1. Introduction

Are multi-player combinatorial games essentially different from two-player combinatorial games? John Nash was recently awarded the Nobel prize in economics in part for his resolution of the analog question about classical or matricial games: matricial games have equilibria regardless of the number of players. From a classical game-theoretic point of view, combinatorial games are a “trivial” sort of zero-sum game: one of the two players has a forced win in any finite two-player deterministic sequential-move perfect-knowledge winner-takeall game.

When a game has more than two players, it is no longer the case that one always has a forced win. In Section 2, we will study Propp's so-called “queer” three-player games [Propp a], in which no player can force a win, but rather one player chooses which of his two opponents will win.

For example, the game of nim is played with several piles of stones. Players take turns removing stones from a single pile of their choice. At least one stone and up to an entire pile may be taken. The player who makes the last move is the sole winner. With a pile of one stone and a pile of two stones, no player can force a win alone (see Figure 1).