Combinatorial Game Theory
who wins, and by how much
Concept

Context — where it appears

The rest of the board, which decides sente, independence, the size of a move and sometimes the winner. Almost every quantity a player calls a property of a position turns out to be a fact about what it is sitting beside.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

How much company equality needs. Each row restricts the quantifier in the definition of equality to the games named, and counts how many of the 22 values born by day two survive as distinct. The bar is the same number drawn; the jump from nine numbers to four games is the whole argument.

Equal in this company

Equality quantifies over every game there is, and the quantifier can be made smaller. Restricted to a company of nine numbers, the twenty-two values born by day two collapse to seventeen; restricted to four games — nought, one, minus one and star — they stay twenty-two, and no three of the four will do. The company that decides equality is tiny, and it has to contain a star.

limits · Universes How hot a background has to be. Every pair of values born by day two that share a reduced canonical form, added to backgrounds of seven temperatures and three means — 609 comparisons in all — with the count of pairs whose outcome the swap changes. Safety is not monotone in the background's temperature, so the threshold the question asks for does not exist; every one of the 48 changes is at a position with a stop exactly on nought.

How hot a background has to be

The reduced canonical form throws away infinitesimals, and the rung below asked for a bound: how hot must the rest of the board be for the discarded part not to matter? There is no such bound. Safety is not monotone in the background's temperature — an eighth is safe, a quarter is not, two is safe again — and the quantity that does decide it is not a temperature but a stop.

sums · Reduced form One number per heap, and one number per state. Sums of Fibonacci Nim components solved in full, against two predictions. Giving each component the number its heap size suggests gets a quarter of the pairs wrong; giving it the Grundy value of its state — the pair of heap size and cap — gets every pair and every triple right.

What restores the theorem

Fibonacci Nim breaks the recipe every impartial game is supposed to obey: one number per heap, exclusive-ored, gets a quarter of two-heap sums wrong. Index the recursion on the pair of heap size and cap instead and the recipe is exact on every pair and every triple — and the number a heap of nine carries turns out to be five rather than one.

limits · Memory Two clauses, and what each is about. Four rulesets against the two clauses of the condition. A ruleset passes both or the one-number-per-component recipe fails on it, and the two clauses fail for different reasons: locality is about the state proposed, isolation is about the rule.

Two clauses and a third question

A component can carry its own rule when two things hold: its moves are a function of what it carries, and a move in it leaves every other component alone. Two rulesets built to fail one clause each are both caught on a named witness. The four real games sort exactly — every one the recipe gets right fails no clause, every one it gets wrong fails one — and the two clauses still miss something, because Fibonacci Nim and a held pass fail the same clause and only one of them can be repaired.

limits · Memory A ko fight decided somewhere else on the board. A ko fight against the number of ko threats each side holds. A threat is a play elsewhere that the opponent must answer, and it lifts the prohibition on recapturing, so a player short of threats runs out of ways to come back. Every cell is a full retrograde labelling and the pattern in them is read off afterwards: the fight goes to whoever is ahead on a quantity that is not on this part of the board at all. Where the labelling never settles, the fight is a no-result whatever anybody holds.

A ko is won somewhere else

The rung below shows the ko rule buying finiteness by deleting one edge. What it buys with the same edge is a fight nobody can settle by looking at it — the prohibition forces a player to spend a threat, threats are counted on the rest of the board, and every decided cell of the sweep goes to whoever is ahead on a quantity that is not in the picture.

applied · Go A game beside its own mirror, and what is left over. Every coin row added to its own negative, played out exactly, with the resulting scores counted. Under the last-move convention every such sum is worth nothing, because the mirroring strategy guarantees the second player the last move. Here the same strategy is available and the score it produces is not nothing: the mirror of a coin conceded is another coin conceded. Gold is the sums that do come to nothing, which are a minority.

Nothing to subtract with

Comparison is defined by contexts and computed by subtraction, and the equivalence between the two is a theorem about groups. A scoring game is not one — sixty-six of eighty-one coin rows do not cancel against their own negatives — and the difference test then fails on a row compared with itself, which every context accepts and nothing certifies.

applied · Scoring Two numbers instead of an expression. The mean and the temperature of a position, set against its brace expression. The pair is readable in a way the expression is not and is half its length, and it is not exact: most of the games born by day three share a pair with some other game. The cost is not abstract — two games written the same way here can be separated by adding an ordinary small position to each, which is exactly the test a table of outcomes fails one level down.

What two numbers cannot tell apart

A thermograph summarises a position in a mean and a temperature — nine characters against the brace form's twenty-two, and readable in a way the expression is not. It is also not exact: 1,454 of the 1,474 games born by day three share a pair with some other game, 291 of them share one pair, and adding a star to two of those gives different winners.

history · Notation Four restrictions, and what each one buys. Four candidate classes of scoring game — every row, the incentive condition at the top, the same condition at every subposition, and the rows that cancel against their own negatives — scored on two families of coin rows for the mean-value bound, for comparison by subtraction, and for cancellation.

The restriction that buys the most

Four candidate classes of scoring game, scored on the same two families and the same three questions. The class everyone expects to be tiny — the rows that cancel against their own negatives — is empty on rows of three and the widest restriction on rows of four, where it holds fifteen rows against the hereditary class's twelve and gets all 225 of its comparisons right against 108 of 144. The trade everyone expected does not exist.

applied · Scoring Cancelling rows add to cancelling rows. Every pair of cancelling coin rows of two, four and five coins from minus two, one and three, grouped by their lengths, with the number of pairs whose sum cancels against the sum of their negatives.

A cancelling pair is a zero

Two cancelling coin rows side by side cancel against their two negatives — all 190 pairs from rows of two, four and five coins, the odd row that cancels without pairing off included. And a cancelling row beside its negative is invisible next to any other row: in 741 tests against every row of one to three coins, neither score of the context moves. A non-cancelling pair moves a score in 452 of 780. Scoring games have no inverses in general; this class has them, and they behave as inverses must.

applied · Scoring

Named alongside it

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

Exhaustive searchEqualityComparisonOutcome classCanonical formDecompositionDifference gameGroupIndependenceScoring gameAdditivityComponent

All concepts