Combinatorial Game Theory
who wins, and by how much
Concept

Freezing point — where it appears

The tax at which neither player will pay to move any longer, which is exactly the position's temperature. It is where the temperature theory hands over to the infinitesimal one, and where the numbers stop deciding anything.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Cooling {5 | 1}, one degree at a time. The same position under a rising tax on moving. Each bar is what Left gets moving first and what Right gets moving first, once every move costs the tax. The bars close as the tax rises, and at the temperature they meet — and from there on the position is worth its mean value and neither player wants to touch it.

Cooling

Charge a tax on every move and a fight becomes a number. The height of tax at which that happens is the temperature — so cooling is not a technique for finding the temperature, it is what the temperature is.

temperature · Temperature The thermograph of {6 | {5 | {4 | 0}}}. Temperature runs up the page and value across it. Each wall is where a player is willing to move once a tax of that much is charged per move; above the temperature at which they meet, neither wants to move and the position is worth its mean value. The height of the meeting point is what is at stake.

A thermograph is built from its options'

The diagram is not measured, it is computed — each wall is an extremum over the options' opposite walls, shifted by the tax. That construction is why a hot follow-up lowers the temperature instead of raising it, and why two fights with the same swing can differ by a factor of two in what is at stake.

temperature · Thermograph Where 1,474 values sit on the scale. The temperature of every value in the pool, counted. The floor is −1 and only numbers are on it; the next rung up is 0, and everything there is a number with an infinitesimal added. Above that the scale is continuous and the counts thin out.

Below zero

Temperature is described as urgency and urgency has no obvious bottom, but the scale stops at −1 and only the numbers are on it. The rung above is exactly zero, and the 337 values born by day three that sit there are the ones a number cannot be told from — flat thermograph, nothing at stake, and an infinitesimal that no number can see. Two independent computations agree on the classification for all 1,474.

temperature · Cold Adding two thermographs. Every pair drawn from the 15 values born by day two that are not numbers — 120 sums — with the two walls added pointwise and compared against the true diagram of the sum. The added walls are always an outer bound and the means always add; the whole diagram is right for 92 of the 120, and the 28 it is wrong for are exactly the pairs in which both components are hot.

When two thermographs can be added

The temperature of a sum is not the sum of the temperatures, and the natural repair is to add the whole diagrams instead. Over every pair of hot values born by day two the added walls always bound the true ones and the mast always comes out right — and the whole diagram is right exactly when at most one of the two components is hot, which is precisely the case a reader has no use for.

temperature · Thermograph How hot a day gets. The hottest value born by each of the first three days, with every temperature that occurs on it. Day one tops out at nought, day two at one, day three at two — a day buys exactly one degree — and the value attaining the maximum is unique each time. Each temperature was computed as the height at which that value's two thermograph walls meet. The temperatures of day three are exactly the half-gaps between the numbers born by day two, which is what puts a hole in the scale at 7/4.

How hot a day gets

A day of construction buys exactly one degree of temperature — nought, then one, then two — and the value attaining the maximum is unique on every day: ∗, then {1 | −1}, then {2 | −2}. The distribution underneath is not tidy at all: it peaks at a half, leans to the right of the peak, and has a hole in it at one and three quarters where nothing is born.

temperature · Cold The same population counted twice. The temperature scale over the positions this site has enumerated, once with every position counted and once with every distinct value counted. The two disagree about how much of the subject is hot, about what the commonest hot temperature is, and about whether a number is the usual thing for a position to be worth.

How hot a real position is

Counted one value at a time, a tenth of the subject is hot. Counted one position at a time — every board this site has enumerated, all 11,397 of them — it is a twentieth, two thirds of the positions are worth numbers outright, and ten of the seventeen rulesets never produce a hot position at all.

temperature · Cold

Named alongside it

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

Mean valueTemperatureThermographHot gameCoolingExhaustive searchInfinitesimalTax on movingCold gameSwitchAll-smallDay three

All concepts