319:
1315:
1313:
1310:
1309:
1287:
1286:
1284:
1281:
1280:
1254:
1253:
1251:
1248:
1247:
1225:
1224:
1222:
1219:
1218:
1196:
1195:
1193:
1190:
1189:
1154:
1151:
1150:
1133:
1128:
1122:
1119:
1118:
1101:
1096:
1090:
1087:
1086:
1070:
1067:
1066:
1037:
1033:
1031:
1028:
1027:
1011:
1008:
1007:
953:
937:
925:
919:belief-system.
879:
876:
875:
847:
844:
843:
827:
824:
823:
807:
804:
803:
784:
781:
780:
758:
755:
754:
738:
735:
734:
701:
602:
599:
598:
573:
570:
569:
536:
533:
532:
490:
479:
476:
475:
446:
443:
442:
366:
363:
362:
326:
323:
322:
276:
273:
272:
229:
226:
225:
209:
206:
205:
198:
193:
141:sequential game
17:
12:
11:
5:
2860:
2850:
2849:
2844:
2827:
2826:
2824:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2788:
2783:
2778:
2773:
2767:
2765:
2761:
2760:
2758:
2757:
2752:
2747:
2742:
2737:
2732:
2726:
2724:
2720:
2719:
2717:
2716:
2711:
2706:
2701:
2696:
2691:
2686:
2681:
2679:Robert Axelrod
2676:
2671:
2666:
2661:
2656:
2654:Olga Bondareva
2651:
2646:
2644:Melvin Dresher
2641:
2636:
2634:Leonid Hurwicz
2631:
2626:
2621:
2616:
2611:
2606:
2601:
2596:
2591:
2586:
2581:
2576:
2571:
2569:Harold W. Kuhn
2566:
2561:
2559:Drew Fudenberg
2556:
2551:
2549:David M. Kreps
2546:
2541:
2536:
2534:Claude Shannon
2531:
2526:
2521:
2516:
2510:
2508:
2502:
2501:
2499:
2498:
2493:
2488:
2483:
2478:
2473:
2471:Nash's theorem
2468:
2463:
2458:
2452:
2450:
2446:
2445:
2443:
2442:
2437:
2432:
2427:
2422:
2417:
2412:
2407:
2402:
2397:
2392:
2387:
2382:
2377:
2372:
2367:
2362:
2357:
2352:
2347:
2342:
2337:
2332:
2330:Ultimatum game
2327:
2322:
2317:
2312:
2310:Dollar auction
2307:
2302:
2297:
2295:Centipede game
2292:
2287:
2282:
2277:
2272:
2267:
2262:
2257:
2252:
2250:Infinite chess
2247:
2242:
2236:
2234:
2228:
2227:
2225:
2224:
2219:
2217:Symmetric game
2214:
2209:
2204:
2202:Signaling game
2199:
2197:Screening game
2194:
2189:
2187:Potential game
2184:
2179:
2174:
2166:
2161:
2156:
2151:
2146:
2140:
2138:
2130:
2129:
2127:
2126:
2121:
2116:
2114:Mixed strategy
2111:
2106:
2101:
2096:
2091:
2086:
2081:
2076:
2071:
2066:
2061:
2056:
2051:
2046:
2040:
2038:
2032:
2031:
2029:
2028:
2023:
2018:
2013:
2008:
2003:
1998:
1993:
1991:Risk dominance
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1943:
1938:
1933:
1928:
1923:
1918:
1913:
1908:
1902:
1900:
1890:
1889:
1887:
1886:
1881:
1876:
1871:
1866:
1861:
1856:
1851:
1846:
1841:
1836:
1834:Graphical game
1831:
1826:
1821:
1816:
1811:
1806:
1801:
1795:
1793:
1789:
1788:
1780:
1779:
1772:
1765:
1757:
1749:
1748:
1738:
1704:
1691:
1665:
1638:
1637:
1635:
1632:
1631:
1630:
1616:
1608:
1605:
1588:
1585:
1565:
1561:
1533:
1530:
1504:
1501:
1487:
1486:
1471:
1468:
1442:
1439:
1425:
1410:
1407:
1401:
1396:
1392:
1368:
1365:
1351:
1336:
1333:
1327:
1322:
1318:
1294:
1291:
1261:
1258:
1232:
1229:
1203:
1200:
1170:
1167:
1164:
1161:
1158:
1136:
1131:
1127:
1104:
1099:
1095:
1074:
1040:
1036:
1015:
994:
993:
987:
986:
983:
980:
976:
975:
972:
969:
965:
964:
961:
958:
952:
949:
945:repeated games
936:
933:
924:
921:
903:
902:
895:
883:
863:
860:
857:
854:
851:
831:
811:
800:
788:
768:
765:
762:
742:
720:
719:
708:
700:
697:
689:
688:
664:
615:
612:
609:
606:
586:
583:
580:
577:
555:
554:
543:
540:
513:Equilibrium 2.
497:
493:
489:
486:
483:
456:
453:
450:
427:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
351:
350:
339:
336:
333:
330:
303:Equilibrium 1.
283:
280:
269:
268:
267:
266:
263:
257:
254:
251:
239:
236:
233:
213:
197:
194:
192:
189:
185:
184:
170:
160:
159:
144:
84:
83:
81:signaling game
78:
74:
73:
71:Bayesian games
67:
63:
62:
59:
55:
54:
50:
49:
44:
40:
39:
35:
34:
24:
23:
15:
9:
6:
4:
3:
2:
2859:
2848:
2845:
2843:
2840:
2839:
2837:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2768:
2766:
2764:Miscellaneous
2762:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2727:
2725:
2721:
2715:
2712:
2710:
2707:
2705:
2702:
2700:
2699:Samuel Bowles
2697:
2695:
2694:Roger Myerson
2692:
2690:
2687:
2685:
2684:Robert Aumann
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2639:Lloyd Shapley
2637:
2635:
2632:
2630:
2627:
2625:
2624:Kenneth Arrow
2622:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2604:John Harsanyi
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2574:Herbert Simon
2572:
2570:
2567:
2565:
2562:
2560:
2557:
2555:
2552:
2550:
2547:
2545:
2542:
2540:
2537:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2511:
2509:
2503:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2453:
2451:
2447:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2411:
2408:
2406:
2403:
2401:
2398:
2396:
2393:
2391:
2388:
2386:
2383:
2381:
2378:
2376:
2375:Fair division
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2350:Dictator game
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2237:
2235:
2233:
2229:
2223:
2222:Zero-sum game
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2192:Repeated game
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2171:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2141:
2139:
2137:
2131:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2109:Pure strategy
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2069:De-escalation
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2045:
2042:
2041:
2039:
2037:
2033:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2011:Shapley value
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1917:
1914:
1912:
1909:
1907:
1904:
1903:
1901:
1899:
1895:
1891:
1885:
1882:
1880:
1879:Succinct game
1877:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1837:
1835:
1832:
1830:
1827:
1825:
1822:
1820:
1817:
1815:
1812:
1810:
1807:
1805:
1802:
1800:
1797:
1796:
1794:
1790:
1786:
1778:
1773:
1771:
1766:
1764:
1759:
1758:
1755:
1746:
1745:Book preview.
1741:
1739:9780262061414
1735:
1731:
1727:
1726:
1721:
1717:
1711:
1709:
1701:
1695:
1676:
1669:
1650:
1643:
1639:
1628:
1624:
1620:
1617:
1614:
1611:
1610:
1604:
1602:
1598:
1594:
1584:
1581:
1563:
1559:
1550:
1528:
1499:
1466:
1437:
1426:
1405:
1399:
1394:
1390:
1363:
1352:
1331:
1325:
1320:
1316:
1289:
1278:
1277:
1276:
1256:
1227:
1198:
1186:
1182:
1165:
1162:
1159:
1134:
1129:
1125:
1102:
1097:
1093:
1072:
1062:
1060:
1059:Bayesian game
1056:
1038:
1034:
1013:
1005:
1001:
992:
988:
984:
981:
978:
977:
973:
970:
967:
966:
962:
959:
957:
956:
948:
946:
942:
932:
930:
923:More examples
920:
916:
913:
909:
900:
896:
881:
858:
855:
852:
829:
809:
801:
786:
766:
763:
760:
740:
732:
731:
730:
727:
725:
717:
713:
709:
706:
705:
704:
696:
694:
686:
682:
678:
673:
670:if a friend,
669:
665:
662:
658:
654:
649:
646:if a friend,
645:
641:
640:
639:
637:
632:
630:
613:
610:
607:
604:
584:
581:
578:
575:
566:
564:
560:
541:
538:
530:
526:
522:
518:
514:
511:
510:
509:
495:
491:
487:
484:
481:
472:
470:
454:
451:
448:
441:
425:
422:
419:
416:
413:
410:
404:
401:
392:
389:
386:
380:
374:
368:
360:
356:
337:
334:
331:
328:
320:
316:
312:
311:Do not accept
308:
304:
301:
300:
299:
297:
281:
278:
264:
261:
260:
258:
255:
252:
237:
234:
231:
211:
203:
202:
201:
188:
182:
178:
174:
171:
168:
165:
164:
163:
157:
153:
149:
145:
142:
138:
134:
133:
132:
130:
126:
121:
119:
115:
111:
107:
103:
102:dynamic games
99:
95:
91:
82:
79:
75:
72:
68:
64:
61:Cho and Kreps
60:
56:
51:
48:
45:
41:
36:
33:
29:
25:
20:
2669:Peyton Young
2664:Paul Milgrom
2579:Hervé Moulin
2519:Amos Tversky
2461:Folk theorem
2172:-player game
2169:
2089:Grim trigger
1970:
1724:
1720:Tirole, Jean
1694:
1682:. Retrieved
1668:
1656:. Retrieved
1647:James Peck.
1642:
1597:jump bidding
1590:
1587:Jump-bidding
1579:
1548:
1488:
1187:
1183:
1063:
1054:
997:
990:
938:
926:
917:
904:
898:
728:
721:
715:
711:
702:
692:
690:
684:
680:
676:
671:
667:
660:
656:
652:
647:
643:
633:
567:
562:
558:
556:
528:
524:
520:
516:
512:
473:
468:
439:
358:
354:
352:
318:
314:
310:
306:
302:
270:
199:
186:
180:
172:
166:
161:
155:
147:
136:
128:
124:
122:
93:
87:
53:Significance
38:Relationship
2786:Coopetition
2589:Jean Tirole
2584:John Conway
2564:Eric Maskin
2360:Blotto game
2345:Pirate game
2154:Global game
2124:Tit for tat
2054:Bid shading
2044:Appeasement
1894:Equilibrium
1874:Solved game
1809:Determinacy
1792:Definitions
1785:game theory
1725:Game Theory
1684:2 September
1004:public good
971:1-C1, 1-C2
699:Gift game 2
685:Do not give
668:Do not give
648:Do not give
196:Gift game 1
177:Bayes' rule
173:Consistency
90:game theory
58:Proposed by
32:game theory
2836:Categories
2430:Trust game
2415:Kuhn poker
2079:Escalation
2074:Deterrence
2064:Cheap talk
2036:Strategies
1854:Preference
1783:Topics of
1658:6 December
1634:References
125:strategies
2614:John Nash
2320:Stag hunt
2059:Collusion
1730:MIT Press
1564:∗
1532:^
1503:^
1470:^
1441:^
1409:^
1367:^
1335:^
1293:^
1260:^
1231:^
1202:^
1135:∗
1103:∗
764:−
579:≥
485:≥
449:≤
420:−
402:−
390:−
332:≤
235:−
43:Subset of
2755:Lazy SMP
2449:Theorems
2400:Deadlock
2255:Checkers
2136:of games
1898:concepts
1722:(1991).
1607:See also
982:1, 1-C2
974:1-C1, 1
912:subgames
822:, where
563:Not give
515:Sender:
307:Not give
305:Sender:
137:strategy
69:Dynamic
66:Used for
2507:figures
2290:Chicken
2144:Auction
2134:Classes
677:Reject.
521:Accept,
129:beliefs
77:Example
1736:
979:Don't
968:Build
963:Don't
960:Build
681:Reject
657:Accept
653:Accept
148:belief
2245:Chess
2232:Games
1678:(PDF)
1652:(PDF)
1551:than
1549:lower
156:types
104:with
1921:Core
1734:ISBN
1686:2016
1660:2021
1621:and
1580:less
1400:<
1326:>
985:0,0
672:Give
661:Give
644:Give
527:and
517:Give
355:Give
317:and
146:The
135:The
127:and
114:dove
110:hawk
92:, a
2505:Key
899:any
611:.99
474:If
88:In
30:in
2838::
2240:Go
1732:.
1718:;
1707:^
1603:.
1395:11
1321:00
1061:.
939:A
716:-1
693:no
582:.5
452:.5
335:.5
131::
2170:n
1776:e
1769:t
1762:v
1742:.
1688:.
1662:.
1629:.
1560:c
1529:c
1500:c
1467:c
1438:c
1424:.
1406:c
1391:c
1364:c
1350:.
1332:c
1317:c
1290:c
1257:c
1228:c
1199:c
1169:]
1166:2
1163:,
1160:0
1157:[
1130:i
1126:C
1098:i
1094:C
1073:i
1039:i
1035:C
1014:i
882:q
862:]
859:1
856:,
853:0
850:[
830:q
810:q
787:p
767:p
761:1
741:p
712:0
614:,
608:=
605:p
585:.
576:p
542:.
539:x
496:2
492:/
488:1
482:p
455:.
426:,
423:1
417:x
414:2
411:=
408:)
405:1
399:(
396:)
393:x
387:1
384:(
381:+
378:)
375:1
372:(
369:x
338:.
329:x
282:,
279:p
238:p
232:1
212:p
143:.
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