Combinatorial Game Theory
who wins, and by how much
Concept

Convention — where it appears

A rule outside the moves that decides who has won once the moves run out — normal play gives the loss to whoever cannot move, misère gives it the win. Everything computed here depends on which one is in force, and a position can change hands when it changes.

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

The money played out, and it never mattered. The bidding rule played move by move with a countable pool of chips, at every way of splitting it. The verdict is constant across the splits and opposite under the two ways of resolving equal bids, so what settles these positions is the tie-break rather than the money.

The auction never gets to the money

The critical fraction is computed and never played. Played out with a countable pool of chips — twelve positions, four pool sizes, every split of the chips, every bid answered — the verdict does not move with the money on a single one of the forty-eight sweeps, and the rule for equal bids settles all forty-eight. The reason is one line long: declining every auction wins, and bidding nothing declines.

limits · Bidding The same number, from a rule that needs no tie-break. The number computed twice: once as the critical share of a pot under the auction, and once as the probability that Left wins when a fair coin decides who moves at each turn. They agree on every position, and only the second derivation survives being played out.

A coin needs no tie-break

The same recursion has a second derivation: a fair coin decides who moves at each turn, a player whose turn it is with no move has lost, and both play to win. Written from those rules it comes out identical on every position — and it needs no rule for equal bids, because there are no bids. The number is a probability, it belongs to Left rather than Right, and the empty position is the one where the coin decides everything.

limits · Bidding

Named alongside it

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

AlternationDecisionDeterminacyDyadic rationalExhaustive searchNormal playOutcome classCounterexampleCountingEnding conditionInfinitesimalRecursion

All concepts