纳什轨迹
Nash Loci
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中文总结 AI 辅助
该研究探讨有限参与者博弈中与固定代数簇相交的纳什轨迹，确定其维数、多重次数及方程，还将其与多重分次相伴簇关联，推广了相关超曲面。
中文摘要 AI 辅助
在有限参与者博弈的纳什均衡研究中，人们常寻求与预先确定的约束相容的均衡，这些约束由参与者或外部主体决定。我们讨论代数轨迹（称为纳什轨迹），即纳什均衡方案与射影空间乘积中固定代数簇相交的博弈对应的轨迹。我们确定了它们在多射影空间中的维数和多重次数，以及两人博弈和小型多人博弈的方程。定义纳什均衡方案的多线性方程使我们能够用格拉斯曼流形（Grassmannians）和普吕克坐标（Plücker coordinates）来描述纳什轨迹。受此启发，我们将纳什轨迹与多重分次相伴簇联系起来，后者是格拉斯曼流形乘积中的子簇，推广了Osserman和Trager提出的多重分次Cayley-Chow超曲面。
英文摘要
In the study of Nash equilibria of finite-player games, one often seeks equilibria that are compatible with predetermined constraints, either determined by the players or by an external agent. We discuss the algebraic loci, called Nash loci, of games whose Nash equilibrium scheme intersects a fixed algebraic variety in a product of projective spaces. We determine their dimensions and multidegrees in multiprojective space, and their equations for two-player games and for small multiple-player games. The multilinear equations defining the Nash equilibrium scheme allow us to describe Nash loci in the language of Grassmannians and Plücker coordinates. Motivated by this fact, we relate Nash loci to multigraded associated varieties, which are subvarieties in products of Grassmannians that generalize the multigraded Cayley-Chow hypersurfaces of Osserman and Trager.
