Eco514-Game Theory Lecture 5: Games with Payoff Uncertainty(2 Marciano siniscalchi September 30, 1999 Introduction This lecture continues our analysis of games with payoff uncertainty. The three main objec tives are: (1)to illustrate the flexibility of the Harsanyi framework(or our version thereof) (2) to highlight the assumptions implicit in the conventional usage of the framework, and the possible departures; 3) to discuss its potential problems, as well as some solutions to the latter Cournot revisited Recall our Cournot model with payoff uncertainty. Firm 2's cost is known to be zero; Firm I s cost is uncertain, and will be denoted by cE, 23. Demand is given by P(Q)=2-Q and each firm can produce qi E 0, 1 We represent the situation as a game with payoff uncertainty as follows: let 32=10, 1 Ti=0J, 213, T2=(Q2) and P2(0)=T. It is easy to see that specifying P1 is not relevant for the purposes of Bayesian Nash equilibrium analysis: what matters there are the beliefs conditional on each ti E T1, but these will obviously be degenerate The following equalities define a Bayesian Nash equilibrium(do you see where these com oIn q1(0) q1(G) 92 1 q1(0)+(1-丌)q1(Eco514—Game Theory Lecture 5: Games with Payoff Uncertainty (2) Marciano Siniscalchi September 30, 1999 Introduction This lecture continues our analysis of games with payoff uncertainty. The three main objec￾tives are: (1) to illustrate the flexibility of the Harsanyi framework (or our version thereof); (2) to highlight the assumptions implicit in the conventional usage of the framework, and the possible departures; (3) to discuss its potential problems, as well as some solutions to the latter. Cournot Revisited Recall our Cournot model with payoff uncertainty. Firm 2’s cost is known to be zero; Firm 1’s cost is uncertain, and will be denoted by c ∈ {0, 1 2 }. Demand is given by P(Q) = 2 − Q and each firm can produce qi ∈ [0, 1]. We represent the situation as a game with payoff uncertainty as follows: let Ω = {0, 1 2 }, T1 = {{0}, { 1 2 }}, T2 = {Ω} and p2(0) = π. It is easy to see that specifying p1 is not relevant for the purposes of Bayesian Nash equilibrium analysis: what matters there are the beliefs conditional on each t1 ∈ T1, but these will obviously be degenerate. The following equalities define a Bayesian Nash equilibrium (do you see where these come from?): q1(0) = 1 − 1 2 q2 q1( 1 2 ) = 3 4 − 1 2 q2 q2 = 1 − 1 2  πq1(0) + (1 − π)q1( 1 2 )  1