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This research provides rigorous mathematical validation that the deviation observed in Kuhn poker is not an inherent bias of Regularized Nash Dynamics, but rather a quantifiable artifact caused by entropy landscape curvature and solver shortfall. By deriving a precise scaling law linking coordinate gaps to these geometric factors, the study confirms that maximum-entropy equilibrium selection holds robustly across both simultaneous and sequential games when stability constraints are managed.
The Curvature Shadow: An Apparent Failure of Maximum-Entropy Equilibrium Selection is a Removable Artifact
arXiv:2607.17543v1 Announce Type: new Abstract: 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 $\mathrm{gap} \approx \sqrt{2\delta/\kappa}$, where $\delta$ 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 $2 \times 10^{-4}$ (under 1 percent relative error). The four matrix games have $\delta \approx 0$ (R-NaD reaches the maximum-entropy member exactly) and therefore 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 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.
Abstract: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 $\mathrm{gap} \approx \sqrt{2\delta/\kappa}$, where $\delta$ 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 $2 \times 10^{-4}$ (under 1 percent relative error). The four matrix games have $\delta \approx 0$ (R-NaD reaches the maximum-entropy member exactly) and therefore 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 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.
11 pages, 4 figures, 3 tables. Diagnostic companion to arXiv:2606.28308. Fully reproducible: a single self-contained notebook regenerates every number, table, and figure
Artificial Intelligence (cs.AI); Computer Science and Game Theory (cs.GT); Machine Learning (cs.LG); Multiagent Systems (cs.MA)
arXiv:2607.17543 [cs.AI]
(or arXiv:2607.17543v1 [cs.AI] for this version)
https://doi.org/10.48550/arXiv.2607.17543 arXiv-issued DOI via DataCite (pending registration)
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