Heretofore, all new applications must also answer one of the three questions below:
Consider a linear Cournot duopoly with the true inverse demand function P = a − Q and zero marginal costs, where P is price, a> 0,and Q = q1 + q2 is the total supply of a good. Now imagine that each firm i ∫ N = {1, 2} perceives the inverse demand function as P = a + bi − Q, where bi ∫ R is the bias in i’s perception. Let G (b1,b2) be the game in which b1 and b2 are common-knowledge.
(a) Show that G (b1,b2) has a unique rationalisable strategy profile. Compute the true payoffs u1 (b1,b2) and u2 (b1,b2) at the rationalisable strategy profile – computed by using P = a − Q.
(b) Consider the meta game Γ =(N, R, R,u1,u2) , where the strategies are choices of b1 and b2,and u1 and u2 are as in (a). Show that Γ is super-modular in a proper order, has a unique Nash equilibrium b∗ ,and that b∗ i > 0 for each i ∈ N . Show that the replicator dynamics for Γ (using the true payoffs) converges to b*
(c) Now consider an evolutionary learning process in which the agents not only develop their perceptions (i.e., b1 and b2) but also learn how to play the game G (b1,b2) given perceptions. Assume that the learning process is a “two-tiered†replicator dynamics in which they learn how to play G (b1,b2) infinitely faster than they change their perceptions, i.e., given any perception-pair (b1,b2), the play converges to the limit of the dynamics for fixed (b1,b2) before they change their perceptions. What is the limit of this “two-tiered†replicator dynamics?
Regards,
Neibruoma Meztress
66 Enchanter
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