Combinatorial Game Theory
who wins, and by how much
Concept

Compound — where it appears

Any way of adding games other than the disjunctive sum: move in every component, or in any set of them, or stop when the first component stops. Each has its own quantity that decides it — remoteness, the outcome of a part, the suspense number — and none of them is the value.

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

Three ways to add the same games. One list of components, added three different ways. Under the disjunctive rule a move is a move in exactly one part; under the conjunctive rule it is a move in every part at once, and play stops as soon as any part runs out; under the selective rule it is a move in any non-empty set of parts. The outcomes are computed by search from each rule's own definition.

Three ways to add the same games

A move in exactly one component is a choice, not a law. Move in every component at once and the game is different; move in any set of them and it is different again. The same two positions, added three ways, give three different answers — and only one of the three has values that add.

sums · Disjunctive sum 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

Named alongside it

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

Conjunctive compoundDecompositionExhaustive searchSelective compoundAdditivityComparisonComponentDisjunctive sumGame lengthGrundy valueHeuristicImpartial

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