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opposite options, then the culprit that defects is free and the culprit who stays quiet serves a long sentence. Ultimately, using backward induction, the last subgame in a finitely repeated
Prisoner's dilemma requires players to play the unique Nash equilibrium (both players defecting). Because of this, all games prior to the last subgame will also play the Nash equilibrium to maximize their single-period payoffs. If a stage-game in a finitely repeated game has multiple Nash equilibria, subgame perfect equilibria can be constructed to play non-stage-game Nash equilibrium actions, through a "carrot and stick" structure. One player can use the one stage-game Nash equilibrium to incentivize playing the non-Nash equilibrium action, while using a stage-game Nash equilibrium with lower payoff to the other player if they choose to defect.
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170:. One then supposes that the last actor will do these actions, and considers the second to last actions, again choosing those that maximize that actor's utility. This process continues until one reaches the first move of the game. The strategies which remain are the set of all subgame perfect equilibria for finite-horizon extensive games of perfect information. However, backward induction cannot be applied to games of
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374:" if one player has the option of ripping the steering wheel from their car they should always take it because it leads to a "sub game" in which their rational opponent is precluded from doing the same thing (and killing them both). The wheel-ripper will always win the game (making his opponent swerve away), and the opponent's threat to suicidally follow suit is not credible.
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has such an optimum strategy for all players. The problem of the relationship between subgame perfection and backward induction was settled by
Kaminski (2019), who proved that a generalized procedure of backward induction produces all subgame perfect equilibria in games that may have infinite length,
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game. The
Prisoner's dilemma gets its name from a situation that contains two guilty culprits. When they are interrogated, they have the option to stay quiet or defect. If both culprits stay quiet, they both serve a short sentence. If both defect, they both serve a moderate sentence. If they choose
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The interesting aspect of the word "credible" in the preceding paragraph is that taken as a whole (disregarding the irreversibility of reaching sub-games) strategies exist which are superior to subgame perfect strategies, but which are not credible in the sense that a threat to carry them out will
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For the entire game Nash equilibria (DA, Y) and (DB, Y) are not subgame perfect equilibria because the move of Player 2 does not constitute a Nash equilibrium. The Nash equilibrium (UA, X) is subgame perfect because it incorporates the subgame Nash equilibrium (A, X) as part of its strategy.
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Determining the subgame perfect equilibrium by using backward induction is shown below in Figure 1. Strategies for Player 1 are given by {Up, Uq, Dp, Dq}, whereas Player 2 has the strategies among {TL, TR, BL, BR}. There are 4 subgames in this example, with 3 proper subgames.
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An extensive-form game with incomplete information is presented below in Figure 2. Note that the node for Player 1 with actions A and B, and all succeeding actions is a subgame. Player 2's nodes are not a subgame as they are part of the same information set.
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For finitely repeated games, if a stage game has only one unique Nash equilibrium, the subgame perfect equilibrium is to play without considering past actions, treating the current subgame as a one-shot game. An example of this is a finitely repeated
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of the original game. Informally, this means that at any point in the game, the players' behavior from that point onward should represent a Nash equilibrium of the continuation game (i.e. of the subgame), no matter what happened before. Every
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The second normal-form game is the normal form representation of the subgame starting from Player 1's second node with actions A and B. For the second normal-form game, the Nash equilibrium of the subgame is (A, X).
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The first normal-form game is the normal form representation of the whole extensive-form game. Based on the provided information, (UA, X), (DA, Y), and (DB, Y) are all Nash equilibria for the entire game.
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proved that any game which can be broken into "sub-games" containing a sub-set of all the available choices in the main game will have a subgame perfect Nash
Equilibrium strategy (possibly as a
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To solve this game, first find the Nash
Equilibria by mutual best response of Subgame 1. Then use backwards induction and plug in (A,X) → (3,4) so that (3,4) become the payoffs for Subgame 2.
157:"equivalent to the assertion that each player is allowed by the rules of the game to remember everything he knew at previous moves and all of his choices at those moves"
166:. Here one first considers the last actions of the game and determines which actions the final mover should take in each possible circumstance to maximize his/her
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The set of subgame perfect equilibria for a given game is always a subset of the set of Nash equilibria for that game. In some cases the sets can be identical.
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Subgame for actions p and q: Player 1 will take action p with payoff (3, 3) to maximize Player 1's payoff, so the payoff for action L becomes (3,3).
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Subgame for actions T and B: Player 2 will take action T to maximize Player 2's payoff, so the payoff for action U becomes (1, 4).
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Player 1 chooses U rather than D because 3 > 2 for Player 1's payoff. The resulting equilibrium is (A, X) → (3,4).
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Subgame 1 is solved and (3,4) replaces all of
Subgame 1 and player one will choose U -> (3,4)Solution for Subgame 1
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infinite actions as each information set, and imperfect information if a condition of final support is satisfied.
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Subgame for actions L and R: Player 2 will take action L for 3 > 2, so the payoff for action D becomes (3, 3).
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The dashed line indicates that player 2 does not know whether player 1 will play A or B in a simultaneous game.
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harm the player making the threat and prevent that combination of strategies. For instance in the game of "
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Thus, the subgame perfect equilibrium through backwards induction is (UA, X) with the payoff (3, 4).
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provides an intuitive example of a game with fewer subgame perfect equilibria than Nash equilibria.
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Zeitschrift fĂĽr die gesamte
Staatswissenschaft/Journal of Institutional and Theoretical Economics
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Selten, R. (1965). Spieltheoretische behandlung eines oligopolmodells mit nachfrageträgheit.
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giving non-deterministic sub-game decisions). Subgame perfection is only used with games of
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Using the backward induction, the players will take the following actions for each subgame:
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A common method for determining subgame perfect equilibria in the case of a finite game is
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Java applet to find a subgame perfect Nash
Equilibrium solution for an extensive form game
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Java applet to find a subgame perfect Nash
Equilibrium solution for an extensive form game
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Subgame for actions U and D: Player 1 will take action D to maximize Player 1's payoff.
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is a subgame perfect equilibrium if it represents a Nash equilibrium of every
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Thus, the subgame perfect equilibrium is {Dp, TL} with the payoff (3, 3).
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has a subgame perfect equilibrium. Perfect recall is a term introduced by
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357:. One game in which the backward induction solution is well known is
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One game in which the backward induction solution is well known is
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Generalized
Backward Induction: Justification for a Folk Algorithm
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Kuhn, Harold William; Tucker, Albert William (2 March 2016).
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The subgame-perfect Nash equilibrium is normally deduced by "
536:. Massachusetts Institute of Technology: MIT OpenCourseWare
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Example of Extensive Form Games with imperfect information
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A subgame perfect equilibrium necessarily satisfies the
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Contributions to the Theory of Games (AM-28), Volume II
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178:because this entails cutting through non-singleton
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483:Strategy : an introduction to game theory
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587:: CS1 maint: location missing publisher (
534:14.12 Economic Applications of Game Theory
513:: CS1 maint: location missing publisher (
555:Takako, Fujiwara-Greve (27 June 2015).
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334:. Subgame perfection can be used with
286:Solution of Subgame Perfect Equilibrium
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353:(because it is not optimal) from that
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708:First-player and second-player win
308:Finding subgame-perfect equilibria
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414:Bellman's principle of optimality
1721:Game theory equilibrium concepts
815:Coalition-proof Nash equilibrium
120:subgame perfect Nash equilibrium
825:Evolutionarily stable strategy
457:. Princeton University Press.
438:An Introduction to Game Theory
68:Evolutionarily stable strategy
1:
753:Simultaneous action selection
1685:List of games in game theory
865:Quantal response equilibrium
855:Perfect Bayesian equilibrium
790:Bayes correlated equilibrium
609:, (H. 2), 301-324, 667-689.
485:(Third ed.). New York.
481:Joel., Watson (2013-05-09).
187:one-shot deviation principle
1154:Optional prisoner's dilemma
885:Self-confirming equilibrium
557:Non-cooperative game theory
116:subgame perfect equilibrium
33:Subgame Perfect Equilibrium
18:Subgame-perfect equilibrium
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1619:Principal variation search
1335:Aumann's agreement theorem
998:Strategy-stealing argument
910:Trembling hand equilibrium
840:Markov perfect equilibrium
835:Mertens-stable equilibrium
440:. Oxford University Press.
1655:Combinatorial game theory
1314:Princess and monster game
870:Quasi-perfect equilibrium
795:Bayesian Nash equilibrium
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1670:Evolutionary game theory
1403:Antoine Augustin Cournot
1289:Guess 2/3 of the average
1086:Strictly determined game
880:Satisfaction equilibrium
698:Escalation of commitment
528:Yildiz, Muhamet (2012).
1675:Glossary of game theory
1274:Stackelberg competition
900:Strong Nash equilibrium
436:Osborne, M. J. (2004).
394:Glossary of game theory
1700:Tragedy of the commons
1680:List of game theorists
1660:Confrontation analysis
1370:Sprague–Grundy theorem
890:Sequential equilibrium
810:Correlated equilibrium
338:games of complete but
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176:incomplete information
1473:Jean-François Mertens
633:. Games 2019, 10, 34.
389:Dynamic inconsistency
361:, but in theory even
340:imperfect information
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145:finite extensive game
1602:Search optimizations
1478:Jennifer Tour Chayes
1365:Revelation principle
1360:Purification theorem
1299:Nash bargaining game
1264:Bertrand competition
1249:El Farol Bar problem
1214:Electronic mail game
1179:Lewis signaling game
723:Hierarchy of beliefs
626:from gametheory.net.
620:from gametheory.net.
332:complete information
93:Extensive form games
1650:Bounded rationality
1269:Cournot competition
1219:Rock paper scissors
1194:Battle of the sexes
1184:Volunteer's dilemma
1056:Perfect information
983:Dominant strategies
820:Epsilon-equilibrium
703:Extensive-form game
530:"12 Repeated Games"
404:Retrograde analysis
27:Game theory concept
1629:Paranoid algorithm
1609:Alpha–beta pruning
1488:John Maynard Smith
1319:Rendezvous problem
1159:Traveler's dilemma
1149:Gift-exchange game
1144:Prisoner's dilemma
1061:Large Poisson game
1028:Bargaining problem
933:Backward induction
905:Subgame perfection
860:Proper equilibrium
347:backward induction
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301:Prisoner's dilemma
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164:backward induction
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1614:Aspiration window
1583:Suzanne Scotchmer
1538:Oskar Morgenstern
1433:Donald B. Gillies
1375:Zermelo's theorem
1304:Induction puzzles
1259:Fair cake-cutting
1234:Public goods game
1164:Coordination game
1038:Intransitive game
968:Forward induction
850:Pareto efficiency
830:Gibbs equilibrium
800:Berge equilibrium
748:Simultaneous game
464:978-1-4008-8197-0
108:
107:
16:(Redirected from
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1695:Topological game
1690:No-win situation
1588:Thomas Schelling
1568:Robert B. Wilson
1528:Merrill M. Flood
1498:John von Neumann
1408:Ariel Rubinstein
1393:Albert W. Tucker
1244:War of attrition
1204:Matching pennies
845:Nash equilibrium
768:Mechanism design
733:Normal-form game
688:Cooperative game
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1345:Minimax theorem
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1043:Mean-field game
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1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
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1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1116:
1114:
1112:
1108:
1102:
1101:Zero-sum game
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1071:Repeated game
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1052:
1050:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1020:
1018:
1016:
1010:
1004:
1001:
999:
996:
994:
991:
989:
988:Pure strategy
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
953:De-escalation
951:
949:
946:
944:
941:
939:
936:
934:
931:
929:
926:
925:
923:
921:
917:
911:
908:
906:
903:
901:
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895:Shapley value
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883:
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871:
868:
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861:
858:
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848:
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838:
836:
833:
831:
828:
826:
823:
821:
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813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
787:
785:
783:
779:
775:
769:
766:
764:
763:Succinct game
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
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691:
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662:
657:
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643:
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632:
628:
625:
622:
619:
616:
614:
611:
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603:
599:
590:
584:
576:
572:
568:
566:9784431556442
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492:9780393918380
488:
484:
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471:
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447:
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419:
415:
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410:
407:
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385:
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367:
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348:
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183:
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177:
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146:
141:
137:
133:
132:dynamic games
129:
125:
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117:
113:
104:
101:
97:
94:
91:
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81:
77:
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69:
66:
62:
59:
56:
52:
47:
44:
40:
36:
31:
19:
1548:Peyton Young
1543:Paul Milgrom
1458:Hervé Moulin
1398:Amos Tversky
1340:Folk theorem
1051:-player game
1048:
973:Grim trigger
904:
606:
556:
550:
538:. Retrieved
533:
523:
482:
453:
446:
437:
368:
351:not credible
344:
322:
297:
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277:
265:
262:
258:
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250:
238:
235:
218:
206:
194:
191:
184:
161:
156:
155:in 1953 and
148:
119:
115:
109:
74:Significance
49:Relationship
1665:Coopetition
1468:Jean Tirole
1463:John Conway
1443:Eric Maskin
1239:Blotto game
1224:Pirate game
1033:Global game
1003:Tit for tat
938:Bid shading
928:Appeasement
778:Equilibrium
758:Solved game
693:Determinacy
676:Definitions
669:game theory
359:tic-tac-toe
318:tic-tac-toe
112:game theory
79:Proposed by
43:game theory
1309:Trust game
1294:Kuhn poker
963:Escalation
958:Deterrence
948:Cheap talk
920:Strategies
738:Preference
667:Topics of
420:References
124:refinement
1493:John Nash
1199:Stag hunt
943:Collusion
583:cite book
575:911616270
559:. Tokyo.
540:April 27,
509:cite book
501:842323069
172:imperfect
54:Subset of
1715:Category
1634:Lazy SMP
1328:Theorems
1279:Deadlock
1134:Checkers
1015:of games
782:concepts
378:See also
247:Figure 2
215:Figure 1
130:used in
89:Used for
1386:figures
1169:Chicken
1023:Auction
1013:Classes
372:chicken
203:Example
168:utility
140:subgame
122:) is a
99:Example
573:
563:
499:
489:
461:
1124:Chess
1111:Games
147:with
134:. A
126:of a
805:Core
589:link
571:OCLC
561:ISBN
542:2021
515:link
497:OCLC
487:ISBN
459:ISBN
355:node
195:The
118:(or
114:, a
1384:Key
174:or
159:.
110:In
41:in
1717::
1119:Go
585:}}
581:{{
569:.
532:.
511:}}
507:{{
495:.
473:^
428:^
363:Go
342:.
189:.
182:.
1049:n
660:e
653:t
646:v
591:)
577:.
544:.
517:)
503:.
467:.
20:)
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