Combinatorial Game Theory
who wins, and by how much
Series

Reduced form — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. What the reduction collapses. Each reduced form with the values that reduce to it. The largest class is the one that reduces to zero and it holds every infinitesimal on the list, which is exactly what the reduction is for — against a hot background, none of them is distinguishable from nothing.

    What is left when the small change is thrown away

    Canonical form answers a demanding question: which positions are interchangeable inside every sum whatever. A player with a hot board does not have every sum — an infinitesimal difference cannot decide anything against a genuine fight — so there is a coarser question with an exact answer. The reduced canonical form takes the 1,474 values born by day three to 61, with 292 of them collapsing to zero, and it is a homomorphism on all 8,100 pairs tested only when a second pass is made.

    part 1 · sums
  2. How hot a background has to be. Every pair of values born by day two that share a reduced canonical form, added to backgrounds of seven temperatures and three means — 609 comparisons in all — with the count of pairs whose outcome the swap changes. Safety is not monotone in the background's temperature, so the threshold the question asks for does not exist; every one of the 48 changes is at a position with a stop exactly on nought.

    How hot a background has to be

    The reduced canonical form throws away infinitesimals, and the rung below asked for a bound: how hot must the rest of the board be for the discarded part not to matter? There is no such bound. Safety is not monotone in the background's temperature — an eighth is safe, a quarter is not, two is safe again — and the quantity that does decide it is not a temperature but a stop.

    part 2 · sums
  3. Add, then reduce again. The arithmetic the homomorphism promises, measured: summing two reduced forms gives a reduced form on 88 per cent of pairs and needs a second reduction on the rest.

    Add, then reduce again

    The homomorphism promises that a sum's reduced form can be computed from its parts', and says nothing about what the operation is. It is addition followed by a second reduction — needed on 431 of 3,600 pairs of day-three values, and on not one of the 1,751 pairs with a cold part. What the second pass removes is an option that only becomes dominated once the two fights are side by side.

    part 3 · sums
  4. Eight ways to name it, and none of them works. Candidate rules for which option the second reduction deletes, scored on every pair where it deletes exactly one. The best reaches four in five and none is exact.

    The option nothing names

    The rung below found the arithmetic on reduced forms to be add and reduce again, needing the second pass on 431 of its sums, and asked whether the option that pass deletes can be named from the parts. Eight rules were scored and the best reaches four in five — and on a pool closed under negation it falls to under half, which says the near-miss is a property of the population. What the second pass does have is a shape and a cheap test that rules it out.

    part 4 · sums
  5. Not a domination, in the order the rung below meant. The second pass's deletions scored as dominations in two orders: the partial order on games, and the order on stops.

    Not a domination, in that order

    The rung below asked which pair the second reduction acts on, taking for granted that the operation is a domination. It is not: on none of the 525 deletions is a surviving option greater than or equal to the deleted one. In the order the reduced form actually works in — both stops at least as good — every deletion with a survivor is a domination, the dominator is unique on all but twelve, and it always comes from the other part.

    part 5 · sums
  6. The test, scored. The recognition test run on every deletion the second reduction makes, against what actually happens.

    A side about to lose its move

    A fifth of the second reduction's work removes the last option a player had on a side, and no rule on the ladder had looked at one — because a deletion with no survivor has no pair in it. The recognisable object is not which option goes but whether the side is one an option can go from, and two comparisons on the parts decide it on all 525.

    part 6 · sums
  7. A cross in the table. Pairs needing a second reduction, by the colder temperature and the gap between the two.

    A cross in the table

    Which pairs need a second reduction had never been asked. Sorted by the two temperatures the answer is a cross — the whole gap-of-a-quarter column and the whole colder-is-three-quarters row — and it is exactly necessary on all 431 with no exception, and wrong 477 times in the other direction.

    part 7 · sums

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