Combinatorial Game Theory
who wins, and by how much
Concept

Infinitesimal comparison — where it appears

Deciding which of two infinitesimals is larger, which no number settles and only a played-out difference can. It is the question the pair of stops declines, and the reason a second theory sits underneath the temperature one.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

How many ups, bracketed. Every position here is all-small, so no number says anything about it and the yardstick has to be ↑ instead. Each bar spans the multiples of ↑ the position lies between: the largest it is at least, and the smallest it is at most. Four of the seven are pinned to a single multiple of ↑; the rest keep a band that comparison cannot narrow, the widest being ∗ at four ups of slack.

How many ups

When every component of a position is smaller than every positive number, no number can decide it. What decides it is a count of ups — and comparison can pin that count down exactly, except when a star is present, when it cannot.

sums · Infinitesimals Tiny and miny: infinitesimals with a scale. Positions that are greater than zero and smaller than every positive number, and which are nevertheless strictly ordered among themselves — the larger the subscript, the smaller the value. Being smaller than everything positive is not one size of thing; it is a whole scale, and up sits above all of it.

Tiny, miny, and the sizes below every size

An empty two-by-four Domineering board is worth less than nothing and more than every negative number. It is not up, not down and not a fraction — it is a miny, and the minies come in sizes, strictly ordered among themselves below a floor no number reaches.

values · Infinitesimals Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal.

The class where nobody runs out first

Three stones in a row — blue, blue, red — and the position is worth exactly up. Clobber cannot produce anything else, because adjacency is symmetric — a player has a move precisely when the opponent does, and a game with that shape can never be worth a whole move to anybody.

values · Infinitesimals 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.

sums · Reduced form

Named alongside it

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

InfinitesimalAll-smallAtomic weightComparisonStar (∗)Up (↑)Canonical formPartial orderTinyMultiples of upClobberDay three

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