Combinatorial Game Theory
who wins, and by how much
Concept

Lattice — where it appears

An order in which every pair has a least value above both and a greatest below both, which the values born by a day turn out to be. The values born by a day form one and the whole class does not, so the structure is a fact about the day rather than about games.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

How often one value is above another. The partial order counted on two successive days. The proportion of pairs that can be compared at all falls sharply, and so does the proportion of values that can be compared with zero — which is the proportion of positions whose winner does not depend on who moves.

How rare it is to be bigger

Values are partially ordered, and 'partially' does most of the work. On day two, 179 of 231 pairs can be compared and 13 of the 22 values can be compared with zero. One day later the shares are 60% and 29%, and the largest set of mutually incomparable values found rises from four to at least twenty-three. Comparison is the exception; confusion is what values normally do to one another.

sums · Comparison The 22 values born by day two, and the order they form. Each value sits above everything it is greater than, joined to what it covers. The order has 36 covering relations and is nine levels deep, and 52 of its 253 pairs are incomparable — and it is still a lattice: every pair has a least upper bound and a greatest lower bound among the same 22 values. Two values are marked, together with their join and their meet.

The simplest game above both

Values sit in a partial order, and a partial order is entitled to be ragged: two things with no least thing above them. The 22 values born by day two are not ragged at all. Every one of their 253 pairs has a least upper bound and a greatest lower bound among the same 22, and the order is distributive on all 10,648 triples — so it is a lattice, and the join of zero and star is one half.

values · Lattice The identity that would join the order to the addition. Every pair of the twenty-two values born by day two, asked whether the join plus the meet equals the sum. It holds on all 201 comparable pairs, where the join is the larger and the meet the smaller and it cannot do otherwise, and on none of the 52 incomparable ones.

Where the order and the sum disagree

Day two is a lattice, and day two is a group, and it is not a lattice-ordered group. The one identity that would join the two structures — the join plus the meet equals the pair — holds on exactly the 201 pairs where it cannot fail and on none of the other 52, and the errors split thirteen high, thirteen low and twenty-six confused.

values · Lattice Fifty-two errors, put to four instruments. The fifty-two discrepancies the lattice identity leaves on day two, counted by what distinguishes them. As values no two are the same; as pairs of stops there are seven; as means three and as temperatures three. Not one of them is a number, and only three are values born by day two.

Fifty-two errors and seven sizes

Day two is a lattice and a group and not a lattice-ordered group, and the fifty-two incomparable pairs it fails on leave fifty-two different error terms. Measured rather than listed, the fifty-two collapse: seven pairs of stops, three means, three temperatures, and a rule that predicts the temperature from the pair on forty-four of them.

values · Lattice The order one day out. Whether the values born by day three still form a lattice. Twice as many pairs are incomparable as at day two, and every incomparable pair still has a least upper bound and a greatest lower bound — so the order becomes more tangled without becoming ragged.

One of four questions

Three rungs of this ladder rest on sweeps of day two — 22 values, 253 pairs. Day three is 1,474 values and over a million pairs, and only one of the four questions can be asked of it. The order can: twice as many pairs are incomparable and every one of 1,606 sampled still has a least upper bound and a greatest lower bound, none of them a value day two already had. The other three compare sums of day-three values, which are born on day six, and sixty of those exhausted an eight-gigabyte heap.

values · Lattice

Named alongside it

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

Partial orderCanonical formComparisonJoinStar (∗)ConfusedDay twoEnumerationGroupInfinitesimalAntichainBirthday

All concepts