Combinatorial Game Theory
who wins, and by how much
Field

Temperature

How much is at stake, measured. Thermographs, cooling, and why a player moves where the game is hottest.
012345601234valuetemperaturetemperature 2mean 3Left's wallRight's wall{5 | 1} — mean 3, temperature 2

What is at stake

Some positions both players are desperate to move in, and some neither player wants to touch. The difference is a number — how much the move is worth — and it turns out to be the most useful single quantity for deciding where to play.

 012345601234valuetemperaturetemperature 2mean 3Left's wallRight's wall{5 | 1} — mean 3, temperature 2

Reading a thermograph

A thermograph is two walls rising from a number line, closing in as the tax on moving increases, and meeting where the position stops being worth fighting over. Everything about a position's hotness is in the shape.

 {6 | 0}t = 3a big fight{2 | 0}t = 1a smaller one{1 | 0}t = 1/2small change{0 | 1}no temperaturesettled — a numbercomponenthow much is at stakethe whole position is worth {{{19/2 | 17/2} | {15/2 | 13/2}} | {{7/2 | 5/2} | {3/2 | 1/2}}}and the first move goes in the hottest part, which is a theorem up to a small error rather than a rule of thumb

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.

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