Combinatorial Game Theory
who wins, and by how much
Concept

Remoteness — where it appears

How long a game lasts when the winner hurries and the loser stalls, which is the quantity the conjunctive compound turns on. Its parity is the outcome, so a component with odd remoteness is a component the mover wins.

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

Two numbers from the same tree. Four subtraction games, each with its Grundy sequence and its remoteness sequence. The Grundy value decides a disjunctive sum and the remoteness decides a conjunctive one; the only thing they always agree about is which heaps are losses for the player to move.

How long it lasts

Move in every component at once and the game ends the moment any one of them does. Grundy values say nothing about that game; what decides it is the remoteness, a second number computed from the same tree that measures how long a component can be made to last. Over 2,268 positions the rule is right every time, and the two numbers determine each other in neither direction.

sums · Remoteness The number nobody needs. The shortened selective compound — move in any non-empty set of components, and the game stops as soon as any one component stops — solved directly on 1,176 three-heap positions across four subtraction sets, with four predictions beside it. The suspense number was introduced for this compound and it is right; so are three cheaper things, and the shortening leaves the winner unchanged.

The number nobody needs

The compound theory carries a third quantity — the suspense number — computed by the remoteness recursion with both preferences reversed, for the compound that stops as soon as any component stops. It governs that compound correctly. So does remoteness, so does the plain Grundy value, and the shortening does not change the winner on any of 1,176 positions.

sums · Remoteness Four rules, asked of compounds made of two different games. Compounds whose two components come from different subtraction games, solved in full and compared with what each rule predicts. The three rules the compound theory supplies are exact on every position; the shortcut a reader carries instead is not.

A compound of two different games

Every rule the compound theory has survives mixing exactly — the minimum-remoteness rule is right on all 5,184 mixed pairs and all 7,560 triples — and the reason is not that the rules are strong. It is that each of them reads one number per component, and a number does not remember which ruleset produced it. The thing mixing damages is the shortcut a reader carries instead.

sums · Remoteness How long a win takes, against how long the argument allows. Ordinary impartial games with the size of their position graphs, the number of rounds the backward labelling takes, and the number of moves the longest win actually lasts. The round a position settles in is the length of the play from it, which is computed here a second way so the two must agree. The rounds are a handful and the positions are many, which is the gap Zermelo's 1913 paper is about — his question was how many moves a forced win needs, and the answer he could prove was the size of the whole graph.

The paper was about how long

Zermelo's 1913 paper is remembered for a theorem it proves in passing. The question it actually asks is how many moves a forced win takes, the answer it can prove is the size of the whole position graph, and the round counter in the procedure is the real answer — a quantity nobody named for another forty years.

history · Determinacy

Named alongside it

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

Exhaustive searchGame lengthConjunctive compoundGrundy valueOutcome classParitySelective compoundSubtraction gameSuspense numberCounterexampleDisjunctive sumImpartial

All concepts