M R 0,20, l.1 Figure 2: A typical case(OR 220.1 Formal definitions The simple games in the previous section emphasize that off-equilibrium beliefs matter: they determine whether or not a given profile is an equilibrium. Moreover, the putative equilibrium profile does not yield a complete specification of beliefs In particular, at any information set, a(behavioral) strategy profile B yields a complete specification of continuation strategies, but may fail to provide information about past play- precisely because, at information sets off the path of play generated by 6, it must necessarily be the case that at least one player j made choices other than those prescribed by B,! The original profile is clearly silent about such alternative choices Note that this problem does not arise in games with observed actions, precisely because past play is observable(so there is no uncertainty about previous moves ). However, it does arise in Bayesian games with observed actions; see the next section for details Kreps and Wilson propose the following definition Definition 1 Fix a general extensive game T. a belief system is a map u △(H\z) such that, for every i∈ N and Ii∈1,p(L1)()=1 It is most helpful to view a belief system as a summary representation of past play leading to information sets. That is, informally speaking, one could imagine that, if the play reaches an information set Ii (where Player i has to move) which is not consistent with the originally expected behavioral strategy profile B, players conclude that a different profile B is actually being played, and base their inferences and forecasts on the latter However, based on simple consistency considerations, B and B should agree at least as far as moves at information sets following I are concerned. But, if this is the case, then all that is required in addition to B in order to verify the sequential rationality of Bi at Ii is the distribution over histories in Ii induced by By-which is precisely the type of information enco oded in u(n)L 2,2 1 q M ￾ ￾ ￾ ￾￾ R ❅ ❅ ❅ ❅ q ❅ r ❆ ❆ ❆ ❆ ❆ 0,2 l ✁ ✁ ✁ ✁ ✁ 3,1 q 1,1 ❆ ❆ ❆ ❆ ❆ 0,2 ✁ ✁ ✁ ✁ ✁ 2 I2 Figure 2: A typical case (OR 220.1) Formal definitions The simple games in the previous section emphasize that off-equilibrium beliefs matter : they determine whether or not a given profile is an equilibrium. Moreover, the putative equilibrium profile does not yield a complete specification of beliefs. In particular, at any information set, a (behavioral) strategy profile β yields a complete specification of continuation strategies, but may fail to provide information about past play— precisely because, at information sets off the path of play generated by β, it must necessarily be the case that at least one player j made choices other than those prescribed by βj ! The original profile is clearly silent about such alternative choices. Note that this problem does not arise in games with observed actions, precisely because past play is observable (so there is no uncertainty about previous moves). However, it does arise in Bayesian games with observed actions; see the next section for details. Kreps and Wilson propose the following definition. Definition 1 Fix a general extensive game Γ. A belief system is a map µ : S i∈N Ii → ∆(H \ Z) such that, for every i ∈ N and Ii ∈ Ii , µ(Ii)(Ii) = 1. It is most helpful to view a belief system as a summary representation of past play leading to information sets. That is, informally speaking, one could imagine that, if the play reaches an information set Ii (where Player i has to move) which is not consistent with the originally expected behavioral strategy profile β, players conclude that a different profile β 0 is actually being played, and base their inferences and forecasts on the latter. However, based on simple consistency considerations, β 0 and β should agree at least as far as moves at information sets following I are concerned. But, if this is the case, then all that is required in addition to β in order to verify the sequential rationality of βi at Ii is the distribution over histories in Ii induced by β 0—which is precisely the type of information encoded in µ(I). 3