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

TL;DR

The paper addresses learning in multiplayer symmetric constant-sum games where equilibrium concepts are non-unique and fail to guarantee robust performance. It introduces the equal-share objective, analyzes two structural conditions under which equal share is achievable, and designs algorithms grounded in no-regret and no-dynamic-regret learning to obtain approximate equal share with provable guarantees. The work provides matching lower bounds to establish sharpness and demonstrates, through experiments on simple multiplayer games, that traditional self-play baselines can fail to secure equal share while the proposed principled approach succeeds. This framework offers a principled path for robust strategy learning in complex multiplayer settings and informs the design of practical AI systems facing diverse opponents.

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.

Theorems & Definitions (19)

• Example 1: Three-player majority vote game
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3
• Theorem 5.1: Stationary opponents
• Theorem 5.2
• Theorem 5.3
• Theorem 5.4
• Theorem 5.5
• proof : Proof of Proposition \ref{['prop:1']}