# The operator approach to entropy games

## TL;DR

This paper introduces an operator approach to entropy games, transforming them into stochastic mean payoff games with entropy-based payments, enabling polynomial-time solutions and comparison of solution algorithms.

## Contribution

It develops a novel operator framework that casts entropy games as stochastic mean payoff games, allowing polynomial-time solutions and algorithmic comparisons.

## Key findings

- Entropy games can be solved in polynomial time with a fixed number of states.
- A policy iteration algorithm effectively solves entropy games.
- The operator approach facilitates comparison with existing spectral algorithms.

## Abstract

Entropy games and matrix multiplication games have been recently introduced by Asarin et al. They model the situation in which one player (Despot) wishes to minimize the growth rate of a matrix product, whereas the other player (Tribune) wishes to maximize it. We develop an operator approach to entropy games. This allows us to show that entropy games can be cast as stochastic mean payoff games in which some action spaces are simplices and payments are given by a relative entropy (Kullback-Leibler divergence). In this way, we show that entropy games with a fixed number of states belonging to Despot can be solved in polynomial time. This approach also allows us to solve these games by a policy iteration algorithm, which we compare with the spectral simplex algorithm developed by Protasov.