Summary
This paper (arXiv:2606.28308) by Luis Leal investigates which member of the Nash equilibrium set different solvers select in two-player zero-sum games. Many such games admit a convex polytope of equilibria sharing the same minimax value but prescribing different behavior, yet standard solvers are typically treated as interchangeable. Using a tabular, exactly solvable testbed of six games with analytically known Nash sets—including a two-dimensional Nash polytope and Kuhn poker—the author shows that equilibrium selection is determined by the algorithm rather than the random seed. Regularized last-iterate methods (R-NaD, magnetic mirror descent) converge to the maximum-entropy equilibrium, while regret-averaging methods (CFR, CFR+, fictitious play) drift toward lower-entropy faces; these findings are confirmed on a set of 180 random games. The chosen equilibrium also has downstream consequences against suboptimal opponents. Two negative results correct common intuitions: removing CFR's orthogonal quadrant projection does not eliminate boundary drift, and R-NaD's selection is anchor-following rather than initialization-independent. The work highlights that solver choice materially shapes which equilibrium is reached in games with non-unique Nash equilibria.
Paper Overview
- Research area: Machine Learning
- Author: Luis Leal
- Published: 2026-06-26
- arXiv: 2606.28308
Abstract (translated)
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.
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:
1. Selection is determined by the algorithm, not the seed, but families differ only on asymmetric Nash sets.
2. Regularized last-iterate methods (R-NaD, magnetic mirror descent) select the maximum-entropy member, while regret-averaging methods (CFR, CFR+, fictitious play) drift toward lower-entropy faces; this is confirmed on a set of 180 random games.
3. The selected member has downstream consequences against suboptimal opponents.
Two negative results correct common intuitions:
- Removing CFR's orthogonal quadrant projection does not eliminate boundary drift.
- R-NaD's selection is anchor-following, not initialization-independent.
*Auto-collected on 2026-06-30.*
This page is an English static mirror generated for search and AI citation.
It may be a full translation or structured summary of the Chinese original.
Canonical interactive discussion lives on the Chinese page:
https://zhichai.net/topic/178208301