# Which Nash Equilibrium? Solver-Dependent Selection on Zero-Sum Nash Polytopes

arXiv: https://arxiv.org/abs/2606.28308 · HTML: https://mlsanity.com/paper/2606.28308/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 21 references that are in the map.
- First posted on arXiv 26 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Mastering the Game of Stratego with Model-Free Multiagent Reinforcement Learning](https://mlsanity.com/paper/2206.15378/) (2022)
- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [Solving Large Imperfect Information Games Using CFR+](https://mlsanity.com/paper/1407.5042/) (2014)
- [A Unified Approach to Reinforcement Learning, Quantal Response Equilibria, and Two-Player Zero-Sum Games](https://mlsanity.com/paper/2206.05825/) (2022)
- [From Poincar'e Recurrence to Convergence in Imperfect Information Games: Finding Equilibrium via Regularization](https://mlsanity.com/paper/2002.08456/) (2020)
- [No-regret learning and mixed Nash equilibria: They do not mix](https://mlsanity.com/paper/2010.09514/) (2020)
- [Re-evaluating Evaluation](https://mlsanity.com/paper/1806.02643/) (2018)
- [An explicit analysis of the entropic penalty in linear programming](https://mlsanity.com/paper/1806.01879/) (2018)
- [Sample-based Approximation of Nash in Large Many-Player Games via Gradient Descent](https://mlsanity.com/paper/2106.01285/) (2021)

## Built on by (most cited first, 2 of 2)

- [The Curvature Shadow: An Apparent Failure of Maximum-Entropy Equilibrium Selection is a Removable Artifact](https://mlsanity.com/paper/2607.17543/) (2026)
- [Steering Equilibrium Selection in Regularized Self-Play via the Reference Policy](https://mlsanity.com/paper/2609.19820/) (2026)

## Abstract (arXiv)

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 -- we find that (i) selection is determined by the algorithm, not the seed, but families differ only on asymmetric Nash sets; (ii) regularized last-iterate methods (R-NaD, magnetic mirror descent) select the maximum-entropy member, the information projection of their uniform reference onto the Nash set -- exactly on the 2-D polytope and at 99.7% of maximum entropy in Kuhn -- while regret-averaging methods (CFR, CFR+, fictitious play) drift to a lower-entropy face; we confirm this on a randomized 180-game ensemble, where R-NaD attains the maximum-entropy member in 100% of converged games while CFR+ sits strictly below it in 94% (paired Wilcoxon p < 10^-27); (iii) the selected member has downstream consequences against sub-optimal opponents that scale with sequential/hidden-information structure but stay bounded -- in Kuhn the max-entropy member is a strictly better hedge, whereas on the matrix games the members differ without either dominating. We also report two negative results correcting common intuitions: removing CFR's positive-orthant (max(R,0)) projection does not eliminate boundary drift; and R-NaD's selection is anchor-following, not initialization-independent. We state the maximum-entropy / I-projection characterization as a strongly data-supported conjecture, checked throughout against analytic ground truth.