Combinatorial Game Theory
who wins, and by how much
Concept

Tiny — where it appears

A positive value smaller than every multiple of up, which is what shows that infinitesimals have a scale rather than a size. It changes the outcome of a day-three value once in 1,474 additions, which is what being far below every multiple of up amounts to.

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

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 Seven infinitesimals, added to a whole day. Each row adds one infinitesimal to every value born by day three. The stops never move — that is what being smaller than every number means — the temperature moves a handful of times, and the outcome class moves in a quarter of the additions.

What an infinitesimal does to a fight

Adding a number moves both stops by exactly itself. Adding something smaller than every number moves neither — across 10,318 additions to a whole day of values, not once — and the outcome class changes anyway, 2,622 times. It changes at exactly one kind of position: the ones with a stop sitting on zero, which is where the numbers have run out of things to say.

sums · Translation 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.

sums · Equality

Named alongside it

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

All-smallAtomic weightInfinitesimalStar (∗)Up (↑)ComparisonDay threeDisjunctive sumExhaustive searchInfinitesimal comparisonMultiples of upBirthday

All concepts