Which Nash Equilibrium? Solver-Dependent Selection in Zero-Sum Games (arXiv 2606.28308)

Forum topic · 小凯 · 2026-06-30

Summary

This post summarizes an arXiv paper (2606.28308) by Luis Leal investigating whether Nash equilibrium solvers in two-player zero-sum games converge to the same equilibrium or select systematically different members of the Nash set. Using a tabular, exactly solvable testbed of six games with analytically known Nash sets—including a two-dimensional Nash polytope and Kuhn poker—the paper finds that equilibrium selection is determined by the algorithm rather than the random seed, though solver families differ only on asymmetric Nash sets. Regularized last-iterate methods (R-NaD, magnetic mirror descent) select the maximum-entropy equilibrium, while regret-averaging methods (CFR, CFR+, fictitious play) drift toward lower-entropy faces of the polytope; this was confirmed on a random set of 180 games. The chosen equilibrium has downstream consequences against suboptimal opponents. Two counterintuitive negative results are reported: removing CFR's orthogonal-quadrant projection does not eliminate boundary drift, and R-NaD's selection is anchor-following rather than initialization-independent.

Paper Overview

Field: Machine Learning
Author: Luis Leal
Published: 2026-06-26
arXiv: 2606.28308

Abstract (Original)

Many two-player zero-sum games admit not a unique Nash equilibrium but a convex set of them: a polytope of profiles that all share the minimax value V* yet prescribe different behaviour. Standard solvers each converge to some equilibrium and are treated as interchangeable. We ask whether they instead select different members of the Nash set, systematically as a function of the algorithm rather than the seed.

Key Findings

Using a tabular, exactly solvable testbed of six games with analytically known Nash sets—including a two-dimensional Nash polytope and Kuhn poker—the paper shows:

(i) Algorithm determines selection, not the seed: equilibrium selection is systematic with respect to the algorithm, though solver families differ only on asymmetric Nash sets.
(ii) Solver families split into two behaviors:
Regularized last-iterate methods (R-NaD, magnetic mirror descent) select the maximum-entropy member of the Nash set.
Regret-averaging methods (CFR, CFR+, fictitious play) drift to lower-entropy faces of the polytope.
This was confirmed on a random collection of 180 games.
(iii) Downstream consequences: the selected equilibrium member matters when playing against suboptimal opponents.

Negative Results (Correcting Common Intuition)

Removing CFR's orthogonal-quadrant projection does not eliminate boundary drift.
R-NaD's selection is anchor-following, not initialization-independent.

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