Requirements 1 through 3 capture the spirit of a perfect Bayesian equilibrium. The crucial new feature of this equilibrium concept is due to Kreps and Wilson (1982): beliefs are elevated to the level of importance of strategies in the definition of equilibrium. Formally, an equilibrium no longer consists of just a strategy for each player but now also includes a belief for each player at each information set at which the player has the move. The advantage of making the players' beliefs explicit in this way is that, just as in earlier chapters we insisted that the players choose credible strategies, we can now also insist that they hold reasonable beliefs, both on the equilibrium path (in Requirement 3) and off the equilibrium path (in Requirement 4, which follows, and in others in Section 4.4).

In simple economic applications—including the signaling game in Section 4.2.A and the cheap-talk game in Section 4.3.A—Requirements 1 through 3 not only capture the spirit but also constitute the definition of perfect Bayesian equilibrium. In richer economic applications, however, more requirements need to be imposed to eliminate implausible equilibria. Different authors have used different definitions of perfect Bayesian equilibrium. All definitions include Requirements 1 through 3; most also include Requirement 4; some impose further requirements.2 In this chapter,

1 Kreps and Wilson formalize this perspective on equilibrium by defining sequential equilibrium, an equilibrium concept that is equivalent to perfect Bayesian equilibrium in many economic applications but in some cases is slightly stronger. Sequential equilibrium is more complicated to define and apply than perfect Bayesian equilibrium, so most authors now use the latter. Some who do so (imprecisely) refer to the equilibrium concept they apply as sequential equilibrium. Kreps and Wilson show that in any finite game (i.e., any game with a finite number of players, types, and possible moves) there exists a sequential equilibrium; this implies that in any finite game there exists a perfect Bayesian equilibrium.

2To give a sense of the issues not addressed by Requirements 1 through 4, suppose players 2 and 3 have observed the same events and then both observe a deviation from the equilibrium by player 1. In a game of incomplete information in which player 1 has private information, should players 2 and 3 hold the same belief about player l's type; in a game of complete information, should players 2 and 3 hold the same belief about earlier unobserved moves by player 1? Similarly, if players 2 and 3 have observed the same events and then player 2 deviates from the equilibrium, should player 3 change his or her belief about player l's type, or about l's unobserved moves?

we take Requirements 1 through 4 to be the definition of perfect Bayesian equilibrium.3

Requirement 4 At information sets off the equilibrium path, beließ are determined by Bay es' rule and the players' equilibrium strategies where possible.

Definition A perfect Bayesian equilibrium consists of strategies and beliefs satisfying Requirements 1 through 4.

It would of course be useful to give a more precise statement of Requirement 4—one that avoids the vague instruction "where possible." We will do so in each of the economic applications analyzed in subsequent sections. For now, we use the three-player games in Figures 4.1.4 and 4.1.5 to illustrate and motivate Requirement 4. (The top, middle, and bottom payoffs are for players 1, 2, and 3, respectively.)

This game has one subgame: it begins at player 2's singleton information set. The unique Nash equilibrium in this subgame between players 2 and 3 is (L,R'), so the unique subgame-perfect Nash equilibrium of the entire game is (D,L,R'). These strategies and the belief p = 1 for player 3 satisfy Requirements 1 through 3. They also trivially satisfy Requirement 4, since there is no information set off this equilibrium path, and so constitute a perfect Bayesian equilibrium.

Now consider the strategies (A,L,L')r together with the belief p = 0. These strategies are a Nash equilibrium—no player wants to deviate unilaterally. These strategies and belief also satisfy Requirements 1 through 3—player 3 has a belief and acts optimally given it, and players 1 and 2 act optimally given the subsequent strategies of the other players. But this Nash equilibrium is not subgame-perfect, because the unique Nash equilibrium of the game's only subgame is (L,R'). Thus, Requirements 1 through 3 do not guarantee that the player's strategies are a subgame-perfect Nash equilibrium. The problem is that player 3's belief (p = 0)

3Fudenberg and Tirole (1991) give a formal definition of perfect Bayesian equilibrium for a broad class of dynamic games of incomplete information. Their definition addresses issues like those raised in footnote 2. In the simple games analyzed in this chapter, however, such issues do not arise, so their definition is equivalent to Requirements 1 through 4. Fudenberg and Tirole provide conditions under which their perfect Bayesian equilibrium is equivalent to Kreps and Wilson's sequential equilibrium.

is inconsistent with player 2's strategy (L), but Requirements 1 through 3 impose no restrictions on 3's belief because 3's information set is not reached if the game is played according to the specified strategies. Requirement 4, however, forces player 3's belief to be determined by player 2's strategy: if 2's strategy is L then 3's belief must be p = 1; if 2's strategy is -R then 3's belief must be p = 0. But if 3's belief is p — 1 then Requirement 2 forces 3's strategy to be R', so the strategies (A, L, L') and the belief p = 0 do not satisfy Requirements 1 through 4.

As a second illustration of Requirement 4, suppose Figure 4.1.4 is modified as shown in Figure 4.1.5: player 2 now has a third possible action, A', which ends the game. (For simplicity, we ignore the payoffs in this game.) As before, if player l's equilibrium strategy is A then player 3's information set is off the equilibrium path, but now Requirement 4 may not determine 3's belief from 2's strategy. If 2's strategy is A' then Requirement 4 puts no restrictions on 3's belief, but if 2's strategy is to play L with

probability q\, R with probability q2, and A! with probability 1 — qi — qi, where q\ + q2 > 0, then Requirement 4 dictates that 3's belief be p = qi/(qi + q2)-

To conclude this section, we informally relate perfect Bayes-ian equilibrium to the equilibrium concepts introduced in earlier chapters. In a Nash equilibrium, each player's strategy must be a best response to the other players' strategies, so no player chooses a strictly dominated strategy. In a perfect Bayesian equilibrium, Requirements 1 and 2 are equivalent to insisting that no player's strategy be strictly dominated beginning at any information set. (See Section 4.4 for a formal definition of strict dominance beginning at an information set.) Nash and Bayesian Nash equilibrium do not share this feature at information sets off the equilibrium path; even subgame-perfect Nash equilibrium does not share this feature at some information sets off the equilibrium path, such as information sets that are not contained in any subgame. Perfect Bayesian equilibrium closes these loopholes: players cannot threaten to play strategies that are strictly dominated beginning at any information set off the equilibrium path.

As noted earlier, one of the virtues of the perfect Bayesian equilibrium concept is that it makes the players' beliefs explicit and so allows us to impose not only Requirements 3 and 4 but also further requirements (on beliefs off the equilibrium path). Since perfect Bayesian equilibrium prevents player i from playing a strategy that is strictly dominated beginning at an information set off the equilibrium path, perhaps it is not reasonable for player j to believe that player i would play such a strategy. Because perfect Bayesian equilibrium makes the players' beliefs explicit, however, such an equilibrium often cannot be constructed by working backwards through the game tree, as we did to construct a subgame-perfect Nash equilibrium. Requirement 2 determines a player's action at a given information set based in part on the player's belief at that information set. If either Requirement 3 or 4 applies at this information set, then it determines the player's belief from the players' actions higher up the game tree. But Requirement 2 determines these actions higher up the game tree based in part on the players' subsequent strategies, including the action at the original information set. This circularity implies that a single pass working backwards through the tree (typically) will not suffice to compute a perfect Bayesian equilibrium.

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