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

This paper examines multiplayer symmetric constant-sum games with more than two players in a competitive setting, such as 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.

Many popular games—like Mahjong, Poker, and online multiplayer games—involve more than two players competing for a fixed reward. In these settings, traditional tools from game theory that work well for two-player games, such as the concept of "Nash equilibrium," often fail to provide meaningful guarantees. As a result, players can end up with highly unequal rewards, even if they play reasonably well.A long-standing question in multiplayer games is: What should the goal be? We focus on a simple but powerful idea called the equal share—securing a fair portion of the total prize. We identify the mathematical conditions under which this goal is achievable and design new algorithms that can reliably secure this outcome. These algorithms are not only theoretically sound but also effective in practice: in our experiments, they consistently achieve fair results, even in challenging scenarios where prior state-of-the-art methods fall short.