Eco514-Game Theory Lecture 13: Repeated Games(2) Marciano siniscalchi October 28. 1999 Introduction [Again, by and large, I will follow OR, Chap. 8, so I will keep these notes to a minimum. Review of key definitions Recall our three payoff aggregation criteria: discounting, i.e (u2)≥1>(2 (also recall that the payoff profile corresponding to a stream (ut)is taken to be(1 8)2t18t-u(a)); limit of means (u21()21分 lim inf下砖-吨0 and overtaking (u)21/:(w )21 lim>(u;-w1)>0 Also recall the definition of machines Definition 1 Fix a normal-form game G. A machine for Player i E N is a tuple Mi= (Q,q,f,n), (i Qi is a finite set(whose elements should be thought of as labels) (ii)qi is the initial state of the machine (iii)fi: Q -; is the action function: it specifies what Player i does at each state; and (iv)T: Qi x AQi is the transition function: if action a E A is played and Player i's machine state is qi E Qi, then at the next stage Player i' s machine state will be Ti(gi, aEco514—Game Theory Lecture 13: Repeated Games (2) Marciano Siniscalchi October 28, 1999 Introduction [Again, by and large, I will follow OR, Chap. 8, so I will keep these notes to a minimum.] Review of key definitions Recall our three payoff aggregation criteria: discounting, i.e. (u t i )t≥1 i (w t i )t≥1 ⇔ X t≥1 δ t−1 (u t i − w t i ) > 0 (also recall that the payoff profile corresponding to a stream (u t ) is taken to be (1 − δ) P t≥1 δ t−1u(a t )); limit of means: (u t i )t≥1 i (w t i )t≥1 ⇔ lim inf t→∞ X T t=1 u t i − w t i T > 0; and overtaking: (u t i )t≥1 i (w t i )t≥1 ⇔ lim inf t→∞ X T t=1 (u t i − w t i ) > 0. Also recall the definition of machines: Definition 1 Fix a normal-form game G. A machine for Player i ∈ N is a tuple Mi = (Qi , q0 i , fi , τi), where: (i) Qi is a finite set (whose elements should be thought of as labels); (ii) q 0 i is the initial state of the machine; (iii) fi : Qi → Ai is the action function: it specifies what Player i does at each state; and (iv) τi : Qi × A → Qi is the transition function: if action a ∈ A is played and Player i’s machine state is qi ∈ Qi , then at the next stage Player i’s machine state will be τi(qi , a). 1