We model an $n$-player game $X$ in normal form 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 $G$ 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 $\mathcal{M}_{\text{global}(G)}$ 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. In the case where 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：我们通过无向离散图模型对一个$n$人正规型博弈$X$进行建模，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性时，图模型$G$的顶点间存在边。我们研究Spohn条件独立（CI）簇$\mathcal{V}_{X,\mathcal{C}}$，它是独立模型$\mathcal{M}_{\text{global}(G)}$与博弈$X$的Spohn簇的交集。我们证明了首位作者与Sturmfels的一个猜想：对于具有二元选择的通用博弈$X$，$\mathcal{V}_{X,\mathcal{C}}$在$\mathcal{M}_{\mathcal{C}}$中的余维数为$n$。当无向图是团的不交并时，我们分析了Spohn CI簇的某些代数几何特征，并证明了仿射普适性定理。