Global Threats in Combinatorial Games: A Computation Model with Applications to Chess Endgames

Summary

ABSTRACT. The end of play in combinatorial games is determined by the normal termination rule: A player unable to move loses. We examine combinatorial games that contain global threats. In sums of such games, a move in a component game can lead to an immediate overall win in the sum of all component games. We show how to model global threats in Combinatorial Game Theory with the help of infinite loopy games. Further, we present an algorithm that avoids computing with infinite game values by cutting off branches of the game tree that lead to global wins. We apply this algorithm to combinatorial chess endgames as introduced by Elkies [4] where this approach allows to deal with positions that contain entailing moves such as captures and threats to capture. As a result, we present a calculator that computes combinatorial values of certain pawn positions which allow the application of Combinatorial Game Theory.

1. Global Wins and Global Threats

Combinatorial game theory (CGT) applies the divide and conquer paradigm to game analysis and game tree search. We decompose a game into independent components (local games) and compute its value as the sum of all local games. The end of play in a sum of combinatorial games is determined by the normal termination rule: A player unable to move loses. Thus, in a sum of games, no single move or game can be decisive by itself. We investigate a class of games where a move in a local game may lead to an overall win in the sum of all local games. We call such a move globally winning.

Examples of globally winning moves are moves that capture a vital opponent piece such as checkmate, moves that promote a piece to a much more powerful one like promoting a king in checkers, or moves that “escape” in games where one side has to try to catch the other side's pieces like in the game Fox and Geese (Winning Ways [2], chapter 20). Figure 1 shows a Fox and Geese position where the fox escapes with his last move from e5 to d4 and obtains “an infinite number of free moves”. If this game were a component of a sum game S, the fox side would never lose in S.