Normal-form proper equilibrium, introduced by Myerson as a refinement of normal-form perfect equilibrium, occupies a distinctive position in the equilibrium analysis of extensive-form games because its more stringent perturbation structure entails the sequential rationality. However, the size of the normal-form representation grows exponentially with the number of parallel information sets, making the direct determination of normal-form proper equilibria intractable. To address this challenge, we develop a compact sequence-form proper equilibrium by redefining the expected payoffs over sequences, and we prove that it coincides with the normal-form proper equilibrium via strategic equivalence. To facilitate computation, we further introduce an alternative representation by defining a class of perturbed games based on an $\varepsilon$-permutahedron over sequences. Building on this representation, we introduce two differentiable path-following methods for computing normal-form proper equilibria. These methods rely on artificial sequence-form games whose expected payoff functions incorporate logarithmic or entropy regularization through an auxiliary variable. We prove the existence of a smooth equilibrium path induced by each artificial game, starting from an arbitrary positive realization plan and converging to a normal-form proper equilibrium of the original game as the auxiliary variable approaches zero. Finally, our experimental results demonstrate the effectiveness and efficiency of the proposed methods.

翻译：正规型适当均衡由Myerson提出，作为正规型完美均衡的一种精炼，在扩展式博弈的均衡分析中占据独特地位，因为其更严格的扰动结构蕴含了序贯理性。然而，正规型表示的规模会随并行信息集的数量呈指数级增长，使得直接求解正规型适当均衡变得不可行。为应对这一挑战，我们通过重新定义基于序列的期望收益，构建了一种紧凑的序列形式适当均衡，并证明其通过策略等价性与正规型适当均衡相一致。为便于计算，我们进一步引入了一种替代表示方法，其基于序列上的$\varepsilon$-置换多面体定义了一类扰动博弈。基于此表示，我们提出了两种可微分的路径跟踪方法用于计算正规型适当均衡。这些方法依赖于人工构造的序列形式博弈，其期望收益函数通过辅助变量引入了对数或熵正则化。我们证明了每个构造博弈均诱导出一条光滑的均衡路径，该路径从任意正实现计划出发，当辅助变量趋于零时收敛至原博弈的一个正规型适当均衡。最后，实验结果验证了所提方法的有效性与计算效率。