Theory of Division Games

Abstract

Lehman [9] generalized the Shannon's switching games to those on Boolean functions. We discuss in this paper, the theory of division games which contain Lehman's games and their reverse games. The theory of Lehman's games is analyzed and rearranged in this paper. Our results for division games contain naturally those for the classical ones as Hex, Bridg-it and their reverse games. Especially for the reverse games, we introduce several concepts dual to those introduced before, and obtain concise reformulations. The most significant feature of this paper is that we adopt the notion "assignment" to deal with generalized games in which the principle of alternate move does not necessarily hold. The classical results will be accordingly purified and divided into four theorems, i. e., Theorems 2, 3, 4 and 5, which indicate the dualities about points and about assignments, with respect to signature and to players. It is also a virtue of this generalization that we can characterize in a game theoretical argument the triple systems and the block designs _W_11 and _W_12 whose automorphism groups are Mathieu groups _W_11 and _W_12.