# Near-Optimal Online Egalitarian learning in General Sum Repeated Matrix Games

## TL;DR

This paper introduces an algorithm for two-player repeated games with unknown rewards that guarantees near-optimal regret bounds, leading to improved bargaining solutions over traditional maximin strategies.

## Contribution

The paper presents a novel algorithm achieving near-optimal regret bounds for egalitarian bargaining in general sum repeated games with unknown rewards.

## Key findings

- Achieves high-probability regret bound of order O(ln T^{1/3} T^{2/3})
- Proves a lower bound of Ω(T^{2/3}) showing near-optimality
- Demonstrates improved bargaining outcomes over maximin strategies

## Abstract

We study two-player general sum repeated finite games where the rewards of each player are generated from an unknown distribution. Our aim is to find the egalitarian bargaining solution (EBS) for the repeated game, which can lead to much higher rewards than the maximin value of both players. Our most important contribution is the derivation of an algorithm that achieves simultaneously, for both players, a high-probability regret bound of order $\mathcal{O}(\sqrt[3]{\ln T}\cdot T^{2/3})$ after any $T$ rounds of play. We demonstrate that our upper bound is nearly optimal by proving a lower bound of $\Omega(T^{2/3})$ for any algorithm.