Combinatorial Game Theory
who wins, and by how much
Concept

Tempo — where it appears

Whose turn it is, treated as a quantity, which a value settles nothing about and which decides positions of equal worth. It is what an infinitesimal measures, and it is the first thing a value throws away.

Named by 18 essays across 5 fields — each of them below, with the objects they name alongside it.

Comparing two positions is playing their difference. To decide whether one position is worth at least another, subtract and see who wins moving second. It is the only definition of comparison the subject has, and it produces a partial order — some pairs come out confused, which no comparison of numbers ever does.

Comparing positions

One position is worth at least another when the second player wins their difference. That is the only definition there is, it is a computation rather than a judgement, and it produces an order in which some pairs are simply not comparable.

sums · Comparison The impartial game inside the scoring one. For every position of a Dots and Boxes board, two questions asked separately: who wins the scoring game, and who wins Nimstring — the same position under the normal-play convention, with no score kept. The bars show how often the two answers agree, grouped by how many boxes are still on the table. Agreement is near-total when there is enough left to be worth controlling and falls away when there is not.

The chains decide it before the boxes do

Under every game of Dots and Boxes there is an impartial game with no score in it, and it settles the question the scoring game keeps asking — who ends up having to open. The rule players learn as folklore falls out of it, and so do the exceptions nobody mentions.

applied · Dots and Boxes 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.

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.

values · Numbers A blocked file, and the tempo it holds. Files of a blocked pawn ending: a White pawn below, a Black pawn above, and a gap between them that either side may close one square at a time. Each file carries the value the game recursion gives it. With only single steps available a file is worth a star or nothing, by the parity of the gap, so the whole position is tempo and no material at all — which is what a chess player means by mutual zugzwang.

A pawn ending is a sum

In a blocked pawn ending the material is level, the files never speak to each other, and whoever has to move is the one in trouble. Chess calls that mutual zugzwang; this site calls it a P-position; and the two vocabularies were built four decades and one subject apart to say the same thing.

applied · Chess 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.

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.

history · Temperature 7 positions of the same value, and how long each of them lasts. Nim positions whose heap sizes all nim-sum to zero. As games they are the same object: each is worth zero, each is a loss for the player to move, and each may be substituted for any other inside any sum without changing a single outcome. The bars are how many moves each one takes, from the shortest legal play to the longest. The value determines everything about who wins and nothing at all about when.

What a value leaves out

A value settles who wins, by how much, and what happens in every sum the position appears in. It says nothing about when. Seven positions here are worth exactly zero and interchangeable everywhere, and they run from two moves long to eighteen.

values · Tempo 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 Where sente stops, by what the answer costs. The largest ambient temperature at which a local move is still answered, sorted by how deep the fight below the answer runs. When the answer ends the fight the crossover is the follow-up's temperature; when the answer starts another fight it is exactly half of it, and a third level does not halve it again.

The answer that starts another fight

A local move is answered while the ambient temperature stays below the follow-up's — and that rule, which this site has carried since the anchor opened, is exact only when the answer ends the fight. When the answer starts another one the crossover is exactly half the follow-up's temperature, on every position tested, and a third level of fight does not halve it again.

temperature · Sente The value does not count the good moves. How many of a Domineering region's placements are best ones, against what the region is worth. Half the values have two regions that disagree about the count, and 52 disagree over whether there is a choice at all.

How many moves are worth making

A value answers who wins and by how much, and the anchor below names the quantities it discards. This is the first of them counted. Over 1,034 Domineering regions and 125 values, 63 values have two regions disagreeing about how many placements are worth making and 52 disagree over whether there is any choice at all — and the count of good moves stays near one and a half however large the region gets.

values · Tempo What a value costs a player. The number of positions a winning strategy has to tell apart, against the number of values among them. The value compresses 4,269 positions into 128 numbers and leaves 3,308 choices to be remembered.

What a strategy has to remember

A value answers who wins and by how much, and it settles neither how many moves achieve it nor whether the best one is unique. Counted over every position reachable inside the catalogue of regions, the gap has a size: 4,269 positions carry 128 values between them, and a player who wants to win rather than to predict has to store 3,308 choices — twenty-six entries for every number the theory supplies.

values · Tempo The list saturates at three. Rules added greedily, each chosen to answer the most decisions given the ones already on the list. Three rules answer 94.5 per cent, and the fourth and fifth answer not one more.

Three rules and a tie-break

An exhaustive table of what a Domineering strategy has to remember is 3,308 lines. Three rules applied in order answer 94.5 per cent of it — leave the opponent fewest replies, then keep the region whole, then take whichever placement comes first — and the fourth and fifth rules answer not one more. The residue is 181 decisions in which every rule scores the candidates the same and one of them is worse.

values · Tempo Two failures, not one. The decisions a Domineering strategy has to store, split by what the two content rules do: answer them, name a worse placement, or leave two candidates standing.

Seventy-two of them were not silence

The rung below said its 181 unanswered decisions were all the rules falling silent and asked whether the position's value picks the placement once the geometry cannot. Seventy-two of the 181 are the rules speaking and being wrong, which is a different failure. On the 109 that really are silence, a rule chosen per value answers more than half — and the star class the rung below singled out is settled outright by leaving the younger position.

values · Tempo Wrong by one, or by nothing. The seventy-two decisions the rules get wrong, by how many replies the named placement misses a best one.

The price of taking the maximum

The seventy-two decisions where a Domineering strategy's rules name the wrong placement are never wrong by more than one reply, and a third of them are the second rule's fault rather than the mobility count's. The repair that follows — keep every placement within one reply of the best — retains a best placement every time and costs thirty decisions for every one it saves.

values · Tempo Every part on its own is worth nothing. The Grundy value of each chain and loop considered as a game by itself, and three real Dots and Boxes boards used to check the turn-by-turn walk against the win-or-lose solver already used here. Every component alone is a second-player win, which is exactly what makes the nim-sum useless.

The parts are worth nothing and the sum is not

Every chain and every loop in Nimstring, taken alone, has Grundy value nought. So the Sprague–Grundy theorem predicts that every position built from them is worth nought — and ninety-six of the two hundred and seven positions checked here are not. The theorem is not being misapplied; it does not apply, because a capture keeps the turn. What replaces it is smaller and sharper: count the short chains, and one long component of any kind reverses the parity.

applied · Dots and Boxes What one king costs a decomposition. Two pawn files and one king a side, solved as a joint position and again as the sum of its files. With the kings unable to move the two answers agree on every configuration, because a king that cannot choose between files is not a shared resource. Give each king a waiting move and the answers come apart, and on some configurations the sum of the parts names the wrong winner rather than merely the wrong value. Independence is a hypothesis about the position and this is the price of assuming it wrongly.

One king, and two files to be in

The whole apparatus needs the files to be independent, and a king is what makes them not. With the kings unable to move the sum of the parts is exact on every configuration; give each king a single waiting move and the sum names the wrong winner on one configuration in six, and on a hundred and twenty-six of two hundred and forty-three with three files.

applied · Chess A bridge circuit, with a link that is not there. The switching graph drawn as a bridge circuit, with an imaginary link from A to B dashed in gold. The graph alone does not split into two edge-disjoint spanning trees, so Short moving second loses; with the imaginary link it does, drawn in blue and red, so Short moving first wins. The green links are the links of the red tree that cross between the two halves the blue tree falls into without the imaginary link — the first moves the trees name.

The first move is a link that is not there

Lehman's criterion answers one question about a switching game — who wins when Short moves second. The other question has the same answer asked of a different graph: add one link from A to B, and Cut is forced to spend its first move deleting it. The trees of that larger graph then name Short's opening, and on every subgraph of seven graphs they name a winner.

applied · Switching What control is worth, to a box. The margin the player who does not have to open nets, beside the formula that predicts it: the total less four boxes for every long chain after the first. Computed against the solver on every endgame of long chains in range.

Four boxes for every chain after the first

Nimstring answers who is forced to open and says nothing about the score. The margin has a formula: the controller nets the total less four boxes for every long chain after the first — two surrendered and two not taken, each time control is kept. Checked against the solver on 175 endgames it is exact on 172, never too generous, and exact wherever it promises the controller anything at all. The three it misses are the three where it promises nothing.

applied · Dots and Boxes The fee the geometry charges. The same endgames solved with the cost of declining changed. Two boxes on a chain and four on a loop are what a single cut and a pair of cuts complete; altering them changes the winner of a large share of positions, which is what says the law depends on them.

Two and four are not conventions

Declining costs two boxes on a chain and four on a loop, and those numbers are read off the geometry rather than chosen: one cut completes the last two boxes of a chain and two cuts complete the last four of a loop. Solved again with the fee changed, 418 endgames give a different winner on up to a third of themselves — so the endgame's law is a law about the fee as much as about the shapes, and the fee is not a free parameter.

applied · Dots and Boxes

Named alongside it

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

DecompositionExhaustive searchHeuristicTemperatureComponentCounterexampleDisjunctive sumDomineeringEndgameEnumerationOutcome classStrategy

All concepts