Combinatorial Game Theory
who wins, and by how much
Theme

The thread: Small things decide

Infinitesimals are smaller than every positive number and are not zero. In a close game they are the whole margin, which is why the theory bothers with them.
Toads and frogs. Toads move right and frogs move left, one square into a gap or hopping over exactly one opponent. A player unable to move loses. It can be played on squared paper by anybody, and its values are immediately stranger than the game looks. Particular games

Toads and Frogs

Toads shuffle right, frogs shuffle left, and either may jump over one of the other. A strip six cells long is worth exactly up. Another six-cell strip is worth exactly down. Nobody has a formula for which.

 Smaller than every positive number, and not zero. Values that sit between zero and every positive number, each compared with zero and with 1/1024. Every relation drawn was computed by playing the difference, and one of them is confusion — neither greater, smaller nor equal. None of these is a number, and in a close game they are the entire margin. Values

Infinitesimals

Some positions are positive — Left wins them whoever moves first — and smaller than every positive number, including a millionth and a millionth of that. They are the values that decide close games, and the smallest of them is a single move's worth of nothing.

 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.

 Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal. Particular games

A game where nobody can be ahead in moves

A blue stone beside a red one is a move for both players at once. So neither player can run out while the other still has something to do — and every value the game produces is smaller than every positive number, by the shape of the rule rather than by inspection.

 Tiny and miny: infinitesimals with a scale. Positions that are greater than zero and smaller than every positive number, and which are nevertheless strictly ordered among themselves — the larger the subscript, the smaller the value. Being smaller than everything positive is not one size of thing; it is a whole scale, and up sits above all of it. Values

Tiny, miny, and the sizes below every size

An empty two-by-four Domineering board is worth less than nothing and more than every negative number. It is not up, not down and not a fraction — it is a miny, and the minies come in sizes, strictly ordered among themselves below a floor no number reaches.

 A green edge is not a number. Green edges may be cut by either player, which makes the position impartial in that part. A single green edge is worth ∗ — a value that is neither positive, negative nor zero, and which no number can equal. Particular games

A green edge on a blue one

Blue over green and green over blue are the same two edges in the other order. One is worth 1∗ and the other ↑∗ — a number with a star on it against something smaller than every positive number — so a stalk with all three colours in it stops being a numeral and starts being a position whose value depends on what is underneath.

 Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal. Values

The class where nobody runs out first

Three stones in a row — blue, blue, red — and the position is worth exactly up. Clobber cannot produce anything else, because adjacency is symmetric — a player has a move precisely when the opponent does, and a game with that shape can never be worth a whole move to anybody.

 Cooling by 1, and heating back. Each row is a position, its temperature, what it becomes when every move is taxed, and what comes back when the tax is refunded. The refund is not an inverse: a position whose temperature was below the tax has already frozen into a number, and heating a number does nothing at all. Temperature

Cooling by exactly one

Cooling is usually met as a way of reading a thermograph: a tax, and two numbers at the height of the tax. Applied as an operator it returns a position instead — and then the obvious question is whether heating gives the position back. Over the 1,474 values born by day three, 27 survive the round trip and 15 of those are numbers that never moved. Cooling throws away almost everything it touches.

 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.

 A hot position, split into its mean and what is left. Each row is a position cooled by exactly its own temperature — the tax at which it stops being worth moving in. The result is the mean value with something small still attached, and the last column is that something, obtained by subtracting the mean from the cooled game rather than by inspection. Temperature

A number and a fight

Charge a position exactly what it is worth fighting over and the fight disappears, leaving the mean value — with something still attached to it. Over all 1,122 hot values born by day three the residue is smaller than every positive number, it is a star in 942 of them, and it is never nothing. So a hot game is its mean plus a fight plus a remainder that no number reports, and the remainder is what decides close games.

 The same strips, with no jumping. Elephants and Rhinos: toads move right and frogs move left, one square into an empty one, and nothing may hop over anything. The pieces keep their order for ever, and the values are computed by the same recursion as the game with the jump in it. Particular games

The same strip without the jump

Delete one clause from Toads and Frogs — the hop over an opponent — and the game is Elephants and Rhinos. Over the same 3,279 strips the values do not become simpler in the way a reader would guess: every value that is a number becomes an integer, against 172 fractions with the jump in, and the count of positions worth fighting over nearly doubles. Removing a move made the game hotter.

 Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal. Particular games

One row of Clobber

Every string of blue, red and empty squares up to eight long — 9,840 rows — carries one of only 111 values, and every one of them is infinitesimal. A third of the rows are worth exactly zero. Six alternating stones are worth zero and eight are worth a form that takes four lines to print, so the values do not simplify as the row grows: they explode, while the row stays trivial to describe.

 What the reduction collapses. Each reduced form with the values that reduce to it. The largest class is the one that reduces to zero and it holds every infinitesimal on the list, which is exactly what the reduction is for — against a hot background, none of them is distinguishable from nothing. Sums and comparison

What is left when the small change is thrown away

Canonical form answers a demanding question: which positions are interchangeable inside every sum whatever. A player with a hot board does not have every sum — an infinitesimal difference cannot decide anything against a genuine fight — so there is a coarser question with an exact answer. The reduced canonical form takes the 1,474 values born by day three to 61, with 292 of them collapsing to zero, and it is a homomorphism on all 8,100 pairs tested only when a second pass is made.

 Where 1,474 values sit on the scale. The temperature of every value in the pool, counted. The floor is −1 and only numbers are on it; the next rung up is 0, and everything there is a number with an infinitesimal added. Above that the scale is continuous and the counts thin out. Temperature

Below zero

Temperature is described as urgency and urgency has no obvious bottom, but the scale stops at −1 and only the numbers are on it. The rung above is exactly zero, and the 337 values born by day three that sit there are the ones a number cannot be told from — flat thermograph, nothing at stake, and an infinitesimal that no number can see. Two independent computations agree on the classification for all 1,474.

 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. Temperature

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.

 Seven infinitesimals, added to a whole day. Each row adds one infinitesimal to every value born by day three. The stops never move — that is what being smaller than every number means — the temperature moves a handful of times, and the outcome class moves in a quarter of the additions. Sums and comparison

What an infinitesimal does to a fight

Adding a number moves both stops by exactly itself. Adding something smaller than every number moves neither — across 10,318 additions to a whole day of values, not once — and the outcome class changes anyway, 2,622 times. It changes at exactly one kind of position: the ones with a stop sitting on zero, which is where the numbers have run out of things to say.

 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.

 What an approximation is worth. Every pair of Clobber rows up to six squares, judged twice: by their up-brackets and by the comparison itself. The bracket is never wrong where it speaks, and most of what it declines to answer has no answer. Particular games

When the bracket decides

A Clobber row's value is an all-small game nobody can hold in their head, so the practical answer is the up-bracket: a pair of integers between which its atomic weight must lie. As an approximation it is worth exactly what it settles — 1,585 of 7,875 pairs of rows are ordered by it, every one of those orders is right, and of the 6,290 it declines, 4,222 have no answer either.

 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. Values

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.

 How hot a background has to be. Every pair of values born by day two that share a reduced canonical form, added to backgrounds of seven temperatures and three means — 609 comparisons in all — with the count of pairs whose outcome the swap changes. Safety is not monotone in the background's temperature, so the threshold the question asks for does not exist; every one of the 48 changes is at a position with a stop exactly on nought. Sums and comparison

How hot a background has to be

The reduced canonical form throws away infinitesimals, and the rung below asked for a bound: how hot must the rest of the board be for the discarded part not to matter? There is no such bound. Safety is not monotone in the background's temperature — an eighth is safe, a quarter is not, two is safe again — and the quantity that does decide it is not a temperature but a stop.

 Two thousand values and no temperature. Every End-Nim row of up to six heaps, with its temperature. Not one of the 5,460 is hot, every one of them is worth an infinitesimal, and the 361 worth numbers are 361 rows worth nought. Particular games

A game with nothing at stake

The rung below explained a coincidence with a claim it did not compute: that End-Nim carries no hot self-negative values. The census says something stronger. Not one of 10,919 rows across three shapes of board is hot, every one of them is worth an infinitesimal, and the 361 rows the earlier census called numbers are 361 rows worth nought. The reason is one line of the rule, and 2,693 distinct values sit underneath it.

 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. Sums and comparison

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.

 Two failures, not one. The decisions a Domineering strategy has to store, split by what the two content rules do: answer them, name a worse placement, or leave two candidates standing. Values

Seventy-two of them were not silence

The rung below said its 181 unanswered decisions were all the rules falling silent and asked whether the position's value picks the placement once the geometry cannot. Seventy-two of the 181 are the rules speaking and being wrong, which is a different failure. On the 109 that really are silence, a rule chosen per value answers more than half — and the star class the rung below singled out is settled outright by leaving the younger position.

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

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.

All themes