This paper describes results and conjectures on two closely related games, SEKI and D-SEKI, played on matrices of non-negative integers. These games were inspired of so-called seki positions in the game Go. This paper should be viewed as an investigation into the structure of SEKI and D-SEKI, without particular practical applications to Go. The two games are two-player turned-based perfect information partizan finite games which can end in a draw, so a priori there are nine possible outcome classes for SEKI and D-SEKI positions. Most of the paper consists of partial characterizations of the matrices that fall into particular outcome classes. The authors pay particular attention to the drawn matrices, called \textit{seki}, and single out among the seki the drawn matrices where each player has a unique non-losing move (\textit{complete seki}). The methods of the paper are combinatorial and elementary and the structure of the games seems interesting and pleasant. The paper contains four conjectures, three about complete seki and a fourth about seki, that seem amenable to both computer-aided brute force analysis and more mathematical methods.