Combinatorial Game Theory
who wins, and by how much
Generator

Comparing two positions is playing their difference

Comparing two positions is playing their difference
Comparing two positions is playing their difference. To decide whether one position is worth at least another, subtract and see who wins moving second. It is the only definition of comparison the subject has, and it produces a partial order — some pairs come out confused, which no comparison of numbers ever does.

To decide whether one position is worth at least another, subtract and see who wins moving second. It is the only definition of comparison the subject has, and it produces a partial order — some pairs come out confused, which no comparison of numbers ever does.

11 essays call comparison. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

The positions it draws

78 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
Clobber 1×3 xxo ↑ L When the ups add
green EE ∗2 N When the ups add
{↑ | ∗} {↑ | ∗} L When the ups add
{↑ | ∗} + ↓ {0 | ↓∗} L When the ups add
{⇑ | ↓} {⇑ | ↓} N When the ups add
{⇑ | ↓} + {⇑ | ↓} ↑ L When the ups add
{∗ | ↓} {∗ | ↓} R The class where nobody runs out first · When the ups add
{∗ | ↓} + {∗ | ↓} {{0 | 0, ↓∗} | 0} R When the ups add
{∗ | ↓} + ↑ {↑∗ | 0} R The class where nobody runs out first · When the ups add
{∗ | ↓} + ∗ {0 | 0, ↓∗} N When the ups add
{∗ | ↓} + Clobber 1×3 xxo {↑∗ | 0} R When the ups add
{∗ | ↓} + green EE {∗3 | {∗3 | 0}} R When the ups add
↑ ↑ L A number and a fight · The class where nobody runs out first · Below zero · Cooling adds and heating does not · Cooling by exactly one · Equal in every company · Turn the board through a right angle · Infinitesimals · Misère play has no negatives · Nobody has to move · Nobody wants to move here · One part that never ends · Three ways to add the same games · Outcomes do not add · The birthday of a sum · The company that is closed · The fight never runs backwards · The simplest game above both · The sum is the object · Toads and Frogs · What a number does to a fight · What a wider pool rescues · What an infinitesimal does to a fight · What can be struck out · What is left when the small change is thrown away · When the ups add · Which part to move in · Who moves last
↑ + ⇑ 3·↑ L When the ups add
↑ + ∗ ↑∗ N Equal in every company · Three ways to add the same games · When the ups add · Which part to move in
↑ − −1/64 1/64↑ L Confused is not the same as unknown
↑ − ∗ ↑∗ N Comparing positions · Confused is not the same as unknown · When the ups add
↑ − ∗2 {0 | ∗3} L When the ups add
↑ − 0 ↑ L How many ups · Comparing positions · Confused is not the same as unknown · Turn the board through a right angle
↑ − 1/1024 −1/1024↑ R Comparing positions
↑ − 1/64 −1/64↑ R Confused is not the same as unknown
↑↑↑∗ 3·↑∗ L When the ups add
↑↑↑∗ + ↑∗ 4·↑ L When the ups add
↑∗ ↑∗ N Infinitesimals · Nobody has to move · Nobody wants to move here · Outcomes do not add · The fight never runs backwards · Toads and Frogs · Two misère outcomes are not enough · What a number does to a fight · What an infinitesimal does to a fight · When the ups add
↑∗ + ↑∗ ⇑ L When the ups add
↑∗ + ↓∗ 0 P When the ups add
↑∗ − ↑ ∗ N Comparing positions
↑∗ − ∗ ↑ L How many ups · Comparing positions · Turn the board through a right angle
↑∗ − 0 ↑∗ N Confused is not the same as unknown · When the ups add
⇑ ⇑ L Infinitesimals · Outcomes do not add · Toads and Frogs · What a number does to a fight · What an infinitesimal does to a fight · When the ups add
⇑ + ↓∗ ↑∗ N When the ups add
⇑ − ↑ ↑ L How many ups · Comparing positions · Confused is not the same as unknown · Turn the board through a right angle
⇑ − ∗ 2·↑∗ L Comparing positions · When the ups add
⇑ − 1/1024 −1/1024⇑ R Comparing positions
⇑ − 1/64 −1/64⇑ R Confused is not the same as unknown
⇑∗ 2·↑∗ L When the ups add
⇑∗ + ⇑∗ 4·↑ L When the ups add
⇓ ⇓ R The sum is the object · When the ups add
⇓ + ⇓∗ 4·↓∗ R When the ups add
−1 − 0 −1 R Comparing positions
−1 − 1 −2 R Confused is not the same as unknown
−1 − 1 | −1 0 | −2 N Confused is not the same as unknown
∗ ∗ N A move that must be answered · A number and a fight · A rule with no promise at all · A self-negative value costs a day · At least five hundred and seventy-one · Below zero · Cooling adds and heating does not · Cooling by exactly one · Equal in every company · Equal in this company · Turn the board through a right angle · Fifty-two errors and seven sizes · How hot a day gets · How rare it is to be bigger · Infinitesimals · Misère play has no negatives · Nobody has to move · Nobody wants to move here · Nothing worth fighting over · Numbers avoid numbers · One part that never ends · Three ways to add the same games · Outcomes do not add · The birthday of a sum · The company that is closed · The fight never runs backwards · The first theorem, and the winner it declines to name · The other way to move a row · The simplest game above both · The thirty that cancel themselves · The values that are their own negatives · Three players and no answer · Toads and Frogs · What a number does to a fight · What a wider pool rescues · What an infinitesimal does to a fight · What can be struck out · What is left when the small change is thrown away · When the ups add · Where the impartial theory stops · Where the order and the sum disagree · Which part to move in · Who moves last · Start at the end and work backwards
∗ + ∗ 0 P Equal in every company · Equal in this company · When the ups add
∗ − ↑ ↓∗ N Comparing positions
∗ − ⇓ 2·↑∗ L When the ups add
∗ − −1/64 1/64∗ L Confused is not the same as unknown
∗ − 0 ∗ N How many ups · Comparing positions · Confused is not the same as unknown · Turn the board through a right angle · How much a list of options can lose · The same fight, eight times over · The numbers it is confused with
∗ − 1/64 −1/64∗ R Confused is not the same as unknown
∗2 ∗2 N A move that must be answered · At least five hundred and seventy-one · Equal in every company · Equal in this company · Turn the board through a right angle · Misère play has no negatives · The fight never runs backwards · The simplest game above both · The sum is the object · The thirty that cancel themselves · The values that are their own negatives · Two misère outcomes are not enough · When the ups add · Where the impartial theory stops · Which part to move in
∗2 + ↓ {∗3 | 0} R When the ups add
∗2 + ∗3 ∗ N The sum is the object · When the ups add
∗2 − ∗ ∗3 N Comparing positions · How much a list of options can lose
∗2 − 0 ∗2 N Comparing positions · Confused is not the same as unknown
∗4 ∗4 N When the ups add
∗4 + green EE ∗6 N When the ups add
0 − ↑ ↓ R Comparing positions
0 − ∗ ∗ N Comparing positions
0 − 0 0 P Comparing positions
1 | −1 − ∗ {1∗ | −1∗} N How rare it is to be bigger
1 | −1 − ∗2 {1∗2 | −1∗2} N How much a list of options can lose
1 | −1 − 0 1 | −1 N Confused is not the same as unknown · How much a list of options can lose · How rare it is to be bigger
1 | −1 − 1 0 | −2 N Confused is not the same as unknown
1 | −1 − 1/2 1/2 | −3/2 N Confused is not the same as unknown
1 − 0 1 L Comparing positions
1/2 − 1/4 1/4 L How many ups · Comparing positions · Turn the board through a right angle · The same fight, eight times over
1∗ − 1 ∗ N Comparing positions
2 | −2 − 0 2 | −2 N Confused is not the same as unknown
2 | 0 − 0 2 | 0 N Confused is not the same as unknown · The same fight, eight times over
2 | 0 − 1 1 | −1 N Confused is not the same as unknown · The same fight, eight times over · The numbers it is confused with
2 | 0 − 2 0 | −2 N The same fight, eight times over · The numbers it is confused with
2 − ∗ 2∗ L How rare it is to be bigger
2 − 1 1 L How rare it is to be bigger
3 | 0 − −1 4 | 1 L Confused is not the same as unknown
3 | 0 − 3/2 3/2 | −3/2 N Confused is not the same as unknown
3 | 0 − 4 −1 | −4 R Confused is not the same as unknown
Clobber 1×3 xxo + Clobber 1×2 xo ↑∗ N When the ups add
green EE + green EE 0 P When the ups add

Where it is called

Changing this generator changes every one of these figures.

Comparing two positions is playing their difference. To decide whether one position is worth at least another, subtract and see who wins moving second. It is the only definition of comparison the subject has, and it produces a partial order — some pairs come out confused, which no comparison of numbers ever does. Sums and comparison

Comparing positions

One position is worth at least another when the second player wins their difference. That is the only definition there is, it is a computation rather than a judgement, and it produces an order in which some pairs are simply not comparable.

 How many ups, bracketed. Every position here is all-small, so no number says anything about it and the yardstick has to be ↑ instead. Each bar spans the multiples of ↑ the position lies between: the largest it is at least, and the smallest it is at most. Four of the seven are pinned to a single multiple of ↑; the rest keep a band that comparison cannot narrow, the widest being ∗ at four ups of slack. Sums and comparison

How many ups

When every component of a position is smaller than every positive number, no number can decide it. What decides it is a count of ups — and comparison can pin that count down exactly, except when a star is present, when it cannot.

 Every position has an exact opposite. A position beside its negative, which is the same game with the players exchanged, and the sum of the two. The sum is worth zero every time — a second-player win — because the second player can answer each move with its mirror image. It is the fact that makes values a group, and it is what lets one position be subtracted from another. Sums and comparison

Turn the board through a right angle

A two-by-four Domineering board is worth something no number can express, and Right is ahead on it. Turn a second board through a right angle, put the two side by side, and the total is exactly zero. Every position has an exact opposite, and that single fact is what makes subtraction — and therefore comparison — possible at all.

 {2 | 0}, added to itself. The value of n copies of one position, for each n, beside n times its mean and the smallest distance between the two. The mean value theorem says that distance stays bounded however many copies are piled up — and the bound is the position's temperature, which is what makes the temperature a second genuine measurement rather than a diagram-reading convenience. Temperature

The same fight, eight times over

The mean is not roughly what a position is worth. It is the number that eight copies of the position stay close to — and the theorem is that the closeness does not decay as the copies pile up. The gap stops at the temperature, and stays there for ever.

 Comparing two positions is playing their difference. To decide whether one position is worth at least another, subtract and see who wins moving second. It is the only definition of comparison the subject has, and it produces a partial order — some pairs come out confused, which no comparison of numbers ever does. Sums and comparison

Confused is not the same as unknown

Two positions can be neither greater, nor smaller, nor equal. That is a fourth relation with its own symbol, it is a fact about the pair rather than a limit of the method, and it is what makes a game worth playing — a position is a first-player win exactly when it is confused with zero.

 The bracket of a sum, against the sum of the brackets. Two all-small positions, the interval of multiples of ↑ each lies between, those two intervals added coordinatewise, and the interval the sum actually lies between. The added one always contains the computed one — greater-than survives addition — so the bracket never widens under a sum. Where it narrows, the parts were each too vague to pin down and the sum is not. Sums and comparison

When the ups add

Atomic weight brackets do not add over a sum — they bound it. Over all 120 pairs from a fifteen-game family the sum's bracket came out exactly the sum of the parts' brackets 56 times, strictly narrower 64 times, and wider never; and the rule separating the two is one line long, because every one of the 54 pairs with a pinned part is exact and only 2 of the other 66 are.

 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. Sums and comparison

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.

 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. 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.

 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. Values

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.

 The fall stops. Comparability on days two, three and four, the last built and corrected. Sums and comparison

A floor, and not a decline

Comparability fell eighteen points from day two to day three and the next day cannot be enumerated. It can be built — and the construction's bias measured one day lower, where the truth is known. Corrected, day four comes to 60.6 per cent against day three's 59.7: the fall was a one-day event.

 The day-four figure, twenty draws at a time. Corrected day-four comparability from twenty seeds of each of two constructions, as dots on a percentage axis, with the range of day three's four slices shaded and the single figure the earlier essay reported marked. The seeds spread over about twelve points and the two constructions agree. Sums and comparison

Twenty draws and a second recipe

Day four's comparability was reported as 60.6 per cent against day three's 59.7, from one built sample and one calibration. Built twenty times with each of two recipes whose biases differ by six points, and calibrated against all 1,474 day-three values rather than a quarter of them, the corrected figure spreads over twelve points from seed to seed and the two recipes agree within one standard error. The floor survives; the decimal was one draw.

The whole library · The position index · The figures that play back