Combinatorial Game Theory
who wins, and by how much
Series

Go — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 1 ko point, no ko rule. A ko fight drawn as a position graph and labelled by retrograde analysis: blue edges are Black's captures, red are White's, and each position carries the verdict for whichever side is to move. Positions the propagation never reaches are drawn — neither player can force a win and the game does not end — and they appear only where the rules permit a repetition.

    The rule that makes Go a finite game

    A ko is a point in Go where a capture can be recaptured for ever, and every set of rules forbids it. That prohibition is not etiquette or tidiness — it is the hypothesis that puts Go inside the class of games every theorem on this site is about, and removing it removes the values.

    part 1 · applied
  2. A ko fight decided somewhere else on the board. A ko fight against the number of ko threats each side holds. A threat is a play elsewhere that the opponent must answer, and it lifts the prohibition on recapturing, so a player short of threats runs out of ways to come back. Every cell is a full retrograde labelling and the pattern in them is read off afterwards: the fight goes to whoever is ahead on a quantity that is not on this part of the board at all. Where the labelling never settles, the fight is a no-result whatever anybody holds.

    A ko is won somewhere else

    The rung below shows the ko rule buying finiteness by deleting one edge. What it buys with the same edge is a fight nobody can settle by looking at it — the prohibition forces a player to spend a threat, threats are counted on the rest of the board, and every decided cell of the sweep goes to whoever is ahead on a quantity that is not in the picture.

    part 2 · applied
  3. Where the two conventions come apart, counted. Every small Go endgame solved under both scoring conventions. The scores agree exactly when the number of neutral points is even and never when it is odd, which is the parity of the stones each side ends up placing. A counted fraction name different winners. And a smaller fraction are played differently, which is the half of the finding a rules argument does not predict: a neutral point is a one-point play under one convention and worth nothing under the other, so the two rule sets disagree about the order of the endgame and not only about its total.

    Two ways to count a finished board

    Territory scoring and area scoring are both in daily use and they are not variants of one rule. Over seventy-nine small endgames they agree exactly on the thirty-nine with an even number of neutral points and on none of the forty with an odd number — sixteen name a different winner, and twelve are played differently, which is not something a convention is supposed to do.

    part 3 · applied

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