Original article (link) posted: 21/10/2005
Repeated Games: Set-Up
We first checked the definitions of the followings; a stage game, a repeated game, a subgame, strategies, histories, a Nash equilibrium, a subgame perfect NE, feasible payoffs, and individually rational payoffs.
Note) Any points in the convex hull of the pure strategy payoffs are feasible when the discount factor is sufficiently large. (The proof is done by using time-averaging strategies. See Sorin(1986))
Then, we investigated APS operator, which captures the similar idea of Bellman operator in a single-agent dynamic optimization problem.
Since this blog is not designed for writing messy equations, I will not cover the mathematical argument about APS operator here. You can check the chapter 5 of Fudenberg and Tirole (1991) ("Dynamic Programming and Self-Generation" in 5.5.4) or the original paper by APS (1990).
Abreu, Pearce and Stachetti (1990) "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring" Econometrica, 58
Sorin (1986) "On Repeated Games with Complete Information" Math. of Operations Research, 11-1