Combinatorial Game Theory
who wins, and by how much
Generator

How much changes hands, against how much is at stake

How much changes hands, against how much is at stake
How much changes hands, against how much is at stake. Two ways of choosing where to move, run against optimal play over every board from a pool of three components. Biggest-first takes the component where the most changes hands, which is the count in every endgame book; hottest-first takes the one with the highest temperature. They disagree on most of these boards, the count costs points more often, and — the difference that matters — the count sometimes loses more than the largest temperature on the board, which is the bound the theory's rule is guaranteed to keep.

Two ways of choosing where to move, run against optimal play over every board from a pool of three components. Biggest-first takes the component where the most changes hands, which is the count in every endgame book; hottest-first takes the one with the highest temperature. They disagree on most of these boards, the count costs points more often, and — the difference that matters — the count sometimes loses more than the largest temperature on the board, which is the bound the theory's rule is guaranteed to keep.

2 essays call sizing-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.

Move where it is hottest. four independent components of one position, ordered by temperature. The temperature is how much a player loses by moving somewhere else instead, so the hottest component is the one to take — and a component that is already a number has no temperature at all, because nobody gains by moving in it. Temperature

Playing the hottest

Given several independent fights, play in the one with most at stake. The rule is simple, it is what strong Go players do without being told, it is provably close to optimal — and it is provably not optimal, which is the interesting part.

 How much changes hands, against how much is at stake. Two ways of choosing where to move, run against optimal play over every board from a pool of three components. Biggest-first takes the component where the most changes hands, which is the count in every endgame book; hottest-first takes the one with the highest temperature. They disagree on most of these boards, the count costs points more often, and — the difference that matters — the count sometimes loses more than the largest temperature on the board, which is the bound the theory's rule is guaranteed to keep. Temperature

Big is not the same as hot

A player sizes a move by how much changes hands when it is played, which is the number in every endgame book. The theory sizes it by temperature. On a plain switch the two agree exactly, so nothing shows; on a move with a follow-up they come apart, and the count gives up more than the guarantee the theory's rule carries.

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