Combinatorial Game Theory
who wins, and by how much
Theme

The thread: How much is at stake

A position is worth something on average and worth something more to move in first, and the second number is the one a player feels. Temperature is that number, and most of what it measures is not where it is expected.
The thermograph of {5 | 1}. 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. Temperature

What is at stake

Some positions both players are desperate to move in, and some neither player wants to touch. The difference is a number — how much the move is worth — and it turns out to be the most useful single quantity for deciding where to play.

 The thermograph of {5 | 1}. 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. Temperature

Reading a thermograph

A thermograph is two walls rising from a number line, closing in as the tax on moving increases, and meeting where the position stops being worth fighting over. Everything about a position's hotness is in the shape.

 Move where it is hottest. four independent components of one position, ordered by temperature. The temperature is how much a player loses by moving somewhere else instead, so the hottest component is the one to take — and a component that is already a number has no temperature at all, because nobody gains by moving in it. Temperature

Playing the hottest

Given several independent fights, play in the one with most at stake. The rule is simple, it is what strong Go players do without being told, it is provably close to optimal — and it is provably not optimal, which is the interesting part.

 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. Temperature

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.

 The endgame, accounted for. Several independent regions, each a fight with a settled value and a size. The account plays them hottest first: add up what each is worth on average, then add the largest amount at stake, subtract the next, and so on down. The exact value of the whole position is computed beside it, and the figure prints both. Temperature

The endgame, accounted for

Add up what each region is worth, then add the biggest thing at stake, subtract the next, and so on down. On a board of simple fights the result is exact — and the moment one region has a fight inside it, the account is out by a point.

 Cutcake: every value an integer. The value of an m by n cake, for every small m and n. Left cuts down, Right cuts across, and neither player ever gains by moving — so nothing is ever at stake, every value is a whole number, and the number says exactly how many spare moves one player has. Particular games

Cutcake, where every value is a whole number

A partizan game in which no position is ever worth a fraction, a star or a fight. Every value is an integer, the integer is a count of spare moves, and the pattern it follows is decided by binary digits.

 Why nobody moves in the number. A hot position added to a number. Left wins the sum whoever moves — but only by moving in the fight. Spending the move on the number instead hands the position back as a first-player win, with Right to move, which throws the win away. The theorem says this is always so, and here it is happening. Values

Numbers avoid numbers

In a position with a number in it and anything else, the number is never the right move. That is a theorem rather than a heuristic, and it is the closest this subject comes to advice a player can carry into a real game.

 {2 | 0} + {2 | 0} — where the temperature goes. Three thermographs on one frame: two positions and their sum. The mean of the sum is the sum of the means, every time. The temperature is not: it is bounded by the hottest of the parts and is often far below it, so the number that says how much is at stake in a whole board cannot be got by adding up the parts. Temperature

Two hot fights that add to a cold number

The mean of a sum is the sum of the means, every time. The temperature is not — it is bounded by the hottest part and is often far below it. Two positions each worth fighting over can add to a plain number that neither player wants to touch.

 {2 | 0}, added to itself. The value of n copies of one position, for each n, beside n times its mean and the smallest distance between the two. The mean value theorem says that distance stays bounded however many copies are piled up — and the bound is the position's temperature, which is what makes the temperature a second genuine measurement rather than a diagram-reading convenience. Temperature

The same fight, eight times over

The mean is not roughly what a position is worth. It is the number that eight copies of the position stay close to — and the theorem is that the closeness does not decay as the copies pile up. The gap stops at the temperature, and stays there for ever.

 A switch, its mean and its temperature. Positions of the form {a | b} with a above b: both players want to move there, so neither is settled. The bar spans the two options, the marked point is the mean the position is worth once the fighting is over, and the temperature is half the gap — which is exactly what moving first is worth. Values

Worth nothing, and worth fighting for

A switch is a position both players want to move in. Its average value can be zero while the difference between getting there first and second is enormous, and that gap is a second number every position carries.

 How much changes hands, against how much is at stake. Two ways of choosing where to move, run against optimal play over every board from a pool of three components. Biggest-first takes the component where the most changes hands, which is the count in every endgame book; hottest-first takes the one with the highest temperature. They disagree on most of these boards, the count costs points more often, and — the difference that matters — the count sometimes loses more than the largest temperature on the board, which is the bound the theory's rule is guaranteed to keep. Temperature

Big is not the same as hot

A player sizes a move by how much changes hands when it is played, which is the number in every endgame book. The theory sizes it by temperature. On a plain switch the two agree exactly, so nothing shows; on a move with a follow-up they come apart, and the count gives up more than the guarantee the theory's rule carries.

 {5 | {4 | 0}} beside one other fight. A local position and a single switch, played out together at each of several ambient temperatures. The middle columns are what optimal play does: whether it opens the local fight, and whether it answers when the opponent opens it. The answer stops being forced at a temperature the local position alone does not name. Temperature

Sente is a fact about the rest of the board

A move that must be answered is called sente, and the word is used as though it described the local position. It does not. The same fight is answered while the rest of the board is quiet and ignored once it is busy, and the crossover — measured by solving the whole board at every temperature — sits at the follow-up's own 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. Temperature

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.

 Is {{5 | 3} | {2 | −4}} double sente?. One local fight with a follow-up on each side, played out inside a sum with a switch whose temperature rises. The two middle columns are what optimal play does when each player opens the fight. Whether the opponent has to answer is settled by the ambient temperature and not by the shape, so the same position is double sente, sente for one player, and gote for both, at three different ambients. Temperature

Double sente is not a property of the position

A fight with a threat on each side is called double sente, as though the phrase named a shape. Swept against a rising ambient temperature, one such fight — { {5 | 3} | {2 | −4}} — is double sente up to an ambient of 1, sente for one player only from 3/2 to 3, and gote for both from 7/2, and the two band edges are the temperatures of the two follow-ups.

 The endgame, accounted for. Several independent regions, each a fight with a settled value and a size. The account plays them hottest first: add up what each is worth on average, then add the largest amount at stake, subtract the next, and so on down. The exact value of the whole position is computed beside it, and the figure prints both. How it was found

The first time it told somebody something

A theory earns its keep when it produces an answer nobody had. Temperature did that for Go endgames — the orthodox account gives a move order that is provably right and is not the one experience offers, and the position it is right about is small enough to check here completely.

 {4 | 0} played out in a stack of 5 coupons. An idealised environment: coupons worth a fixed step less each, which either player may take instead of moving in the game. The rows are the line optimal play takes over the whole board, in order. What the game turned out to be worth is set beside its mean value, and the coupon the players stopped at beside its temperature — two quantities measured from the play, and two computed from the thermograph. Temperature

An environment made of coupons

Beside the game sits a stack of coupons worth 4, 3, 2, 1, 0, and a player may always take the top one instead of moving. Play the whole thing out and two quantities the theory computes are measured instead: {4 | 0} comes out worth exactly 2, its mean value, and the coupons stop at 2, its temperature. For {10 | {9 | 1}}, whose temperature is 1, they stop at 7/2 — because what the stopping coupon measures is the hottest temperature anywhere in the tree.

 Every position of both games, and where the fights are. Every colouring of every graph in the library, under both rules, with the temperature of each computed. A position with a positive temperature is one both players want to move in. Col has none anywhere; Snort has them on every graph, and the hottest grows with the graph. Particular games

One rule makes it cold, the other hot

Col forbids painting next to your own colour and Snort forbids painting next to your opponent's. Over all 540 colourings of six graphs, not one Col position has anything at stake and 61 Snort positions do — the difference between a game that is a count and a game that is a series of fights, produced by one word.

 the temperature that bounds the loss. 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. What it costs

A rule that is never right and cannot be far wrong

Playing the hottest component is not optimal — over 440 measured lines it costs something on 17 of them. What makes it worth having is that the cost is bounded by the temperature, provably, and that the same test run with the ordering reversed breaks the bound on 54.

 The thermograph of {{5 | {3 | 1}} | 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. The marks partway up the walls are the bends: the heights at which the option holding a wall up stops holding it up. Each stands at the temperature of a follow-up. Every one of them is below the temperature of the position itself, which is where the two walls meet. Temperature

A thermograph with two bends

A wall bends where the option holding it up stops holding it up, and most drawn thermographs bend once. { {5 | {3 | 1}} | 0} bends twice, at 1 and 3/2, and neither height is its temperature of 7/4 — every bend lies strictly below where the walls meet. Over a census of 17,255 hot positions two levels deep, a second bend in one wall never happened once.

 The thermograph of {5 | 1}. 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. The two marks on the base line are the stops — what each player gets by moving first and playing the fight out with no tax charged at all. Values

Where the fight stops

Keep taking the biggest thing on offer until somebody is left facing a number, and the number reached is a stop. Every position has two of them, one for each player moving first, and between them they say how a fight ends — except for the part of it no number can reach.

 Cooling by 1, and heating back. Each row is a position, its temperature, what it becomes when every move is taxed, and what comes back when the tax is refunded. The refund is not an inverse: a position whose temperature was below the tax has already frozen into a number, and heating a number does nothing at all. Temperature

Cooling by exactly one

Cooling is usually met as a way of reading a thermograph: a tax, and two numbers at the height of the tax. Applied as an operator it returns a position instead — and then the obvious question is whether heating gives the position back. Over the 1,474 values born by day three, 27 survive the round trip and 15 of those are numbers that never moved. Cooling throws away almost everything it touches.

 Small Domineering boards and what they are worth. Every value here was computed from the moves rather than looked up. Even on boards this small the values are switches and infinitesimals rather than numbers, which is the ordinary situation for a partizan game and the reason the theory needs more than arithmetic. Particular games

The values of every small board

Thirty Domineering rectangles, every value computed from the moves rather than looked up. The 1×n row obeys a formula and the 2×n row does not: its outcomes run L N N R three times over and then 2×13 comes out worth exactly 0, and its temperatures climb to 19/16 and fall back without settling.

 A hot position, split into its mean and what is left. Each row is a position cooled by exactly its own temperature — the tax at which it stops being worth moving in. The result is the mean value with something small still attached, and the last column is that something, obtained by subtracting the mean from the cooled game rather than by inspection. Temperature

A number and a fight

Charge a position exactly what it is worth fighting over and the fight disappears, leaving the mean value — with something still attached to it. Over all 1,122 hot values born by day three the residue is smaller than every positive number, it is a star in 942 of them, and it is never nothing. So a hot game is its mean plus a fight plus a remainder that no number reports, and the remainder is what decides close games.

 Amazons on one line. A one-dimensional Amazons board: an amazon slides along the row and shoots along the row, and the square the arrow lands on is burnt for the rest of the game. The whole board fits in a sentence, and the values it produces are already of several different kinds. Particular games

Amazons on one line

A board one square high is small enough to evaluate completely: every strip from two to ten squares with one amazon a side is 37,886 positions taking 81 distinct values, and every one of them is an integer, a switch, a number plus a star, or a bare star. Not one is a fraction — and forcing the arrow onto the square just vacated, which takes a freedom away rather than adding one, produces 1,196 that are.

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