Combinatorial Game Theory
who wins, and by how much
Series

Comparison — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Comparing two positions is playing their difference. To decide whether one position is worth at least another, subtract and see who wins moving second. It is the only definition of comparison the subject has, and it produces a partial order — some pairs come out confused, which no comparison of numbers ever does.

    Comparing positions

    One position is worth at least another when the second player wins their difference. That is the only definition there is, it is a computation rather than a judgement, and it produces an order in which some pairs are simply not comparable.

    part 1 · sums
  2. Comparing two positions means playing a third. Pairs of positions with the relation between them, and the game whose solution decided it. There is no way to compare two games by looking at them: the question “is G at least H?” is answered by playing G − H and asking who wins, which is a search, and its cost is counted here beside each answer.

    Comparing two positions means playing a third

    There is no way to look at two games and see which is better. The question "is G at least H?" is answered by building G − H and asking who wins it — so the most basic operation in the theory is a decision problem, and every canonical form is built out of them.

    part 2 · sums
  3. Comparing two positions is playing their difference. To decide whether one position is worth at least another, subtract and see who wins moving second. It is the only definition of comparison the subject has, and it produces a partial order — some pairs come out confused, which no comparison of numbers ever does.

    Confused is not the same as unknown

    Two positions can be neither greater, nor smaller, nor equal. That is a fourth relation with its own symbol, it is a fact about the pair rather than a limit of the method, and it is what makes a game worth playing — a position is a first-player win exactly when it is confused with zero.

    part 3 · sums
  4. How often one value is above another. The partial order counted on two successive days. The proportion of pairs that can be compared at all falls sharply, and so does the proportion of values that can be compared with zero — which is the proportion of positions whose winner does not depend on who moves.

    How rare it is to be bigger

    Values are partially ordered, and 'partially' does most of the work. On day two, 179 of 231 pairs can be compared and 13 of the 22 values can be compared with zero. One day later the shares are 60% and 29%, and the largest set of mutually incomparable values found rises from four to at least twenty-three. Comparison is the exception; confusion is what values normally do to one another.

    part 4 · sums
  5. The fall stops. Comparability on days two, three and four, the last built and corrected.

    A floor, and not a decline

    Comparability fell eighteen points from day two to day three and the next day cannot be enumerated. It can be built — and the construction's bias measured one day lower, where the truth is known. Corrected, day four comes to 60.6 per cent against day three's 59.7: the fall was a one-day event.

    part 5 · sums
  6. The day-four figure, twenty draws at a time. Corrected day-four comparability from twenty seeds of each of two constructions, as dots on a percentage axis, with the range of day three's four slices shaded and the single figure the earlier essay reported marked. The seeds spread over about twelve points and the two constructions agree.

    Twenty draws and a second recipe

    Day four's comparability was reported as 60.6 per cent against day three's 59.7, from one built sample and one calibration. Built twenty times with each of two recipes whose biases differ by six points, and calibrated against all 1,474 day-three values rather than a quarter of them, the corrected figure spreads over twelve points from seed to seed and the two recipes agree within one standard error. The floor survives; the decimal was one draw.

    part 6 · sums

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