Combinatorial Game Theory
who wins, and by how much
Ladder

Temperature — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    rung 1 · temperature
  2. mean 3tax 0gap 4tax 1/2gap 3tax 1gap 2tax 3/2gap 1tax 2frozen — a numbertax 5/2frozen — a numbervaluethe walls meet at a tax of 2, which is the temperatureheating a number by t makes a switch of temperature t, and cooling it by t gives the number back

    Cooling

    Charge a tax on every move and a fight becomes a number. The height of tax at which that happens is the temperature — so cooling is not a technique for finding the temperature, it is what the temperature is.

    rung 2 · temperature
  3. {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.

    rung 3 · temperature

All ladders