Combinatorial Game Theory
who wins, and by how much
Concept

Option list — where it appears

The moves one player has from a position, taken as a set rather than in any order. It is a partial order rather than a ranking, and the reduction keeps exactly its maximal elements.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

One option list, as the order it is. The four options above, with an arrow from each option to every option it is at least as good as. Deleting keeps the one nothing points at and removes the rest, so the reduction takes three of them — a number read off the shape and not off the values.

How much a list of options can lose

Deleting a dominated option is the reduction with no surprises, and how many options it takes is decided by the shape of the order rather than by the values in it: the survivors are the maximal elements, and the count is the length of the list less the number of them. The essay separating the two reductions closed by predicting that the longest chain would give the number. It is a lower bound, exact on 3,859 of the 7,315 four-option lists and wrong on the rest.

values · Dominance How many options a value needs. The canonical form is the smallest form of its value, so the number of options it carries is a property of the value. Three days of the construction, with the widths that occur and the widest value of each.

How wide a form can get

Bypassing a reversible option replaces it with a whole option list, so a form grows in the middle of its own reduction. Whether it can come out wider than it went in is the question that leaves standing, and over 64,515 forms built from day-two options the answer is no, not once — the growth is real, it is transient, and the widest canonical form reached is exactly as wide as the widest form that reaches it.

values · Reversibility What a value costs to write down. Every one of the 1,474 values born by day three, grouped by the width of its canonical form, with the number of symbols the form takes when it is written out. Each count was obtained by walking the canonical form and counting its nodes, so a subposition appearing twice is counted twice — which is what writing it out does. The widest values of the day are not the longest to write.

What a value costs to write down

The canonical form is the smallest form of its value, and it is smallest in the one currency the reduction happens to spend: options. Counted in symbols it is nothing of the kind — the widest value born by day three is not the longest, the longest has six options rather than seven, and every canonical form on the day except the seven integers writes some position out twice.

values · Reversibility The switch formulas, off the hypothesis they were stated for. Values born by day three with exactly one option a side, split by whether both options are numbers. On the twenty-one that satisfy the textbook hypothesis the midpoint and half-gap formulas are exact; on the 146 that do not, the same formulas read off the two stops instead hold about three quarters of the time.

A fight with no midpoint

The mean of {a | b} is the midpoint and the temperature is half the gap — on the twenty-one values of day three where a and b are numbers. One hundred and forty-six others have the same shape and not the hypothesis, and the repair that suggests itself, reading the two stops instead of the two options, holds on about three quarters of them and no more.

values · Switches Which description of a surviving option is right. The two candidate readings of what the reduction keeps, scored over every Domineering option list on six boards. Taking the most room is right on under half the lists, which is what a description with no content scores on lists this short. Leaving the opponent fewest replies is right on nine in ten.

Which option the reduction keeps

Domination deletes an option when another is at least as good, so what survives is the top of an order. On a board that order is made of moves, and two descriptions of the surviving move suggest themselves. Over 1,586 Domineering option lists one of them is right 47% of the time and the other 90%, and the one that wins is not the one a player would guess.

values · Dominance One-sided, all three. The three option tests with their disagreements split by direction. None ever refuses a comparison that holds.

Wrong in one direction only

The rung below asked for the simplified comparison test the dead-ending hypothesis is supposed to license, and predicted it would agree with the quantifier on the dead-ending rulesets and not on Toads and Frogs. Written three ways and scored on 492 pairs, it agrees best on the ruleset that is not dead-ending — and never once refuses a comparison that holds, which makes it a sound filter and not a test.

limits · Dead-ending How many turns are choices. Every position of each game with both sides to move, classified by whether the turn is a choice at all: no move, exactly one move, several moves that all lead to the same verdict, and several that do not. Only the last is a turn at which the alternation is doing any work.

Eleven moves and one decision

A prefix has one quantifier a turn, so a game of eleven moves is eleven alternations. Counted on the boards themselves, a Toads and Frogs strip of eleven moves has twenty-six turns with exactly one move available and one turn anywhere at which the choice changes the answer; a Clobber board has a hundred and fourteen turns and none. Nim, the game everybody calls solved, decides at four turns in five.

complexity · Alternation

Named alongside it

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

Canonical formCounterexampleExhaustive searchReductionComparisonDay threeOutcome classSimplificationAntichainBirthdayDay twoDominated option

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