Eyespace Values in Go

Most of the application of combinatorial game theory to Go has been focussed on late endgame situations and scoring. However, it is also possible to apply it to any other aspect of the game that involves counting. In particular, life-and-death situations often involve counting eyes. Assuming all surrounding groups are alive, a group that has two or more eyes is alive, and a group that has one eye or less is dead. This naturally raises the question of which game-theoretical values can occur for an eyemaking game. We define games that provide a theoretical framework in which this question can be asked precisely, and then give the known results to date. For the single-group case, eyespace values include 0, 1, 2, R 1 2 , R 1 1 2 , R 3 4 , R 1 1 4 , R 1∗, and several ko-related loopy games, as well as some seki-related values. The R 1 2 eye is well-understood in traditional Go theory, and R 1 1 2 only a little less so, but R 3 4 , R 1 1 4 , and R 1∗ may be new discoveries, even though they occur frequently in actual games. For a battle between two or more opposed groups, the theory gets more complicated.