395:, and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors. A second interpretation imagines the game players standing for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents.
98:
A can assume competitor B enters the market. From there, Competitor A compares the payoffs they receive by entering and not entering. The next step is to assume
Competitor B does not enter and then consider which payoff is better based on if Competitor A chooses to enter or not enter. This technique can identify dominant strategies where a player can identify an action that they can take no matter what the competitor does to try to maximize the payoff.
262:
illustrates this situation, a simplified form of the game studied by
Chiappori, Levitt, and Groseclose (2002). It assumes that if the goalie guesses correctly, the kick is blocked, which is set to the base payoff of 0 for both players. If the goalie guesses wrong, the kick is more likely to go in if it is to the left (payoffs of +2 for the kicker and -2 for the goalie) than if it is to the right (the lower payoff of +1 to kicker and -1 to goalie).
435:, a behavioral outlook on traditional game-theoretic hypotheses. The result establishes that in any finite extensive-form game with perfect recall, for any player and any mixed strategy, there exists a behavior strategy that, against all profiles of strategies (of other players), induces the same distribution over terminal nodes as the mixed strategy does. The converse is also true.
327:
is g(1) + (1-g)(0). Equating these yields g= 2/3. Similarly, the goalie is willing to randomize only if the kicker chooses mixed strategy probability k such that Lean Left's payoff of k(0) + (1-k)(-1) equals Lean Right's payoff of k(-2) + (1-k)(0), so k = 1/3. Thus, the mixed-strategy equilibrium is (Prob(Kick Left) = 1/3, Prob(Lean Left) = 2/3).
451:
opponent plays, the outcome distribution of the mixed and behavioral strategy must be equal. This equivalence can be described by the following formula: (Q^(U(i), S(-i)))(z) = (Q^(β(i), S(-i)))(z), where U(i) describes Player i's mixed strategy, β(i) describes Player i's behavioral strategy, and S(-i) is the opponent's strategy.
326:
The kicker's mixed-strategy equilibrium is found from the fact that they will deviate from randomizing unless their payoffs from Left Kick and Right Kick are exactly equal. If the goalie leans left with probability g, the kicker's expected payoff from Kick Left is g(0) + (1-g)(2), and from Kick Right
173:
In applied game theory, the definition of the strategy sets is an important part of the art of making a game simultaneously solvable and meaningful. The game theorist can use knowledge of the overall problem, that is the friction between two or more players, to limit the strategy spaces, and ease the
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Without perfect information (i.e. imperfect information), players make a choice at each decision node without knowledge of the decisions that have preceded it. Therefore, a player’s mixed strategy can produce outcomes that their behavioral strategy cannot, and vice versa. This is demonstrated in the
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Perfect recall is defined as the ability of every player in game to remember and recall all past actions within the game. Perfect recall is required for equivalence as, in finite games with imperfect recall, there will be existing mixed strategies of Player I in which there is no equivalent behavior
426:
assigns at each information set a probability distribution over the set of possible actions. While the two concepts are very closely related in the context of normal form games, they have very different implications for extensive form games. Roughly, a mixed strategy randomly chooses a deterministic
97:
It is helpful to think about a "strategy" as a list of directions, and a "move" as a single turn on the list of directions itself. This strategy is based on the payoff or outcome of each action. The goal of each agent is to consider their payoff based on a competitors action. For example, competitor
473:
game. With perfect recall and information, the driver has a single pure strategy, which is , as the driver is aware of what intersection (or decision node) they are at when they arrive to it. On the other hand, looking at the planning-optimal stage only, the maximum payoff is achieved by continuing
261:
In a soccer penalty kick, the kicker must choose whether to kick to the right or left side of the goal, and simultaneously the goalie must decide which way to block it. Also, the kicker has a direction they are best at shooting, which is left if they are right-footed. The matrix for the soccer game
334:
Chiappori, Levitt, and
Groseclose try to measure how important it is for the kicker to kick to their favored side, add center kicks, etc., and look at how professional players actually behave. They find that they do randomize, and that kickers kick to their favored side 45% of the time and goalies
221:
to each pure strategy. When enlisting mixed strategy, it is often because the game does not allow for a rational description in specifying a pure strategy for the game. This allows for a player to randomly select a pure strategy. (See the following section for an illustration.) Since probabilities
205:
provides a complete definition of how a player will play a game. Pure strategy can be thought about as a singular concrete plan subject to the observations they make during the course of the game of play. In particular, it determines the move a player will make for any situation they could face. A
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Outcome equivalence combines the mixed and behavioral strategy of Player i in relation to the pure strategy of Player i’s opponent. Outcome equivalence is defined as the situation in which, for any mixed and behavioral strategy that Player i takes, in response to any pure strategy that Player I’s
464:
game formulated by
Piccione and Rubinstein. In short, this game is based on the decision-making of a driver with imperfect recall, who needs to take the second exit off the highway to reach home but does not remember which intersection they are at when they reach it. Figure describes this game.
384:
During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic", since they are weak Nash equilibria, and a player is indifferent about whether to follow their equilibrium strategy probability or deviate to some other probability. Game theorist
376:). Further, games can have both pure strategy and mixed strategy equilibria. An easy example is the pure coordination game, where in addition to the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both players play either strategy with probability 1/2.
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In equilibrium, the kicker kicks to their best side only 1/3 of the time. That is because the goalie is guarding that side more. Also, in equilibrium, the kicker is indifferent which way they kick, but for it to be an equilibrium they must choose exactly 1/3 probability.
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are equilibria where at least one player is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies, see
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are continuous, there are infinitely many mixed strategies available to a player. Since probabilities are being assigned to strategies for a specific player when discussing the payoffs of certain scenarios the payoff must be referred to as "expected payoff".
322:
This game has no pure-strategy equilibrium, because one player or the other would deviate from any profile of strategies—for example, (Left, Left) is not an equilibrium because the Kicker would deviate to Right and increase his payoff from 0 to 1.
61:
on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship.
73:
for playing a game, telling a player what to do for every possible situation. A player's strategy determines the action the player will take at any stage of the game. However, the idea of a strategy is often confused or
165:, or games in which players have incomplete information about one another, the strategy set is similar to that in a dynamic game. It consists of rules for what action to take for any possible private information.
94:" is an example of a valid strategy, and as a result every move can also be considered to be a strategy. Other authors treat strategies as being a different type of thing from actions, and therefore distinct.
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at both intersections, maximized at p=2/3 (reference). This simple one player game demonstrates the importance of perfect recall for outcome equivalence, and its impact on normal and extended form games.
132:
comprises a single move by each player—and each player's move is made without knowledge of the other's, not as a response—so each player has the finite strategy set {rock paper scissors}.
181:. Including all such strategies makes for a very large strategy space and a somewhat difficult problem. A game theorist might instead believe they can limit the strategy set to: {Reject any offer ≤
410:
the other was equally likely to play each strategy. This interpretation weakens the descriptive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to
109:) is a set of strategies for all players which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.
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Of course, one can regard a pure strategy as a degenerate case of a mixed strategy, in which that particular pure strategy is selected with probability
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is a mixed strategy in which the player assigns a strictly positive probability to every pure strategy. (Totally mixed strategies are important for
431:, while a behavior strategy can be seen as a stochastic path. The relationship between mixed and behavior strategies is the subject of
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play a pure strategy of Rock in each play of the game, even though over time the probabilities are those of the mixed strategy.
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lean to that side 57% of the time. Their article is well-known as an example of how people in real life use mixed strategies.
158:, the strategy set for the second player would consist of every possible rule for which offers to accept and which to reject.
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A famous example of why perfect recall is required for the equivalence is given by
Piccione and Rubinstein (1997) with their
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has a bounded continuum of strategies in the strategy set {Cut anywhere between zero percent and 100 percent of the cake}.
146:, games that are played over a series of time, the strategy set consists of the possible rules a player could give to a
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675:(1973). "Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points".
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562:"Testing Mixed-Strategy Equilibria when Players Are Heterogeneous: The Case of Penalty Kicks in Soccer"
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describes alternative ways of understanding the concept. The first, due to
Harsanyi (1973), is called
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strategy set if they have a number of discrete strategies available to them. For instance, a game of
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is any one of the options which a player can choose in a setting where the optimal outcome depends
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Later, Aumann and
Brandenburger (1995), re-interpreted Nash equilibrium as an equilibrium in
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For instance, strictly speaking in the
Ultimatum game a player can have strategies such as:
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Reject offers of ($ 1, $ 3, $ 5, ..., $ 19), accept offers of ($ 0, $ 2, $ 4, ..., $ 20)
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with that of a move or action, because of the correspondence between moves and
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Complete plan on how a game player will behave in every possible game situation
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364:. However, many games do have pure strategy Nash equilibria (e.g. the
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for every finite game. One can divide Nash equilibria into two types.
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are Nash equilibria where all players are playing pure strategies.
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Kak, Subhash (2017). "The Absent-Minded Driver
Problem Redux".
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635:(1991). "Comments on the interpretation of Game Theory".
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defines what strategies are available for them to play.
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is the set of pure strategies available to that player.
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A strategy set is infinite otherwise. For instance the
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Chiappori, P. -A.; Levitt, S.; Groseclose, T. (2002).
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717:(1995). "Epistemic Conditions for Nash Equilibrium".
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316: Payoff for the Soccer Game (Kicker, Goalie)
406:an equilibrium in beliefs would have each player
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154:on how to play the game. For instance, in the
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618:. Oxford: Basil Blackwell. pp. 909–924.
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609:"What is Game Theory Trying to accomplish?"
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229:and every other strategy with probability
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460:strategy. This is fully described in the
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402:, rather than actions. For instance, in
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614:. In Arrow, K.; Honkapohja, S. (eds.).
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69:is typically used to mean a complete
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537:Game Theory. In: Palgrave Macmillan
380:Interpretations of mixed strategies
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890:First-player and second-player win
246:trembling hand perfect equilibrium
30:For other uses of "Strategy", see
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516:Game Theory: Lecture 1 Transcript
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997:Coalition-proof Nash equilibrium
193:in ($ 0, $ 1, $ 2, ..., $ 20)}.
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1007:Evolutionarily stable strategy
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762:Shimoji, Makoto (2012-05-01).
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539:. London: Palgrave Macmillan.
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494:Evolutionarily stable strategy
357:Mixed strategy Nash equilibria
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935:Simultaneous action selection
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353:Pure strategy Nash equilibria
1867:List of games in game theory
1047:Quantal response equilibrium
1037:Perfect Bayesian equilibrium
972:Bayes correlated equilibrium
518:ECON 159, 5 September 2007,
455:Strategy with perfect recall
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1336:Optional prisoner's dilemma
1067:Self-confirming equilibrium
768:Games and Economic Behavior
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1801:Principal variation search
1517:Aumann's agreement theorem
1180:Strategy-stealing argument
1092:Trembling hand equilibrium
1022:Markov perfect equilibrium
1017:Mertens-stable equilibrium
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591:10.1257/00028280260344678
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197:Pure and mixed strategies
32:Strategy (disambiguation)
1852:Evolutionary game theory
1585:Antoine Augustin Cournot
1471:Guess 2/3 of the average
1268:Strictly determined game
1062:Satisfaction equilibrium
880:Escalation of commitment
569:American Economic Review
347:proved that there is an
185:, accept any offer >
1857:Glossary of game theory
1456:Stackelberg competition
1082:Strong Nash equilibrium
169:Choosing a strategy set
1903:Strategy (game theory)
1882:Tragedy of the commons
1862:List of game theorists
1842:Confrontation analysis
1552:Sprague–Grundy theorem
1072:Sequential equilibrium
992:Correlated equilibrium
616:Frontiers of Economics
303: 0, 0
242:equilibrium refinement
238:totally mixed strategy
217:is an assignment of a
1655:Jean-François Mertens
343:In his famous paper,
289: 0, 0
57:on their own actions
1784:Search optimizations
1660:Jennifer Tour Chayes
1547:Revelation principle
1542:Purification theorem
1481:Nash bargaining game
1446:Bertrand competition
1431:El Farol Bar problem
1396:Electronic mail game
1361:Lewis signaling game
905:Hierarchy of beliefs
489:Haven (graph theory)
471:Absent-minded Driver
462:Absent-Minded Driver
440:Absent-Minded Driver
107:strategy combination
105:(sometimes called a
90:, "always play move
1832:Bounded rationality
1451:Cournot competition
1401:Rock paper scissors
1376:Battle of the sexes
1366:Volunteer's dilemma
1238:Perfect information
1165:Dominant strategies
1002:Epsilon-equilibrium
885:Extensive-form game
715:Brandenburger, Adam
677:Int. J. Game Theory
446:Outcome equivalence
404:rock paper scissors
130:rock paper scissors
1811:Paranoid algorithm
1791:Alpha–beta pruning
1670:John Maynard Smith
1501:Rendezvous problem
1341:Traveler's dilemma
1331:Gift-exchange game
1326:Prisoner's dilemma
1243:Large Poisson game
1210:Bargaining problem
1115:Backward induction
1087:Subgame perfection
1042:Proper equilibrium
689:10.1007/BF01737554
370:Prisoner's dilemma
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1796:Aspiration window
1765:Suzanne Scotchmer
1720:Oskar Morgenstern
1615:Donald B. Gillies
1557:Zermelo's theorem
1486:Induction puzzles
1441:Fair cake-cutting
1416:Public goods game
1346:Coordination game
1220:Intransitive game
1150:Forward induction
1032:Pareto efficiency
1012:Gibbs equilibrium
982:Berge equilibrium
930:Simultaneous game
546:978-1-349-95121-5
535:(22 March 2017).
520:Open Yale Courses
427:path through the
424:behavior strategy
418:Behavior strategy
366:Coordination game
320:
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137:cake cutting game
16:(Redirected from
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1877:Topological game
1872:No-win situation
1770:Thomas Schelling
1750:Robert B. Wilson
1710:Merrill M. Flood
1680:John von Neumann
1590:Ariel Rubinstein
1575:Albert W. Tucker
1426:War of attrition
1386:Matching pennies
1027:Nash equilibrium
950:Mechanism design
915:Normal-form game
870:Cooperative game
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902:
900:Graphical game
897:
892:
887:
882:
877:
872:
867:
861:
859:
855:
854:
846:
845:
838:
831:
823:
815:
814:
793:
774:(1): 441–447.
754:
711:Aumann, Robert
702:
673:Harsanyi, John
664:
645:(4): 909–924.
633:Rubinstein, A.
621:
596:
552:
545:
524:
504:
503:
501:
498:
497:
496:
491:
486:
479:
476:
456:
453:
447:
444:
433:Kuhn's theorem
419:
416:
381:
378:
340:
337:
318:
317:
313:
312:
309:
308:
305:
304:
301:
298:
294:
293:
290:
287:
284:
280:
279:
276:
272:
271:
268:
258:
255:
253:
252:Mixed strategy
250:
215:mixed strategy
198:
195:
170:
167:
156:ultimatum game
114:
111:
26:
9:
6:
4:
3:
2:
1915:
1904:
1901:
1900:
1898:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1829:
1827:
1825:Miscellaneous
1823:
1817:
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1788:
1786:
1782:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1760:Samuel Bowles
1758:
1756:
1755:Roger Myerson
1753:
1751:
1748:
1746:
1745:Robert Aumann
1743:
1741:
1738:
1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1703:
1701:
1700:Lloyd Shapley
1698:
1696:
1693:
1691:
1688:
1686:
1685:Kenneth Arrow
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1665:John Harsanyi
1663:
1661:
1658:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1635:Herbert Simon
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1572:
1570:
1564:
1558:
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1514:
1512:
1508:
1502:
1499:
1497:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1477:
1474:
1472:
1469:
1467:
1464:
1462:
1459:
1457:
1454:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1436:Fair division
1434:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1411:Dictator game
1409:
1407:
1404:
1402:
1399:
1397:
1394:
1392:
1389:
1387:
1384:
1382:
1379:
1377:
1374:
1372:
1369:
1367:
1364:
1362:
1359:
1357:
1354:
1352:
1349:
1347:
1344:
1342:
1339:
1337:
1334:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1309:
1307:
1304:
1302:
1299:
1298:
1296:
1294:
1290:
1284:
1283:Zero-sum game
1281:
1279:
1276:
1274:
1271:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1253:Repeated game
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1232:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1203:
1202:
1200:
1198:
1192:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1170:Pure strategy
1168:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1141:
1138:
1136:
1135:De-escalation
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1107:
1105:
1103:
1099:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1077:Shapley value
1075:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
969:
967:
965:
961:
957:
951:
948:
946:
945:Succinct game
943:
941:
938:
936:
933:
931:
928:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
862:
860:
856:
852:
844:
839:
837:
832:
830:
825:
824:
821:
809:
804:
797:
789:
785:
781:
777:
773:
769:
765:
758:
750:
746:
742:
738:
733:
728:
724:
720:
716:
712:
706:
698:
694:
690:
686:
682:
678:
674:
668:
660:
656:
652:
648:
644:
640:
639:
634:
628:
626:
617:
610:
606:
600:
592:
588:
583:
578:
574:
570:
563:
556:
548:
542:
538:
534:
528:
521:
517:
514:
509:
505:
495:
492:
490:
487:
485:
482:
481:
475:
472:
466:
463:
452:
443:
441:
436:
434:
430:
425:
415:
413:
409:
405:
401:
396:
394:
393:
388:
377:
375:
371:
367:
363:
358:
354:
350:
346:
336:
332:
328:
324:
314:
310:
306:
302:
299:
295:
291:
288:
281:
277:
274:
273:
266:
263:
249:
247:
243:
239:
234:
232:
228:
223:
220:
216:
211:
209:
204:
203:pure strategy
194:
192:
188:
184:
180:
175:
166:
164:
163:Bayesian game
159:
157:
153:
149:
145:
140:
138:
133:
131:
127:
122:
120:
110:
108:
104:
99:
95:
93:
89:
85:
81:
77:
72:
68:
63:
60:
56:
52:
48:
44:
40:
33:
19:
1730:Peyton Young
1725:Paul Milgrom
1640:Hervé Moulin
1580:Amos Tversky
1522:Folk theorem
1233:-player game
1230:
1169:
1155:Grim trigger
1101:
796:
771:
767:
757:
722:
719:Econometrica
718:
705:
680:
676:
667:
642:
638:Econometrica
636:
615:
599:
572:
568:
555:
536:
527:
515:
508:
470:
467:
461:
458:
449:
439:
437:
423:
421:
411:
407:
399:
397:
392:purification
390:
383:
356:
352:
342:
339:Significance
333:
329:
325:
321:
260:
257:Illustration
237:
235:
230:
226:
224:
214:
212:
208:strategy set
207:
202:
200:
190:
186:
182:
178:
176:
172:
160:
144:dynamic game
141:
134:
125:
123:
119:strategy set
118:
116:
113:Strategy set
106:
102:
100:
96:
91:
87:
66:
64:
58:
54:
50:
46:
42:
36:
1847:Coopetition
1650:Jean Tirole
1645:John Conway
1625:Eric Maskin
1421:Blotto game
1406:Pirate game
1215:Global game
1185:Tit for tat
1120:Bid shading
1110:Appeasement
960:Equilibrium
940:Solved game
875:Determinacy
858:Definitions
851:game theory
575:(4): 1138.
349:equilibrium
278:Lean Right
219:probability
117:A player's
39:game theory
1491:Trust game
1476:Kuhn poker
1145:Escalation
1140:Deterrence
1130:Cheap talk
1102:Strategies
920:Preference
849:Topics of
808:1702.05778
605:Aumann, R.
533:Aumann, R.
500:References
297:Kick Right
174:solution.
84:most games
1675:John Nash
1381:Stag hunt
1125:Collusion
788:0899-8256
727:CiteSeerX
697:154484458
577:CiteSeerX
513:Ben Polak
429:game tree
408:believing
374:Stag hunt
286:Kick Left
275:Lean Left
206:player's
76:conflated
71:algorithm
65:The term
1897:Category
1816:Lazy SMP
1510:Theorems
1461:Deadlock
1316:Checkers
1197:of games
964:concepts
683:: 1–23.
607:(1985).
478:See also
412:actually
244:such as
67:strategy
55:not only
1568:figures
1351:Chicken
1205:Auction
1195:Classes
749:2171725
659:2938166
400:beliefs
292:+2, -2
270:Goalie
786:
747:
729:
695:
657:
579:
543:
442:game.
372:, the
368:, the
300:+1, -1
283:Kicker
189:; for
126:finite
47:action
1306:Chess
1293:Games
803:arXiv
745:JSTOR
693:S2CID
655:JSTOR
612:(PDF)
565:(PDF)
161:In a
152:agent
148:robot
142:In a
49:, or
987:Core
784:ISSN
541:ISBN
51:play
43:move
41:, a
1566:Key
776:doi
737:doi
685:doi
647:doi
587:doi
248:.)
150:or
82:in
59:but
37:In
1899::
1301:Go
782:.
772:75
770:.
766:.
743:.
735:.
723:63
721:.
713:;
691:.
679:.
653:.
643:59
641:.
624:^
585:.
573:92
571:.
567:.
236:A
233:.
213:A
201:A
101:A
45:,
1231:n
842:e
835:t
828:v
811:.
805::
790:.
778::
751:.
739::
699:.
687::
681:2
661:.
649::
593:.
589::
549:.
522:.
231:0
227:1
191:x
187:x
183:x
92:X
88:X
34:.
20:)
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