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 $\mathcal{V}_{X,\mathcal{C}}$, which is the intersection of the independence model $\mathcal{M}_{\mathcal{C}}$ with the Spohn variety of the game $X$. We prove a conjecture by the first author and Sturmfels that $\mathcal{V}_{X,\mathcal{C}}$ is of codimension $n$ in $\mathcal{M}_{\mathcal{C}}$ 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.

翻译：标准形式的$n$人博弈$X$可以通过无向离散图模型来建模，其中离散随机变量代表玩家，其状态空间为纯策略集合。当关联玩家之间存在依赖关系时，图模型的顶点之间便存在一条边。我们研究斯彭条件独立（CI）簇$\mathcal{V}_{X,\mathcal{C}}$，它是独立模型$\mathcal{M}_{\mathcal{C}}$与博弈$X$的斯彭簇的交集。我们证明了第一作者与Sturmfels的一个猜想：对于具有二元选择的泛型博弈$X$，$\mathcal{V}_{X,\mathcal{C}}$在$\mathcal{M}_{\mathcal{C}}$中的余维数为$n$。我们证明了对于具有二元选择的泛型博弈$X$，全混合CI均衡集（即斯彭CI簇在开概率单纯形上的限制）是一个光滑的半代数流形。如果无向图是若干团的非交并，我们分析了斯彭CI簇的某些代数几何特征，并证明了仿射普适性定理。