Coordination in Noncooperative Multiplayer Matrix Games via Reduced Rank Correlated Equilibria
Jaehan Im, Yue Yu, David Fridovich-Keil, Ufuk Topcu
TL;DR
The paper tackles coordination in large-scale noncooperative multiplayer matrix games by introducing Reduced Rank Correlated Equilibria (RRCE), which approximates the correlated equilibrium through the convex hull of multiple Nash equilibria, thereby reducing joint-action considerations from $O(m^n)$ to $O(mn)$. It presents a two-phase method: first compute several Nash equilibria to obtain joint-action distributions, then form RRCE as a convex combination with weights $\bm{\gamma}$ by solving $\min_{\bm{\gamma}\ge 0, \mathbf{1}^T \bm{\gamma}=1} J\left(\sum_{k=1}^d [\bm{\gamma}]_k \bm{z}^k\right)$. Numerical experiments in an air traffic management setting show RRCE can handle up to $2^{21}$ joint actions, delivering fairness and average-delay improvements over Nash that are comparable to CE while offering substantial computational speedups. The results demonstrate that RRCE provides scalable coordination while maintaining high-quality outcomes, with a maximum observed optimality gap to CE of $0.066\%$ and up to $99.5\%$ improvement in fairness over purely non-coordinated Nash solutions. The work highlights RRCE as a practical pathway to applying correlated-equilibrium-like coordination in large systems, and points to future directions in enlarging the Nash hull and refining the convex-hull approximation for even greater scalability and robustness.
Abstract
Coordination in multiplayer games enables players to avoid the lose-lose outcome that often arises at Nash equilibria. However, designing a coordination mechanism typically requires the consideration of the joint actions of all players, which becomes intractable in large-scale games. We develop a novel coordination mechanism, termed reduced rank correlated equilibria, which reduces the number of joint actions to be considered and thereby mitigates computational complexity. The idea is to approximate the set of all joint actions with the actions used in a set of pre-computed Nash equilibria via a convex hull operation. In a game with n players and each player having m actions, the proposed mechanism reduces the number of joint actions considered from O(m^n) to O(mn). We demonstrate the application of the proposed mechanism to an air traffic queue management problem. Compared with the correlated equilibrium-a popular benchmark coordination mechanism-the proposed approach is capable of solving a problem involving four thousand times more joint actions while yielding similar or better performance in terms of a fairness indicator and showing a maximum optimality gap of 0.066% in terms of the average delay cost. In the meantime, it yields a solution that shows up to 99.5% improvement in a fairness indicator and up to 50.4% reduction in average delay cost compared to the Nash solution, which does not involve coordination.