Securing Equal Share: A Principled Approach for Learning Multiplayer Symmetric Games

TL;DR

This paper introduces a principled approach for learning in multiplayer symmetric games by focusing on securing an equal share of the payoff, addressing fundamental challenges in equilibrium concepts and providing algorithms with theoretical guarantees.

Contribution

It proposes a new objective of equal share in multiplayer symmetric games, along with efficient algorithms and theoretical analysis to achieve it.

Findings

Algorithms provably attain approximate equal share

Lower bounds justify the theoretical limits

Experimental results show success over prior meta-algorithms

Abstract

This paper examines multiplayer symmetric constant-sum games with more than two players in a competitive setting, including examples like Mahjong, Poker, and various board and video games. In contrast to two-player zero-sum games, equilibria in multiplayer games are neither unique nor non-exploitable, failing to provide meaningful guarantees when competing against opponents who play different equilibria or non-equilibrium strategies. This gives rise to a series of long-lasting fundamental questions in multiplayer games regarding suitable objectives, solution concepts, and principled algorithms. This paper takes an initial step towards addressing these challenges by focusing on the natural objective of equal share -- securing an expected payoff of C/n in an n-player symmetric game with a total payoff of C. We rigorously identify the theoretical conditions under which achieving an equal share is tractable and design a series of efficient algorithms, inspired by no-regret learning, that provably attain approximate equal share across various settings. Furthermore, we provide complementary lower bounds that justify the sharpness of our theoretical results. Our experimental results highlight worst-case scenarios where meta-algorithms from prior state-of-the-art systems for multiplayer games fail to secure an equal share, while our algorithm succeeds, demonstrating the effectiveness of our approach.