论文标题
依赖平衡的几何形状
Geometry of Dependency Equilibria
论文作者
论文摘要
$ n $ person游戏由相同格式的$ n $张量指定。我们将其平衡视为该张量空间中的点。依赖性平衡是由条件概率的线性约束来定义的，因此是由张量条目中的确定性序言定义的。这些方程式削减了Spohn品种，以引入依赖性平衡的哲学家命名。其中的纳什平衡是等级第一的张量。我们研究了Spohn品种的实际代数几何形状。这种品种是合理的，除了$ 2 \ times 2 $游戏，当它是椭圆曲线时。对于$ 3 \ times 2 $游戏，它是第二学位的Del Pezzo表面。我们使用定向的矩阵来表征收益区域及其边界，并在统计中开发了与贝叶斯网络的连接。
An $n$-person game is specified by $n$ tensors of the same format. We view its equilibria as points in that tensor space. Dependency equilibria are defined by linear constraints on conditional probabilities, and thus by determinantal quadrics in the tensor entries. These equations cut out the Spohn variety, named after the philosopher who introduced dependency equilibria. The Nash equilibria among these are the tensors of rank one. We study the real algebraic geometry of the Spohn variety. This variety is rational, except for $2 \times 2$ games, when it is an elliptic curve. For $3 \times 2$ games, it is a del Pezzo surface of degree two. We characterize the payoff regions and their boundaries using oriented matroids, and we develop the connection to Bayesian networks in statistics.
