Eco514-Game Theory Lecture 12: Repeated Games(1 Marciano siniscalchi October 26. 1999 Introduction By and large, I will follow OR, Chap. 8, so I will keep these notes to a minimum. ] The theory of repeated games is a double-edged sword. On one hand, it indicates how payoff profiles that are not consistent with Nash equilibrium in a simultaneous-move game might be achieved when the latter is played repeatedly, in a manner consistent with Nash or even subgame-perfect equilibrium On the other hand, it shows that essentially every individually rational payoff profile can be thus achieved if the game is repeated indefinitely(and a similar, but a bit more restrictive result holds for finitely repeated games ). Thus, the theory has little predictive power To make matters worse, the expression"repeated-game phenomenon"is often invoked to account for the occurrence of a non-equilibrium outcome in real-world strategic interactions But, as usual, the theory refers to a clearly specified, highly stylized situation, which may or may not approximate the actual strategic interaction OR emphasize the structure of the equilibria supporting these outcomes, rather than the set of attainable outcomes themselves, as the most interesting product of the theor Payoff Aggregation Criteria Definition 1 Consider a normal-form game G=(N, (Ai, uiieN). The T-repeated game T≤∞) induced by G is the perfect-information game I=(N,A,H,Z,P,(≥)l∈eN) where: (i)A=UeN Ai is the set of actions available to players in G (i)h is the set of sequences of length at most T of elements of lien a (iii)P(h)=M (iv)i satisfies weak separability: for any sequence(a')a and profiles a, aE Ilen Ai (a)≥u4(a) implies(a1,…,a-1,a,a2+1,)≥;(a2Eco514—Game Theory Lecture 12: Repeated Games (1) Marciano Siniscalchi October 26, 1999 Introduction [By and large, I will follow OR, Chap. 8, so I will keep these notes to a minimum.] The theory of repeated games is a double-edged sword. On one hand, it indicates how payoff profiles that are not consistent with Nash equilibrium in a simultaneous-move game might be achieved when the latter is played repeatedly, in a manner consistent with Nash or even subgame-perfect equilibrium. On the other hand, it shows that essentially every individually rational payoff profile can be thus achieved if the game is repeated indefinitely (and a similar, but a bit more restrictive result holds for finitely repeated games). Thus, the theory has little predictive power. [To make matters worse, the expression “repeated-game phenomenon” is often invoked to account for the occurrence of a non-equilibrium outcome in real-world strategic interactions. But, as usual, the theory refers to a clearly specified, highly stylized situation, which may or may not approximate the actual strategic interaction.] OR emphasize the structure of the equilibria supporting these outcomes, rather than the set of attainable outcomes themselves, as the most interesting product of the theory. Payoff Aggregation Criteria Definition 1 Consider a normal-form game G = (N,(Ai , ui)i∈N ). The T-repeated game (T ≤ ∞) induced by G is the perfect-information game Γ = (N, A, H, Z, P,(i)i∈N ) where: (i) A = S i∈N Ai is the set of actions available to players in G; (ii) H is the set of sequences of length at most T of elements of Q i∈N Ai ; (iii) P(h) = N (iv) i satisfies weak separability: for any sequence (a t ) T t=1 and profiles a, a0 ∈ Q i∈N Ai , ui(a) ≥ ui(a 0 ) implies (a 1 , . . . , at−1 , a, at+1 , . . .) i (a 1 , . . . , at−1 , a0 , at+1 , . . .). 1