How Fast Do Equilibrium Payoff Sets Converge in Repeated Games?

We provide tight bounds on the rate of convergence of the equilibrium payoff sets for repeated games under both perfect and imperfect public monitoring. The distance between the equilibrium payoff set and its limit vanishes at rate (1 − δ) 1/2 under perfect monitoring, and at rate (1 − δ) 1/4 under imperfect monitoring. For strictly individually rational payoff vectors, these rates improve to 0 (i.e., all strictly individually rational payoff vectors are exactly achieved as equilibrium payoffs for delta high enough) and (1 − δ) 1/2 , respectively

Rate of Convergence of Empirical Measures and Costs in Controlled Markov Chains and Transient Optimality

The purpose of this paper is two-fold. First, bounds on the rate of convergence of empirical measures in Controlled Markov Chains are obtained under some recurrence conditions. These include bounds obtained through Large Deviations and Central Limit Theorem arguments. These results are then applied to optimal Control Problems. Bounds on the rate of convergence of the empirical measures that are uniform over different sets of policies are derived, resulting in bounds on the rate of convergence of the costs. Finally, new optimal control problems that involve not only average cost criteria but also measures on the transient behavior of the cost, namely the rate of convergence, are introduced and applied to a problem in telecommunications. The solution to these problems rely on the bounds introduced in previous Sections