Combinatorial games modeling seki in GO

Discrete Mathematics. 2014. Vol. 329. P. 19–32.

Gurvich V., Gol'berg A., Andrade D. V., Borys K., Rudolf G.

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 sekior 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}$is\; a\; complete\; seki.\; The\; game\; SEKI\; is\; closely\; related\; to\; the\; seki\; (shared\; life)\; positions\; in\; the\; classical\; game\; of\; GO.$

Keywords: Combinatorial games, Draw, Pass, GO