# The Curvature Shadow: An Apparent Failure of Maximum-Entropy Equilibrium Selection is a Removable Artifact

arXiv: https://arxiv.org/abs/2607.17543 · HTML: https://mlsanity.com/paper/2607.17543/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 8 references that are in the map.
- First posted on arXiv 20 Jul 2026.

## Builds on (most cited first, 4 of 4)

- [Mastering the Game of Stratego with Model-Free Multiagent Reinforcement Learning](https://mlsanity.com/paper/2206.15378/) (2022)
- [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)
- [Which Nash Equilibrium? Solver-Dependent Selection on Zero-Sum Nash Polytopes](https://mlsanity.com/paper/2606.28308/) (2026)

## Built on by (most cited first, 1 of 1)

- [Steering Equilibrium Selection in Regularized Self-Play via the Reference Policy](https://mlsanity.com/paper/2609.19820/) (2026)

## Abstract (arXiv)

In two-player zero-sum games whose Nash equilibria form a convex set, regularized solvers such as Regularized Nash Dynamics (R-NaD) empirically select the maximum-entropy member: the information projection (I-projection) of a uniform reference onto the Nash set. On a panel of small games this match is exact, with one apparent exception: in Kuhn poker R-NaD lands at bluff coordinate 0.180 while the maximum-entropy member sits at 0.201, a coordinate gap of about 0.021, even though R-NaD attains 99.7 percent of the maximum entropy. We ask whether this gap is a genuine selection bias or an artifact, and answer it quantitatively. We show that for selection on a one-dimensional Nash manifold the coordinate gap factorizes as gap ≈ √2δ/kappa, where δ is the entropy shortfall of the solver and kappa is the curvature of the entropy landscape at its peak. Across five games this relation holds to within 2 × 10-4 (under 1 percent relative error). The four matrix games have δ ≈ 0 (R-NaD reaches the maximum-entropy member exactly) and therefore no gap regardless of curvature; only the sequential game (Kuhn) has δ > 0. A causal sweep of the magnet strength drives δ → 0 and the gap toward zero along the predicted curve (fitted scaling exponent 0.50, R2 > 0.999999, against the exact prediction of 1/2), until the dynamics destabilize at a stability floor: behavior consistent with a removable shortfall and inconsistent with a fixed bias. We quantify the curvature half of the law from measured curvatures and flag a moving-target pitfall in the natural Tsallis-entropy experiment. The Kuhn gap is thus the curvature shadow of a small, removable entropy shortfall on an unusually flat peak; the I-projection account is upheld up to a flatness-limited residual.