For T=5, we get (0)=0625;91()=0.375;=075 (by comparison, a is the equilibrium quantity for both firms if Firm 1's cost is always c=0.) Textbook analysis Recall that, for any probability measure q E A(Q)and player i E N, we defined the event lai= w: pi(wlti w))=q. In this game, [p2]2=Q: that is, at any state of the world, Player 2's beliefs are given by p2. By way of comparison, it is easy to see that it cannot be the case that Ipili=@(why? regardless of how we specify pi For notational ease(and also as a "sneak preview"of our forthcoming treatment of interactive epistemology), we introduce the belief operator. Recall that, for every iE N and wES, ti(@)denotes the cell of the partition Ti containing w Definition 1 Given a game with payoff uncertainty G=(N, Q2, (Ai, ui, Ti)ieN), Player i's belief operator is the map Bi: 22-2 defined by VECO, B(E)=wEQ: Pi(Elt;(w))=1] A more appropriate name for Bi( would perhaps be certainty operator, but we shall follow traditional usage. Ifw E B (E), we shall say that "At w, Player i is certain that(or believes that)E is true. "Certainty is thus taken to denote probability one belief Now return to the Cournot game and take the point of view of Player 1. Since Ip2]2=Q2 it is trivially true that B1(]2)=Q; in words, at any state w E Q, Player 1 is certain that Player 2's beliefs are given by p2(i.e. by T ). By the exact same argument, at any state w, Player 2 is certain that Player 1 is certain that Player 2s beliefs are given by p2: that is, B2(B1(P22)=Q2). And so on and so forth The key point is that Harsanyi's model of games with payoff uncertainty, together with a specification o f the players'priors, easily generates infinite hierarchies of interactive beliefs that is "beliefs about beliefs edy Although you may not immediately"see"these hierarchies, they are there-and they are ilv retrieved uncertainty concerning Firm I's payoffs, but no uncertainty about Firm 2s beliee s there is We can summarize the situation as follows: in the setup under considerationFor π = 1 2 , we get q1(0) = 0.625; q1( 1 2 ) = 0.375; q2 = 0.75 (by comparison, 2 3 is the equilibrium quantity for both firms if Firm 1’s cost is always c = 0.) Textbook analysis Recall that, for any probability measure q ∈ ∆(Ω) and player i ∈ N, we defined the event [q]i = {ω : pi(ω|ti(ω)) = q}. In this game, [p2]2 = Ω: that is, at any state of the world, Player 2’s beliefs are given by p2. By way of comparison, it is easy to see that it cannot be the case that [p1]1 = Ω (why?) regardless of how we specify p1. For notational ease (and also as a “sneak preview” of our forthcoming treatment of interactive epistemology), we introduce the belief operator. Recall that, for every i ∈ N and ω ∈ Ω, ti(Ω) denotes the cell of the partition Ti containing ω. Definition 1 Given a game with payoff uncertainty G = (N, Ω,(Ai , ui , Ti)i∈N ), Player i’s belief operator is the map Bi : 2Ω → 2 Ω defined by ∀E ⊂ Ω, Bi(E) = {ω ∈ Ω : pi(E|ti(ω)) = 1} A more appropriate name for Bi(·) would perhaps be certainty operator, but we shall follow traditional usage. If ω ∈ Bi(E), we shall say that “At ω, Player i is certain that (or believes that) E is true.” Certainty is thus taken to denote probability one belief. Now return to the Cournot game and take the point of view of Player 1. Since [p2]2 = Ω, it is trivially true that B1([p2]2) = Ω; in words, at any state ω ∈ Ω, Player 1 is certain that Player 2’s beliefs are given by p2 (i.e. by π). By the exact same argument, at any state ω, Player 2 is certain that Player 1 is certain that Player 2’s beliefs are given by p2: that is, B2(B1([p2]2) = Ω). And so on and so forth. The key point is that Harsanyi’s model of games with payoff uncertainty, together with a specification of the players’ priors, easily generates infinite hierarchies of interactive beliefs, that is “beliefs about beliefs...” Although you may not immediately “see” these hierarchies, they are there—and they are easily retrieved. We can summarize the situation as follows: in the setup under consideration, there is uncertainty concerning Firm 1’s payoffs, but no uncertainty about Firm 2’s beliefs. 2