Combinatorial Game Theory
who wins, and by how much
Series

Stops — the series

4 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. The two marks on the base line are the stops — what each player gets by moving first and playing the fight out with no tax charged at all.

    Where the fight stops

    Keep taking the biggest thing on offer until somebody is left facing a number, and the number reached is a stop. Every position has two of them, one for each player moving first, and between them they say how a fight ends — except for the part of it no number can reach.

    part 1 · values
  2. The fight never runs backwards. Each of the 22 values born by day two, drawn from its right stop to its left stop — what Right gets moving first, and what Left gets moving first, once the fight has been played out to a number. Every bar runs the same way. The left stop is never below the right one, which is what "both players are trying to improve their own position" amounts to, and the cold rows, where the two coincide, are drawn as a single point.

    The fight never runs backwards

    Left's stop is never below Right's — in every one of 1,780 distinct values, computed twice by two independently written routes, with nothing that disagreed anywhere. The inequality is what makes a mean value well defined and a fight a fight; and where it collapses to equality, 433 of the 460 cold positions turn out not to be numbers at all.

    part 2 · values
  3. The numbers each position is confused with. Each row is a position. The bar runs from its right stop to its left stop; the filled part is the set of numbers the position is genuinely confused with, computed one comparison at a time. The two coincide except at the ends, and a position whose stops meet is confused with nothing at all even when it is not a number.

    The numbers it is confused with

    A position is confused with a number when neither is at least as good as the other, and the set of such numbers is an interval. It is exactly the open interval between the two stops: over 36,850 comparisons the rule is wrong nowhere it speaks, and the 2,596 comparisons it declines are precisely the ones at an endpoint, where the position and the number differ by an infinitesimal.

    part 3 · values
  4. Which end of the interval is open. Every value born by day 3 compared with each of its own two stops — 2,948 comparisons, each one a search over the difference. The two rows are mirror images because the day is closed under negation, and the small number in each row is the exception class: the 352 values whose two stops coincide, for which the left stop is the right stop and the law has nothing to bite on.

    Which end of the interval is open

    The confusion interval is open at both ends, and the two ends are not the same kind of open. At its own left stop a position can be below the number, confused with it or equal to it, and — over 2,948 comparisons — above it exactly thirty-three times, every one of them a value whose two stops are the same number and whose left end is therefore also its right one.

    part 4 · values

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