AI 中文总结
研究两人零和博弈中R-NaD选择均衡时出现的差距问题，通过量化分析表明该差距可分解为与熵不足和曲率相关的形式，经实验验证此关系，发现库恩博弈的差距是可消除熵不足的曲率阴影，支持I-投影解释。
AI 中文摘要
在纳什均衡构成凸集的两人零和博弈中，正则化求解器（如正则化纳什动力学，R-NaD）凭经验选择最大熵成员，即均匀参考在纳什集上的信息投影（I-投影）。在一组小博弈中，除了一个明显例外，这种匹配是精确的：在库恩扑克中，R-NaD落在虚张声势坐标0.180处，而最大熵成员位于0.201处，坐标差距约为0.021，尽管R-NaD达到了最大熵的99.7%。我们研究这个差距是真正的选择偏差还是伪像，并进行了定量解答。我们表明，在一维纳什流形上进行选择时，坐标差距可分解为$\mathrm{gap} \approx \sqrt{2\delta/\kappa}$，其中$\delta$是求解器的熵不足，$\kappa$是熵景观峰值处的曲率。在五个博弈中，这种关系的误差在$2 \times 10^{-4}$以内（相对误差低于1%）。四个矩阵博弈的$\delta \approx 0$（R-NaD精确地达到最大熵成员），因此无论曲率如何都没有差距；只有序贯博弈（库恩）的$\delta > 0$。对磁体强度的因果扫描使$\delta \to 0$，差距沿着预测曲线趋近于零（拟合缩放指数为0.50，$R^2 > 0.999999$，与精确预测的1/2相符），直到动力学在稳定性下限处不稳定：这种行为与可消除的不足一致，与固定偏差不一致。我们从测量的曲率中量化了该定律的曲率部分，并指出了自然Tsallis熵实验中的移动目标陷阱。因此，库恩差距是在异常平坦峰值上一个小的、可消除的熵不足的曲率阴影；I-投影解释在平坦度限制的残差范围内成立。
英文摘要
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 $\mathrm{gap} \approx \sqrt{2δ/κ}$, where $δ$ is the entropy shortfall of the solver and $κ$ 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 $δ\approx 0$ (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 $δ\to 0$ 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.
Comments11 pages, 4 figures, 3 tables. Diagnostic companion to arXiv:2606.28308. Fully reproducible: a single self-contained notebook regenerates every number, table, and figure
