Combinatorial Game Theory
who wins, and by how much
Concept

Environment — where it appears

The rest of the board, treated as a supply of moves of known sizes rather than as particular positions. Modelling it as a falling stack of coupons gives an ambient temperature at every moment, which turns questions of timing into questions with numbers in them.

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

The move that stops the other one. Six local fights, each placed beside an environment of known temperature. The last two columns are the band over which taking the local move beats spending the move outside, and the band over which the opponent's move in the same fight has to be answered. They are not the same band.

What a move nobody makes is worth

If Right's move in a fight has to be answered, then Left's move in the same fight prevents an exchange Right was going to get for nothing. That is a reverse sente, and pricing it is the awkward case: over sixty measurements the gain matches the local temperature once and the follow-up's swing twice, and the band over which the move is worth taking is not the band over which the move it reverses is sente.

temperature · Sente When the players stop taking coupons. Every pair of fights from a pool of nine, played beside a coupon stack, with the coupon standing when somebody first plays on the board. Sixty of the eighty-one leave exactly when the coupon falls to the board's temperature.

When to leave the environment

A Go player's question is not which fight to take but when to stop taking the small stuff. Put two fights beside a stack of coupons and the orthodox answer — leave when the coupon falls to the hottest temperature on the board — is exact on sixty of eighty-one pairs. All twenty-one departures have a fight with a follow-up in them, and every pair of plain switches leaves on time.

temperature · Coupons The same temperature, and four different departures. Positions with a temperature of one whose follow-ups are worth different amounts, with the coupon at which the players leave the environment. The departure tracks the follow-up.

How big the answer is

The rung below found every early departure from a coupon stack caused by a position with a follow-up, and could not say more: its follow-ups were all of a similar size, so the class it measured was one bit. A pool graded by follow-up size answers it. With the position's own temperature held at one, the departure runs from coupon 1 to coupon 3.5 as the follow-up's temperature runs from 1 to 4 — and over the whole grid the players leave at the larger of the two temperatures.

temperature · Coupons A plateau, not a point. The rule's score as the coefficient is varied on a fine grid. It is constant across the open unit interval and drops at exactly one.

The worst value in its own interval

The rung below scored a component by its temperature less its hottest answer's and asked what rate the answer should really be charged at. Every weight strictly between nought and one scores the same and beats the rung below's choice of one at every board size — because a ranking rule's score is a step function of its own coefficient, and one is exactly where two components tie.

temperature · Coupons

Named alongside it

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

Follow-upTemperatureEnumerationMean valueSenteStrategyAmbientMove selectionSwitchThermographAmbient temperatureApproximation

All concepts