Combinatorial Game Theory
who wins, and by how much
Concept

Decreasing quantity — where it appears

Something about a position that goes down at every move and cannot go down for ever, which is what makes a recursion on it legitimate. A game with one is a game that ends; a game whose only such quantity is an ordinal ends with nothing bounding when.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

Two ways to be certain and ignorant at once. Ten positions from two games that both terminate for reasons no bound comes out of. Sylver Coinage's proof counts something that goes down and can be counted; the hydra's counts an ordinal, which cannot, and the last column shows what that difference is worth.

Two ways to end with no bound

Sylver Coinage and the hydra are both guaranteed to finish and neither will say when. The difference is that one of them carries its own bound: every move in Sylver removes at least one gap, the gaps can be counted in a moment, and over ten openings the longest play uses every single one. The hydra has no decreasing quantity a solver can hold — three hydras of five nodes each take seven chops, twenty-one, and a number past two hundred and seventy-nine that this machine never reaches.

limits · Termination 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 A fortress, and the counter that gives it a label. A pawn ending where the defender's king shuffles for ever and the attacker needs time. Down the rows, how many moves of preparation the breakthrough needs; across the columns, how many moves the rule allows before declaring a draw. With no breakthrough the position is drawn whatever the rule says, and drawn as a residue the backward induction never reaches. With a breakthrough and no rule the attacker wins despite the cycle. Where the march is longer than the counter allows, the rule turns a won position into a drawn one.

A position with no value, and the rule that gives it one

A fortress is a cycle in the position graph, so the recursion defining a value has nowhere to bottom out and the propagation never reaches it. Chess has a rule for that — count fifty moves and call it drawn — and the rule does not merely tidy the theory up. On eleven cells of the sweep it takes away a win.

applied · Chess

Named alongside it

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

Exhaustive searchGame lengthOutcome classTerminationAdditivityBackward inductionBoundChessConjunctive compoundCounterexampleDisjunctive sumDraw

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