2024/02/20 by Irem Portakal, Portakal, Irem, Javier Sendra–Arranz +1
Computer Science · #14A10 #14Q05 #62R01 #91A06 #91A80 #Algebraic Geometry (math.AG) #Computer Science and Game Theory (cs.GT) #Constraint Satisfaction and Optimization #FOS: Computer and information sciences #FOS: Mathematics #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2402.13246
openalex publication_date 2024/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An n-player game X in normal form can be modeled via undirected discrete graphical models where the discrete random variables represent the players and their state spaces are the set of pure strategies. There exists an edge between the vertices of the graphical model whenever there is a dependency between the associated players. We study the Spohn conditional independence (CI) variety VX,C, which is the intersection of the independence model MC with the Spohn variety of the game X. We prove a conjecture by the first author and Sturmfels that VX,C is of codimension n in MC for a generic game X with binary choices. We show that the set of totally mixed CI equilibria i.e., the restriction of the Spohn CI variety to the open probability simplex is a smooth semialgebraic manifold for a generic game X with binary choices. If the undirected graph is a disjoint union of cliques, we analyze certain algebro-geometric features of Spohn CI varieties and prove affine universality theorems.