Nash solvers have preferences when the value is identical

Same value, different behavior

A lot of AI work treats “finding a Nash equilibrium” like hitting a single target. Solve the game. Get the policy. Move on.

That is often too simple.

In many two-player zero-sum games, there is not one Nash equilibrium. There is a whole convex set of them, a Nash polytope. Every point in that set has the same minimax value, V*. But the policies can behave differently.

The new arXiv work, “Which Nash Equilibrium? Solver-Dependent Selection on Zero-Sum Nash Polytopes,” asks a narrow but important question: when standard solvers land somewhere inside that set, are they picking randomly, or does each algorithm have a predictable bias?

The authors tested six tabular games with analytically known Nash sets, including a two-dimensional Nash polytope and Kuhn poker. Their finding is clean: selection was driven by the algorithm, not by the random seed. That only mattered on asymmetric Nash sets, where different equilibrium points actually prescribe different behavior.

This is the kind of result that sounds academic until you imagine an agent trained to be “optimal” in a negotiation, auction, security game, or poker-like setting. The value may be identical against a perfect adversary. The actual behavior can still change.

Regularization pulls toward the center

The most useful split is between solver families.

The authors report that regularized last-iterate methods, specifically R-NaD and magnetic mirror descent, selected the maximum-entropy member of the Nash set. More precisely, they matched the information projection of a uniform reference onto the Nash set. That held exactly on the two-dimensional polytope and reached 99.7% of maximum entropy in Kuhn poker.

Regret-averaging methods behaved differently. CFR, CFR+, and fictitious play drifted toward a lower-entropy face of the set. On a randomized ensemble of 180 games, R-NaD reached the maximum-entropy member in 100% of converged games, while CFR+ sat strictly below it in 94%. The paired Wilcoxon result was reported at p < 10^-27.

That does not mean CFR is “bad.” CFR is a core tool for imperfect-information games for a reason. The point is narrower: if a game has many equilibria, a solver is not a neutral transport mechanism. It brings an implicit selection rule.

The paper also knocks down two easy explanations. Removing CFR’s positive-orthant projection, the max(R, 0) step, did not remove the boundary drift. And R-NaD’s behavior was not initialization-independent. It followed its anchor.

So the regularizer is not just a training trick. It is a preference over equilibrium behavior.

Equilibrium choice matters most against the imperfect

Against an optimal opponent, every Nash point has the same value. That is the definition doing its job.

The interesting part is what happens when the opponent is not optimal. The authors found downstream consequences against sub-optimal opponents. Those effects grew with sequential and hidden-information structure, though they stayed bounded. In Kuhn poker, the maximum-entropy equilibrium was a strictly better hedge. In simpler matrix games, equilibrium members differed, but neither dominated across the board.

That distinction matters for builders. Most deployed agents do not face perfectly rational adversaries. They face users, customers, counterparties, attackers, bots, and other models. They face weird policies. Half-trained policies. Human shortcuts. Distribution shift.

If two equilibria have equal minimax value, I care which one fails more gracefully when reality is messy. Maximum entropy is not automatically better, but it is a plausible default in hidden-information settings because it avoids unnecessary commitment. A lower-entropy face may be sharper, more specialized, and sometimes useful. But you should know you picked it.

The authors frame the maximum-entropy and information-projection result as a strongly data-supported conjecture, not a theorem for all cases. That is the right level of confidence. Six exactly solvable games plus a randomized 180-game ensemble is meaningful evidence. It is not a universal law.

For a builder, the move is simple: stop treating “Nash equilibrium found” as a complete spec. If you are training game-playing agents, negotiation policies, market simulators, or safety eval adversaries, run at least two solver families and compare the selected policies, not just the value. Test against sub-optimal opponent sets. If R-NaD and CFR+ land on behaviorally different equilibria, that difference is product-relevant. The catch most readers miss: your solver may be making a policy choice you never wrote down.