热力学极限下若干社会困境博弈的纳什均衡映射、哈密顿动力学与达尔文进化比较
Nash equilibrium mapping vs Hamiltonian dynamics vs Darwinian evolution for some social dilemma games in the thermodynamic limit
AI总结：
本文比较了社会困境博弈在热力学极限下的三种分析方法，发现纳什均衡映射与数值模拟吻合，而达尔文进化在特定条件下给出假阳性，哈密顿动力学则完全失效。
摘要：
合作如何在热力学极限或无限玩家极限下演化并显现，是社会困境博弈中一个备受推测的问题。已有多种解析方法被提出用于分析社会困境的热力学极限。在本工作中，我们比较了两种解析方法，即达尔文进化与纳什均衡映射，并与数值基于智能体的方法进行对比。为完整性起见，我们也给出了另一种解析方法——哈密顿动力学的结果。与涉及所有个体收益最大化的哈密顿动力学不同，达尔文进化中单个玩家的收益相对于其与最近邻的相互作用被最大化。当收益满足条件 a + d = b + c（其中 a、d 表示对称社会困境博弈收益矩阵中的对角元素，b、c 表示非对角元素）时，哈密顿动力学方法相比纳什均衡映射完全失败，而达尔文进化方法则对博弈磁化强度（合作者与背叛者分数之差）给出假阳性结果。当 a + d ≠ b + c 或考察每个玩家的平均收益时，达尔文进化方法同样失败，与哈密顿动力学方法类似。另一方面，对于所考察的社会困境，即鹰鸽博弈和公共品博弈，纳什均衡映射与数值基于智能体的方法在博弈磁化强度和每个玩家的平均收益方面均吻合良好。因此，本文揭示了达尔文进化方法相对于纳什均衡映射和数值基于智能体方法的不一致性。
英文摘要：
How cooperation evolves and manifests itself in the thermodynamic or infinite player limit of social dilemma games is a matter of intense speculation. Various analytical methods have been proposed to analyze the thermodynamic limit of social dilemmas. In this work, we compare two analytical methods, i.e., Darwinian evolution and Nash equilibrium mapping, with a numerical agent-based approach. For completeness, we also give results for another analytical method, Hamiltonian dynamics. In contrast to Hamiltonian dynamics, which involves the maximization of payoffs of all individuals, in Darwinian evolution, the payoff of a single player is maximized with respect to its interaction with the nearest neighbor. While the Hamiltonian dynamics method utterly fails as compared to Nash equilibrium mapping, the Darwinian evolution method gives a false positive for game magnetization -- the net difference between the fraction of cooperators and defectors -- when payoffs obey the condition a + d = b + c, wherein a,d represents the diagonal elements and b,c the off-diagonal elements in a symmetric social dilemma game payoff matrix. When either a + d =/= b + c or when one looks at the average payoff per player, the Darwinian evolution method fails, much like the Hamiltonian dynamics approach. On the other hand, the Nash equilibrium mapping and numerical agent-based method agree well for both game magnetization and average payoff per player for the social dilemmas in question, i.e., the Hawk-Dove game and the Public goods game. This paper thus brings to light the inconsistency of the Darwinian evolution method vis-a-vis both Nash equilibrium mapping and a numerical agent-based approach.
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