In two-player zero-sum games, when the 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. This match is exact across a panel of small games, with one apparent exception: in Kuhn poker, R-NaD lands at bluff coordinate 0.180 while the maximum-entropy member is at 0.201, resulting in a coordinate gap of about 0.021, despite R-NaD achieving 99.7 percent of the maximum entropy. We investigate whether this gap is a genuine selection bias or an artifact, and provide a quantitative answer.
We show that for selection on a one-dimensional Nash manifold, the coordinate gap factorizes as: $$\text{gap} \approx \sqrt{\frac{2\delta}{\kappa}}$$, where $\delta$ is the entropy shortfall of the solver and $\kappa$ is the curvature of the entropy landscape at its peak. This relation holds to within $2 \times 10^{-4}$ across five games (with under 1 percent relative error). The four matrix games have $\delta \approx 0$ (R-NaD reaches the maximum-entropy member exactly), hence no gap regardless of curvature; only the sequential game (Kuhn) has $\delta > 0$. A causal sweep of the magnet strength drives $\delta \to 0$ and the gap toward zero along the predicted curve (fitted scaling exponent 0.50, $R^2 = 0.999999$, against the exact prediction of 1/2), until 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. Thus, the Kuhn gap is 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.
Blogger's Review: This paper provides an in-depth analysis of the mechanisms behind maximum entropy selection in zero-sum games, particularly the discrepancies observed in Kuhn poker. The research reveals a quantitative relationship between entropy shortfall and curvature, offering new insights into more complex game-theoretic models, which is of significant theoretical and practical importance. The contributions to the understanding and application of entropy cannot be overlooked.