Approximating the Set of Nash Equilibria for Convex Games
Zachary Feinstein, Niklas Hey, Birgit Rudloff
Operations Research · 2025 · Volume 73 · Issue 5
Abstract
In Feinstein and Rudloff (2024) , it was shown that the set of Nash equilibria for any noncooperative N player game coincides with the set of Pareto optimal points of a certain vector optimization problem with nonconvex ordering cone. To avoid dealing with a nonconvex ordering cone, an equivalent characterization of the set of Nash equilibria as the intersection of the Pareto optimal points of N multi-objective problems (i.e., with the natural ordering cone) is proven. So far, algorithms to compute the exact set of Pareto optimal points of a multi-objective problem exist only for the class of linear problems, which reduces the possibility of finding the true set of Nash equilibria by those algorithms to linear games only. In this paper, we will consider the larger class of convex games. Because typically only approximate solutions can be computed for convex vector optimization problems, we first show, in total analogy to the result above, that the set of ϵ -approximate Nash equilibria can be characterized by the intersection of ϵ -approximate Pareto optimal points for N convex multi-objective problems. Then, we propose an algorithm based on results from vector optimization and convex projections that allows for the computation of a set that, on one hand, contains the set of all true Nash equilibria and is, on the other hand, contained in the set of ϵ -approximate Nash equilibria. In addition to the joint convexity of the cost function for each player, this algorithm works provided the players are restricted by either shared polyhedral constraints or independent convex constraints. Funding: This work was supported by Austrian Science Funds (FWF) [W1260-N35]. Supplemental Material: The computer code and data that support the findings of this study are available within this article’s supplemental material at https://doi.org/10.1287/opre.2023.0541 .
Keywords
Optimization, convex game, set of Nash equilibria, Pareto optimality, approximation, algorithm
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