Combinatorial games modeling seki in GO

Abstract

The game SEKI is played on an (m×n)-matrix A with non-negative integer entries. Two players R (for rows) and C (for columns) alternately reduce a positive entry of A by 1 or pass. If they pass successively, the game is a draw. Otherwise, the game ends when a row or column contains only zeros, in which case R or C wins, respectively. If a zero row and column appear simultaneously, then the player who made the last move is the winner. We will also study another version of the game, called D-SEKI, in which the above case is defined as a draw.An integer non-negative matrix A is a seki or d-seki if the corresponding game results in a draw, regardless of whether R or C begins. Of particular interest are the matrices in which each player loses after every option except pass. Such a matrix is called a complete seki or a complete d-seki. For example, each matrix with entries in {0,1} that has the same sum (at least 2) in each row and column is a complete d-seki, and each such matrix with entries in {0,1,2} is a complete seki. The game SEKI is closely related to the seki (shared life) positions in the classical game of GO.