Games and Economic Behavior · 2026 · Adam Harris、Scott McCallum、Michael S. Harré
中文摘要
我们借助突变论考察非对称二人二策略正规形式博弈的 logit 量子响应均衡的拓扑结构。我们给出三项主要结果: 其一, 为基本动力学构造了一个新的光滑势函数, 相对于该势函数, logit 均衡表现为驻点, 全体驻点构成一张参数化超曲面; 其二, 证明了该流形内部的所有突变至多为三阶, 即“折叠与尖点”; 其三, 发现了一种新的拓扑现象——“褶环”(pleated loop), 它对应于一对方向相反的叉式分岔。我们以具有唯一纳什均衡的博弈 (如囚徒困境) 为例展示这一现象, 这类博弈在有限精度参数下存在多个 logit 均衡。这些结果拓展了随机决策理论与突变论同经济博弈交叉领域的研究。文中通篇讨论了将上述结果应用于 McKelvey 与 Palfrey 提出的 logit 解追踪程序 (tracing procedure) 中分岔问题的途径。
Abstract
We report on the topological structure of logit quantal response equilibria for asymmetric, two-player, two-choice games in normal form, via catastrophe theory. We present three main outcomes: first, a novel, smooth potential function for the underlying dynamics, relative to which logit equilibria arise as stationary points comprising a parameterised hyper-surface; secondly, a proof that all catastrophes within this manifold are of order at most three, i.e. “folds and cusps”; thirdly, discovery of a new topological phenomenon, the “pleated loop”, corresponding to a pair of opposed pitchfork bifurcations. This we exhibit for games, such as Prisoner's Dilemma, that have a unique Nash equilibrium, with multiple logit equilibria for finite precision parameters. These results extend work at the intersection of stochastic decision theory and catastrophe theory with economic games. Their application to the problem of bifurcations in the tracing procedure for logit solutions, proposed by McKelvey and Palfrey, is discussed throughout.