How rare it is to be bigger

A reader arriving at game values from arithmetic brings an expectation with them: that values are like numbers with some awkward cases. Most pairs compare, some do not, and the ones that do not are the interesting exceptions.

The proportions are the other way round, and they get worse fast.

What comparison means here #

Two positions are compared by playing a third. $G \geq H$ holds exactly when Left wins $G - H$ moving second, and the four possible outcomes of that difference give four relations: $G > H$, $G < H$, $G = H$, and confused with — neither above nor below.

The fourth is not a failure of the method. It is an answer, it is the answer most of the time, and it is what makes a game worth playing: a position is a first-player win exactly when it is confused with zero, which is the outcome classification stated as an order relation.

Every count on this page is that computation run many times over. Nothing about a pair is inspected, nothing is inferred from a shape, and the tally of confused answers is a tally of searches that finished and reported a first-player win.

The counts #

Both days are counted the same way and both counts exclude a value paired with itself, which matters when they are set beside the lattice essay’s figures — that one counts 253 pairs on day two because it includes the diagonal, and a value is trivially comparable with itself.

Day two. Twenty-two values, 231 unordered pairs. 179 comparable, 52 not: 77% comparable, which still looks like an order with exceptions in it.

Day three. Every fourth value of the enumeration, 369 of them, 67,896 pairs. 40,558 comparable, 27,338 not: 60%. Two in five pairs of values born by day three cannot be put in order at all.

The share comparable with zero falls faster. On day two, 13 of the 22 values are above or below zero — a win for one player whoever moves — and 9 are not. One day later it is 106 of 369, which is 29%. Seven values in ten born by day three are games somebody wins by moving first, and the fraction is climbing.

The width, which is the sharper measurement #

A partial order can have a high proportion of comparable pairs and still be shallow, or a low one and still be almost a line. The quantity that says how far from a line it is, is the width: the size of the largest set of values no two of which compare.

Greedily collected, day two supplies four such values — $0$, $\ast$, $\{1 \mid -1\}$ and $\ast 2$ — and the day-three sample supplies at least twenty-three. Both numbers are lower bounds, since greedy collection is not optimal; the growth between them is what matters.

Four is a number a reader can hold. Twenty-three is not, and there is no reason to expect it to stop. The 1,474 values born by day three already contain more distinct forms than anybody can hold, and the width is growing at least as fast as the count. What it says is that the day-three values are not an order with a few sideways bits in it; they are an order most of whose elements are sideways to most of the others.

Which values are the sociable ones #

The share is an average, and the spread behind it is more informative than the average.

Among the 22 values born by day two, the two extremes are $-2$ and $2$, each comparable with 21 of the other 21 — everything, since a value at the end of the order is above or below everything. At the other end sit $0$, $\ast$ and $\{1 \mid -1\}$, comparable with 12, 12 and 10 respectively.

That the switch is the least sociable is expected: a position both players want to move in is confused with a whole interval of numbers, and the interval is where its neighbours are. That zero is nearly as unsociable is not, and is worth a second look. Zero is comparable with every number and confused with every position that is a first-player win, and by day two those are already half the catalogue.

The day-three sample has the same shape stretched out. The most sociable value there compares with 365 of the other 368; the least compares with 51. A range from 14% to 99% is not a distribution with a meaningful centre, and quoting the 60% without it would be quoting an average over two different populations.

Why it goes this way and not the other #

The trend has an explanation and it is worth having, because a reader might reasonably expect the opposite: more values should mean more chances for one to sit above another.

More values does mean more comparable pairs in absolute terms — 40,558 against 179. What falls is the share, and the reason is that the new values arriving each day are overwhelmingly of a kind that compares with very little.

A value born on a late day is a position with deep options on both sides. For it to be above another value, Left has to win the difference moving second, which means answering every Right move in either component — a demanding condition when both components are deep. Two shallow values are easy to compare because one of them usually runs out of moves first. Two deep ones are not, and the difference game between them can be longer than either.

The infinitesimals are the clearest case of it. $\uparrow$, $\downarrow$, $\ast$, $\uparrow\!\ast$ and their neighbours all sit strictly between every negative number and every positive one, so none of them is separated from another by anything a number can see; the comparisons between them are confused or fall into narrow chains, and there are more of them on every day. They are also all confused with nought except the ups and downs, which is what pulls the second measurement down so fast.

The spine of the order is disappearing #

There is a second measurement that belongs beside the two above, and it is the one that says what is being lost rather than how much.

The numbers are the totally ordered part of the universe. Any two of them compare, every one of them compares with every other value in the fixed way the stops describe, and they are the only family of which either sentence is true. They are the spine the rest of the order hangs from.

On day two, 7 of the 22 values are numbers — 32% of the day. On day three, 15 of 1,474 — one per cent.

Numbers are born at one per day per gap, so their count grows by a constant factor while the values around them grow by squaring. One day of the construction takes the totally ordered part from a third of everything to a rounding error, and it goes on doing so: whatever day four holds, it holds thirty-one numbers.

That is the shape of the thing the counts are measuring. It is not that the order is acquiring exceptions. It is that the part of it which behaves like arithmetic is a fixed, slowly growing family sitting inside a universe that is exploding around it, and by day three a value picked at random is essentially certain not to be one.

The obvious explanation for the falling share, and why it is wrong #

Having noticed that, the tempting next step is to explain the fall in comparability by it: the comparable pairs are the ones with a number in them, the numbers are vanishing, so the share must fall. It is a tidy argument and the day-two counts refute it.

Of day two’s 179 comparable pairs, 99 have a number in them and 80 have none. And among the 105 pairs where neither value is a number, 80 are comparable — a share of 76%, against 77% for the day as a whole. Removing the numbers entirely leaves the comparability of day two almost exactly where it was.

So the numbers are not carrying the day-two figure, and their disappearance cannot be what pulls the day-three figure down. At day three they are 1% of the values and could not carry it in any case: whatever 60% is measuring, it is measuring what non-numbers do to one another, and the drop from 77% to 60% is a drop in that quantity rather than a change in the mixture.

Two facts, then, and they are independent. The spine is vanishing as a fraction, which is a statement about what kinds of value exist. And comparability among the values that were never on the spine is falling on its own, which is a statement about how those values relate — and it is the one the trend is actually about.

The second is also the harder of the two to explain, and this page does not explain it. The section below offers a mechanism — deep options on both sides make a difference game hard for either player to win moving second — and a mechanism is not a rate. What the numbers establish is that the fall cannot be dismissed as bookkeeping about which values happen to be in the pool.

It does sharpen the question the last section leaves open. Asking whether comparability settles above zero is asking about a population from which the arithmetic-like values have already been removed, since they were a third of the first day counted and are a hundredth of the second. Whatever limit there is, the numbers will not be in it.

The order is still not a mess #

It would be easy to read the counts as a statement that the values have no useful structure. They have a great deal, and two facts one rung over say so.

Every pair of day-two values, comparable or not, has a least upper bound and a greatest lower bound among the day-two values — all 253 pairs, with no exceptions — so the order is a lattice, and the simplest game above both is a game a reader can be handed. Incomparability and structure are not opposites, and the same is true of the infinitesimals, which are strictly ordered among themselves while being confused with almost everything else.

And the order is not merely a lattice: on day two it is distributive, on all 10,648 triples. That is a strong property, one that a partial order owes nobody, and it holds on a day where a fifth of the pairs are confused.

There is one identity it does not satisfy, and where it fails is exactly the measurement this page is about.

That is a sharper statement of what confusion costs than any share. A comparable pair contributes nothing new: its join and meet are the two values back again. An incomparable pair produces a join and a meet that are genuinely other values, and the amount by which they overshoot is a game rather than a number.

What a player loses when comparison fails #

The practical content of the measurement is about substitution.

If $G \geq H$, then $G$ may be substituted for $H$ in any sum without making things worse for Left. That is the whole use of the order, and it is what makes equality in every company the relation the theory is built on: a player who knows a position is at least as good as another may reason about the simpler one. When two positions are confused, no substitution is licensed in either direction — the difference is a first-player win, so there is some company in which each is better than the other.

At 60% comparability that licence is available for three pairs in five, which sounds usable. At 29% comparability with zero it is available for fewer than one position in three when the question is the one that matters most: is this position good for me, full stop.

A confused answer is a complete answer — the search finished and reported a first-player win — and it licenses nothing. That is the whole practical content of the counts: not that the theory is unsure about three pairs in five, but that for three pairs in five there is no substitution to be had in either direction, and a player who wants one has to look at the company as well as the components.

What this does to the picture of a value #

There is a mental image the counts should displace.

A number is a point on a line, and a line is totally ordered: any two points compare, and the comparison is the whole relationship between them. A reader who imagines game values as points on a line with occasional smudges is carrying a picture in which comparison is the norm.

The measurement says the picture should be inverted. Most pairs of values are sideways to one another. What a value is, geometrically, is an interval between its stops with a fine structure at the ends — and two intervals that overlap are confused, which is why overlap is the normal condition once there are enough values for the intervals to be dense.

Taking that seriously changes what a question about values is for. “Is this position better than that one?” is usually the wrong question, and it usually has no answer. “Which of these may be substituted for the other?” is the same question stated so that the answer neither is informative rather than a failure. And “who wins this, in this company?” — the question a sum actually asks — is decided by a comparison with zero, which is available for fewer than a third of the values on day three.

Equality is the exception to all of it, and the exception is worth naming because it is the relation the whole order is built on. Two equal positions are interchangeable everywhere, with no reasoning about company at all; $\{0 \mid 1\}$, $\{0, -1 \mid 1\}$ and $\{0 \mid 1, 2\}$ are three forms of one half and any of them may stand in for any other in any sum whatever. It is the one relation among values that behaves exactly as a reader expects, and among 253 pairs of day-two values it holds on 22 — the diagonal, and nothing else.

What the solver computed, and how #

Day two is enumerated completely: the 22 values, all 231 unordered pairs, each compared by building the difference, reducing it and reading its outcome class.

Day three is 1,474 values and 1,086,275 pairs, which is more comparisons than a build should make. The sample is every fourth value of the enumeration — spread across it rather than taken from its beginning — and the pair count is reported beside the answer, so what is drawn is a measured share of a stated sample and not an estimate presented as a census.

The antichain is collected greedily: walk the values in order and keep each one that is incomparable with everything kept so far. That gives a lower bound on the width and is labelled as one; computing the true width is a maximum-antichain problem and is not attempted.

The check that could have failed is the day-two figure against the lattice essay one rung over, which counted 253 pairs including the diagonal. Both counts are of the same object and the difference is whether a value is paired with itself; stating which is which is the difference between two numbers agreeing and two numbers looking as though they disagree.

The one comparison that stays cheap #

Against all of this, one comparison remains easy, and it is worth naming because it is the reason the theory is usable at all.

Comparing a value with a number is decided by the two stops: $G$ is above every number below $RS$, below every number above $LS$, and confused with the numbers strictly in between. So the whole of a value’s relationship with the entire number line is two numbers, computed once, and no search is needed after that — which is what adding a number to a fight does, read as a statement about order rather than about play.

That is why the stops are worth computing and why a thermograph is worth drawing. The order restricted to numbers is a line, the position sits across an interval of it, and the interval is a complete answer to an infinite family of comparisons.

What has no such shortcut is comparing two values that are both not numbers. There the difference has to be built and played, the answer is confused three times in five, and there is no summary that predicts which. The measurement on this page is a measurement of exactly that gap: cheap against numbers, expensive against everything else, and everything else is most of what there is.

Where the model stops #

One sample and two days. The day-three share is a sample statistic, and while the sample is spread across the enumeration it is not random — a systematic relationship between an enumeration index and comparability would bias it, and no such relationship has been ruled out.

Two days is also a short trend. The direction is clear and the rate is not: 77% to 60% is one step, and a reader would be entitled to ask whether the share settles somewhere above zero or heads for it. Day four is not reachable here.

And the width numbers are bounds, not values. Four and twenty-three are what a greedy walk found; the true widths are at least that and could be considerably more, which makes the growth a lower bound on a lower bound.

Where the ladder goes next #

The rungs below establish what comparison is, that it is a search, and that its fourth answer is a fact about the pair. This rung counts how often each answer comes up and finds the proportions the wrong way round from what a reader expects. The rung above is the trend: whether comparability has a limit above zero, which needs a day this evaluator cannot enumerate.

Two neighbours are worth the trip. The simplest game above both is the structure that survives the confusion, and it is a stronger structure than the counts here would suggest. And confused is not the same as unknown is the rung that makes the fourth relation a relation rather than a shrug.