Combinatorial Game Theory
who wins, and by how much
Generator

One rule, two rules, and the position neither of them settles

One rule, two rules, and the position neither of them settles
One rule, two rules, and the position neither of them settles. Ko fights of one, two and three points, under no repetition rule, the simple ko rule and superko. Each cell reports whether retrograde analysis reached every position in that graph or left some of them drawn. The simple rule settles one point and two and leaves three drawn, which is the triple ko; superko settles all three.

Ko fights of one, two and three points, under no repetition rule, the simple ko rule and superko. Each cell reports whether retrograde analysis reached every position in that graph or left some of them drawn. The simple rule settles one point and two and leaves three drawn, which is the triple ko; superko settles all three.

3 essays call ko-rules. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

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. Out in the world

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.

 a cycle of three: what the backward analysis settles. A position graph in which the moves can lead back to where they started. The labels are the order in which a backward analysis settles each position, starting from the ones where a player has already run out of moves. Positions the analysis never reaches are drawn — and there is no test for that; being unreachable is what a draw is. Where it stops

An outcome with no value behind it

Retrograde analysis labels positions in rounds, outward from the ones already lost. Whatever is still blank when nothing more can be deduced is a draw — and there is no separate test for a draw, because a draw is exactly the residue the method never reaches.

 a loop with a way out under three rules for never ending. One graph, one labelling, and three ways of reading the residue the labelling never reaches. A draw is not a computed outcome here — it is what is left over — so declaring infinite play a win for one side is a legal alternative that costs no extra search and changes who wins. Where it stops

When never ending is a win

Retrograde analysis labels a position a win when somebody can force the opponent to be stuck, and leaves everything else blank. Calling the blanks draws is a rule from outside the game — and two other rules are available. The labelling does not change under any of them; only the residue does, and on a three-cycle that residue is every position on the board.

The whole library · The position index · The figures that play back