Combinatorial Game Theory
who wins, and by how much
Concept

Confusion interval — where it appears

The open interval between a position's two stops, which is exactly the set of numbers the position is confused with. Its two endpoints are decided by an infinitesimal instead, so the interval is open and the ends belong to another theory.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

values · Stops 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.

values · Stops

Named alongside it

The objects these essays reach for when they reach for this one.

ComparisonConfusedDay threeExhaustive searchInfinitesimalStar (∗)StopsTemperatureAll-smallAtomic weightDifference gameDyadic rational

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