On Numbers and Endgames: Combinatorial Game Theory in Chess Endgames
Noam D. Elkies · 1997
In an investigation of the applications of CGT to chess, we construct novel mutual Zugzwang positions, explain the pawn endgame above, show positions containing non-integer values (fractions, switches, tinies, and loopy games), and pose open problems concerning the values that may be realized by positions on either standard or nonstandard chessboards. 1. Introduction It was already noted in Winning Ways [Berlekamp et al. 1982, p. 16] that combinatorial game theory (CGT) does not apply directly to chess, because the winner of a chess game is in general not determined by who makes the last move, and indeed a game may be neither won nor lost at all but drawn by infinite play. Still, CGT has been effectively applied to other games such as Dots-and-Boxes and Go, which are not combinatorial games in the sense of Winning Ways. The main difficulty with doing the same for chess is that the 8 x 8 chessboard is too small to decompose into many independent subgames, or rather that some of the chess pieces are so powerful and influence such a large fraction of the board's area that even a decomposition into two weakly interacting subgames (say a kingside attack and a queenside counteroffensive) generally breaks down in a few moves. Another problem is that CGT works best with “cold” games, where having the move is a liability or at most an infinitesimal boon, whereas the vast majority of chess positions are “hot“: Zugzwang positions (where one side loses or draws but would have done better if allowed to pass the move) are already unusual, and positions of mutual Zugzwang, where neither side has a good or even neutral move, are much rarer.