Combinatorial Game Theory
who wins, and by how much
Series

Temperature — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The thermograph of {5 | 1}. Temperature runs up the page and value across it. Each wall is where a player is willing to move once a tax of that much is charged per move; above the temperature at which they meet, neither wants to move and the position is worth its mean value. The height of the meeting point is what is at stake.

    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.

    part 1 · temperature
  2. Cooling {5 | 1}, one degree at a time. The same position under a rising tax on moving. Each bar is what Left gets moving first and what Right gets moving first, once every move costs the tax. The bars close as the tax rises, and at the temperature they meet — and from there on the position is worth its mean value and neither player wants to touch it.

    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.

    part 2 · temperature
  3. 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.

    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.

    part 3 · temperature
  4. {2 | 0}, added to itself. The value of n copies of one position, for each n, beside n times its mean and the smallest distance between the two. The mean value theorem says that distance stays bounded however many copies are piled up — and the bound is the position's temperature, which is what makes the temperature a second genuine measurement rather than a diagram-reading convenience.

    The same fight, eight times over

    The mean is not roughly what a position is worth. It is the number that eight copies of the position stay close to — and the theorem is that the closeness does not decay as the copies pile up. The gap stops at the temperature, and stays there for ever.

    part 4 · temperature
  5. The endgame, accounted for. Several independent regions, each a fight with a settled value and a size. The account plays them hottest first: add up what each is worth on average, then add the largest amount at stake, subtract the next, and so on down. The exact value of the whole position is computed beside it, and the figure prints both.

    The first time it told somebody something

    A theory earns its keep when it produces an answer nobody had. Temperature did that for Go endgames — the orthodox account gives a move order that is provably right and is not the one experience offers, and the position it is right about is small enough to check here completely.

    part 5 · history
  6. 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.

    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.

    part 6 · temperature
  7. What is left when the copies pair off. For each position, the difference between n copies and n times the mean, reduced to canonical form. The first two are drawn and the last column says what the sequence does after them: half of these settle into a short cycle and the rest produce a new leftover every time, all of them the same bounded size.

    What is left when the copies pair off

    A pile of n copies stays within a bounded distance of n times the mean, and the distance never grows. The difference is a game rather than a number, and what it actually is has a much better answer: for a plain switch it alternates between one fight and nothing at all, and for a fight with a follow-up it is different every time — bounded in size and unbounded in complexity.

    part 7 · temperature
  8. {2 | {1 | 0}}, added to itself. The value of n copies of one position, for each n, beside n times its mean and the smallest distance between the two. The mean value theorem says that distance stays bounded however many copies are piled up — and the bound is the position's temperature, which is what makes the temperature a second genuine measurement rather than a diagram-reading convenience.

    The residues as a sequence

    Four of the eight residue sequences never repeat, and a recurrence is not a description. There is a closed form and it is not for the game: the stops and the temperature of the n-th residue are periodic with period one, two or four on every sequence in the pool, while three of them produce a different game at every n.

    part 8 · temperature

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