The rate was the alphabet

The entry fee was the cap found that the excess — how much larger the cheapest position exhibiting a value is than the value’s birthday — is an artefact of the sweep’s size cap and not a property of a ruleset. It closed by naming what to measure instead:

The rung above is the quantity that survives a cap. Everything measured on this ladder has been a distance between two numbers, and the distance is censored; what is not censored is a rate — how many new values a ruleset produces per extra square, which is 74, 215, 583, 1,472 for Toads and Frogs and is a property of the rules rather than of the sweep.

The rate does survive a cap. It is not a property of the rules.

Every rate is close to its alphabet #

Seven rulesets in the census come in sizes rather than as a handful of boards — Toads and Frogs, Clobber, Toppling Dominoes, Push, Shove, and Hackenbush with and without green edges — and every one of them adds new values geometrically. Toads and Frogs adds 1, 2, 3, 8, 12, 28, 59, 167; Shove adds 2, 6, 14, 32, 72, 160, 348, 768; Hackenbush adds 2, 4, 8, 16, and so on to 1,024.

Fit a rate to each and the seven come out between 1.83 and 3.00 — a spread of 1.64 times, which is nothing. And they sort by something that has no connection to how the games are played:

• Hackenbush, written over two symbols, grows at exactly 2.00.
• Toppling Dominoes, also two symbols, at 1.85.
• Hackenbush with green, three symbols, at exactly 3.00.
• Push, Shove and Toads and Frogs, all three symbols, at 2.37, 2.30 and 2.04.
• Clobber, three symbols, at 1.83.

The ceiling is the alphabet, and it has to be: a ruleset with $k$ symbols has at most $k^n$ positions of size $n$, so it cannot exhibit new values faster than $k$ times as many per square. Every rate here is at or under its own ceiling, and two of them are exactly on it.

So the rate is bounded by how a position is written down. Green Hackenbush’s rate is three because it has three colours of edge; Hackenbush’s is two because it has two. That is a fact about the alphabet a game’s positions are recorded in, which is a choice of notation and not a fact about the game — and the notation is not the position is a thread this site runs on for exactly this reason.

What does separate them #

The quantity that survives the cap and says something about the games is the one hiding underneath: the yield, how large a share of the positions of a size are worth something no smaller position is worth.

Every Hackenbush string of ten edges or fewer — all 1,024 of them at that size — is worth something no shorter string is worth. Fewer than one Clobber row in a hundred is: 55 new values out of 6,560 rows of eight squares. Push is at 9 per cent, Shove at 18, Toads and Frogs at 2.6, Toppling Dominoes at 53.

That is a spread of 119 times, against the rate’s 1.64. A yield is measured inside one size rather than across two, so no size cap can censor it — which is exactly what the rung below was asking for, arrived at by measuring the rate and finding the interesting variation underneath it.

The reading is not subtle. A ruleset with a high yield is one whose positions are their values: change the drawing and the number changes with it. A ruleset with a low yield is one where enormously many different-looking positions collapse onto the same value, which is the ordinary condition and is what makes a value theory worth having at all.

Which way each yield is moving #

The direction matters more than the level, because a direction is a statement about what happens next.

Six of the seven yields fall. Toads and Frogs goes from a half to a fortieth; Clobber from a half to less than a hundredth; Push from a half to an eleventh. Each extra square adds more positions than it adds values, so the ruleset is spending its position space on repeats — which is what running out of new values looks like from inside a census.

One is flat. Hackenbush’s yield is exactly one at every size to ten edges, and there is a reason: a Hackenbush string is a binary numeral, so two different strings are two different numbers, and the map from positions to values is injective by construction rather than by luck. Hackenbush is a numeral is the whole explanation, and it was written long before this measurement existed.

This is the same reading the site applies to its own subject one level up. The values nobody’s game produces found the theory handing down 1,474 values born by day three and the rulesets reaching under a tenth of them; a falling yield is that observation running forward in time rather than across a census. A ruleset saturates its own value space, and the yield is the instrument that says how fast.

Green Hackenbush, and where it stops #

The rung below’s second question was whether green Hackenbush is genuinely different from the other rulesets or merely measured over fewer sizes. It is genuinely different, and the difference has an end that can be dated.

Every green Hackenbush string of at most seven edges is worth something no shorter string is worth — 3, 9, 27, 81, 243, 729, 2,187, each one a new value. At eight edges it stops: 6,513 new values from 6,561 strings, so 48 strings repeat something already reached.

So green Hackenbush is injective up to seven edges and not beyond, and both halves of that are worth having. The injectivity is why its rate is exactly its alphabet — a rate of three means every word is a value — and the failure at eight is where the alphabet stops being able to deliver. A third symbol adds a green edge, which is a Nim heap of one, and a string mixing green with the numbers is a number plus a nimber; there are only so many small nimbers, so the collisions arrive as soon as the string is long enough for two arrangements to give the same pair. Green Hackenbush is where that construction is set out.

Why a ruleset collapses, and why two do not #

The yields split the seven rulesets into two kinds, and the split is not about how complicated the games are.

The two that do not collapse are the two Hackenbushes, and in both the value is read off the drawing rather than searched for. A blue-red string is a binary numeral: the first colour change fixes the integer part and every edge after it is a bit. Change any edge and a bit changes, so two strings of the same length are never worth the same thing, and the yield is one by construction. Adding green edges adds a nimber to the number, and for a while that stays injective too because the small nimbers are distinct enough to keep the pairs apart.

The five that collapse are the five where the value comes out of a recursion over options. A Clobber row of eight squares has 6,560 arrangements and 55 new values, because the recursion is a many-to-one map and nothing about the drawing constrains where two rows land. The other way to move a row is Shove, whose yield of 18 per cent is the highest of the five, and it is the one of the five whose values are closest to being readable off the board.

So the yield is a measure of how far a ruleset’s notation is from its value, and the ordering it produces — Hackenbush, green Hackenbush, Toppling Dominoes, Shove, Push, Toads and Frogs, Clobber — is very nearly the ordering of how directly each game’s positions can be read. That is a quantity the ladder has wanted since wider costs less tried to get at it through the width of a canonical form and found width pointing the wrong way.

It also explains why the excess two rungs below looked as though it separated the rulesets. A ruleset whose positions are their values exhibits every value at its birthday and has no excess; a ruleset that collapses has to search a long way for the position that happens to land on a given value, so its cheapest exhibit is large. The excess was reading the yield through a censored measurement, which is why raising the cap kept moving it.

An alphabet is a choice, and it shows #

One consequence is worth stating plainly, because it makes the rate a worse quantity than it first appears rather than merely a duller one.

The alphabet is not fixed by a game. Toads and Frogs is written over three symbols because a square holds a toad, a frog or nothing; the same positions could be written over two by recording only the occupied squares and their colours in a different encoding, and the count of positions of a given size would change with it. Nothing about how the game plays would change at all.

So a rate near three does not mean this game is rich. It means somebody wrote this game’s positions in three symbols. Two rulesets with the same rate can be as far apart as Clobber and Toads and Frogs, and two with different rates can be as close as Hackenbush and Toppling Dominoes. A quantity that moves when the notation changes and stays still when the game does is measuring the wrong object, however honestly it is measured.

What a rate is good for after all #

It would be easy to read this page as saying the rung below’s proposal failed, and that is not quite right.

The rate did what it was asked to do: it is a quantity a size cap cannot censor, and it can be quoted honestly at any cap. What it turned out to have is no variation worth reading — every ruleset’s rate is pinned to its alphabet, so the number carries one bit of information, namely how many symbols the ruleset was written in.

That is a familiar shape and this site keeps meeting it. A quantity that is well defined, cheap and stable, and that turns out to be a restatement of something already known, is the commonest failure mode of a measurement — commoner than a quantity that is wrong. A bound with one number too many is the same discovery on a different ladder, where a bound’s third term turned out to be doing no work.

The correct move when this happens is not to discard the measurement but to divide it by its ceiling, which is what the yield is: the rate over the alphabet, or equivalently the share of the position space being used. That quotient is where every one of the two orders of magnitude lives.

A rate and a level are censored differently #

The move from a level to a rate is what makes this measurement survive a size cap, and it is worth stating the general reason, because it applies to every capped sweep on this site.

A level is what the sample contains. Any average taken over a capped sweep is an average over the positions the cap admitted, and the cap admits the small ones — so a level is systematically biased wherever the quantity correlates with size, which is nearly always.

A rate is a difference between adjacent levels. Both are censored, and if the censoring is similar at adjacent sizes, most of the bias cancels. What is left is the change from one size to the next, which is what the rate is measuring in the first place.

That is why the rate is honest and the level was not, and it is not a property of these particular quantities. Differencing removes a bias that is common to the terms, which is the standard reason to look at increments rather than at totals when a sample is truncated.

The catch is worth naming too. A rate is only honest where the censoring is stable, so the rate near the cap is as unreliable as the level — the last size in a sweep is the one whose censoring differs most from its neighbour’s. So a rate should be read from the middle of a sweep and not from its edge, and a sweep with three sizes in it has one usable increment.

Which is the reason this page’s rate is quoted across the whole range rather than at the end, and why a widening would improve it by adding interior points rather than by reaching further.

What this does not say #

Seven rulesets, not fifteen. The census two rungs below compares fifteen rulesets, and eight of them do not come in sizes at all — Domineering, NoGo and Col are a handful of small boards rather than a family — so they cannot have a rate. What is compared here is the subset with a size parameter, which is a real restriction and is why Domineering, the site’s most-used partizan game, is absent.

The sizes are small. Toads and Frogs stops at eight squares, Clobber at eight, Push and Shove at eight, green Hackenbush at eight. The rates are fitted over between eight and ten points, and a rate fitted over eight points of a sequence that might not be geometric forever is an estimate.

The yield’s fall is measured, not extrapolated. Six yields fall over the sizes swept and nothing here says they keep falling, or what they fall to. A yield that flattens at a positive number would mean a ruleset with an infinite supply of values, and nothing in this sweep can tell that from a yield on its way to nought.

And a value here is a canonical form, not a game. Two positions count as exhibiting the same value when their canonical forms are equal, which is the right notion and is also the reason the yields are as low as they are: many forms one value is the essay about how much collapsing that does.

The convention, named #

Normal play throughout, and every value computed by the recursion and reduced to canonical form.

A ruleset’s size is the number of pieces a player can see — an edge in Hackenbush, a square in a Toads and Frogs strip, a coin in a row. It is the size of the exhibit and deliberately not a measure of how hard the position is to evaluate.

A ruleset’s alphabet is how many symbols a position of that ruleset is written in: two for Hackenbush (blue, red) and Toppling Dominoes, three for Toads and Frogs (toad, frog, empty), Clobber, Push, Shove and green Hackenbush.

A value is added at size $n$ when some position of $n$ pieces is worth it and no position of fewer is. The yield at size $n$ is the number added divided by the number of positions of that size.

The rate is fitted as the slope of a least-squares line through the logarithms of the added counts, over every size the sweep reaches — so it is a summary of the whole sequence rather than the ratio of its last two terms.

Where the ladder goes next #

The realisability anchor has six rungs: the cheapest way to show a value, the birthday as a floor, which values no game produces, whether width accounts for the excess, that the entry fee was the cap, and now what a rate measures.

The rung above is the yield’s limit. Six of the seven yields fall and the question a falling sequence always raises is what it falls to — and unlike the rate, the answer would say something: a yield tending to nought means a ruleset with finitely many values, and one tending to a positive number means a ruleset whose value space grows as fast as its position space forever. Toppling Dominoes is the case to run, because its yield has fallen from one to a half over eight sizes and has fallen much less than the others; two or three more sizes would say whether it is heading for a floor or merely falling slowly.

Two neighbours are worth the trip. The values nobody’s game produces is the same gap measured across a census rather than along a size, and the two together are the clearest statement of how much of the theory the games decline to use. And hackenbush is a numeral is why one row of the table here is flat, and it is worth reading beside a measurement it predicted without being asked to.