Combinatorial Game Theory
who wins, and by how much
Concept

Simplification — where it appears

Reducing a form by removing options without changing its value, which is what produces the canonical form. One half of it can only ever remove an option and the other can put several back, so the two behave nothing alike.

Named by 6 essays across 2 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 How old a sum is. Every unordered pair of the twenty-two values born by day two, with nought dropped because adding it settles nothing — 231 sums. The birthday of each sum was read off its own canonical form and compared with the sum of the two parts' birthdays, which is the bound. The bound holds everywhere and is attained 163 times.

The birthday of a sum

Two values born by days m and n have a sum born by day m + n at the latest, which is the bound that stops a board made of many small parts from being unboundedly complicated. Over 231 pairs of day-two values the bound holds every time and is exact 163 times — and every pair it misses by three days or more has a sum that is a number or a nimber, so the slack is not noise but a measure of how much cancelled.

values · Numbers Not a domination, in the order the rung below meant. The second pass's deletions scored as dominations in two orders: the partial order on games, and the order on stops.

Not a domination, in that order

The rung below asked which pair the second reduction acts on, taking for granted that the operation is a domination. It is not: on none of the 525 deletions is a surviving option greater than or equal to the deleted one. In the order the reduced form actually works in — both stops at least as good — every deletion with a survivor is a domination, the dominator is unique on all but twelve, and it always comes from the other part.

sums · Reduced form The test, scored. The recognition test run on every deletion the second reduction makes, against what actually happens.

A side about to lose its move

A fifth of the second reduction's work removes the last option a player had on a side, and no rule on the ladder had looked at one — because a deletion with no survivor has no pair in it. The recognisable object is not which option goes but whether the side is one an option can go from, and two comparisons on the parts decide it on all 525.

sums · Reduced form

Named alongside it

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

Canonical formDay threeDominated optionExhaustive searchBirthdayDay twoDisjunctive sumOption listPartial orderReductionDyadic rationalEnumeration

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