Kernels and Games

Abstract

The present chapter is almost independent of the preceding material. Given a homogeneous binary relation B, we shall look for solutions x of the equation homogeneous, binary relation B, we shall look for solutions x of the equation $\bar x = Bx$. These Boolean eigenvectors have nice application in combinatorial games. We proceed step by step and investigate in Sect. 8.1 the cases $\bar x \subset Bx$ (x absorbant or externally stable) and $Bx \subset \bar x$(x stable) separately. Subsets x that have both properties are kernels of B. In Sect. 8.2 we discuss when kernels exist. Here again, the powerful tool comes to bear that we used already in the first part of Sect. 6.3; its lattice-theoretic nature will be clarified in Appendix A.3. In Sect. 8.3 the results on kernels will be applied to game theory, in particular to nim-type games and to chess endings.