Combinatorial Game Theory
who wins, and by how much
The collection

Every essay — page 29

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Values

What a position is worth — numbers, and the things that are not numbers, and how to find the simplest one.

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*.

8 figures · 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.

7 figures · 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.

8 figures · 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.

8 figures · 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.

7 figures · 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.

7 figures · Realisability 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.

7 figures · Dominance The numbers each position is confused with. Each row is a position. The bar runs from its right stop to its left stop; the filled part is the set of numbers the position is genuinely confused with, computed one comparison at a time. The two coincide except at the ends, and a position whose stops meet is confused with nothing at all even when it is not a number.

The numbers it is confused with

A position is confused with a number when neither is at least as good as the other, and the set of such numbers is an interval. It is exactly the open interval between the two stops: over 36,850 comparisons the rule is wrong nowhere it speaks, and the 2,596 comparisons it declines are precisely the ones at an endpoint, where the position and the number differ by an infinitesimal.

8 figures · Stops 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.

7 figures · 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.

8 figures · Reversibility Which end of the interval is open. Every value born by day 3 compared with each of its own two stops — 2,948 comparisons, each one a search over the difference. The two rows are mirror images because the day is closed under negation, and the small number in each row is the exception class: the 352 values whose two stops coincide, for which the left stop is the right stop and the law has nothing to bite on.

Which end of the interval is open

The confusion interval is open at both ends, and the two ends are not the same kind of open. At its own left stop a position can be below the number, confused with it or equal to it, and — over 2,948 comparisons — above it exactly thirty-three times, every one of them a value whose two stops are the same number and whose left end is therefore also its right one.

8 figures · Stops 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.

8 figures · Numbers The identity that would join the order to the addition. Every pair of the twenty-two values born by day two, asked whether the join plus the meet equals the sum. It holds on all 201 comparable pairs, where the join is the larger and the meet the smaller and it cannot do otherwise, and on none of the 52 incomparable ones.

Where the order and the sum disagree

Day two is a lattice, and day two is a group, and it is not a lattice-ordered group. The one identity that would join the two structures — the join plus the meet equals the pair — holds on exactly the 201 pairs where it cannot fail and on none of the other 52, and the errors split thirteen high, thirteen low and twenty-six confused.

8 figures · Lattice 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.

6 figures · 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.

7 figures · Switches

All ladders · Every object named here · The position index · Figures that play back · Search