Invariant games

In the context of 2-player removal games, we define the notion of invariant game for which each allowed move is independent of the position it is played from. We present a family of invariant games which are variations of Wythoff’s game. The set of P -positions of these games are given by a pair of complementary Beatty sequences related to the irrational quadratic number αk = (1; 1, k). We also provide a recursive characterization of this set. We assume that the reader has some knowledge in combinatorial game theory. Basic definitions can be found in [2]. The set of nonnegative (resp. positive) integers is denoted by N (resp. N≥1). Given an infinite sequence S = (An, Bn)n≥0 of nonnegative integers with (A0, B0) = (0, 0), a 2-player removal game on two heaps having S as set of P-positions can always be defined. Indeed, the following na¨ıve rules can be chosen: from any position (x, y) not in S, there is a unique allowed move (x, y) → (0, 0). And from any position (An, Bn...

• Type: Journal
• Year: 2010
• Where: TCS
• Authors: Éric Duchêne, Michel Rigo
• Keywords: 2-player Removal Game, Game, Invariant Game, TCS 2010, Theoretical Computer Science