From now on, the basic unit of our analysis will be a pair(B, u) consisting of a behavioral strategy profile and a belief system; such a pair is called an assessment Focus on games with perfect recall. In keeping with conventional usage, we define se- quentially rational assessments. First, define the conditional outcome function O(B, ur)to be the distribution over Z induced by the assessment(6, u) conditional upon starting the game at I: formally, for every z E Z, if no h E I is a subhistory of z, then O(6, uD)(a)=0; otherwise, by perfect recall, there can be only one such subhistory h(if there were two, then one would have to be a subhistory of the other, and an information set cannot contain a pair of nested histories). Thus, z=(h, a,...,a), and we let O,|D(2)=H(D)(h)(a.a(h,n2…,a)(a4+) (recall that, with some abuse of notation, for h E Ii Ii, Bi (h)=B,() Definition 2 Fix a general extensive game r. An assessment(6, u)for r is sequentially rational iff, for every player i E N, every information set Ii E Ii, and every behaviora strategy B∈B ∑U(2)O(.川41)≥∑U(2)O(1,B2D z∈Z It should be obvious that some relationship between belief systems and behavioral strate- gies in an assessment(B, u)should be maintained In particular, at least at information sets consistent with B, u should be derived from B by means of Bayes'rule tle But we may reasonably require more than this. For instance, consider a modification of the game in Figure 2 in which Player 1 chooses between L and C, and if 1 chooses C, a third player 1'chooses between M and R; then Player 2 has to choose observing only Player 1's initial move. That is, Player 1s choice among L, M and R is split between two players, 1 and 1, choosing consecutively Consider a behavioral strategy profile in which Player 1 chooses L and Player 1'chooses M. Then, even after a deviation by Player 1, we would probably still want to require that u((C, M), ( C, R))((C, M))=1; that is, a deviation by Player 1 should not affect Player 2s beliefs about the choices of Player 1. However, strictly speaking, Bayes'rule does not apply to this information set, because it lies off the path induced by the behavioral strategy profile under consideration Note incidentally that if 1 and 1' were the same player, this argument would be less convincing; one would have to invoke some kind of trembles or mistakes, whereas the argu ment in the case where 1 and 1' are distinct is essentially based on independence. As will be clear, however, the standard definition of sequential equilibrium does not discriminate between these two casesFrom now on, the basic unit of our analysis will be a pair (β, µ) consisting of a behavioral strategy profile and a belief system; such a pair is called an assessment. Focus on games with perfect recall. In keeping with conventional usage, we define se￾quentially rational assessments. First, define the conditional outcome function O(β, µ|I) to be the distribution over Z induced by the assessment (β, µ) conditional upon starting the game at I: formally, for every z ∈ Z, if no h ∈ I is a subhistory of z, then O(β, µ|I)(z) = 0; otherwise, by perfect recall, there can be only one such subhistory h (if there were two, then one would have to be a subhistory of the other, and an information set cannot contain a pair of nested histories). Thus, z = (h, a1 , . . . , aK), and we let O(β, µ|I)(z) = µ(I)(h) K Y−1 `=1 βP(h,a1,...,a`) (h, a1 , . . . , a` )(a `+1) (recall that, with some abuse of notation, for h ∈ Ii ∈ Ii , βi(h) = βi(Ii)). Definition 2 Fix a general extensive game Γ. An assessment (β, µ) for Γ is sequentially rational iff, for every player i ∈ N, every information set Ii ∈ Ii , and every behavioral strategy β 0 i ∈ Bi , X z∈Z Ui(z)O(β, µ|Ii) ≥ X z∈Z Ui(z)O((β 0 i , β−i)µ|Ii) It should be obvious that some relationship between belief systems and behavioral strate￾gies in an assessment (β, µ) should be maintained. In particular, at least at information sets consistent with β, µ should be derived from β by means of Bayes’ rule. But we may reasonably require more than this. For instance, consider a modification of the game in Figure 2 in which Player 1 chooses between L and C, and if 1 chooses C, a third player 1 0 chooses between M and R; then Player 2 has to choose observing only Player 1’s initial move. That is, Player 1’s choice among L, M and R is split between two players, 1 and 10 , choosing consecutively. Consider a behavioral strategy profile in which Player 1 chooses L and Player 10 chooses M. Then, even after a deviation by Player 1, we would probably still want to require that µ({(C, M),(C, R)})((C, M)) = 1; that is, a deviation by Player 1 should not affect Player 2’s beliefs about the choices of Player 1 0 . However, strictly speaking, Bayes’ rule does not apply to this information set, because it lies off the path induced by the behavioral strategy profile under consideration. [Note incidentally that if 1 and 10 were the same player, this argument would be less convincing; one would have to invoke some kind of trembles or mistakes, whereas the argu￾ment in the case where 1 and 1 0 are distinct is essentially based on independence. As will be clear, however, the standard definition of sequential equilibrium does not discriminate between these two cases.] 4