Forward Induction Wilson

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Econometrica , Vol. 77, No. 1 (January, 2009), 128

ON FORWARD INDUCTION

BY SRIHARI GOVINDAN AND ROBERT WILSON 1

A players pure strategy is called relevant for an outcome of a game in extensive form with perfect recall if there exists a weakly sequential equilibrium with that outcome for which the strategy is an optimal reply at every information set it does not exclude. Theoutcome satises forward induction if it results from a weakly sequential equilibriumin which players beliefs assign positive probability only to relevant strategies at eachinformation set reached by a prole of relevant strategies. We prove that if there aretwo players and payoffs are generic, then an outcome satises forward induction if everygame with the same reduced normal form after eliminating redundant pure strategieshas a sequential equilibrium with an equivalent outcome. Thus in this case forwardinduction is implied by decision-theoretic criteria.

K EYWORDS : Game theory, equilibrium renement, forward induction, backward in-

duction.

THIS PAPER HAS TWO PURPOSES . One is to provide a general denition of forward induction for games in extensive form with perfect recall. As a renementof weakly sequential equilibrium, forward induction restricts the support of aplayers belief at an information set to others strategies that are optimal repliesto some weakly sequential equilibrium with the same outcome, if there are anythat reach that information set.

The second purpose is to resolve a conjecture by Hillas and Kohlberg ( 2002,

Sec. 13.6), of which the gist is that an invariant backward induction outcomesatises forward induction. A backward induction outcome is invariant if everygame representing the same strategic situation (i.e., they have the same re-duced normal form) has a sequential equilibrium with an equivalent outcome.For a game with two players and generic payoffs, we prove that an invariantbackward induction outcome satises forward induction.

The denitions and theorem are entirely decision-theoretic. None of thetechnical devices invoked in game theory, such as perturbations of playersstrategies or payoffs, is needed. 2

Sections 1 and 2 review the motivations for backward induction and forward

induction. Sections 3 and 4 provide general denitions of forward inductionand invariance. The formulation and proof of the theorem are in Sections 5and 6. Section 7 examines an alternative version of forward induction, and

1This work was funded in part by a grant from the National Science Foundation of the UnitedStates. We are grateful for superb insights and suggestions from a referee.

2We retain Kreps and Wilsons (1982) denition of sequential equilibrium that consistent be-liefs are limits of beliefs induced by sequences of completely mixed strategies converging toequilibrium strategies. However, Kohlberg and Reny ( 1997, Theorem 3.2) established that thisproperty is implied by elementary assumptions that represent the fundamental property of non-cooperative games, namely that no players strategy choice affects any other players choice.

2009 The Econometric Society DOI: 10.3982/ECTA6956
http://www.econometricsociety.org/http://www.econometricsociety.org/http://dx.doi.org/10.3982/ECTA6956http://dx.doi.org/10.3982/ECTA6956http://www.econometricsociety.org/http://www.econometricsociety.org/


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2 S. GOVINDAN AND R. WILSON

Section 8 concludes. Appendix A proves a technical lemma and Appendix Bdescribes a denition of forward induction for a game in normal form.

1. INTRODUCTION

We consider a nite game in extensive form specied by a game tree and anassignment of players payoffs to its terminal nodes. Throughout, we assumeperfect recall, so the game tree induces a decision tree for each player. A purestrategy for a player species an action at each of his information sets, and amixed strategy is a distribution over pure strategies. A mixed strategy inducesa behavioral strategy that mixes anew according to the conditional distributionamong actions at each information set. Kuhn ( 1953, Theorem 4) establishedfor a game with perfect recall that each behavioral strategy is induced by a

mixed strategy, and vice versa, inducing the same distribution on histories of play.

1.1. Backward Induction

Economic models formulated as games typically have multiple Nash equi-libria. Decision-theoretic criteria are invoked to select among Nash equilib-ria. For a game in extensive form with perfect recall, the primary criterion isbackward induction. Backward induction is invoked to eliminate Nash equi-libria that depend on implausible behaviors at information sets excluded byother players equilibrium strategies. Thus backward induction requires that aplayers strategy remains optimal after every contingency, even those that donot occur if all players use equilibrium strategies.

We assume here that backward induction is implemented by sequential equi-librium as dened by Kreps and Wilson ( 1982, pp. 872, 882). A sequential equi-librium is a pair of proles of players behavioral strategies and beliefs. Here we dene a players belief to be a conditional probability system (i.e., satisfyingBayes rule where well dened) that at each of his information sets species adistribution over pure strategies that do not exclude the information set frombeing reached. A players belief is required to be consistent in that it is a limit of the conditional distributions induced by proles of completely mixed or equiv-alent behavioral strategies converging to the prole of equilibrium strategies.

Kreps and Wilsons exposition differs in that for a players belief at an infor-mation set they used only the induced distribution over nodes in that informa-tion set. This restriction cannot be invoked here since the purpose of forwardinduction is to ensure that the support of a players belief at an information setis conned to others optimal strategies wherever possible, both before this in-formation set as in their formulation in terms of nodes, and also subsequentlyin the continuation of the game. Thus we use throughout the more generalspecication that a players belief is over strategies.


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ON FORWARD INDUCTION 3

The dening feature of a sequential equilibrium is the requirement that inthe event an information set is reached, the player acting there behaves ac-cording to a strategy that in the continuation is optimal given his belief about

Natures and other players strategies. A weakly sequential equilibrium as de-ned by Reny ( 1992, p. 631) is the same as a sequential equilibrium except thatif a players strategy excludes an information set from being reached, then hiscontinuation strategy there need not be optimal. Section 3 provides a formaldenition of weakly sequential equilibrium, which is then used in our denitionof forward induction.

1.2. Forward Induction

Kohlberg and Mertens ( 1986, Sec. 2.3) emphasized that rening Nash equi-

librium to sequential equilibrium is not sufcient to ensure that behaviors are justied by plausible beliefs. One of their chief illustrations is their Proposition6 that a stable set of Nash equilibria contains a stable set of the game obtainedby deleting a strategy that is not an optimal reply to any equilibrium in theset. Kohlberg and Mertens labeled this result forward induction, but they andother authors do not dene the criterion explicitly. The main idea is the oneexpressed by Hillas and Kohlberg ( 2002, Sec. 42.13.6) in their survey article:Forward induction involves an assumption that players assume, even if theysee something unexpected, that the other players chose rationally in the past,to which one can add and other players will choose rationally in the future.This addendum is implicit because rationality presumes that prior actions arepart of an optimal strategy. See also Kohlberg ( 1990) and van Damme ( 2002).

Studies of particular classes of two-player games with generic payoffs revealaspects of what this idea entails. Outside-option games (as in Example 2.1 be-low) were addressed by van Damme ( 1989, p. 485). He proposed as a minimalrequirement that forward induction should select a sequential equilibrium in which a player rejects the outside option if in the ensuing subgame there isonly one equilibrium whose outcome he prefers to the outside option. Signal-ing games (as in Example 2.2 below) were addressed by Cho and Kreps ( 1987,p. 202). They proposed an intuitive criterion that was rened further by Banksand Sobel ( 1987, Sec. 3) to obtain criteria called divinity and universal divinity.These are obtained from iterative application of criteria called D1 and D2 byCho and Kreps ( 1987, p. 205).

Briey, a sequential equilibrium satises the intuitive criterion if no type of sender could obtain a payoff higher than his equilibrium payoff were he tochoose a nonequilibrium message and the receiver responds with an actionthat is an optimal reply to a belief that imputes zero probability to Natureschoice of those types that cannot gain from such a deviation regardless of thereceivers reply. The D1 criterion requires that after an unexpected message,the receivers belief imputes zero probability to a type of sender for which thereis another type who prefers this deviation for a larger set of those responses of


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the receiver that are justied by beliefs concentrated on types who could gainfrom the deviation and response. See also Cho and Sobel ( 1990) and surveys by van Damme ( 2002), Fudenberg and Tirole ( 1993, Sec. 11), Hillas and Kohlberg

(2002), and Kreps and Sobel ( 1994).Battigalli and Siniscalchi ( 2002, Sec. 5) derived the intuitive criterion froman epistemic model. They said that a player strongly believes that an event istrue if he remains certain of this event after any history that does not contradictthis event. They consider a signaling game and a belief-complete space of play-ers types; for example, one containing all possible hierarchies of conditionalprobability systems (beliefs about beliefs) that satisfy a coherency condition.Say that a player expects an outcome if his rst-order beliefs are consistent with this outcome, interpreted as a probability distribution on terminal nodesof the game tree. They showed in Proposition 11 that an outcome of a sequen-tial equilibrium satises the intuitive criterion under the following assumptionabout the epistemic model:

The sender (1) is rational and (2) expects the outcome and believes that (2a) the receiveris rational and (2b) the receiver expects the outcome and strongly believes that (2b.i) thesender is rational and (2b.ii) the sender expects the outcome and believes the receiver isrational.

The key aspect of this condition is the receivers strong belief in the sendersrationality. This implies that the receiver sustains his belief in the senders ra-tionality after any message for which there exists some rational explanation forsending that message.

These contributions agree that forward induction should ensure that aplayers belief assigns positive probability only to a restricted set of strategiesof other players. In each case, the restricted set comprises strategies that satisfyminimal criteria for rational play.

Two prominent contributions apply such criteria iteratively. McLennan(1985, p. 901) and Reny ( 1992, p. 639) proposed different algorithms for it-erative elimination of beliefs that are implausible according to some criterion.McLennan dened the set of justiable equilibria iteratively by excluding a se-quential equilibrium that includes a belief for one player that assigns positiveprobability at an information set to an action of another player that is not opti-mal in any sequential equilibrium in the restricted set obtained in the previous

iteration. Reny dened the set of explicable equilibria iteratively by excludinga belief that assigns positive probability to a pure strategy that is not a bestresponse to some belief in the restricted set obtained in the previous iteration.Essentially, these procedures apply variants of Pearces ( 1984) iterative proce-dure for identifying rationalizable strategies to the more restrictive context of sequential equilibria.

1.3. Synopsis

In Section 2 we illustrate further the motivation for forward induction viatwo standard examples from the literature. Our analyses of these examples


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anticipate the theorem in Section 6 by showing that the result usually obtainedby forward induction reasoning is implied by the decision-theoretic criterioncalled invariance. Invariance requires that the outcome should be unaffected

by whether a mixed strategy is treated as a pure strategy.In Section 3 we propose a general denition of forward induction. Its keycomponent species relevant pure strategies, that is, those that satisfy minimalcriteria for rational play resulting in any given outcomethe induced probabil-ity distribution on terminal nodes of the game tree and thus on possible pathsof equilibrium play. Our denition says that a pure strategy is relevant if thereis some weakly sequential equilibrium with that outcome for which the strategyprescribes an optimal continuation at every information set the strategy doesnot exclude. 3 We then say that an outcome satises forward induction if it re-sults from a weakly sequential equilibrium in which each players belief at an

information set reached by relevant strategies assigns positive probability onlyto relevant strategies.In Section 6 we prove for general two-player games with generic payoffs

that backward induction and invariance imply forward induction. Thus for suchgames forward induction is implied by standard decision-theoretic criteria.

2. EXAMPLES

In this section we illustrate the main ideas with two standard examples.These examples illustrate how forward induction can reject some sequentialequilibria in favor of others. Each example is rst addressed informally usingthe forward induction reasoning invoked by prior authors. The literature provides no formal denition of forward induction and we defer the statement of our denition to Section 3, but the main idea is evident from the context. Eachexample is then analyzed using the decision-theoretic criterion called invari-ance to obtain the same result. Invariance is dened formally in Section 4 andinvoked in Theorem 6.1, but in these examples and the theorem it is sufcientto interpret invariance as requiring only that the outcome resulting from a se-quential equilibrium is not affected by adding a redundant pure strategy, thatis, a pure strategy whose payoffs for all players are replicated by a mixture of other pure strategies.

2.1. An Outside-Option Game

The top panel of Figure 1 displays the extensive and normal forms of a two-player game consisting of a subgame with simultaneous moves that is precededby an outside option initially available to player I. The component of Nash

3See Govindan and Wilson (2007) for a version that obtains results for the weaker concept of relevant actions, rather than strategies. We are indebted to a referee who provided an exampleof an irrelevant strategy that uses only relevant actions.


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FIGURE 1.Two versions of a game with an outside option.

equilibria in which player I chooses his outside option includes an equilibriumin which player IIs strategy has probability 2 / 3 of his left column and thereforeplayer I is indifferent about deviating to his top row in the subgame, whereasthere is no such equilibrium justifying deviating to the bottom row. Alterna-tively, player I might anticipate that II will recognize rejection of the outsideoption as a signal that I intends to choose the top row and therefore II shouldrespond with the left column.

To apply forward induction, one excludes from the support of IIs belief thedominated strategy in which I rejects the outside option and chooses the bot-tom row in the subgame. If this restriction is imposed, then after I rejects theoutside option, II is sure that I will play the top row and, therefore, IIs opti-mal strategy is the left column, and anticipating this, player I rejects the outsideoption.

As in Hillas (1994, Figure 2), one can invoke invariance to obtain this con-clusion. The bottom panel of Figure 1 shows the expanded extensive form afteradjoining the redundant strategy in which, after tentatively rejecting the out-side option, player I randomizes between the outside option and the top row of the subgame with probabilities 3 / 4 and 1/ 4. Player II does not observe whichstrategy of player I led to rejection of the outside option. In the unique sequen-tial equilibrium of this expanded game, player I rejects both the outside optionand the redundant strategy, and then chooses the top row of the nal subgame.

A lesson from this example is that an expanded game has imperfect infor-mation in the sense that II has imperfect observability about whether I chosethe redundant strategy. This is signicant for II because I retains the option tochoose the bottom row in the subgame iff he rejected the redundant strategy.


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Even though subgame perfection could sufce in the original and expandedgame for this simple example, in general one needs sequential equilibrium toanalyze the expanded gameas will be seen later in Figure 4 of Section 7. Ad-

dition of redundant strategies can alter equilibrium strategies, but if invarianceis satised, then the induced probabilities of actions along equilibrium paths of the original game are preserved and thus so too is the predicted outcome. Onecould explicitly map equilibria of an expanded game into induced behavioralprobabilities of actions in the original game, but we omit this complication.

2.2. A Signaling Game

The top panel of Figure 2 displays the two-player two-stage signaling gameBeerQuiche studied by Cho and Kreps ( 1987, Sec. II) and discussed further

by Kohlberg and Mertens ( 1986, Sec. 3.6.B) and Fudenberg and Tirole ( 1993,Sec. 11.2).Consider sequential equilibria with the outcome QQ-R; that is, both types W

and S of player I (the sender) choose Q and player II (the receiver) responds

FIGURE 2.Two versions of the BeerQuiche game.


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to Q with R and to B with a probability of F that is 1/ 2. The equilibria in thiscomponent are sustained by player IIs belief after observing B that imputes toIs type W a greater likelihood of having deviated than to type S. In all these

equilibria, B is not an optimal action for type W. But in the equilibrium for which player II assigns equal probabilities to W and S after observing B andmixes equally between F and R, type S is indifferent between Q and B. If IIrecognizes this as the source of Is deviation, then he infers after observing Bthat Is type is S and therefore chooses R. Alternatively, if player Is type isS, then he might deviate to B in hopes that this action will credibly signal histype, since his equilibrium payoff is 2 from Q but he obtains 3 from player IIsoptimal reply R if the signal is recognized, but type W has no comparable in-centive to deviatethis is the speech suggested by Cho and Kreps ( 1987,pp. 180181) to justify their intuitive criterion.

One applies forward induction by excluding from the support of IIs belief after observing B those strategies that take action B when Is type is W. In fact,the sequential equilibria in which both types of I choose Q do not survive thisrestriction on IIs belief because IIs optimal response to B is then R, whichmakes it advantageous for player Is type S to deviate by choosing B. Thussequential equilibria with the outcome QQ-R do not satisfy forward induction.This leaves only sequential equilibria with the outcome BB-R in which bothtypes of player I choose B and II chooses R after observing B.

As in Example 2.1, one can obtain this same conclusion by invoking invari-ance. The bottom panel of Figure 2 shows the extensive form after adjoining a

mixed action X for type S of player I that produces a randomization betweenB and Q with probabilities 1 / 9 and 8/ 9. Denote by BQ player Is pure strategythat chooses B if his type is W and chooses Q if his type is S, and similarly forhis other pure strategies. The normal form of this expanded game is shown inTable I with all payoffs multiplied by 10 (we intentionally omit the pure strat-egy BX to keep the analysis simple). Now consider the following extensive formthat has the same reduced normal form. Player I initially chooses whether ornot to use his pure strategy QQ, and if not then subsequently he chooses amonghis other pure strategies BB, BQ, QB, and QX. After each of these ve pure

TABLE ISTRATEGIC FORM OF THE BEER QUICHE G AME WITH THE REDUNDANT STRATEGY QX

B: F F R RW S Q: F R F R

B B 9, 1 9, 1 29, 9 29, 9B Q 0, 1 18, 10 2, 0 20, 9Q B 10, 1 12, 0 28, 10 30, 9Q Q 1, 1 21, 9 1, 1 21, 9Q X 2, 1 20, 8 4, 2 22, 9


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strategies, the extensive form in the bottom panel ensues, but with Is actiondictated by his prior choice of a pure strategy. That is, nature chooses Is typeto be W or S, the selected pure strategy dictates the subsequent choice of B

or Q, and then player II (still having observed only which one of B or Q waschosen) chooses F or R. At player Is information set where, after rejectingQQ, he chooses among his other pure strategies, a sequential equilibrium re-quires that he assigns zero probability to BQ because it is strictly dominatedby QX in the continuation. At player IIs information set, after observing B, asequential equilibrium requires that his behavioral strategy is an optimal replyto some consistent belief about those strategies of player I that reach this in-formation set. But every mixture of Is pure strategies BB, QB, and QX impliesthat, given his choice of B, the induced conditional probability that his type isS exceeds 9/ 10. Therefore, player IIs reply to B must be R in every sequential

equilibrium of this game. Hence the sequential equilibria with outcome QQ-Rare inconsistent with invariance, in agreement with failure to satisfy forwardinduction.

A similar analysis applies to the game in Figure 3, which resembles gamesconsidered in studies of signaling via costly educational credentials in labormarkets as in Spence ( 1974) and Kreps (1990, Sec. 17.2). In this game, forwardinduction rejects the outcome of the pooling equilibrium in which both typesof player I move right and II responds to right with up (and to left with prob-ability of up 1/ 2), and accepts the outcome of the separating equilibrium in which only Is top type moves right. The most common use of forward induc-tion in economic models is to reject pooling equilibria in favor of a separatingequilibrium, although often what is actually assumed is a weaker implicationof forward induction.

The way in which invariance is invoked in Example 2.2 is indicative of theproof of Theorem 6.1 in Section 6.

FIGURE 3.A signaling game with pooling and separating equilibria.


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3. DEFINITION OF FORWARD INDUCTION

In this section we propose a general denition of forward induction for agame in extensive form with perfect recall.

Our denition of forward induction relies on the solution concept called weakly sequential equilibrium by Reny ( 1992, p. 631). Recall from Section 1.1that a weakly sequential equilibrium is the same as a sequential equilibrium ex-cept that a players strategy need not be optimal at information sets it excludes.See Reny ( 1992, Sec. 3.4) for an expanded justication of weakly sequentialequilibrium as the right concept for analysis of forward induction.

Our denition differs from Renys in that we interpret players beliefs asspecifying distributions over others strategies. Beliefs over strategies typicallyencode more information than necessary to implement sequential rationality,that is, as in Kreps and Wilson ( 1982), the conditional distribution over nodes

in an information set sufces to verify optimality. However, it is only from abelief specied as a conditional distribution over strategies that one can ver-ify whether a players belief recognizes the rationality of others strategies. AsExamples 2.1 and 2.2 illustrate, the purpose of forward induction as a rene-ment is to reject outcomes that deter player Is deviation by the threat of IIsresponse that is optimal for II only because his belief does not recognize Isdeviation as part of an optimal strategy for some equilibrium with the sameoutcome. To reject such outcomes, it is sufcient that the support of IIs belief is conned to Is pure strategies that are optimal replies at information setsthey do not exclude.

The following denition is the analog of the denitions in Kreps and Wilson(1982) and Reny ( 1992).

DEFINITION 3.1Weakly Sequential Equilibrium: A weakly sequential equilibrium is a pair (b ) of proles of players behavioral strategies and beliefs. At each information set h n of player n, his behavioral strategy species a dis-tribution bn( | h n) over his feasible actions, and his belief species a distribu-tion n( | h n) over proles of Natures and other players pure strategies thatenable hn to be reached. These proles are required to satisfy the followingconditions:

(i) Consistency : There exists a sequence {bk } of proles of completely mixedbehavioral strategies converging to b and a sequence { k } of completely mixedequivalent normal-form strategies such that for each information set of eachplayer the conditional distribution specied by is the limit of the conditionaldistributions obtained from { k }.4

(ii) Weak Sequential Rationality : For each player n and each information seth n that bn does not exclude, each action in the support of bn( | h n) is part of apure strategy that is an optimal reply to n( | h n) in the continuation from hn .

4In this denition, Natures strategy is not perturbed. Also, the belief n( | h n) might entailcorrelation, as observed by Kreps and Ramey (1987).


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A sequential equilibrium is dened exactly the same except that each playersactions must be optimal at all his information sets, including those excluded byhis equilibrium strategy.

We interpret forward induction as a property of an outcome of the game,dened as follows.

DEFINITION 3.2Outcome of an Equilibrium: The outcome of an equilib-rium of a game in extensive form is the induced probability distribution overthe terminal nodes of the game tree.

A key feature in the denition of forward induction is the concept of a relevant strategy.

DEFINITION

3.3Relevant Strategy: A pure strategy of a player is relevant

for a given outcome if there is a weakly sequential equilibrium with that out-come for which the strategy at every information set it does not exclude pre-scribes an optimal continuation given the players equilibrium belief there.

Thus a relevant strategy is optimal for some expectation about others equi-librium play with that outcome and his beliefs at events after their devia-tions. For instance, in Example 2.2 of a signaling game, the strategy QB of thesender I in which type W chooses Q and type S chooses B is relevant for theoutcome QQ-R because it is an optimal reply to the weakly sequential equilib-rium with that outcome in which the receiver II responds to B by using F and R with equal probabilities. But the strategies BB and BQ are irrelevant becauseB is not an optimal reply for Is type W to any weakly sequential equilibrium with outcome QQ-R.

For the standard examples in Section 2 it is sufcient to interpret forwardinduction as requiring merely that player IIs belief at the information set ex-cluded by Is equilibrium strategy imputes positive probability only to the nodereached by Is nonequilibrium relevant strategy. For general games, however,a stronger requirement is desirable.

We propose a general denition of forward induction that identies thoseoutcomes resulting from the conjunction of rational play and belief that othersplay is rational, and thus minimally consistent with Battigalli and Siniscalchis(2002) epistemic model of strong belief in rationality. Because a relevant strat-egy is optimal, hence rational, in some weakly sequential equilibrium with thesame outcome, the relevant strategies are the minimal set for which one canrequire the support of one players belief to recognize the rationality of otherplayers strategiesindeed that is the lesson from the standard examples inSection 2. Our proposed denition of a forward induction outcome there-fore requires that the outcome results from a weakly sequential equilibriumin which every player maintains the hypothesis that other players are usingrelevant strategies throughout the game, as long as that hypothesis is tenable.


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Thus forward induction is applied only to information sets reached by prolesof relevant strategies:

DEFINITION 3.4Relevant Information Set: An information set is relevantfor an outcome if it is not excluded by every prole of strategies that are relevant for that outcome. 5

Then we dene a forward induction outcome as follows. 6

DEFINITION 3.5Forward Induction: An outcome satises forward induc-tion if it results from a weakly sequential equilibrium in which at every infor-mation set that is relevant for that outcome the support of the belief of theplayer acting there is conned to proles of Natures strategies and other play-ers strategies that are relevant for that outcome.

Section 7 compares this denition with Renys alternative interpretation. Applied to the standard examples in Section 2, our denition yields the con-clusions obtained from forward induction reasoning in the literature. Forinstance, in Example 2.2 of a signaling game, the outcome QQ-R does not sat-isfy forward induction because the denition requires that after observing B,player II assigns zero probability to Is irrelevant strategies BB and BQ, andthus assigns positive probability only to Is relevant strategy QB that enablesIIs information set B to be reached.

For signaling games in general, forward induction implies the intuitive crite-rion, D1, D2 (whose iterative version denes universal divinity), and Cho andKreps (1987, Sec. IV.5) strongest criterion, called never weak best response(NWBR) for signaling games. These implications are veried by showing thata strategy s of the sender is irrelevant if s prescribes that his type t sends amessage m that is not sent by any type in some weakly sequential equilibrium with the given outcome, and the pair (t m) satises any of these criteria. Forinstance, the criterion NWBR excludes the strategy s from the receivers belief if the continuation strategy m at information set t yields exactly the senderstype-contingent payoff from the given outcome for some beliefs and optimalresponses of the receiver only when some other type t that could send m wouldget a type-contingent payoff that is higher than from the designated outcomefor the same or a larger set of the receivers optimal responses. But this condi-tion implies that there is no weakly sequential equilibrium with the same out-come for which m is an optimal action for type t . Were there such an equilib-rium, the receiver could use any such response at the off-the-equilibrium-path

5This differs from Kuhns (1953, Denition 6) and Renys ( 1992, p. 631) denition of an infor-mation set that is relevant for a pure strategy because the information set is not excluded by thatstrategy.

6For readers who prefer normal-form analysis, Denition B.1 in Appendix B proposes a slightlystronger denition applied to the normal form of the game that is equivalent to the denitionproposed here when payoffs are generic and that enables an analog of Theorem 6.1.


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information set m, but then type t could obtain a superior payoff by sending m.Thus, m cannot be an optimal continuation by type t in any weakly sequentialequilibrium with the given outcome, and therefore s is an irrelevant strategy.

For general games in extensive form, forward induction implies a versionof NWBR that Fudenberg and Tirole ( 1993, p. 454) attributed to Kohlbergand Mertens ( 1986, Proposition 6). A pure strategy that is an inferior replyto every equilibrium with a given outcome chooses an inferior action at someinformation set that intersects a path of equilibrium play. Such a strategy isirrelevant for that outcome according to Denition 3.3 and, therefore, if theoutcome satises forward induction according to Denition 3.5, then it resultsfrom a weakly sequential equilibrium in which, at every information set that isrelevant for that outcome, the support of the belief of the player acting thereassigns zero probability to this irrelevant strategy.

4. DEFINITION OF INVARIANCE

In this section we dene invariance as a property of a solution concept. First we dene relations of equivalence between games and between outcomes of equivalent games.

Recall that a players pure strategy is redundant if its payoffs for all play-ers are replicated by a mixture of his other pure strategies. From the normalform of a game one obtains its reduced normal form by deleting redundantstrategies. Thus the reduced normal form is the minimal representation of theessential features of the strategic situation.

DEFINITION 4.1Equivalent Games: Two games are equivalent if their re-duced normal forms are the same up to relabeling of strategies.

As specied in Denition 3.2, the outcome of an equilibrium of a game inextensive form is the induced probability distribution on terminal nodes andthus on the paths through the tree. Associated with each outcome is a set of proles of Natures and players mixed strategies that result in the outcome,and in turn each such prole can be replicated by a prole of mixed strategiesin the reduced normal form. Hence we dene equivalent outcomes as follows.

DEFINITION 4.2Equivalent Outcomes of Equivalent Games: Outcomesof two equivalent games are equivalent if they result from the same prole of mixed strategies of their reduced normal form.

Trivially, the outcome of any Nash equilibrium is equivalent to the outcomeof a Nash equilibrium of any equivalent game. For any solution concept that isa renement of Nash equilibrium, we dene invariance as follows.

DEFINITION 4.3Invariant Outcome: An outcome is invariant for a solutionconcept if every equivalent game has an equivalent outcome of an equilibriumselected by the solution concept.


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This denition is used in Section 6 where the solution concept is sequen-tial equilibrium. Existence of invariant outcomes of sequential equilibria forgeneric games in extensive form with perfect recall is implied by the existence

of stable sets as dened by Mertens ( 1989).7

5. FORMULATION

In this section we introduce notation used in the proof of Theorem 6.1 inSection 6.

Let be a game in extensive form with perfect recall. For each player n letH n be the collection of his information sets, and let S n Bn , and n be his setsof pure, behavioral, and mixed strategies. A pure strategy chooses an action ateach information set in H n , a behavioral strategy chooses a distribution over

actions at each information set, and a mixed strategy chooses a distributionover pure strategies. We say that a pure strategy enables an information setif the strategys prior actions do not exclude the information set from beingreached, and similarly for behavioral and mixed strategies.

Let P be the outcome of a sequential equilibrium of the game . Say thata pure strategy or an information set is P -relevant if it is relevant for the out-come P . Let (P) and B(P) be the sets of Nash equilibria of representedas proles of mixed and behavioral strategies, respectively, that result in theoutcome P . Also, let BM (P) be the set of weakly sequential equilibria whoseoutcome is P , where each (b ) BM (P) consists of a prole b B(P) of players behaviorial strategies and a prole of players consistent beliefs. Asin Denition 3.1, in each weakly sequential equilibrium (b ) the belief of aplayer n at his information set hn H n is a probability distribution n( | h n)over Natures and other players pure strategies that reach h n . A pure strategysn S n for player n is optimal in reply to (b ) if at each information set of nenabled by sn , the action specied there by sn is optimal given (b ).

Given an outcome P , say that a P -path through the game tree is one thatterminates at a node in the support of P . Actions on P -paths are called equi-librium actions. Let H n(P) be the collection of player ns information sets thatintersect P -paths. Obviously, every equilibrium in B(P) prescribes the samemixture at each information set in H n(P) . Let S n(P) S n comprise those purestrategies sn of player n such that sn chooses an equilibrium action at every in-formation set in H n(P) that sn enables. Note that if (P) , then the supportof n is contained in S n(P) . Moreover, every strategy in S n(P) is optimal againstevery equilibrium in (P) . Partition the complement T n(P) S n \ S n(P) intosubsets Rn and Qn of ns pure strategies that are P -relevant and P -irrelevant,respectively. Note that S n(P) may also contain P -irrelevant strategies.

7In Govindan and Wilson (2006) we proved that if a solution concept satises invariance and acondition called strong backward induction, then it selects sets of equilibria that are stable in the weaker sense dened by Kohlberg and Mertens (1986, Sec. 3.5).


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Dene an equivalence relation among player ns pure strategies as follows.Two strategies are equivalent if they prescribe the same action at each infor-mation set in H n(P) . Let E n(P) be the set of equivalence classes. Denote a

typical element of E n(P) by E n and let E n(sn) be the equivalence class thatcontains sn . Let E n (P) be the subcollection of equivalence classes that containstrategies in S n(P) . Thus, any strategy that is used in some equilibrium in (P)belongs to some equivalence class in E n (P) , while any strategy that is in T n(P)does not. (In Example 2.2, for P = QQ R and player n = I, S n(P) = {QQ },Rn(P) = { QB}, E n(P) = { QQ QB BQ BB}, and E n (P) = { QQ }.)

If Rn is not empty, then for each probability (0 1) let t n be a mixedstrategy of n of the form [1 ]sn + n , where sn is a strategy in S n(P) and nis a mixed strategy whose support is Rn . Since sn is a best reply against everyequilibrium in (P) , t n is an approximate best reply against equilibria in (P) when is small, a fact we need in the next section. Dene a game G in normalform by adding to the normal form G of the redundant pure strategy t n foreach player n for whom Rn is not empty. In particular, t n is added iff there issome information set in H n(P) where some nonequilibrium action is part of a P -relevant strategy. (In Example 2.1, the redundant strategy t n for player Iis the one shown in Figure 1 with parameter = 1/ 4; in Example 2.2, it is thestrategy QX with parameter = 1/ 9.)

Next we dene a game in extensive form with perfect recall whose normalform is equivalent to G , and thus also is equivalent to . A path of playin consists of choices by players in an initial stage, followed by a path of play in a copy of . In the subsequent play of , no player is informed aboutchoices made by other players in the initial stage of . The rules of are thefollowing. If Rn is empty, then in player n chooses among all his equivalenceclasses in E n(P) in the initial stage. If Rn is not empty, then in the initial stage herst chooses whether to play an equivalence class in E n (P) or not. If he decidesto play something in E n (P) , then he chooses one of these equivalence classes;if he chooses not to, then he proceeds to a second information set where hechooses to play either the redundant pure strategy t n or an equivalence classamong those not in E n (P) . After these initial stages for all players, evolvesthe same as does, that is, a copy of follows each sequence of choices inthe initial stage. In the information sets in are expanded to encompassappropriate copies of to represent that no player ever observes what otherschose in the initial stage; thus, the information revealed in is exactly thesame as in . The information set h n H n in has in for each E n E n(P)an expanded copy hn(E n) and a copy hn(t n ) . Nature makes the choice at theexpansions of those information sets in H n(P) (but not at expansions of thosein H n \ H n(P) ) according to the equivalence class chosen in the initial stage orat all expansions of information sets in H n if t n was chosen. That is, if n choosest n at the second information set, then Nature automatically implements theentire strategy; but if he chooses some equivalence class E n in E n(P) , thenNature implements actions prescribed by E n at each h n(E n) when h n H n(P)


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and leaves it to him to choose at those that are not, if and when they occur. (If is the BeerQuiche game and = 1/ 9, then t n = QX and is the game in which player I chooses whether to play QQ, and if not then he chooses among

BB, BQ, QB, and QX, as described in the text of Example 2.2.)One can interpret player ns choice of an equivalence class from, say, E n (P)

as equivalent to making an initial commitment to take a specied equilibriumaction at every information set in H n(P) , leaving the actions at ns other infor-mation sets unspecied until those information sets are reached.

A pure strategy sn S n can be implemented in by rst choosing E n(sn)in the initial stage and then making the choices prescribed by sn at all hn H n \ H n(P) , and any strategy in that begins by choosing some equivalenceclass E n in the rst stage ends up implementing some sn E n . Observe too thatthe redundant pure strategy t

n, when available, ends up implementing the mix-

ture given by t n . Thus, it is obvious that G is obtained from the normal form of by deleting some redundant pure strategies in the latter that are duplicatesof other pure strategies. Hence, by a slight abuse of notation, we view G asthe normal form of . The game is now easily seen to be equivalent to .

Suppose hn H n \ H n(P) . If an equivalence class E n contains a pure strat-egy sn that enables h n in , then in the corresponding strategy sn that is,choosing E n in the initial stage and then making sns choices at all h n /H n(P) enables h n(E n) . Conversely, if E n does not contain such an sn , then there is aninformation set vn H n(P) that precedes hn , is enabled by E n , and where E n

makes a choice different from the one that leads to hn . Thus, in , Natureschoice at vn(E n) prevents hn(E n) from being reached. Therefore, to analyzethe game , we need to consider only information sets h n(E n) where E n con-tains a strategy that enables h n in . For simplicity in this section and the next,by an information set hn(E n) in of player n, we mean an hn H n \ H n(P)and an E n that contains a strategy that enables hn in .

Now assume the game has two players. We use m to denote the opponentof player n. Suppose that E n and E n are two equivalence classes that containstrategies that enable some h n /H n(P) . The information that n has at h n(E n)

andh

n(E

n)

aboutm

s choices are the same at both information sets. Therefore,a pure strategy of m in G enables one if and only if it enables the other. Inparticular, in a sequential equilibrium (b ) of , player ns belief at h n(E n)is independent of E n and can thus be denoted n( | h n) .

Likewise, suppose m is a mixed strategy of m in G that enables some infor-mation set h n(E n) of n. Then m induces a conditional distribution m over thepure strategies of m in G that enable h n(E n) . Let m and m be the equivalentstrategies in G. It is easily checked that m is the conditional distribution in-duced by m over the pure strategies that enable h n , and an action a n at h n(E n)in is optimal against m iff it is optimal against m in .


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6. STATEMENT AND PROOF THE THEOREM

In this section we show for two-player games with generic payoffs that aninvariant backward induction outcome satises forward induction.

The notion of genericity we invoke is the following. Let G be the space of all games generated by assigning payoffs to the terminal nodes of a xed two-player game tree. In Govindan and Wilson ( 2001) we showed that there existsa closed lower-dimensional subset G 0 such that for each game not in G 0 thereare nitely many outcomes of Nash equilibria. For technical reasons, in Ap-pendix A we construct another closed lower-dimensional subset denoted G 1.Now a game is generic if it is in the complement of both G 0 and G 1. With this,the formal statement of our theorem is the following:

THEOREM 6.1: An outcome of a two-player game with perfect recall and generic payoffs satises forward induction if it is invariant for the solution concept sequential equilibrium .

PROOF : Assume that is a two-player game in extensive form with per-fect recall and generic payoffs. Assume also that P is an invariant sequentialequilibrium outcome of , that is, each game equivalent to has a sequentialequilibrium whose outcome is equivalent to P . Because is equivalent to thegame dened in Section 5, has a sequential equilibrium (b ) whoseoutcome is equivalent to P . Because is consistent, there exists a sequence{b

} of proles of completely mixed behavioral strategies that converges as

0 to b and a corresponding equivalent sequence { } of proles of com-pletely mixed strategies in the normal form G that converges to some prole and such that the belief prole is the limit of the beliefs derived from thesequence { }.

Since b induces an outcome that is equivalent to P , the strategy , whichis equivalent to b , has its support in S n(P) for each n: indeed, a strategy inT n(P) , or the strategy t n when available, chooses a nonequilibrium action atsome h n H n(P) in that it enables. Therefore, under b each player n in theinitial stage assigns positive probability only to choices of equivalence classesin E n (P) .

Corresponding to the sequence { }, there is an equivalent sequence { }of proles of mixed strategies in the normal form of for which there is anequivalent sequence {b } of proles of behavioral strategies in the extensiveform of . Let be the prole of beliefs induced by . Denote selectedlimit points of these sequences by , b , and . By construction, (P) ,b B(P) , and is consistent. It follows from our remarks at the end of the previous section that for each n and hn / H n(P) , n( | h n) is equivalentto n( | h n) , and an action at hn is optimal in against n( | h n) iff it is optimalin at hn(E n) against n( | h n) for the corresponding copies in .


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Next we argue that (b ) is a weakly sequential equilibrium of . Let h nbe an information set of player n that bn enables. We need to show that thechoice made by bn at hn is optimal against n( | h n) . If hn belongs to H n(P) ,

then n( | h n) is derived from m and obviously b

n chooses optimally at hn .Suppose that hn / H n(P) . Let an be an arbitrary action at hn that is chosen

with positive probability by bn . Since hn is enabled by bn there exists a purestrategy sn in the support of that enables h n and chooses an there. Since

is equivalent to , in b player n with positive probability chooses E n(sn) andthen makes the choices prescribed by sn . Sequential rationality of an at h n(E n)implies its optimality against n( | h n) . Hence, from the previous paragraph anis optimal against n( | h n) in . Since a n was arbitrary, this shows that (b )is a weakly sequential equilibrium of .

For some sequence 0, ( b ) converges to some limit point

( b ). Clearly, (P) , b B(P) , is consistent, and (b ) BM (P)is a weakly sequential equilibrium of because BM (P) is a closed set.It remains to prove the forward induction property for the belief prole .

For each n for whom Rn is not empty, and each , let { n } be the sequenceof conditional distributions over T n T n(P) {t n } induced by the sequence{ } and let n be a limit point. The sequence { n } and therefore its limit aredetermined by choices made after n chooses in the initial stage to avoid equiv-alence classes in E n (P) . Therefore, the probability of t n is nonzero under n iff t n is chosen with positive probability at ns second information set in the initialstage, and the probability of sn T n(P) is nonzero under n iff n chooses E(s n)

with positive probability at this stage and then implements the choices of sn with positive probability after this choice. Express n as a convex combination nt n + [ 1 n] n where the support of n is contained in T n(P) . Then sequen-tial rationality at the initial stage after rejecting equivalence classes in E n (P)and at subsequent information sets have the following two implications. First, n is nonzero only if t n is at least as good a reply as each sn T n(P) against bm .Second, if n < 1, then a strategy sn T n(P) belongs to the support of n onlyif it is at least as good a reply against bm as the other strategies in T n , and foreach h n /H n(P) enabled by sn in , the choice prescribed by sn at h n is optimalat h n(E n(sn)) given the belief n( | h n) . If an information set h m(E m) of playerm is enabled by n but not by n , then the beliefs m( | h m) are derived from n .

The sequence { n } induces a corresponding sequence of equivalent strate-gies in G that induces a sequence of conditional distributions over T n(P) .Because t n = [ 1 ]sn + n , the limit of the sequence of strategies in Gthat is equivalent to the sequence { n } is [1 ] nsn + n + [ 1 n] n .Therefore, the limit of the sequence of conditional distributions over T n(P) is n = n + [ 1 n] n , where n = n/ [ n + (1 n)]. Obviously if an in-formation set hm of player m is enabled by n but not by n , then the beliefs m( | h m) are those derived from n .

Passing to a subsequence if necessary, the limit n of the sequence n canbe expressed as a convex combination n = n + [ 1 n] n where n and n


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are the limits of n and n , respectively. As in the previous paragraph, if aninformation set hm of player m is enabled by n but not by n , then the beliefs m( | h m) are those derived from n .

CLAIM 6.2: (i) n > 0. (ii) If n < 1, then the support of n consists of strategiesin Rn . In particular , for each sn in its support and each information set h n that sn enables , the choice of sn at hn is optimal given (b ).

PROOF OF CLAIM : We prove (ii) rst. Suppose n < 1. Let sn be a strategyin T n(P) that is not optimal in reply to (b ). We show that sn is not in thesupport of n for all sufciently small , which proves the second statement.Let h n be an information set that sn enables where its action is not optimal. If h n H n(P) , then every strategy in E n(sn) is suboptimal against bm . Because sn ,

the strategy that belongs to S n(P) and to the support of t n for all , is optimalagainst bm for all and because b is the limit of b , for sufciently small

the strategy t n does better against b than every strategy in the equivalenceclass E n(sn) . At the second information set in the initial stage of where ndecides among the redundant strategy t n and equivalence classes not in E n (P) ,sequential rationality implies that he chooses the equivalence class E n(sn) withzero probability for all small . As we remarked above, this implies that forsuch , the probability of sn is zero in n .

If h n /H n(P) , then there exists another strategy sn in the equivalence classE n(sn) that agrees with sn elsewhere but prescribes an optimal continuation at

h n . Obviously, for all small , sn is a better reply than sn in reply to (b

) . Butthen sequential rationality at the copy hn(E n(sn)) of hn in for such small implies that he would choose according to sn there and not sn . Again, theprobability of sn under n is zero for small . Thus every strategy in the supportof n is optimal in reply to (b ) and therefore is a P -relevant strategy.

It remains to show that n = 0. Suppose to the contrary that n = 0. LetS n be the set of strategies in the support of either n or n . Let H m be thecollection of information sets in H m enabled by strategies in S n . Because (b )is a weakly sequential equilibrium, we obtain the following properties for eachinformation set h m in H m of player m: if hm is enabled by n , then the actionprescribed by bm at hm is optimal against n ; if h m is enabled by n and not by n , then the action prescribed by bm is optimal against n . Therefore, for eachsmall > 0 there exists a perturbation () of , where only ms payoffs areperturbed, such that m is optimal against n() [ 1 ] n + n in () andthat () converges to as goes to zero. As we argued above, n is optimalagainst m in . Therefore, for all small , ( m n()) is an equilibrium of () . Since is generic, it belongs to some component C of the open setG \ G 1, where G 1 is the set constructed in Appendix A . Since G \ G 1 has nitelymany connected components, C is open in G \ G 1 and, hence, in G . Therefore,the sequence () is in C for all small . By Lemma A.1 in Appendix A , there


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exists a strategy n such that (i) the support of n equals S n and (ii) m is abest reply against n in . Therefore, for all 0 1, ( m (1 ) n + n) isan equilibrium of . But because the strategies in the support of n choose a

nonequilibrium action at some h n H n(P) that they enable, all these equilibriaresult in different outcomes. This is impossible because the payoffs in aregeneric and therefore has only nitely many equilibrium outcomes, as shownin Govindan and Wilson ( 2001). Thus n = 0. Q.E.D.

Now we prove that P satises forward induction by showing that (b ) in-duces beliefs that assign positive probability only to P -relevant strategies. Leth n be an information set of n that is enabled by bn . If hn H n(P) , then obvi-ously ns belief over the continuation strategies of m is the one derived from m , and strategies in the support of m are obviously P -relevant. If h n /H n(P) ,

then the only strategies of m

that enable h

n are those in T

m(P)

. If there is nostrategy in Rm that enables h n , then there is nothing to prove. Otherwise, thesubset of strategies in Rm that enable hn is not empty and then the strategy m , which by the above claim has Rm as its support, enables h n . Therefore, n( | h n)is derived from m , and in this case, too, the restriction on beliefs imposed byforward induction holds. Thus P satises forward induction. Q.E.D.

Theorem 6.1 resolves a conjecture by Hillas and Kohlberg ( 2002, Sec. 13.6).Its remarkable aspect is that backward induction and invariance sufce for forward inductionif there are two players and payoffs are generic. No furtherassumption about rationality of behavior or plausibility of beliefs is invoked;neither are perturbations of strategies invoked as in studies of perfect equilib-ria and stable sets of equilibria, and for signaling games there is no reliance onCho and Kreps ( 1987, p. 181) auxiliary scenario in which the sender makes aspeech that the others intransigent belief ignores the fact that a deviation would be rational provided merely that the receiver recognizes and acts on itsimplications by excluding irrelevant strategies from the support of his belief.

Invariance excludes one particular presentation effect by requiring that theoutcome should not depend on whether a mixed strategy is treated as an addi-tional pure strategy. One interpretation of forward induction is that it excludesanother presentation effect by requiring that the outcome does not depend onirrelevant strategies. Indeed, van Damme ( 2002, p. 1555) interpreted forwardinduction as akin to the axiom called independence of irrelevant alternativesin social choice theory. In the case of a game, the analog of social choice is theoutcome (the probability distribution on terminal nodes) and the irrelevantalternatives are players irrelevant strategies.

Our proof of Theorem 6.1 relies on the assumption that has two playersand generic payoffs. Indeed, the conclusion of Claim 6.2 relies on genericity. 8

8That extensions to nongeneric payoffs are problematic can be seen in studies of signalinggames where signals are costless to the sender, as in Chen, Kartik, and Sobel (2008).


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Moreover, the proof does not sufce for the BeerQuiche game in Section 2.2if the two types of player I are treated as two different players; in particular,the game has a sequential equilibrium in which both types choose Q. The

intuitive reason why the proof does not apply to an N -player game is that at aplayers information set that does not intersect paths of equilibrium play, hisbeliefs might need to evaluate the relative likelihood of one opponent choos-ing a relevant strategy compared to another opponent choosing an irrelevantstrategy. Asking for a sequential equilibrium in the game does not imposeany discipline on such considerations. We surmise that additional decision-theoretic criteria must be imposed to obtain a version of Theorem 6.1 for agame with more than two players.

7. RENYS INTERPRETATION OF FORWARD INDUCTION

An implication of Theorem 6.1 is that there is no conict between backwardand forward induction if one adopts the decision-theoretic principle of invari-ance. This conclusion depends on our denitions of relevant strategies andforward induction outcomes; for example, we interpret forward induction asa renement of weakly sequential equilibrium that ensures the outcome doesnot depend on one player believing the other is using an irrelevant strategy ata relevant information set.

In this section we compare our denitions with the principle alternative, rep-resented by the discussion in Reny ( 1992, Sec. 4). He invoked best responsemotivated inferences as an instance of forward induction logic and con-cluded from an example that it can conict with backward induction.

Although he does not propose an explicit denition, the main ingredientsdiffer from our formulation as follows. Our denitions are narrowwe inter-pret forward induction as a property of an outcome of a weakly sequentialequilibrium and ask only that the outcome results from one in which the sup-port of a players belief at a relevant information set is conned to relevantstrategies, which we limit to those strategies that are optimal replies to some weakly sequential equilibrium. Renys view applies forward induction reason-ing directly to players strategies rather than to outcomes, and applies it tomore information sets and more strategies. At every information set not ex-cluded by a players own strategy, he asks only that the support of the playersbelief is conned to those strategies that reach that information set for whichthere are some beliefs of the other player that would justify using them.

The implications of Renys expanded view of forward induction reasoningare illustrated by the motivating example in his Figure 3. The top panel of Fig-ure 4 shows a game in which players I and II alternately choose whether to endthe game. Reny argued that this example shows a tension between forward andbackward induction. He observed that Is choice of the pure strategy D strictlydominates Ad . He inferred from this that forward induction should requirethat if I rejects D, then II must believe that Is strategy is surely Aa and hence


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FIGURE 4.Top panel: Renys example of a game between players I and II. Bottom panel: Thegame modied so that player I can choose the redundant strategy x() after rejecting D.

IIs only optimal reply is Ad . But backward induction requires each player tochoose d, and before that D, which contradicts the seeming implication of forward induction that IIs strategy should be Ad . From this Reny concluded thatIIs backward induction strategy is rendered irrational and thus the inap-propriateness, indeed the inapplicability of the usual backward programmingargument in the presence of best response motivated inferences (Reny ( 1992,p. 637), italics as in original).

Our analysis of this example differs in two respects. First, Is only relevantstrategy is to choose D initially, so forward induction according to our deni-tion has no implications for IIs beliefs. This is so because our denitions iden-tify outcomes that result from the conjunction of rational play and beliefs thatother players are playing rationally; hence we apply them only to informationsets reached by rational play as represented by relevant strategies. In contrast,


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8. CONCLUSION

Theorem 6.1 offers an explanation of why forward induction is a desirable re-nement of sequential equilibrium in two-player games with generic payoffs. If an outcome does not satisfy forward inductionthat is, depends on one playerbelieving the other is using an irrelevant strategythen there is an equivalentgame in which this outcome results only from Nash equilibria and not from anysequential equilibrium.

Failure of an economic model to predict an outcome that satises forwardinduction could motivate reconsideration of whether the essential features of the strategic situation are well represented by the specic extensive form usedin the model, or if one has condence in the model, then this prediction mightbe rejected because for the same model there necessarily exists another pre-diction that does satisfy forward induction.

Because the theorem is restricted to games with two players and generic pay-offs, it does not establish that our denitions of relevant strategies and forwardinduction outcomes are surely the right ones for general games. But it suggeststhat similar denitions might enable forward induction reasoning to be jus-tied by decision-theoretic criteria.

APPENDIX A: T ECHNICAL L EMMA

Given an extensive form with two players, for each player n let S n and n bens sets of pure and mixed strategies, and let S = S 1 S 2 and = 1 2 bethe product sets of proles. Let G be the Euclidean space of games generatedby assigning payoffs to the players at the terminal nodes of the given extensiveform.

L EMMA A.1: There exists a closed , lower-dimensional , semi-algebraic set G 1 of G such that G \ G 1 has nitely many connected components . For each connected component C , the following holds : if for some game C and prole the set of proles of pure strategies that are the players optimal replies to isT = T 1 T 2 S , then for every game C there exists a prole with the same support as and such that in the set of pure optimal replies to is T .

PROOF : Let X = G and let p : X G be the natural projection. Foreach pair R = R1 R2 and T = T 1 T 2 of subsets of S , let X(R T ) be the setof ( ) in X such that, for each n, Rn is the support of n and T n is the set of ns pure optimal replies in to the mixed strategy m of the other player. Bythe generic local triviality theorem in Bochnak, Coste, and Roy ( 1998) thereexists a closed, lower-dimensional, semi-algebraic subset G 1 of G such G \ G 1has nitely many connected components. Moreover, for each connected com-ponent C of G \ G 1 there exist (i) a semi-algebraic ber F , (ii) for each pair(R T), a subset F(R T) of F , and (iii) a homeomorphism h : C F p 1(C)


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with the properties that (a) p h( f) = for all C , f F and (b) h mapsC F(R T) homeomorphically onto p 1(C) X(R T) for each (R T ).

Suppose T is the set of proles of pure optimal replies to in a game C .

Let R be the support of . Then ( ) belongs to X(R T ). Therefore, thereexists f F(R T) such that h( f) = ( ). For each C , let (f ) be theunique mixed strategy in for which h( f ) = ( (f)) . Then the supportof (f ) is R and the set of proles of pure optimal replies in to (f )is T . Q.E.D.

APPENDIX B: F ORWARD INDUCTION IN THE NORMAL FORM

The classical view in game theory is that the normal form of a game is suf-cient to capture all strategically signicant aspects. Hence the question arisesas to whether we can state a comparable version of forward induction for agame in normal form. Here we provide one such denition.

The following three components of Denition 3.5 for a game in extensiveform need to be rephrased in terms of the normal form: (i) weakly sequen-tial equilibria, (ii) relevant strategies, and (iii) restriction of beliefs to thoseinduced by relevant strategies whenever possible. As will be seen below, if thesequential rationality requirement in the denition of weakly sequential equi-libria is strengthened slightly (and only for nongeneric games), then the corre-sponding denition of forward induction has a normal-form counterpart.

Given a game G in normal form, let be a prole of players mixed strate-gies and let b be an equivalent prole in behavioral strategies for an extensive-

form game with that normal form. Reny ( 1992, Proposition 1) showed that is a normal-form perfect equilibrium of G iff in there exists a sequence b of completely mixed proles converging to b such that for each player n and eachinformation set h n that bn does not exclude, the action prescribed by bn at h nis optimal against b n for all small . Thus the difference between weakly se-quential equilibrium and normal-form perfect equilibrium is analogous to thatbetween sequential equilibrium and extensive-form perfect equilibrium: onerequires optimality only in reply to the limit, while the other requires optimal-ity in reply to the sequence as well. Reny also showed that weakly sequentialequilibria coincide with normal-form perfect equilibria for generic extensive-form games. Therefore, a perfect equilibrium seems to be the right normal-form analog of a weakly sequential equilibrium.

Suppose is a set of Nash equilibria of G. (To x ideas, could be theset (P) of equilibria inducing an outcome P in an extensive-form version of the game, but to allow applications to nongeneric games, we allow multipleoutcomes.) In the extensive-form case, we said that a strategy was relevantif it was optimal against a pair of proles of strategies and beliefs inducingthe given outcome. But as noted above, if we insist on optimality along thesequence, then the appropriate normal-form denition of a relevant strategybecomes: a strategy is relevant if it is optimal against a sequence of -perfectequilibria converging to an equilibrium in .


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Finally, we turn to belief restrictions. The idea in the extensive-form caseis that if an information set hn of player n is reached by a prole of relevantstrategies of his opponents, then he assigns zero probability to continuations

that are enabled only by proles that contain an irrelevant strategy for one of the other players. Let R n(h n) be the set of proles of relevant strategies of ns opponents that reach such an h n . If we use a sequence of normal-formproles to generate players beliefs and their continuation strategies, then thebelief restriction says that ns belief at h n and the continuation strategies of hisopponents should be obtained from the limit of the sequence of conditionaldistributions over R n(h n) induced by the sequence . That is, the beliefs atall information sets of all players that are reached by relevant strategies can begenerated from the sequence of conditional distributions conned to relevantstrategies.

Because we insist on optimality along the sequence, what we obtain is a per-fect equilibrium with a restriction on the form of its representation as a lexi-cographic probability system, as in Blume, Brandenburger, and Dekel ( 1991,Propopositions 4, 7). The restriction is that any prole that includes an irrelevant strategy for some player should occur later in the lexicographic sequencethan those proles that include only relevant strategies. This implements thebasic requirement that each player believes the other is using a relevant strat-egy as long as that hypothesis is tenable. Thus, we are led to the followingdenition:

DEFINITION

B.1Normal-Form Forward Induction: A set of Nash equilib-ria satises normal-form forward induction if it contains a perfect equilibrium whose lexicographic representation has all proles of relevant strategies occur-ring before all proles that include irrelevant strategies.

In general, this is a stronger requirement than the one in the text, but for ageneric two-player game in extensive form with perfect recall it can be shownthat the set of weakly sequential equilibria inducing an outcome P satisesthe above denition iff P satises forward induction as dened in the text.The reason for this equivalence is similar to the reason that weakly sequentialequilibria and normal-form perfect equilibria coincide for generic extensive-form games as established by Reny ( 1992, Proposition 1). An implication is thatthe analog of Theorem 6.1 is true with this denition of forward induction, thatis, the set of Nash equilibria resulting in an invariant sequential equilibriumoutcome of a two-player game in extensive form with perfect recall and genericpayoffs satises normal-form forward induction.

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