L R t1(w) al(w) pl(w) t2(w)a2(w)p2(w t1 0 22 R 0.4 ,10.0 B0.0 tiT 0 L 0.5 B Figure 1: A game and a model for it The right-hand table includes all the information required by Definition 1. In particular note that it implicitly defines the partitions Ti, i=1, 2, via the possibility correspondences As previously advertised, at each state w E S2 in a model, players actions and beliefs are completely specified. For instance, at wl, the profile(t, R)is played, Player 1 is certain that Player 2 chooses L (note that this belief is incorrect), and Player 2 is certain that Player 1 chooses T(which is a correct belief). Thus, given their beliefs, Player 1 is rational (T is a best reply to L) and Player 2 is not(R is not a best reply to T c. Moreover, note that, at w2, Player 2 believes that the state is w2(hence, that Player 1 nooses T) with probability and that it is w(hence, that Player 1 chooses B with probability & At w2 Player 2 chooses L, which is her unique best reply given her beliefs Thus, we can also say that at wn Player 1 assigns probability one to the event that the state is really w2, and hence that (i) Player 2s beliefs about Player 1's actions are given b (T: 6. B); and that(i)Player 2 chooses L. Thus, at w Player 1 is"certain"that Player 2 is rational. Of course, note that at w/ Player 2 is really not rational We can push this quite a bit further. For instance, type t? of Player 2 assigns probability d to w3, a state in which Player 1 is not rational (she is certain that the state is w3, hence that Player 2 chooses L, but she plays B). Hence, at wl, Player 1 is"certain"that Player 2 assigns probability 6 to the"event "that she (i) believes that 2 chooses L, and (i) play B-hence. she is not rational. this is a statement involving three orders of beliefs it also corresponds to an incorrect belief: at wl, Player 2 is certain that Player 1 chooses T and is of type tl-hence, that she is rational! We are ready for formal definitions of "rationality"and"certainty "Recall that, given any belief a-i E A(A-i) for Player i, ri(a-i) is the set of best replies for i given a-i First, a preliminary notion Definition 2 Fix a game G=(N, (Ai, uiieN) and a model M=( Q, (Ti, ai, pilieN) for G The first-order beliefs function a-i: Q-A(A-i for Player i is defined by vu∈9,a-i∈A-:a-()(a-)=p1({u:Wj≠i,a(u)=a3H(u) That is, the probability of a profile a_i E A-i is given by the(conditional) probability of all states where that profile is played. Notice that the function a-i( is T--measurable, just like ai(). Also note that this is a belief about players jti, held by player iL R T 1,1 0,0 B 0,0 1,1 ω t1(ω) a1(ω) p1(ω) t2(ω) a2(ω) p2(ω) ω1 t 1 1 T 0 t 1 2 R 0.4 ω2 t 1 1 T 0.5 t 2 2 L 0.5 ω3 t 2 1 B 0.5 t 2 2 L 0.1 Figure 1: A game and a model for it The right-hand table includes all the information required by Definition 1. In particular, note that it implicitly defines the partitions Ti , i = 1, 2, via the possibility correspondences ti : Ω ⇒ Ω. As previously advertised, at each state ω ∈ Ω in a model, players’ actions and beliefs are completely specified. For instance, at ω1, the profile (T,R) is played, Player 1 is certain that Player 2 chooses L (note that this belief is incorrect), and Player 2 is certain that Player 1 chooses T (which is a correct belief). Thus, given their beliefs, Player 1 is rational (T is a best reply to L) and Player 2 is not (R is not a best reply to T). Moreover, note that, at ω2, Player 2 believes that the state is ω2 (hence, that Player 1 chooses T) with probability 0.5 0.5+0.1 = 5 6 , and that it is ω3 (hence, that Player 1 chooses B) with probability 1 6 . At ω2 Player 2 chooses L, which is her unique best reply given her beliefs. Thus, we can also say that at ω1 Player 1 assigns probability one to the event that the state is really ω2, and hence that (i) Player 2’s beliefs about Player 1’s actions are given by ( 5 6 ,T; 1 6 ,B); and that (ii) Player 2 chooses L. Thus, at ω1 Player 1 is “certain” that Player 2 is rational. Of course, note that at ω1 Player 2 is really not rational! We can push this quite a bit further. For instance, type t 2 2 of Player 2 assigns probability 1 6 to ω3, a state in which Player 1 is not rational (she is certain that the state is ω3, hence that Player 2 chooses L, but she plays B). Hence, at ω1, Player 1 is “certain” that Player 2 assigns probability 1 6 to the “event” that she (i) believes that 2 chooses L, and (ii) plays B—hence, she is not rational. This is a statement involving three orders of beliefs. It also corresponds to an incorrect belief: at ω1, Player 2 is certain that Player 1 chooses T and is of type t 1 1—hence, that she is rational! We are ready for formal definitions of “rationality” and “certainty.” Recall that, given any belief α−i ∈ ∆(A−i) for Player i, ri(α−i) is the set of best replies for i given α−i . First, a preliminary notion: Definition 2 Fix a game G = (N,(Ai , ui)i∈N ) and a model M = (Ω,(Ti , ai , pi)i∈N ) for G. The first-order beliefs function α−i : Ω → ∆(A−i) for Player i is defined by ∀ω ∈ Ω, a−i ∈ A−i : α−i(ω)(a−i) = pi ({ω 0 : ∀j 6= i, aj (ω 0 ) = aj}|ti(ω)) That is, the probability of a profile a−i ∈ A−i is given by the (conditional) probability of all states where that profile is played. Notice that the function α−i(·) is Ti-measurable, just like ai(·). Also note that this is a belief about players j 6= i, held by player i. 3