(2) For all t∈T; and for all a∈A, E{.()(- )}≥0 Note that it is of course sufficient to check for profitable deviations to pure actions. The product in parentheses is the probability that the action profile a-i will be played in state wE Q. This is the product of the probability that each a;,j+i, gets played, given that, in state w, player j i is of type t;(w) Next, the expression in curly braces is the difference in expected payoff from playing a and a! in state w Finally, these differences are weighted by the posterior probability that the state is actu- ally w, given that the true state must lie in ti Observe that, it is also possible to formulate the best-reply property for type-contingent randomized actions ai =(ai. ieN. As long as condition(1)above holds, it can be shown that the resulting definition is actually equivalent to the one given above: I will ask you to provide the details Where does this come from? We now take a step back and try to provide a rationale for this seemingly complex construc- tion(due largely to John Harsanyi, a recent Nobel laureate). It turns out that, indeed, this framework greatly simplifies the problem of describing strategic interaction in games with payoff uncertainty To keep the exposition simple(and in view of the fact that auctions will be the subject of a forthcoming lecture), let us consider a simple two-firm Cournot model with uncertainty about firn1’ s cost. Denote the latter variable by c E 10,5. Recall the setup: demand is given by P(Q) 2-Q and each firm can produce qi E [ 0, 1]. Firm 2's cost is zero It is probably easiest to think in standard equilibrium terms. Suppose that we are told that a given tuple(1.0, 11, 2 )is "an equilibrium outcome "of the game. Let us ask ourselves exactly what this implies At first blush, there is little action on Firm I's side: she(? )expects Firm 2 to produce g2 units, as per the equilibrium prescription, and best-responds to this: 1. c= BR1(92, c) (1-)-是g,forc∈{0.是 Now let us concentrate on Firm 2. Clearly, the latter cannot compute a best reply to his opponent's actions without formulating a conjecture about the relative likelihood of the Iso note that, strictly speaking, what is defined above is Bayesian Nash equilibrium for the mixed extension of the original game C(2) For all ti ∈ Ti and for all a 0 i ∈ Ai , X ω∈ti pi(ω) pi(ti)    X a−i∈A−i Y j6=i αj,tj (ω)(aj ) ! " X ai∈Ai αi,ti (ai)ui(ai , a−i , ω) ! − ui(a 0 i , a−i , ω) #   ≥ 0 Note that it is of course sufficient to check for profitable deviations to pure actions.1 The product in parentheses is the probability that the action profile a−i will be played in state ω ∈ Ω. This is the product of the probability that each aj , j 6= i, gets played, given that, in state ω, player j 6= i is of type tj (ω). Next, the expression in curly braces is the difference in expected payoff from playing ai and a 0 i in state ω. Finally, these differences are weighted by the posterior probability that the state is actu￾ally ω, given that the true state must lie in ti . Observe that, it is also possible to formulate the best-reply property for type-contingent randomized actions αi = (αi,ti )i∈N . As long as condition (1) above holds, it can be shown that the resulting definition is actually equivalent to the one given above: I will ask you to provide the details. Where does this come from? We now take a step back and try to provide a rationale for this seemingly complex construc￾tion (due largely to John Harsanyi, a recent Nobel laureate). It turns out that, indeed, this framework greatly simplifies the problem of describing strategic interaction in games with payoff uncertainty. To keep the exposition simple (and in view of the fact that auctions will be the subject of a forthcoming lecture), let us consider a simple two-firm Cournot model with uncertainty about Firm 1’s cost. Denote the latter variable by c ∈ {0, 1 2 }. Recall the setup: demand is given by P(Q) = 2 − Q and each firm can produce qi ∈ [0, 1]. Firm 2’s cost is zero. It is probably easiest to think in standard equilibrium terms. Suppose that we are told that a given tuple (q1,0, q1, 1 2 , q2) is “an equilibrium outcome” of the game. Let us ask ourselves exactly what this implies. At first blush, there is little action on Firm 1’s side: she (?) expects Firm 2 to produce q2 units, as per the equilibrium prescription, and best-responds to this: q1,c = BR1(q2, c) = (1 − c 2 ) − 1 2 q2, for c ∈ {0, 1 2 }. Now let us concentrate on Firm 2. Clearly, the latter cannot compute a best reply to his opponent’s actions without formulating a conjecture about the relative likelihood of the 1Also note that, strictly speaking, what is defined above is Bayesian Nash equilibrium for the mixed extension of the original game G. 4