AI 中文总结
本文证明在丰富的C²博弈类中，不同纯策略纳什均衡通常具有不同福利值，福利最大化均衡若存在则唯一，并推广至一般收益聚合器及扰动情形。
AI 中文摘要
我们证明，在具有开放策略集的足够丰富的 $C^2$ 博弈类中，不同的纯策略纳什均衡通常具有不同的功利主义福利值。这里，“通常”既指通有性又指有限普遍性，而“丰富”指该类允许一个有限维扰动族，该扰动族能在任意两个不同策略组合处独立地沿任意方向移动收益值和一阶导数（例如，次数至多为三的多项式扰动）。因此，若存在福利最大化均衡，则它必是唯一的。该结果推广到每个固定的 $C^1$ 收益聚合器，只要其对收益向量的导数处处非零。在完整的 $C^2$ 效用空间中，我们还处理了依赖于策略、收益和收益导数的标量准则，但需满足在收益水平或跨玩家导数方向上的非消失条件。最后，从合适的有限维空间中附加任意小的扰动，使得福利分离在增广族中成为通有性质，且对每个固定基础博弈的几乎每个扰动，分离均成立。
英文摘要
We show that distinct pure-strategy Nash equilibria typically have different utilitarian welfare values in sufficiently rich classes of $C^2$ games with open strategy sets. Here, typical means both generic and finitely prevalent, while rich means that the class admits a finite-dimensional family of perturbations that can independently move payoff values and first derivatives in any direction at any two distinct strategy profiles (for example, polynomial perturbations of degree at most three). Consequently, a welfare-maximizing equilibrium, if one exists, is unique. The result extends to each fixed $C^1$ payoff aggregator whose derivative with respect to the payoff vector is everywhere nonzero. In the full $C^2$ utility space, we also treat scalar criteria depending on strategies, payoffs, and payoff derivatives, subject to a nonvanishing condition in payoff-level or cross-player derivative directions. Finally, adjoining arbitrarily small perturbations from a suitable finite-dimensional space makes welfare separation typical in the augmented family, with separation holding for almost every perturbation of each fixed base game.
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