Combinatorial Game Theory
who wins, and by how much
Concept

Uniqueness — where it appears

The fact that a value's canonical form does not depend on the order the reduction steps are taken in. The reduction is confluent, so a form reduced in any order reaches the same place, which is what makes a canonical form a name for a value.

Named by 16 essays across 5 fields — each of them below, with the objects they name alongside it.

The same game, written twice. A position as it arises and the same position reduced. Left would never move to −1 when 0 is available, so that option is dominated and can go. The two games are equal — checked, not assumed — and the second is the canonical form.

Canonical form

Two positions are worth the same when neither player can tell them apart inside any larger game. Deciding that could be an infinite search. Instead there is a normal form — delete what nobody would play, bypass what backfires — and equality becomes a comparison of two small trees.

values · Canonical form The mex, and the rules that cannot replace it. Six candidate rules for the value of an impartial position, each a function of its options' values, run over the same subtraction game. The top strip is the truth. Every candidate but the mex assigns zero to a position somebody wins, or a non-zero value to a position somebody loses, and the circle marks the first heap where each one does it — which is why two people reaching for the same rule four years apart is evidence about the rule rather than about them.

Two people, four years apart, one theorem

Roland Sprague proved it in 1935 and Patrick Michael Grundy proved it in 1939, neither knowing of the other. That looks like coincidence until the alternatives are examined — and the rule they both reached turns out to be the only one that can work at all.

history · Sprague–Grundy The days this site can compute, and the ones it cannot. Zero on the first day, ±1 on the second, and thereafter the simplest number in every remaining gap — the construction run by the game recursion, which produces only fractions with a power of two underneath however long it goes on. Below it, three objects the same recursion reaches when the stopping rule is removed, each written with its option set and the exact reason this site's machinery cannot hold it. They are named rather than drawn, which is the honest half of a figure-first collection.

The numbers came out of the game

The construction is always taught numbers first and games second, and the discovery ran the other way. Conway arrived at the number system from positions, which is why the definition quantifies over sets of previously built objects rather than over cuts — and why it produces a genuinely different collection at every finite stage.

history · Numbers The same game, written twice. A position as it arises and the same position reduced. Three of the options are dominated — a sibling is at least as good for the player who owns them — so they can go. The two games are equal — checked, not assumed — and the second is the canonical form.

Two hundred and fifty-six ways to write twenty-two things

Every game whose options come from the four born on day one — there are 256 of them, and between them they carry 22 values. The reduction that collapses one to the other has choices in it at every step, and uniqueness is the claim that none of the choices matters.

values · Canonical form One position, three ways of writing it, and only one of them adds. The same positions as a sentence about who wins, as a description of the position itself, and in the notation Winning Ways introduced. The first two columns carry identical information and support no operation whatever. The third column can be added — and the sums below it are values that no manipulation of the first two columns could reach, because two of these pairs start from the same two outcomes and finish differently.

The notation was the argument

Up, star and the brace form are not abbreviations for case analyses. They are the claim that these objects add — and the arithmetic they support is arithmetic that no table of outcomes could ever produce, because two positions with identical outcomes can have different sums.

history · Notation Nim-multiplication below 16, and every field axiom checked. The nim-product, defined by taking the least value the product is not forced to be — the same manoeuvre as the mex rule, applied to a product rather than to a move. The result is that these values are not merely a group under nim-addition but a field: every axiom is checked over the whole table here, including an inverse for every non-zero value, and the sizes at which the axioms fail are reported rather than avoided.

The nimbers multiply

Nim-addition is exclusive-or and everybody meets it first. There is also a multiplication, defined by the same take-the-least-value-not-forced manoeuvre as the mex — and it makes the nimbers below sixteen a field, with every axiom checked here and an inverse for every non-zero value.

impartial · Nim 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 Two operators that undo the same tax. Heating leaves every number alone; the warming operator leaves every number alone except an integer, which comes back with a star on it. That single clause is the whole difference between them, and it is what the Go endgame literature needs, because a chilled integer is usually a fight that has been frozen. The rows shown are the ones whose four entries fit in sixteen characters — a warmed day-three value runs to fifty-two, and the clause is legible only in the short ones.

The operator that puts the star back

Chilling is not invertible: it freezes, and 400 values born by day three collapse onto 29. Both heating and Norton's warming operator are exact right inverses of it — each lands back where it started, on all 400 — and they pick different preimages, differing on 396 of them and differing by exactly a star on 335. The clause that separates them is one line long and it is about the integers.

temperature · Chilling The reduction that puts options back. How the two reductions change the width of a form. Domination only ever removes an option. Bypassing a reversible option substitutes the answer's whole option list, so it can leave the form wider than it started — and the finished canonical form can be wider than the form it came from.

The reduction that puts options back

Canonical form is presented as simplification, and half of it is. Deleting a dominated option takes one away. Bypassing a reversible one substitutes the answer's whole option list, so it can leave the form wider than it started — and 60 of 32,428 forms end up with a canonical form wider than they are.

values · Reversibility What deleting is worth on its own. The reduction split into its two halves and each measured. Deleting a dominated option removes exactly one option and can do nothing else; bypassing a reversible one substitutes an option list and can widen the form. The counts say how much of the reduction the monotone half accounts for.

The reduction that always shrinks

Canonical form is two reductions and they are not the same kind of operation. Deleting a dominated option removes one option and can do nothing else; bypassing a reversible one substitutes a whole option list. Over the 256 forms born by day two, deleting alone finishes 225 of them and accounts for 480 of the 520 options that come off — and the 31 it cannot finish are almost all the ones with a star in them.

values · Dominance 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 What a day of canonical forms costs, written out and written once. Three costs for the values born by each of the first three days: every node written every time it occurs, every distinct subposition of a single form, and every distinct subposition of any form of the day. The last is one node per value, and the gap between the first and the last widens as the construction goes on.

The same position, written once

Writing out the canonical forms of day three takes 24,940 nodes. Naming each distinct subposition once inside each form takes 10,102, and naming each distinct subposition once across the whole day takes exactly 1,474 — one per value, because nothing appears inside a canonical form that is not itself a value of the day.

values · Reversibility Star's fibre, described. The fourteen antichains whose mirror value is star, with the two conditions that pick them out of the ninety-six.

A mex with no impartial game in it

The rung below described the zero fibre of the mirror map and left star's fourteen undescribed. Star's fibre is 'some element is at least nought, and none is at least star' — and the two rules are one rule: the mirror value is the least nimber no element of the set reaches. That is a mex, in a construction built entirely from partizan values.

sums · Negation Two solutions to one set of equations. The winning condition written as a single predicate and solved twice: once as the least solution of its own equations and once as the greatest. The least says Left can force a win; the greatest says Left cannot be forced to lose, which admits the positions where Left can keep the game going for ever. On a graph with no cycle in it the two coincide and the equations determine an answer. Where they differ, the difference is exactly the set the backward propagation never reaches — so a draw is not a leftover of the algorithm, it is the equations failing to have one answer.

The gap between two answers

A draw is usually described as what the backward labelling never reached, which makes it sound like a shortfall of the algorithm. Written as one predicate the winning condition is an equation, the equation is monotone, and it has a least solution and a greatest one — and the set the two disagree about is exactly the drawn set, on every game checked.

history · Determinacy Every region of three positions, counted. The 262,144 graphs on three positions reduced to the regions that are genuinely three positions with a cycle in them, and then split by whether the two-position vocabulary has a name for both of their sides.

Four thousand nine hundred regions with no name

Two positions give 256 regions and ten names cover every side of all of them. Three positions give 262,144 graphs, 110,934 genuine loopy regions — and 4,931 of those have a side that no name in the two-position vocabulary reproduces, with 3,990 of them named on one side and blank on the other. The count the earlier essay left open comes back in the affirmative.

history · Notation The guess, and what it covered. Two attempts to name the leftover sides out of the old vocabulary: every pair of the six stoppers, and every two-position region that is a stopper, each with small finite games added. Both cover nothing, and the count of distinct leftovers is what remains.

The names are not built out of the old ones

The guess was that a three-position region's missing names would be sums of two loopy ones — on plus over, and that family. Built and tried, every pair of the six stoppers covers none of the 4,931 regions that need one, and so does every two-position stopper there is, all seventy-nine of them with small games added. Thirteen names have to be invented, and forty-eight cover the whole census against ten at two positions.

history · Notation

Named alongside it

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

Canonical formExhaustive searchStar (∗)ComparisonDominated optionNotationReductionBirthdayEnumerationEquivalenceOutcome classReversible option

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