An LP Approach for Solving Two-Player Zero-Sum Repeated Bayesian Games

TL;DR

This paper develops linear programming methods to compute security strategies in two-player zero-sum repeated Bayesian games with private types, providing explicit strategies based on sufficient statistics for finite and discounted cases.

Contribution

It introduces a novel LP-based framework for deriving security strategies using sufficient statistics in repeated Bayesian games, including finite horizon and discounted scenarios.

Findings

Explicit LPs for finite horizon security strategies.

Sufficient statistics include belief and regret measures.

Performance bounds for approximate strategies.

Abstract

This paper studies two-player zero-sum repeated Bayesian games in which every player has a private type that is unknown to the other player, and the initial probability of the type of every player is publicly known. The types of players are independently chosen according to the initial probabilities, and are kept the same all through the game. At every stage, players simultaneously choose actions, and announce their actions publicly. For finite horizon cases, an explicit linear program is provided to compute players' security strategies. Moreover, based on the existing results in [1], this paper shows that a player's sufficient statistics, which is independent of the strategy of the other player, consists of the belief over the player's own type, the regret with respect to the other player's type, and the stage. Explicit linear programs are provided to compute the initial regrets, and the security strategies that only depends on the sufficient statistics. For discounted cases, following the same idea in the finite horizon, this paper shows that a player's sufficient statistics consists of the belief of the player's own type and the anti-discounted regret with respect to the other player's type. Besides, an approximated security strategy depending on the sufficient statistics is provided, and an explicit linear program to compute the approximated security strategy is given. This paper also obtains a bound on the performance difference between the approximated security strategy and the security strategy.