Published online by Cambridge University Press: 27 June 2025
1. What are Combinatorial Games? What are they Good For?
Roughly speaking, the family of combinatorial games consists of two-player games with perfect information (no hidden information as in some card games), no chance moves (no dice) and outcome restricted to (lose, win), (tie, tie) and (draw, draw) for the two players who move alternately. Tie is an end position such as in tic-tac-toe, where no player wins, whereas draw is a dynamic tie: any position from which a player has a nonlosing move, but cannot force a win. Both the easy game of Nim and the seemingly difficult chess are examples of combinatorial games. We use the shorter terms game and games below to designate combinatorial games.
Amusing oneself with games may sound like a frivolous occupation. But the fact is that the bulk of interesting and natural mathematical problems that are hardest in complexity classes beyond NP, such as Pspace, Exptime and Expspace, are two-player games; occasionally even one-player games (puzzles) or even zeroplayer games (Conway's “Life“). Two of the reasons for the high complexity of two-player games are outlined below. Before that we note that in addition to a natural appeal of the subject, there are applications or connections to various areas, including complexity, logic, graph and matroid theory, networks, errorcorrecting codes, surreal numbers, on-line algorithms and biology.
But when the chips are down, it is this “natural appeal” that compels both amateurs and professionals to become addicted to the subject. What is the essence of this appeal? Perhaps it is rooted in our primal beastly instincts; the desire to corner, torture, or at least dominate our peers.
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