Combinatorial Game Theory
who wins, and by how much
Concept

Born on day — where it appears

The stage of the construction at which a value first appears, which measures how complicated it is. It bounds how wide a value's canonical form can be, and the bound is loose enough to be almost useless in practice.

Named by 17 essays across 2 fields — each of them below, with the objects they name alongside it.

The numbers, by the day they are born. Zero on the first day, ±1 on the second, and thereafter the simplest number in each remaining gap. Every number reachable in finitely many days is a fraction with a power of two underneath, and every such fraction appears — which is a strange thing for a construction with no arithmetic in it to produce.

The day a number is born

Start with a position in which neither player can move, apply one rule, and the numbers appear — but only the fractions with a power of two underneath, and only in a particular order. That order is what "simplest" means.

values · 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 The values born by day three that are their own negatives. Every game satisfies G + (−G) = 0, so a game equal to its own negative satisfies G + G = 0 — it has order two. The nimbers do, and they are not the only ones: a switch symmetric about zero is unchanged by negation, and so is anything whose Left options are the negatives of its Right options. Each row carries the value, whether it is a nimber, and its outcome.

The values that are their own negatives

Every game satisfies G + (−G) = 0, so a game equal to its own negative satisfies G + G = 0 — it has order two in a group whose elements otherwise have infinite order. The nimbers do. So does ±1, on sight. Over the 1,474 values born by day three there are 30 of them and only four are nimbers, every one of the 900 sums of two is another, and the equality test and a symmetry of the written form agree 1,474 times out of 1,474.

sums · Negation How old a form is, and how old its value is. Every one of the 256 forms born by day two, placed by the depth it is written at and by the birthday of the value it carries. Nothing sits above the diagonal, because a form cannot be younger than the value in it; the diagonal holds the forms written at exactly their value's birthday, and everything below it is a position written older than it needs to be. The count in each cell was obtained by canonicalising all 256 forms and measuring both depths.

How old a value is

A form's depth bounds the birthday of the value inside it, and reducing to canonical form attains the bound — for all 22 values born by day two, with no exception. Twenty-four of the 256 forms are older than what they are worth. The same reduction that makes the bound tight is what puts day three within reach: 98 option sets a side instead of four million, 9,604 forms, 1,474 values, a quarter of a second.

values · Numbers Every day-three value, moved by every quarter. Four claims counted over 3,000 translations — 120 values, each moved by every quarter from −3 to 3: that adding a number leaves the temperature alone, that it shifts both stops by exactly itself, that the interval between the negated stops is where the two players are confused, and that the only failures of the last are on its endpoints.

What a number does to a fight

Adding a number to a position moves everything and changes nothing: over three thousand translations the temperature never once shifted and both stops moved by exactly the number added, every time. What the number decides is whether the fight is worth having — and the interval where the two players are confused is exactly the open interval between the negated stops, right in all 2,890 cases away from its endpoints and wrong in 110 that are all on them.

sums · Translation How often one value is above another. The partial order counted on two successive days. The proportion of pairs that can be compared at all falls sharply, and so does the proportion of values that can be compared with zero — which is the proportion of positions whose winner does not depend on who moves.

How rare it is to be bigger

Values are partially ordered, and 'partially' does most of the work. On day two, 179 of 231 pairs can be compared and 13 of the 22 values can be compared with zero. One day later the shares are 60% and 29%, and the largest set of mutually incomparable values found rises from four to at least twenty-three. Comparison is the exception; confusion is what values normally do to one another.

sums · Comparison Options handed to Left in 1 | −1. A position, and one candidate option after another added to it. Where the gift is one the player would never take the value does not move at all; where it is one they would, it does. The last column is the value of the enlarged form, computed by the same recursion as the original.

An option nobody would take

Every reduction of a form deletes. The gift horse principle adds: a move may be handed to a player for nothing, provided it is one they would never choose. Over all 484 additions to the values born by day two, 283 leave the value exactly where it was and the 201 that move it are precisely the ones the condition forbids — with the boundary at *not better*, which is a weaker demand than *worse*.

values · Gift horse The 22 values born by day two, and the order they form. Each value sits above everything it is greater than, joined to what it covers. The order has 36 covering relations and is nine levels deep, and 52 of its 253 pairs are incomparable — and it is still a lattice: every pair has a least upper bound and a greatest lower bound among the same 22 values. Two values are marked, together with their join and their meet.

The simplest game above both

Values sit in a partial order, and a partial order is entitled to be ragged: two things with no least thing above them. The 22 values born by day two are not ragged at all. Every one of their 253 pairs has a least upper bound and a greatest lower bound among the same 22, and the order is distributive on all 10,648 triples — so it is a lattice, and the join of zero and star is one half.

values · Lattice What each move is worth to the player making it. For each position: every incentive, whether they are all strictly negative, whether the position is a number, and its temperature. The middle two columns are two different computations of the same fact.

Nobody wants to move here

A position is a number exactly when every move loses ground for the player making it. The test never mentions numbers, it disagrees with the ordinary one on none of the 1,474 values born by day three — and the reason a position fails it is not that somebody wants to move. It is that somebody cannot afford to wait.

values · Incentives 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 The values the construction hands down, and the values games produce. The two lists counted against each other. The construction produces 1,474 values by day three; the eleven thousand positions swept here produce 1,193, and only 116 of those are on the construction's list. A value's birthday and a value's reachability have nothing to do with each other.

The values nobody's game produces

The construction hands down 1,474 values by day three. Seventeen rulesets on this site, swept to eleven thousand positions, produce 1,193 — and only 116 of those are on the construction's list. Two of the twenty-two values born by day two are produced by no position of any game here, and 1,077 of the values that are produced are born later than day three. A value's birthday and a value's reachability have almost nothing to do with each other.

values · Realisability 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 The trend, running backwards. The excess grouped by the width of the value's canonical form. Values with wide option lists are exhibited nearer their birthdays than narrow ones, which is the opposite of what the proposal predicted.

Wider costs less

The rung below found the cheapest exhibit of a value never smaller than its birthday, exactly equal on two thirds, and the ruleset explaining 40 per cent of the rest. The variable it proposed for the remainder was the width of the form. Width and excess correlate at −0.39: the wider the value, the closer to its birthday it is exhibited, and inside a ruleset the relation cannot even agree on a sign.

values · Realisability Self-negative values, day by day. How many values each day of the construction supplies and how many of them are their own negatives. Day four cannot be counted; a corner of it supplies 571.

At least five hundred and seventy-one

The rung below dated the self-negative values — the earliest of temperature t is born the day after t — and left the count to a day-four census nobody can run. The construction settles it instead: a value is its own negative exactly when its form is a mirror, so the subgroup can be built from subsets of the day below rather than sifted out of the day above. Day four supplies at least 571 against day three's 26, and the share of a day that is self-negative keeps falling.

sums · Negation The excess is not a flat fee. The excess fitted against the birthday inside each ruleset with enough values to fit a line. A fee would have a slope of nought; every slope here but one is negative, so the excess is largest on the values born earliest.

The entry fee was the cap

Two rungs measured how much bigger a position has to be than the value it exhibits, and attributed what was left to the ruleset — Toads and Frogs paying 2.25 squares on everything, green Hackenbush paying nothing. Neither number is a property of the rules. Inside every ruleset the excess falls as the birthday rises, because the sweep's size cap censors exactly the values that would pay most — and three squares past the cap, Toads and Frogs exhibits values born later than the strip is long.

values · Realisability The collapse happens twice. How many subsets, antichains and values there are. Domination takes 1,793 subsets to 96 antichains and the rest of the reduction takes those to 30 values.

What identifies two subsets

Every subset of a day gives a self-negative value by mirroring it, and 1,793 subsets of day two give thirty values. The collapse happens in two stages with different characters: domination takes the 1,793 to 96 antichains and is a theorem, and the rest of the reduction takes 96 to 30 and is concentrated almost entirely on two values — nought, which has an exact description, and star, which has none.

sums · Negation

Named alongside it

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

Canonical formStar (∗)ComparisonExhaustive searchBirthdayEnumerationConfusionEqualityNumbersUp (↑)Dominated optionNormal play

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