Combinatorial Game Theory
who wins, and by how much
Series

Equality — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The context that tells them apart. Two positions put into the same company, one context at a time. Each column is a game X; each cell is the outcome class of that side added to X. Equality means every column agrees, for every X there is — so a single disagreeing column is a disproof, and agreement across a bounded list of contexts is evidence rather than proof. The proof is that the difference is zero.

    Equal in every company

    Two games are equal when no third game can tell them apart — a quantifier over every position there is, discharged by one finite test. A search over 184 contexts separates all 5,790 unequal pairs it is handed and still calls two different games the same, which is exactly why G − H = 0 is a theorem and an exhaustive search is not.

    part 1 · sums
  2. Indistinguishable, until something is added. 567 pairs of unequal games that the 184 contexts cannot tell apart, negated, added to each of the 22 day-two values, and used as options of larger games. Adding a game separates them 1527 times out of 12474; the other two operations never do. Equality is broken by none of them.

    The same parts, different wholes

    Equality is worth having because equal parts make equal wholes: a game may be replaced by an equal one inside any sum and any larger position. The relation that 184 contexts define — no context tells the two apart — is exact on day three and merges a third of a sample of day four. Put 567 of its merged pairs through negation, addition and forming options, and it survives two of the three. Adding the same game to both separates them 1,527 times.

    part 2 · sums
  3. Wider stalls, deeper does not. Merged pairs in a sample of 495 day-four values under universes of sums of one to five day-two values (3044, 567, 212, 178, 147 pairs) and under the 985 day-three values, which merge none.

    A universe grows by days, not by sums

    A relation built from a sample of contexts can be repaired two ways: allow sums of more values, or allow values born a day later. On a sample of day four the first stalls — 147 pairs of different games stay merged from five summands to seven, through 25,696 contexts — while the 985 values of day three, a seventh as many, separate every pair. The reason is four lines long: the context that tells two games apart is always minus one of their options.

    part 3 · sums

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