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

Technical Summary: The Curvature Shadow in Regularized Nash Dynamics

Problem Statement
In two-player zero-sum games where the set of Nash equilibria (NE) forms a convex polytope rather than a single point, regularized solvers such as Regularized Nash Dynamics (R-NaD) are empirically observed to converge to the maximum-entropy equilibrium. This member is theoretically equivalent to the information projection (I-projection) of a uniform reference strategy onto the NE set. However, an apparent exception exists in Kuhn poker: while R-NaD converges to an exact Nash equilibrium (zero exploitability), the resulting bluff frequency (0.180) deviates from the theoretical maximum-entropy coordinate (0.201) by approximately 0.021. This occurs despite the converged profile attaining 99.7% of the maximum possible entropy. The central question is whether this coordinate gap represents a genuine, systematic selection bias in R-NaD or an artifact arising from the interaction between a small residual entropy shortfall and the local geometry of the entropy landscape.

Methodology
The study employs a diagnostic approach using a tabular extensive-form engine with exact counterfactual values and no sampling or function approximation. The analysis focuses on five games with one-dimensional Nash segments: Kuhn poker (sequential, imperfect information) and four matrix games (single information set per player).

1. Theoretical Decomposition: The authors derive a local geometric identity for the selection gap on a one-dimensional Nash manifold. By expanding the entropy function $H(\sigma(c))$ around its maximum $c^*$, they establish that for a point with an entropy shortfall $\delta = H^* - H(\sigma(c))$, the coordinate gap $|c - c^*|$ factorizes as:
   $$\text{gap} \approx \sqrt{2\delta / \kappa}$$
   where $\kappa$ is the local curvature of the entropy landscape at its peak. This decomposition separates the "shortfall" (why the solver stops short of maximum entropy) from the "curvature" (how much that shortfall is amplified into a coordinate offset).

2. Hypothesis Testing: The paper contrasts two hypotheses:

   • $H_{flat}$: The gap is an artifact of a small, removable entropy shortfall $\delta$ amplified by flatness ($\kappa$). As $\delta \to 0$, the gap vanishes.
   • $H_{bias}$: R-NaD has a systematic bias, meaning the gap persists at a non-zero value even as $\delta \to 0$.

3. Experimental Verification:

   • Static Comparison: R-NaD is run on all five games to measure the landed coordinate, entropy shortfall $\delta$, and curvature $\kappa$.
   • Causal Sweep (Kuhn): The magnet strength parameter $\eta$ in R-NaD is swept from 0.80 down to 0.10. This variation drives the entropy shortfall $\delta$ toward zero while keeping the dynamics stable, allowing observation of the gap's behavior relative to the predicted $\sqrt{2\delta/\kappa}$ curve.
   • Curvature Quantification: The study quantifies $\kappa$ for each game using least-squares quadratic fits on the entropy landscape and verifies the scaling relationship.

Key Contributions and Results

• Exact Factorization of the Gap: The study confirms that the coordinate gap factorizes exactly as $\sqrt{2\delta/\kappa}$ across all tested games. The relationship holds to within $2 \times 10^{-4}$ (relative error < 1%).
• Resolution of the Kuhn Exception:
  • In the four matrix games, R-NaD reaches the maximum-entropy member exactly ($\delta \approx 0$), resulting in zero gap regardless of the curvature (even in games flatter than Kuhn).
  • In Kuhn poker, a small entropy shortfall exists ($\delta \approx 8.3 \times 10^{-4}$). Due to the moderately flat peak ($\kappa \approx 4.0$), this small shortfall is amplified into the observed 0.021 coordinate gap.
  • The gap is thus identified as the "curvature shadow" of a removable shortfall, not a selection bias.
• Causal Validation via Magnet Sweep: By weakening the magnet strength $\eta$ in Kuhn poker, the authors drive $\delta \to 0$. The resulting gap decreases along the predicted curve toward zero. A log-log fit of gap vs. $\delta$ yields a scaling exponent of 0.5013 ($R^2 > 0.999999$), matching the theoretical prediction of $1/2$ for $H_{flat}$ and contradicting $H_{bias}$ (which would predict a slope approaching 0). The gap only fails to reach zero when $\eta$ drops below a stability floor ($\eta \approx 0.15$), causing dynamical instability (limit cycles).
• Curvature Dependence: The study demonstrates that flatness is necessary but not sufficient for a gap; a non-zero shortfall is also required. A hypothetical shortfall on a sharp peak ($\kappa \approx 20$) would produce a gap $\approx 2.24$ times smaller than on Kuhn's flatter peak.
• Tsallis Entropy Caveat: The paper warns against using Tsallis entropy as a "curvature dial" without adjustment, as changing the entropy index $q$ shifts the target maximum-entropy coordinate $c^*_q$. Comparing a Tsallis-regularized solver to a Shannon-based target would conflate the moving target with the curvature effect.

Significance and Claims
The paper claims to provide a defensible account of the apparent counterexample in Kuhn poker, upholding the I-projection conjecture (that R-NaD selects the maximum-entropy member) up to a "flatness-limited residual." The primary significance is the demonstration that the observed deviation is not a fundamental failure of the selection mechanism but a geometric artifact of the entropy landscape's curvature acting on a small, removable shortfall.

The study explicitly limits its scope:

• It does not prove the I-projection conjecture for all games or all initialization conditions, but validates it for the uniform reference regime.
• It does not identify the precise mechanism causing the entropy shortfall in sequential games (hypothesized to be related to reach-weighted regularization), only that it exists.
• It does not perform a causal sweep of curvature $\kappa$ itself (due to the lack of a stable sequential family with tunable curvature), relying instead on measured curvatures and the derived geometric identity.

The work concludes that the "Kuhn gap" is a removable artifact, and the I-projection account remains valid provided the entropy shortfall is driven to zero, subject to the dynamical stability constraints of the solver.