Repeated Games with Perfect Monitoring

We present a relatively general model of repeated games first, which will be later specialized for each different repeated game with different monitoring structure. Stage game is a standard strategic (normal) form game G = {N,A, g} , where N = {1, 2, ..., n} be the set of players, Ai is player i’s finite or compact action set ¡ A = Q i∈N Ai ¢ , and g : A→ <n is the payoff functions of n players, with gi (a) being player i’s payoff. We assume that gi (a) is continuous when Ai is compact. Feasible payoff set is denoted by V = convex full of {g (a) |a ∈ A}. Players play the stage game repeatedly over time. Time is discrete and denoted by t = 1, 2, .... Player i observes a signal hi,t ∈ Yi in the end of period t. Player i’s signal is generated by fi : A→ 4Yi every period given the action profile chosen in the period. Player i’s period t history is hi = (hi,1, ...., hi,t−1), which is player i’s information accumulated by the end of period t− 1. Let Ht i ¡ = Y t−1 i ¢ be the set of all possible player i’s period t history at period t (H1 i = ∅ is null history in the beginning of the game) andHi = ∪∞t=1Ht i be all possible histories of player i. Player i’s (pure) strategy σi is a mapping from Hi to Ai. I assume that strategies are pure unless noted otherwise. Let Σi be the set of player i’s strategies. Take any strategy profile e σ, which generates a sequence of action profiles ¡ea1,ea2, ...¢ . Then player i0s discounted average payoff from the strategy profile e σ is given by Vi (e σ) = (1− δ)P∞t=1 δt−1gi ¡eat¢ .1 where δ is a common discount factor. Another way to define average payoffs would be to use a variety of limit of arithmetic