Wider costs less

The birthday is a floor asked what the cheapest position exhibiting a given value costs, and found an inequality where a correlation had been expected: the smallest exhibit is never smaller than the value’s birthday, and is exactly equal to it on 476 of 728 values. On the 252 that cost more, the ruleset accounts for about 40 per cent of the excess — Toads and Frogs pays 2.25 squares over the birthday on average, End-Nim pays 1.08, Hackenbush with green pays nothing at all.

That page named the variable it had not tried:

The natural second variable is the width of the form rather than its depth — a value with many options at some day needs a position with many moves available there, and nothing in the chain argument charges for that.

The argument is sound and the measurement refuses it. Width and excess correlate at −0.39.

What width is taken to mean #

The sentence asks for the width at the level where the form is widest, not at the root. So the measure here walks the whole canonical form and takes the largest option list it finds anywhere in it — Left’s options and Right’s together at a single position.

That is the reading the argument needs, and it is a quantity this site has already had to measure once. The reduction that puts options back established that canonicalisation is not a contraction — bypassing a reversible option replaces it with a whole list, so a form can come out wider than it went in — and how wide a form can get measured what widths are reachable and found the birthday to be a hopeless bound on them. A value whose root has two options but which, three moves down, has a position with eight, still requires a game position with eight moves available at that point. Taking the root’s width instead gives a weaker relation in the same direction, so nothing here turns on the choice.

The measure runs from 2 to 16 across the census. The narrowest possible non-number is 2 — one option each way, which is what a switch is — and 185 of the 728 values are of that shape. The widest is $\ast 8$, whose canonical form has all sixteen smaller nimbers as options, eight a side.

The floor those two pages sit under is worth restating, because everything here is measured against it.

The trend, and how strong it is #

The table above is the whole finding in one column. Values whose widest list is 2 are exhibited on average two squares above their birthday; at width 4 it is 1.52; at width 6 it is 0.19; at width 8 it is 0.24. Between the extremes the excess falls by a factor of eight.

The eight widest values in the collection have an excess of nought, every one. $\ast 8$ is born on day eight, is exhibited on an eight-counter End-Nim heap or an eight-edge green Hackenbush stalk, and its sixteen options cost nothing extra to arrange. That is not what a chain argument would predict, and it is not a fluke of one game: two different rulesets exhibit the widest values at their birthdays independently.

The account, scored #

Sorting the values by width accounts for 19 per cent of the variation in the excess, against the ruleset’s 40. Sorting them by both accounts for 67.

That last number is the one a reader will want to seize on, and it should not be. Sorting by two variables produces thirty groups where sorting by one produces eleven, and a partition into more groups accounts for more variation whatever the groups mean; carried far enough, one group per value accounts for all of it. The 67 per cent is the honest arithmetic of a partition and not evidence that width has explanatory work in it.

The way to find out whether it does is to hold the ruleset fixed and look inside.

Inside a ruleset the sign changes #

Six rulesets supply ten values or more, and four of them have any variation in the excess at all:

• End-Nim, 214 values, correlation −0.64;
• Clobber, 44 values, correlation +0.67;
• Partizan subtraction, 24 values, +0.25;
• Toppling Dominoes, 84 values, +0.08.

A variable that runs strongly one way in the largest population and strongly the other way in the third largest is not measuring a property of values. It is measuring something about each game, and the four numbers are four facts about four games.

Hackenbush with green and Toads and Frogs are the two with nothing to measure — the first exhibits all 297 of its values exactly and the second exhibits none of them exactly but at a nearly constant excess of 2.25 with every value the same width. Both are informative in their own way: the excess in this census is much more nearly a fixed charge per ruleset than a function of the value in it.

The confound, in one number #

Width and birthday correlate at 0.71, which settles the mechanism. A wide form is a late form: to have eight options at a level, the level has to exist, and each level costs a day. So the values with wide forms are the values with large birthdays, and the excess is measured against the birthday — a value born on day eight that needs eight squares has an excess of nought, however complicated it is.

The negative correlation is then not mysterious at all. It is the floor doing its work. The birthday is a lower bound on the exhibit, wide values have high birthdays, and a high floor leaves less room for an excess above it. Width predicts the size of the exhibit perfectly well — at 0.36 — and predicts almost nothing about the gap, because the birthday has already claimed that information.

And the rulesets line up with it. The game with the widest forms in the census, Hackenbush with green, is the game that realises every value it produces at exactly its birthday, because a green stalk of $n$ edges is $\ast n$ and there is nothing to arrange. So the marginal correlation is partly a comparison between games dressed up as a statement about values, which is the same confusion the rung below’s 40 per cent was designed to detect and which comes back here in a different costume.

Who asked, and what they were asking about #

The question of which values a game can produce is realisability, and it is one of the few questions in this subject that came from the games rather than from the theory. Conway’s construction says what the values are; it says nothing about which of them any particular ruleset can put on a board, and a value that no game reaches is a value the theory carries for nothing.

The birthday arrives as a measure of complication in On Numbers and Games, where it counts days of construction, and its use as a cost is later and more practical: how old a value is is where this site establishes it as a measure and shows it not to bound the width. The pairing of the two — birthday as a floor on the exhibit, width as the natural correction — is the obvious next move for anybody who has both quantities in hand, which is why it was worth making and worth checking.

The answer here is a small piece of evidence for an old suspicion about this subject: a value’s complexity as an object and its cost as a position are not the same currency, and the exchange rate between them is set by the game rather than by the value. How long a row a value needs reached the same conclusion from the other side, with a birthday failing as a bound on the length of a Toppling Dominoes row.

What is left unexplained, then #

Two thirds of the excess is still unaccounted for, and this page has removed the leading candidate rather than replaced it. It is worth being precise about what remains.

Within a ruleset, the excess barely varies. Toads and Frogs pays 2.25 with almost no spread; partizan subtraction pays 4.00; Toppling Dominoes 2.83. These read as entry fees: a fixed cost of writing anything at all in that game, paid by every value it produces, and not modulated by which value it is.

And where a game does vary, it varies for reasons about the game. End-Nim’s negative relation inside itself is the same confound one level down: its wide values are its large nimbers, which arrive on single heaps at exactly their birthdays, while its narrow values are the assembled positions — several heaps, whose value is a fight — and those pay for the assembly. The relation is about heaps and rows, not about option lists.

One more measurement supports the reading and costs nothing, since it is already in the table: sorting the values by birthday alone accounts for 16 per cent of the excess, which is less than width’s 19 and in the same confounded way. Every variable available here that says this value is complicated explains a little of the excess with the wrong sign attached, and the reason is always the same — the floor rises with complication faster than the exhibit does.

So the honest statement is not the excess is 40 per cent ruleset and 60 per cent unknown. It is that the excess is very nearly all ruleset — a per-game constant plus a little noise — and the 40 per cent figure is low because a single mean per ruleset does not capture the few values inside each game that are genuinely dearer.

Why a per-ruleset constant is the tempting explanation #

The residue this page cannot account for invites one particular reading, and it is worth naming the reading before adopting it, because it is the reading a sweep of this shape almost always suggests and it is the one most likely to be an artefact.

The pattern is: a quantity varies within each ruleset, the variation is partly explained, and what is left correlates with which ruleset a position came from. The natural conclusion is that the remainder is a property of the rules — an entry fee, a constant overhead, something the game charges before it can exhibit anything.

That conclusion has a competitor and the two are hard to tell apart from inside one sweep. A bounded sweep does not see every ruleset’s positions equally. Each game is swept to a size that its own branching allows, so the values reached differ from ruleset to ruleset not because the games differ but because the sweeps stopped in different places. Any quantity that varies with what was reached will then look as though it varies with the ruleset.

The test that distinguishes them is not a wider pool of rulesets; it is looking at the shape of the residue inside each one. A genuine per-ruleset constant is flat: every value that ruleset produces pays it, whatever the value’s birthday. A censored sample is not flat, and it slopes in a predictable direction — the values near the cap are the ones whose cheapest witnesses were nearly excluded, so the measured excess falls as the birthday rises.

So the constant is refutable from data already in hand, without widening anything, by plotting the residue against the birthday within each ruleset rather than averaging it. That is the check this page does not run and the rung above does, and it is the reason the entry fee is named here as a hypothesis rather than reported as a finding.

What this does not say #

Four limits.

One measure of width. The widest option list anywhere in the form is one reading of a vague word. Others exist: the number of distinct positions at the widest level, which correlates at −0.18 with the excess; the total number of positions in the form, which gives −0.17. All three point the same way and none of them rescues the proposal, but width is not a single quantity and this page has not exhausted it.

Correlations on a census, not on a sample. These are exhaustive figures over the whole population of values realisable in the sweep, so there is no sampling error to report and no significance to test. What there is instead is a selection: the population is the values that this set of eleven rulesets at these sizes produces, and it is heavily weighted toward two games.

A negative correlation is not a proof that width is irrelevant. It is a demonstration that width, unadjusted, points the wrong way and that adjusting it by ruleset leaves an inconsistent sign. A model with the birthday partialled out and enough values per game might find a genuine positive residue; the census is not large enough per game to say.

And the excess is measured in squares, edges or counters, which are not the same unit. A Toads and Frogs square and an End-Nim counter cost different amounts of game, and comparing excesses across rulesets is comparing quantities that only look alike. That is one more reason the ruleset dominates every account here — part of what it accounts for is the choice of unit.

The convention, named #

Normal play, canonical forms, computed by the recursion.

A value’s birthday is the number of days of the construction it takes to appear, and its exhibit is the smallest position in any of the eleven rulesets whose value it is; the excess is the second less the first. Numbers are excluded, because the floor does not hold for them — an integer $n$ is born on day $n$ and a six-counter Shove strip can be worth 21.

Width is the largest option list anywhere in the canonical form, Left’s and Right’s counted together at a single position. It is a property of the value, since the canonical form is unique.

Accounted for means the share of the variance in the excess removed by replacing each value’s excess with its group’s mean. It is arithmetic, not inference, and the group count is printed beside it every time for that reason.

One sentence survives the whole census. A value’s complication and a position’s size are different currencies, and the exchange rate is set by the game — which is why the ruleset explains what it explains and why no property of the value has yet explained the rest.

Where the ladder goes next #

The realisability anchor has four rungs to here: which values any game reaches, what the cheapest way to reach one costs, that the cost is bounded below by the birthday and equal to it two thirds of the time, and now that the width of the form does not explain the rest.

The rung above takes the entry fee this page proposes — a constant per ruleset, Toads and Frogs paying 2.25 squares on everything and green Hackenbush paying nought — and finds that neither number is a property of the rules. The entry fee was the cap shows the excess falling as the birthday rises inside every ruleset, which is the signature of a censored sample rather than of a fee: the sweep’s size cap removes exactly the values that would have paid most, so a ruleset that reaches deep values cheaply looks expensive and one that cannot reach them at all looks free. Three squares past the cap, Toads and Frogs exhibits values born later than the strip is long.

That is a caution about every per-ruleset constant on this anchor, this page’s included. A quantity measured under a size cap and reported as a property of a game is a quantity the cap chose.

The rate was the alphabet then supplies what a censored sweep cannot censor: not how much a value costs but how many new values a ruleset produces per extra square. That rate is honest, and what it measures turns out to be the notation rather than the game — every ruleset grows at close to the number of symbols its positions are written in, and the seven of them span less than a factor of two. What separates the rulesets is the yield, which spans a hundred and nineteen.

So the anchor ends by splitting one question into two that behave completely differently: a growth rate that is a fact about the alphabet and is nearly the same everywhere, and a yield that is a fact about the game and is not. This page’s variable — the width of the form — turns out to belong to neither.

Two neighbours are worth the trip. How wide a form can get is where the width is established as a quantity and shown not to be bounded by the birthday, and it is the page that makes this one’s variable measurable at all. And the values nobody’s game produces is the census on the other side of the same question — not what a value costs to exhibit but whether anything exhibits it.