This paper is a significant step forward in understanding dependency equilibria within the framework of real algebraic geometry encompassing both pure and mixed equilibria. We start by breaking down the concept for a general audience, using concrete examples to illustrate the main results. In alignment with Spohn's original definition of dependency equilibria, we propose three alternative definitions, allowing for an algebro-geometric comprehensive study of all dependency equilibria. We give a sufficient condition for the existence of a pure dependency equilibrium and show that every Nash equilibrium lies on the Spohn variety, the algebraic model for dependency equilibria. For generic games, the set of real points of the Spohn variety is Zariski dense. Furthermore, every Nash equilibrium in this case is a dependency equilibrium. Finally, we present a detailed analysis of the geometric structure of dependency equilibria for $(2\times2)$-games.

翻译：本文在实代数几何框架下对依赖均衡（包括纯均衡与混合均衡）的理解迈出了重要一步。我们首先面向一般读者解析这一概念，并通过具体算例阐明主要结果。依据Spohn对依赖均衡的原始定义，我们提出了三种替代定义，从而实现对所有依赖均衡的代数几何综合研究。我们给出了纯依赖均衡存在的充分条件，并证明每个纳什均衡都位于Spohn簇（依赖均衡的代数模型）上。对于一般博弈，Spohn簇的实点集是Zariski稠密的。此外，在此情形下每个纳什均衡都是依赖均衡。最后，我们对$(2\times2)$博弈中依赖均衡的几何结构进行了细致分析。