of possible costs for Firm 1), for a bet X: 10, 51-R to be profitable to Firm 1 conditional on every ti(w)=w it must be the case that X(O)>0 and X()>0; but then no such bet can be acceptable to Firm 2 Justifying Harsanyi Models There are two related problems with the models of payoff uncertainty we have proposed The first has to do with generality. We have seen that we can construct rather involved models of interactive beliefs in the presence of payoff uncertainty. In particular, our models generate complicated hierarchies of beliefs rather easily However, let us ask the reverse question. Given some collection of hierarchies of beliefs can we always exhibit a model which generates in in some state? The second question has to do with one implicit assumption of the model. In order to make statements about Player 1's beliefs concerning Player 2s beliefs, Player 1 must be assumed to"know"p2 and T2. In order to make statements about Player 1s beliefs about Player 2's beliefs about Player 1's beliefs, we must assume that Player 2"knows"pi and Ti and that Player 1"knows"this More concisely: the model itself must be "common knowledge". But we cannot even formulate this assumption Both issues are addressed in a brilliant paper by Mertens and Zamir(Int. J. Game Theory, 1985). The basic idea is as follows Let us focus on two-player games. First, let us fix a collection Q2 of payoff parameters This set is meant to capture "physical uncertainty, or more generally any kind of uncertainty that is not related to players' beliefs a player's beliefs about Q2 are by definition represented by points in A(Q0).Now, here's the idea: if we wish to represent a player's beliefs about: (1)Q2 and(2) her opponent's beliefs about oo it is natural to consider the set g2=90×△(9°) and describe a player's beliefs about(1)and(2)above as points in A(@). The idea readily generalizes: suppose we have constructed a set S as above: then and we represent beliefs about(1)S and(2)the opponent's beliefs about 2 as points in A(Q+). Of course these are complicated spaces(even if Q20 is finite), but in any case the definitions are straightforward We wish to describe a player's beliefs by an infinite sequence e=(p, p2,...)of probability measures such that, for each k>1, PE 4(Q2k-). That is, each player's beliefs are describedof possible costs for Firm 1), for a bet X : {0, 1 2 } → R to be profitable to Firm 1 conditional on every t1(ω) = ω it must be the case that X(0) > 0 and X( 1 2 ) > 0; but then no such bet can be acceptable to Firm 2. Justifying Harsanyi Models There are two related problems with the models of payoff uncertainty we have proposed. The first has to do with generality. We have seen that we can construct rather involved models of interactive beliefs in the presence of payoff uncertainty. In particular, our models generate complicated hierarchies of beliefs rather easily. However, let us ask the reverse question. Given some collection of hierarchies of beliefs, can we always exhibit a model which generates in in some state? The second question has to do with one implicit assumption of the model. In order to make statements about Player 1’s beliefs concerning Player 2’s beliefs, Player 1 must be assumed to “know” p2 and T2. In order to make statements about Player 1’s beliefs about Player 2’s beliefs about Player 1’s beliefs, we must assume that Player 2 “knows” p1 and T1, and that Player 1 “knows” this. More concisely: the model itself must be “common knowledge”. But we cannot even formulate this assumption! Both issues are addressed in a brilliant paper by Mertens and Zamir (Int. J. Game Theory, 1985). The basic idea is as follows. Let us focus on two-player games. First, let us fix a collection Ω 0 of payoff parameters. This set is meant to capture “physical uncertainty,” or more generally any kind of uncertainty that is not related to players’ beliefs. A player’s beliefs about Ω0 are by definition represented by points in ∆(Ω0 ). Now, here’s the idea: if we wish to represent a player’s beliefs about: (1) Ω 0 and (2) her opponent’s beliefs about Ω 0 , it is natural to consider the set Ω 1 = Ω0 × ∆(Ω0 ) and describe a player’s beliefs about (1) and (2) above as points in ∆(Ω1 ). The idea readily generalizes: suppose we have constructed a set Ω k as above: then Ω k+1 = Ωk × ∆(Ωk ) and we represent beliefs about (1) Ω k and (2) the opponent’s beliefs about Ω k as points in ∆(Ωk+1). Of course these are complicated spaces (even if Ω 0 is finite), but in any case the definitions are straightforward. We wish to describe a player’s beliefs by an infinite sequence e = (p 1 , p2 , . . .) of probability measures such that, for each k ≥ 1, p k ∈ ∆(Ωk−1 ). That is, each player’s beliefs are described 7