This paper extends the concept of face games, introduced by González-Díaz and Sánchez-Rodríguez (Games Econ Behav 62:100–105, 2008) for convex games, to the general class of balanced games. Each face of the core is the core of a face game and contains the best stable allocations for a coalition provided that the members of the complement coalition get their miminum worth inside the core. Since face games are exact we investigate several properties of the exact envelope of a balanced game that allow us to characterize exactness, convexity and decomposability of a game in terms of its face games. The close connection between extreme points of the core and extreme points of the face games is analyzed. In particular, we show that the marginal vectors that belong to the core and the lexinal vectors must be marginal vectors and lexinal vectors, respectively, of the single player face games. Finally, we present several subclasses of games where face games could provide some insight on the core structure.

中文翻译：

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平衡游戏的核心边界：面子游戏

本文将 González-Díaz 和 Sánchez-Rodríguez (Games Econ Behav 62:100–105, 2008) 为凸游戏引入的面子游戏概念扩展到一般类平衡游戏。核心的每个面都是面子游戏的核心，并且包含联盟的最佳稳定分配，前提是互补联盟的成员在核心内获得最低价值。由于面部游戏是精确的，我们研究了平衡游戏的精确包络的几个属性，这些属性使我们能够根据面部游戏来表征游戏的精确性、凸性和可分解性。分析了核心极值点与人脸游戏极值点的紧密联系。特别地，我们证明了属于核心的边缘向量和词法向量必须是边缘向量和词法向量，分别是单人面对的游戏。最后，我们展示了几个游戏的子类，其中面部游戏可以提供对核心结构的一些见解。