# Entropy-Regularized Stochastic Games

## TL;DR

This paper introduces entropy-regularized stochastic games, incorporating a regularization term to balance rationality and environmental uncertainty, and provides algorithms for computing optimal strategies with practical applications.

## Contribution

It extends stochastic game theory by adding entropy regularization, proving existence of game values, and developing convex optimization algorithms for strategy computation.

## Key findings

- Existence of value in entropy-regularized N-stage and discounted stochastic games.
- Markovian and stationary strategies suffice to attain the value.
- Numerical example demonstrates the impact of regularization on payoff.

## Abstract

In two-player zero-sum stochastic games, where two competing players make decisions under uncertainty, a pair of optimal strategies is traditionally described by Nash equilibrium and computed under the assumption that the players have perfect information about the stochastic transition model of the environment. However, implementing such strategies may make the players vulnerable to unforeseen changes in the environment. In this paper, we introduce entropy-regularized stochastic games where each player aims to maximize the causal entropy of its strategy in addition to its expected payoff. The regularization term balances each player's rationality with its belief about the level of misinformation about the transition model. We consider both entropy-regularized $N$-stage and entropy-regularized discounted stochastic games, and establish the existence of a value in both games. Moreover, we prove the sufficiency of Markovian and stationary mixed strategies to attain the value, respectively, in $N$-stage and discounted games. Finally, we present algorithms, which are based on convex optimization problems, to compute the optimal strategies. In a numerical example, we demonstrate the proposed method on a motion planning scenario and illustrate the effect of the regularization term on the expected payoff.