Author s address: Robert Wilson Stanford Business School, Stanford CA 94305-5015 USA. Tel: 415-723-8620, Fax: 415-725-7979 Internet: RW@yen.stanford.edu
Admissibility and Stability Robert Wilson 1 1 Stanford University, Stanford CA 94305, USA Abstract: Admissibility is a useful criterion for selecting among equilibria, but I argue that enforcing admissibility dilutes the power of stability criteria to select among equilibrium components, and this accounts for anomalous examples of stable sets. Therefore, admissibility should be invoked only when selecting equilibria within a component that is immune to payoff perturbations. Keywords: Game theory, stability, admissibility, payoff perturbations The purpose of this essay is to examine the role of admissibility in formulating criteria for stability of equilibria. Recall that admissibility excludes an equilibrium that uses a dominated pure strategy. In particular, each pure strategy used (i.e., with positive probability) must be a best response to some completely mixed strategy. Reinhard Selten s (1975) criterion of perfection ensures admissibility by requiring that a pure strategy is used only if it is a best response to a sequence of completely mixed strategies. Admissibility was adopted subsequently by Kohlberg and Mertens (1986) and Mertens (1989) as a primary desideratum in their identification of stable sets of equilibria. My thesis here is that admissibility plays two roles, only the first of which advances the program of defining stability for games in extensive form. The first role is that in selecting among equilibria within a component, admissibility strengthens the criterion of stability. However, in its second role in selecting among equilibrium components, admissibility weakens the criterion of stability as examples will illustrate. I argue, therefore, that admissibility criteria should be deleted when selecting among equilibrium components; that is, the selected components should be essential, in the sense that they are stable with respect to all perturbations of normal-form payoffs. This lends credence to Kohlberg and Mertens original definitions of hyperstable and fully stable sets of equilibria, provided they are restricted to lie in essential components. It suggests modifying Mertens (1989) homological definition of stability to invoke payoff perturbations rather than strategy perturbations. 1
Section 1 summarizes technical aspects that are the basis in Section 2 for a comparison of stability criteria derived from perturbations of strategies and payoffs. All examples are collected in the Appendix. For a general survey of this topic see van Damme (1991). 1 Background To simplify, we consider only games in extensive form with two players, called I and II. Each player has a finite number of pure strategies; each has perfect recall; and all the data of the game are common knowledge. Thus the normal form of a game has two matrices with rows labeled by I s pure strategies and columns labeled by II s pure strategies. For each pair of their pure strategies, the corresponding entry in one matrix describes I s payoff, and in the other matrix II s payoff, each expressed in terms of that player s von Neumann-Morgenstern utility. We use u k to denote player k s payoff ij from I s pure strategy i and II s pure strategy j. Recall that each pure strategy assigns a feasible action to each information set of that player; that is, to each event in which that player takes an action. Each player s feasible strategies are his mixed strategies, i.e., the probability distributions over his pure strategies. If I uses the strategy x =(x i ) and II uses the strategy y =(y j ),where x 0 and Pi x i = 1 and similarly for y,thenk s expected payoff is PiPj x iu k y ij j. A Nash equilibrium is a pair (x; y) of strategies for the two players such that each player uses only pure strategies that are optimal responses to the other player s strategy; that is, x i X X > 0 only if u I y ij j =max u Ì y j j ; ` and similarly for player II. j Selten (1965, 1975) argued convincingly that only a subset of the Nash equilibria predicts behaviors by rational players. He proposed restricting equilibria to those that are perfect. One version of his 1975 (normal-form) definition says that an equilibrium (x; y) is perfect if it is the limit of a sequence of completely mixed strategies (i.e., every pure strategy is used), say (x h ;y h ) 0 converging to (x; y), suchthat x i X X > 0 only if u I =max ij yh u Ì j j yh ; j ` j and similarly for player II. Because each used pure strategy is an optimal response to a completely mixed strategy, it must be undominated. A perfect equilibrium therefore satisfies admissibility. 2 j j
One interpretation of Selten s construction is that each strategy used in a perfect equilibrium must remain optimal against some tremble by the other player; that is, small probabilities of using strategies that the equilibrium predicts will otherwise be unused. His other, equivalent definition emphasizes this aspect: an equilibrium (x; y) of a game G is perfect if there is a sequence (x h ;y h ; G h ) converging to (x; y; G), for which each (x h ;y h ) is an equilibrium of the perturbed game G h. In this version, G h is the strategy perturbation of G obtained by adjoining the constraints x h h and y h h,where ( h ; h ) 0 and lim h!1( h ; h ) = 0. This interpretation in terms of perturbations of the game had a profound effect on the subsequent development of theories of equilibrium refinements. The prominent feature of Selten s formulation is that it considers perturbations of mixed strategies. This approach was well-adapted to the specification of Kreps and Wilson s (1982) definition of sequential equilibrium, adapted from Selten s 1965 definition of subgame-perfect equilibrium, to address the games without proper subgames that motivate Selten s 1975 article. This criterion imposes the requirement of sequential rationality; that is, at each information set a player s continuation strategy is optimal with respect to a probability distribution (over possible histories) consistent with the game and Bayes Rule. A perfect equilibrium meets this requirement, using the limit of the conditional probabilities induced by the sequence of perturbed games. The perfect equilibria are therefore the sequential equilibria satisfying a strong form of admissibility. 1 Indeed, in generic extensive games the sets of perfect and sequential equilibria coincide (Blume and Zame, 1994). One version of this fact is that allowing all perturbations of payoffs (rather than strategies) produces exact coincidence between the resulting weakly perfect equilibria and the sequential equilibria. The different style of reasoning involved in using payoff perturbations to select equilibria is illustrated in the Appendix via the well-known Beer-Quiche game. In their study of hyperstability, Kohlberg and Mertens allowed all perturbations of normal-form payoffs. As they recognized, this enlarges the set of allowed perturbations. Each perturbation of strategies has the same net effect as a corresponding perturbation 1 Admissibility and normal-form perfection of equilibria are equivalent in two-person games. The analogous extensive-form definition of perfection allows use of dominated strategies. In Mertens (1995) example, the dominant-strategy equilibrium is not extensiveform perfect, and remarkably, every extensive-form perfect equilibrium uses a dominated strategy. Similar examples led van Damme (1984) to propose an alternative definition of quasi-perfection. 3
of payoffs. This can be seen by interpreting the trembles as chance moves whose small probabilities perturb the expected payoffs from each pure strategy specifying intended actions at information sets. Kohlberg and Mertens advanced Selten s agenda towards identifying essential or strictly perfect equilibria, those immune to all perturbations in a neighborhood of the game, not just a single sequence of strategy perturbations. Because such ideal equilibria need not exist, this step required reconsideration of the unit of analysis. 2 They demonstrated that each game s equilibria are partitioned into a finite number of closed connected components; and for generic extensive-form games, equilibria in the same component have the same outcomes, namely, they differ only off the equilibrium path and therefore their probability distributions over histories are the same. Combining these two results indicates that an equilibrium outcome immune to payoff perturbations is identified generically by a component that is essential; that is, every nearby game has an equilibrium near the component. 3 They also proved that every game has essential components. 4 Moreover, some are invariant in the sense that they are essential in every game with the same reduced normal form obtained by deleting redundant strategies (those strategies whose payoffs for all players are convex combinations of other strategies payoffs). Kohlberg and Mertens showed that an invariant essential component has several desirable properties. Invariance: Because an invariant component depends only on the reduced normal form, it does not depend on which among the many strategically equivalent extensive forms is used. Backward Induction: It contains perfect and even proper equilibria. 5 Such equilib- 2 Kohlberg and Mertens (1986, Appendix D) show that an equilibrium that is extensiveform perfect in every extensive game with the same reduced normal form is strictly perfect, provided all best responses are used. For many interesting games, however, this proviso is not satisfied. 3 Genericity is mostly immaterial here because a nongeneric case has a component that is the union of components induced by nearby games; a case where none of these subcomponents is essential would be doubly rare. 4 That such a component exists follows from their demonstration that the projection of the graph of the Nash equilibrium correspondence is homotopic to a homeomorphism. This is called the rubber sphere theorem, because it shows that when the space of games is mapped onto the surface of a sphere via a one-point compatification, the equilibrium graph is mapped to a deformation of a sphere of larger radius. Consequently, above every game lies a component for which every nearby game has an equilibrium nearby. 5 To be proper, the sequence justifying a perfect equilibrium must assign a probability of lower order to one pure strategy that is inferior to another (Myerson, 1978). 4
ria satisfy Selten s 1965 criterion (each strategy remains optimal in every subgame) and Kreps and Wilson s generalization to extended subgames in games with imperfect information. Proper equilibria are perfect, and therefore use only admissible strategies. Moreover, a proper equilibrium induces a sequential equilibrium in every extensive form with the same reduced normal form. Iterated Dominance: Its projection onto the equilibrium graph of a smaller game, obtained by deleting dominated strategies, contains an invariant essential component. Forward Induction: Its projection onto the equilibrium graph of a smaller game, obtained by deleting a strategy that is suboptimal at all its equilibria, contains an invariant essential component. 6 Further, within an invariant essential component is a minimal closed set (called hyperstable) with analogous properties; inside that is a minimal closed set (called fully stable) immune to perturbations of any finite set of pure or mixed strategies; and inside that is a minimal closed set (called stable) immune to perturbations of pure strategies. The latter (though omitting minimality) allows a homological definition proposed by Mertens (1989). In the next section we re-examine Kohlberg and Mertens and Mertens rejection of all these except stable sets, on the grounds that only stable sets equilibria exclude inadmissible strategies. We shall see that this has unfortunate consequences, because some stable sets reside in inessential components. This brief review omits a vast literature emanating from Selten s insights, but it includes some main themes. Extensive games require equilibrium refinements. Intuitive criteria such as admissibility, backward induction, and consistent conditional probabilities can be founded on consideration of strategy perturbations. Combining these with iterative elimination of dominated or suboptimal strategies meets further criteria, such as rationalizability (Pearce, 1984) and forward induction. To convey the scope of extensions not described above, we mention only the models of reputation effects in finitely-repeated and centipede games. These models are derived from Selten s (1978) study of the chain-store game. In repeated games (without 6 This property is motivated by games involving signaling. These games typically have multiple equilibria reflecting possible inferences from disequilibrium behaviors. The motive for forward induction insists that the conditional probabilities validating a sequential equilibrium should be zero for histories dependent on strategies that are suboptimal at every equilibrium in the component. Thus, one can prune those branches ofthegametreethatoccuronlywhensuchsuboptimalstrategiesareused. SeeBanks and Sobel (1987) and Cho and Kreps (1987). 5
or with stopping options), a perturbation of a strategy that repeats a particular behavior can attract imitation, leading to an equilibrium in which the perturbed strategy is the only one used, except in the last stages of the game. These models are relevant to the subsequent discussion because typically (as in the repeated chain-store and prisoners dilemma games) their striking feature is that it is a strategy that is inadmissible in the stage game that attracts imitation. 2 A Critique of Admissibility We now examine the role of admissibility in constructing equilibrium refinements such as those described above. First I suggest via examples that the known deficiencies of stable sets derive primarily from the imposition of admissibility criteria. The justification offered for admissibility is that it is a cornerstone of single-person decision theory, but this rationale applies only to selection among equilibria, not to selection among components (e.g., Kohlberg and Mertens, p. 1014; Mertens, 1989, p. 577(c)). Applying admissibility criteria to component selection is therefore misplaced. This still leaves ample latitude to select equilibria within an invariant essential component using criteria of admissibility or perfection. I conclude with cautionary remarks: in some contexts admissibility is unduly restrictive even in selecting among equilibria within an essential component. In the construction of stable sets, admissibility is ensured by allowing only strategy perturbations, rather than all payoff perturbations. This formulation might seem legitimate in view of the fact that differences between weak-perfect or sequential (justified by payoff perturbations) and perfect (justified by strategy perturbations) equilibria are nongeneric. In fact, however, the difference between the existential quantifier ( there exist perturbations yielding nearby equilibria ) used for these equilibrium selections, and the universal quantifier ( all perturbations yield nearby equilibria ) used for component selections can be substantial. In fact, generic games can have stable sets (immune to strategy perturbations) inside components that are not essential (immune to payoff perturbations). For this reason, I see no justification for restricting the allowable payoff perturbations when selecting among components. In the Appendix, three examples illustrate how the exclusion of some payoff perturbations allows implausible components to survive the criterion of stability. The first is van Damme s (1989) generic example in which there is a stable set that fails to induce an equilibrium in a subgame. Mertens (1989) revised definition produces a larger stable set that remedies this deficiency. Nevertheless, one sees easily that 6
these stable sets lie in an inessential component, as one can see from a simple payoff perturbation. The second example is the nongeneric game Trivial Pursuit in which there is a stable set whose outcome for one player is inferior to his stable outcome in a continuation that he has the option to elect. Again, a simple payoff perturbation suffices to eliminate this stable set lying in an inessential component. The third example is Cho and Kreps (1987) ingenious example of a stable set that motivates their conclusion that if there is an intuitive story to go with the full strength of stability, it is beyond our powers to offer it here (p. 220). But once again, a simple payoff perturbation suffices to show that this stable set resides in an inessential component. I conclude from such examples that in devising criteria for selecting among components, it is better to use all payoff perturbations. Of necessity, this approach may select essential components with equilibria using inadmissible strategies. The second step, therefore, is to use criteria such as admissibility to select, say, stable sets (or proper or perfect equilibria) within the selected component. 7 This two-step procedure was implicit in Kohlberg and Mertens original work, but unfortunately they abandoned it when they attempted to construct a definition that obtains all desirable properties in a single step. Besides the above examples, the deficiency of the single-step approach is evidenced by Mertens (1995) second example (which he calls pretty damning as to the behavior of stable sets in at least some non-generic games ) of a game with perfect information in which the unique stable set includes all admissible equilibria (which includes a twodimensional set), of which only one is subgame-perfect. Moreover, this subgame-perfect equilibrium yields the only stable outcome derived from all games in the neighborhood of payoff perturbations except those in a slice satisfying a single equality condition. Such examples indicate that a second step of selection within an essential component is inescapable. It is important to realize, however, that a two-step procedure has its own deficiencies. To illustrate, the Appendix presents a generic extensive-form game with an invariant essential component containing two stable sets (in the strong sense of Mertens 7 An example of the effectiveness of this two-step procedure even when admissibility is not an issue is van Damme s (1987, p. 119) generic extensive game with a two proper equilibria, one of which he argues is not sensible. A simple payoff perturbation suffices to show that this equilibrium s component is inessential. 7
definition) and two proper equilibria, reflecting the fact that its projection map (from its neighborhood in the equilibrium graph to the space of games) has degree 2. Neither of these stable sets is immune to payoff perturbations, even though they reside in a component that is immune. We turn now to the selection of equilibria within an (invariant) essential component and ask: Are the strategy perturbations the best set to invoke in selecting among equilibria? Clearly, Selten s pure-strategy perturbations are a minimal set sufficient to ensure admissibility and consistency. 8 On these grounds one can say that Selten s construction is exactly right. On the other hand, I think it is prudent to realize that admissibility is justified only if one is quite sure about the validity of some particular extensive form of the game. As Fudenberg, Kreps, and Levine (1988) prove, every pure-strategy equilibrium is the limit of strict equilibria of nearby elaborations ; that is, games in which players may have differing information about which perturbation applies. Fudenberg and Maskin (1986), Myerson (1986), van Damme (1987) illustrate particular examples. Further, Bagwell (1995) provides an example in which perturbations of the observability of actions induce payoff perturbations that justify an inadmissible equilibrium in the unique essential component containing the subgame-perfect equilibrium. Bagwell s Stackelberg game is described in the Appendix. In all these cases, the inadmissible equilibria are justified by payoff perturbations, but not by strategy perturbations. 3 Conclusion In sum, I see no harm and substantial advantages to selecting invariant essential components based on consideration of all payoff perturbations. Doing so eliminates anomalous stable sets in the known examples. As a second step, one can select a stable set or a proper or perfect equilibrium within the essential component, provided there is assurance that admissibility is an appropriate criterion. This second step is problematic, however, whenever potentially relevant perturbations of the extensive form generate payoff perturbations not induced by strategy perturbations. 8 These comments apply to normal-form perturbations; for extensive-form versions, one can also use the perturbations used by van Damme (1984) to define quasi-perfection. A somewhat larger set that includes mixed-strategy perturbations is apparently a nearly maximal set ensuring admissibility. One subset of mixed-strategy perturbations ensuring admissibility is used by Kohlberg and Mertens (1986, p. 1025) to construct proper equilibria; however, they show by example (p. 1026, Fig. 9) that the full set fails to ensure admissibility. 8
Acknowledgement: Research support was provided by NSF grant SES9207850. Srihari Govindan, John Hillas, and Eric van Damme kindly pointed out errors in a previous draft. 9
References Bagwell, Kyle (1995), Commitment and Observability in Games, Games and Economic Behavior, 8: 271-80. Banks, Jeffrey, and Joel Sobel (1987), Equilibrium Selection in Signaling Games, Econometrica, 55: 647-61. Blume, Lawrence, and William Zame (1994), The Algebraic Geometry of Perfect and Sequential Equilibrium, Econometrica, 62: 783-94. Cho, In-Koo, and David Kreps (1987), Signaling Games and Stable Equilibria, Quarterly Journal of Economics, 102: 179-221. Fudenberg, Drew, David Kreps, and David Levine (1988), On the Robustness of Equilibrium Refinements, Journal of Economic Theory, 44: 354-80. Fudenberg, Drew, and Eric Maskin (1986), The Folk Theorem in Repeated Games with Discounting and with Incomplete Information, Econometrica, 54: 533-54. Kohlberg, Elon, and Jean-François Mertens (1986), On the Strategic Stability of Equilibria, Econometrica, 54: 1003-38. Kreps, David (1990), A Course in Economic Theory. Princeton:PrincetonUniversityPress. Kreps, David, and Robert Wilson (1982), Sequential Equilibria, Econometrica, 50: 863-94. Mertens, Jean-François (1989), Stable Equilibria A Reformulation, Mathematics of Operations Research, 14: 575-625. Mertens, Jean-François (1995), Two Examples of Strategic Equilibrium, Games and Economic Behavior, 8: 378-88. Myerson, Roger (1978), Refinement of the Nash Equilibrium Concept, International JournalofGameTheory,7: 73-80. Myerson, Roger (1986), Multistage Games with Communication, Econometrica, 54: 323-58. Pearce, David (1984), Rationalizable Strategic Behavior and the Problem of Perfection, Econometrica, 52: 1029-51. Selten, Reinhard (1965), Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit, Zeitschrift fur die gesamte Staatswissenschaft, 121: 301-24. Selten, Reinhard (1975), Re-examination of the Perfectness Concept for Equilibrium Points in Extensive Games, International Journal of Game Theory, 4: 25-55. Selten, Reinhard (1978), The Chain-Store Paradox, Theory and Decision, 9: 127-59. 10
van Damme, Eric (1984), A Relation between Perfect Equilibria in Extensive-Form Games and Proper Equilibria in Normal-Form Games, International Journal of Game Theory, 13: 1-13. van Damme, Eric (1987), Stability and Perfection of Nash Equilibria. Berlin: Springer-Verlag. van Damme, Eric (1989), Stable Equilibria and Forward Induction, Journal of Economic Theory, 48: 476-96. van Damme, Eric (1991), Refinements of Nash Equilibrium, in J.J. Laffont (1992), Advances in Economic Theory: Sixth World Congress. New York: Cambridge University Press. 11
Appendix Examples Example 1: The Beer-Quiche Game The distinction between strategy and payoff perturbations is immaterial in generic cases. To illustrate, Figure 1 shows a version of Kreps Beer-Quiche game (1990, p. 465; Kohlberg and Mertens, 1986, p. 1031). In this game, player I chooses Left or Right and then player II chooses Across or Up; however, only I knows whether chance has chosen the payoffs along the Top or Bottom (with probabilities p and 1 0 p,where 0 < p < 1=2). The stable component contains the pure strategy equilibrium in which I surely chooses Left, and II chooses Up after Left and Across after Right. The unstable component is reversed: I surely chooses Right, and II chooses Across after Left and Up after Right. The latter violates forward induction because I s unused strategy that chooses Left after Top and Right after Bottom is suboptimal in that component; if it is deleted then II s action after Left becomes suboptimal, and if that is revised then I surely prefers Left. In terms of strategy perturbations this conclusion is evident because II s strategy cannot be optimal whenever the conditional probability of a deviation to Left is less after Top than Bottom. In terms of payoff perturbations the reasoning is altered: a perturbation that makes II s unused optimal strategy (i.e., Up invariably) superior requires II to use this strategy exclusively, in which case I prefers Left after Bottom. 9 Example 2: van Damme s Game Figure 2 depicts the generic extensive-form game studied by van Damme (1989) in which player I initially chooses either Up or to play a simultaneous-move subgame with player II. This game has a stable set with the payoff (2,2) obtained when player I chooses Up. This outcome cannot arise from an essential component: a negative perturbation of II s payoff from (Up, Middle) yields a game in which no equilibrium uses Up. The effect of suchaperturbationonii sbest-responseregionswithinthesimplexofi sstrategiesis shown in Figure 3. 9 Other than the implicit corollary seemingly implied by the elaborate proof in Kreps and Wilson (1982, strengthened by the results of Blume and Zame, 1994) of the generic coincidence of perfect and sequential equilibria, I know no direct demonstration of the generic equivalence of these two styles of reasoning. 12
From his study of this game, van Damme argued that the outcome Up violates a plausible interpretation of forward induction; moreover, no equilibrium in a minimal stable set induces an equilibrium in the subgame after I chooses Right. Mertens (1989) definition provides a larger stable set that does include subgame equilibria. Even so, it is stable with respect to strategy perturbations but not payoff perturbations. Example 3: Trivial Pursuit Figure 4 depicts the game Trivial Pursuit. In the continuation after player II initially chooses Right, were this known to I, I s strictly dominant strategy is Up, ensuring a payoff of 1, and the unique stable payoffs are (1,2). Now consider the full game with the parameter satisfying 0 <d<1. This adds an initial move (Up) by II from which he gets a certain payoff of 1 and I gets less both of which are inferior to their payoffs from the stable outcome after Right were it anticipated by I. However, there is a stable set in which all equilibria require both players to randomize their initial choices including a positive probability of Up initially by II. This stable set is not in the essential component, however, because every negative perturbation of either of II s payoffs from Up yields a game in which II surely chooses Right initially, as shown in Figure 5. This game is nongeneric only to the extent that II s payoffs from Up initially are replicated by a mixture of his payoffs from Right, so Up is a redundant strategy for II. Nevertheless, the normal form of this game cannot be reduced by deleting Up, because I s payoffs are affected. Example 4: Cho and Kreps Signaling Game Figure 6 depicts graphically the relevant data for a signaling game like Example 1 except that there are three types of player I and player II has three actions. They consider an equilibrium outcome in which all types of I choose Right (R). In the figure, the left simplex comprises II s possible beliefs about I s type after observing I s alternative action Left (L), and shows its partition into II s best response regions. The right simplex shows the partition of II s simplex of behavioral strategies after Left into I s best response regions; e.g., at the top the notation (L,R,R) indicates that types 1, 2, and 3 prefer Left, Right, and Right respectively if II is sufficiently likely to choose his Action 1 after Left. Cho and Kreps (1987) show that this configuration enables the outcome from the specified equilibrium [in which I uses the strategy (R,R,R)] to be stable. One sees easily, however, that this equilibrium is in an inessential component. A positive payoff perturbation of any one of II s optimal strategies ensures that it must be used exclusively, in which 13
case the corresponding type prefers Left. For instance, a positive payoff perturbation of Action 1 requires that II uses only Action 1 in response to Left, but then I s Type 1 prefers Left. Example 5: The Game Figure 7 depicts the generic extensive-form game, and Figure 8 shows the players best responses. The component in which I uses only strategy a is invariant and essential; in fact the projection from its neighborhood has degree 2. The degree is reflected in the presence of the two proper equilibria where II uses e or f, each of which is contained in a Mertens-stable set: the interval [e,de] or the interval [f,df]. Neither of these stable sets is immune to payoff perturbations. For instance, those perturbed games with two equilibria near f and def have no equilibria near [e,de]. Example 6: Bagwell s Stackelberg Game This example illustrates that even in the games of perfect information that motivate Selten s 1965 criterion of subgame perfection, the sole essential component can contain inadmissible equilibria justified as the limits of equilibria of nearby games with imperfect observability. Figure 9 displays the extensive form of Bagwell s (1995) Stackelberg game. First I chooses Up or Down, then chance reveals up or down to II (with a probability p of an erroneous report), and then II chooses Up or Down. If the error probability is zero then this is a game with perfect information: the unique subgame-perfect equilibrium has I choosing Up and II choosing Up after up and Down after down. This subgame equilibrium lies in the unique essential component that requires II to choose Down after down with conditional probability at least one-half. (The inessential component has I choosing Down and then II choosing Down, provided II s conditional probability of Down after up is at least one-half.) However, any perturbation that allows a positive probability of erroneous reports has only one equilibrium close to the essential component, and the equilibrium it is close to is not the perfect equilibrium, but rather the inadmissible equilibrium where II mixes equally between Up and Down after down. This is a robust feature of games with imperfectly observed actions. The important feature of the observability perturbation above is that it induces a payoff perturbation that cannot be mimicked by a strategy perturbation. Indeed, a perturbationofi sstrategiesiscapableofexcludingonlythedominatedstrategiesof II that choose Up invariably or Down invariably. In contrast, the payoff perturbation induced by a positive error probability is one that makes these strategies undominated. 14
Unlike strategy perturbations, which envision nonrational trembles, payoff perturbations account for slight chances of inaccurate observations. This is similar to the effect of reputational considerations in repeated games, where again the relevant perturbation is one that converts a dominated strategy into an undominated strategy. 15
2,0 3,0 0,1 I 1,1 p II II 1,0 3,1 2,1 I 1-p 0,0 Figure 1: The Beer-Quiche game. 16
2,2 Up I II Left Middle Right Top 0,0 3,2 0,3 I Bottom 3,3 0,2 3,0 Figure 2: Van Damme s example of a generic game with a stable set in an inessential component. 17
I s Strategies II s Best Responses Up II s Strategies I s Best Responses Middle Top Up Left Middle Right Bottom Bottom Top Left Right Dashed boundaries show effects of perturbing II s payoff, leaving only the equilibrium (Bottom, Left). Figure 3: Van Damme s example depicted graphically, showing the effect of perturbing player II s payoff. 18
0,1 1,2 d,1 1,0 I II 0,0 Up II Right 0,2 Figure 4: The game Trivial Pursuit with a stable set in an inessential component. 19
I s Strategies II s Best Responses II s Strategies I s Best Responses Up Right-Up Up Right-Right Right Up Right Up A negative perturbation of II s payoff from Up leaves only the equilibrium (Up, Right-Up). Right-Up Right-Right Figure 5: The best-responses for Trivial Pursuit, showing a perturbation of II s payoff from Up that eliminates the inessential component. 20
II s beliefs after I chooses L II s best action after I chooses L II s behavioral strategy after L I s best responses Type 1 Action 1 (L,R,R) Action 1 (R,R,R) 3 2 (R,L,R) (R,L,L) (R,R,L) Type 2 Type 3 Action 2 Action 3 Figure 6: The data for Cho and Kreps example of a stable set in an inessential component. 21
a 3,6 4,3 d e I II b 1,1 f 2,4 Player I II c 2,4 1,1 Figure 7: The generic extensive-form game. 22
a d c b (de) (def) (df) f d e a b c e f Figure 8: The players best-response correspondences for the game. 23
5,2 5,2 Up Up 3,1 Down 1-p p up down Down 3,1 Up II I II 6,3 6,3 Down 4,4 p up 1-p down 4,4 Figure 9: Bagwell s Stackelberg game with imperfect observation by II of I s action. 24