When the bracket decides

One row of Clobber evaluated every row up to a length and ran into the thing that makes the game hard to write about: the values are all-small games with canonical forms running to several options a side, and quoting one tells a reader nothing.

The practical answer is the up-bracket: the largest $n$ with $G \ge n \cdot \uparrow$ and the smallest $n$ with $G \le n \cdot \uparrow$, both found by comparison. It is a pair of integers, it is honest — it is exactly what comparison can establish — and it is an approximation.

That essay closed by naming the measurement it had not made: how often two Clobber rows are ordered by their brackets, and how often the brackets overlap and decide nothing. An approximation is worth what it decides, and this is the count.

The three things that can happen #

Two rows, two brackets, and three outcomes.

The brackets are disjoint. One row’s lower end sits above the other’s upper end, so the first is at least as large as something the second is at most — and the order is forced. This is the approximation deciding, and whether it decides correctly is a thing to check rather than assume.

The brackets overlap and the truth decides anyway. The approximation is silent and the real comparison is not. This is what the approximation costs.

The brackets overlap and the truth is silent too. The two rows are genuinely confused, and the approximation has lost nothing at all.

The third is the one that turns a percentage into an argument. An approximation silent exactly where the truth is silent is not an approximation; it is the answer.

The counts #

One hundred and twenty-six rows, up to six squares, giving 7,875 pairs.

One thousand five hundred and eighty-five pairs have disjoint brackets. All 1,585 orders agree with the true comparison — no disagreements, which is what being a bound rather than a guess means and which had to be checked, because a bracket that lied would take the whole method down with it.

Six thousand two hundred and ninety pairs overlap. Of those, 2,068 are decided by the true comparison and 4,222 are genuinely confused.

So the truth decides 3,653 pairs and the bracket decides 1,585 of them, which is 43 per cent. And of the 6,290 the bracket declines, two thirds had nothing to decide.

The soundness check, and why it is the important one #

Fifteen hundred and eighty-five orders, and none of them wrong.

That zero is the load-bearing number in the whole essay and it is worth saying why. A bracket is defined as a pair of bounds: the lower end is an $n$ with $G \ge n\cdot\uparrow$ established by an actual comparison, and the upper end an $n$ with $G \le n\cdot\uparrow$ likewise. So if two brackets are disjoint, transitivity does the rest, and the order the brackets give is a theorem.

A single disagreement would therefore not be a wrong prediction. It would be a broken implementation — a comparison returning the wrong answer, a multiple of up built wrongly, an off-by-one in the search for the ends — and the effect would be silent everywhere else in the sweep. The 1,585 agreements are what establish that the bracket code is computing the object the argument is about, and the argument is what establishes that they had to agree.

That is the shape of every check on this site that returns a clean zero. It is not evidence that a claim is true; it is evidence that the machinery is computing the claim.

What a player does with a band #

Suppose a board has two Clobber regions and a player has to choose which to move in.

If the two brackets are disjoint the choice is settled by the theory: one region is worth strictly more ups than the other, and the whole board is a sum of infinitesimals in which the larger atomic weight wins. That is the 43 per cent.

If the brackets overlap, the player has two options and neither is a mistake on the evidence available. Running the real comparison settles it in 2,068 of 6,290 cases and returns confused in the rest — and in a confused pair the choice genuinely depends on the rest of the board, so no amount of local computation would have helped.

So the practical reading of the sweep is that a bracket answers a player’s question about a third of the time, that the answer is reliable when it comes, and that most of the remaining silence is not the bracket’s fault. A player who wants more has to compare, and comparing is a search.

What is being approximated #

The atomic weight is the number of ups a position is worth, and it is a number the calculus produces rather than something comparison hands over. The bracket is the evidence a reader can check: it says the answer lies between these two integers, and it says so by exhibiting comparisons.

For a position with no star in it the bracket is a single integer and the approximation is exact. For a position with a star it is not, because a star is confused with $\uparrow$ and with $\downarrow$ and comparison cannot see past it — the bracket widens by two either way, and the atomic weight sits in the middle where comparison cannot reach.

Taking the star-free half on its own is what turns that from an observation about five rows into a claim with nothing left over.

How often a row is pinned #

One hundred and two of the 254 rows up to seven squares have a bracket of width nought — an exact atomic weight, established by comparison alone.

The proportion by length runs 100, 50, 75, 37.5, 50, 37.5 and 36 per cent. It falls, and it falls unevenly — the rows of three do better than the rows of two.

That unevenness is the clue that length is the wrong variable. What decides whether a row is pinned is whether its value has a star in it, and star-ness depends on the parity of things in the row rather than on how many squares it has. The widths that occur are nought, one, two, three, four and five, and the commonest by far are nought and four — 102 and 84 of the 254 — which is exactly the two-population shape a star-or-no-star explanation predicts.

Nine thousand rows, and a third of them nought #

The one-row census reaches further than the pairs do — 9,840 rows over every length up to eight, counting empty squares as well as stones, carrying 111 values between them, and 3,568 of the rows are worth exactly zero.

That distribution explains a good deal of what the bracket does. A row worth zero is pinned trivially, at nought either way, and there are thousands of them; the pinned proportion is therefore held up by the zeroes and the interesting cases are elsewhere. Meanwhile 111 values over 9,840 rows means the average value is carried by nearly ninety different rows, and thirty-six per cent of all rows are worth nothing at all, so the collapse in this game is enormous.

A bracket is a summary of a value, and in a game where 9,840 positions carry 111 values the summarising has largely been done already by the game itself. The census in full is where those figures are drawn and where the ten commonest values are counted off against each other; what the bracket adds is a further collapse, from 111 values down to a handful of bracket shapes, and the question this essay answers is how much of the order survives the second collapse.

Why width four is the second peak #

Of the 254 rows up to seven squares, 102 have brackets of width nought and 84 have width four. The other four widths — one, two, three and five — account for the remaining sixty-eight between them.

Width nought is a value with no star in it: comparison pins it exactly.

Width four is what a single star does. A star is confused with $\uparrow$ and with $\downarrow$, so a value carrying one is confused with the two multiples of up either side of its atomic weight, and both ends of the bracket move out by two. Two plus two is four, and the count of 84 is the count of rows whose values carry exactly one star’s worth of that confusion.

The widths in between — one, two, three — are the values where the star interacts with something else, and there are far fewer of them. So the histogram is bimodal, and the two modes are no star and one star, which is a fact about the values rather than about the rows.

The values themselves are worth putting on the axis, because the widths they land at are not the widths their names suggest.

Widening the axis from four to six changes none of them. That is the point of a bracket being made of comparisons: its ends are the largest and smallest multiples of up the position can be tested against successfully, and testing against more distant ones cannot move a bound that was already established.

That is the sort of statement a bracket is good for. It is a poor summary of a single value and an excellent summary of a population, because the thing it throws away — where exactly inside the band the answer sits — is the same for every member of a class.

The rows themselves #

A Clobber row is a string of blue and red stones. A move takes a stone of the mover’s colour, moves it onto an adjacent stone of the opponent’s, and removes the one that was there — so every move removes exactly one stone, and a player has a move precisely when a stone of theirs is adjacent to one of the opponent’s.

Every move removes a stone, so the game ends after at most as many moves as there are stones — which is the termination the whole recursion needs, available here by inspection.

That condition is symmetric, which is why every Clobber position is all-small: a move for one player is a move for the other, all the way down. So every value here is an infinitesimal, no number says anything about any of them, and the whole apparatus of stops and temperatures reports nought throughout.

The widest canonical form among the 254 belongs to $\texttt{xoxo}$, at four options. Four squares, and the value is already an expression a reader will not carry.

What silence costs and what it saves #

The honest way to price an approximation is against the alternative, which here is running the comparison.

Comparing two Clobber rows means deciding who wins their difference, which is a search over a game tree built from both. For rows of six that is affordable; for rows of twenty it is not, and it is exactly the regime in which somebody would want a bracket.

The bracket is cheaper only in a specific sense: computing it is itself a sequence of comparisons — against $\uparrow$, $\Uparrow$, and so on — but those are comparisons against fixed small positions, and they are done once per row rather than once per pair. With $n$ rows the bracket costs $n$ batches of comparisons and answers $n^2$ questions, of which it gets 43 per cent right.

So the trade is real and it is not free. A reader who wants the whole order pays for the whole order, and the difference between those two prices is the same difference that separates deciding a winner from computing a value.

Each end of a bracket is one such comparison, run against a multiple of up and settled by playing the difference out: is $\uparrow$ above $\ast$, is $\Uparrow$ above $\uparrow$, is $\uparrow\!\ast$ above $\ast$. The first comes back confused, which is the star doing the whole of its work; the second comes back above; the third is confused again. A bracket is the record of which of those came back positive, and the confused ones are exactly where its ends stop moving inward — which is why comparison alone cannot reach an atomic weight on a position with a star in it.

The 986 that are equal #

Of the 7,875 pairs, 986 are pairs of rows with the same value.

That is a large number and it deserves a moment. One hundred and twenty-six rows carry far fewer than 126 distinct values, so the game collapses heavily — and every pair of rows sharing a value is a pair the bracket also puts in the same place, since equal games have equal brackets.

Those pairs are counted among the confused in the table, because a comparison that comes back equal is a comparison the bracket cannot distinguish from confused on the evidence of two integers. That is a genuine limitation of reporting a bracket rather than a value: it merges equality with incomparability, and the two are very different for a player.

Silence is a third answer and it has to be designed in #

A reading that can decline is a different object from a reading that always answers, and the difference is worth stating because it changes what the hit rate means.

A rule that always answers has two outcomes — right and wrong — and one number describes it. A rule that may decline has three, and the three trade against each other: making it decline more often raises its accuracy on the cases it does answer and lowers its coverage. So a single percentage cannot describe it, and quoting one invites the reader to compare it against a rule of the other kind, which is not a comparison at all.

The pair that does describe it is coverage and accuracy where it speaks, and only the second is allowed to be the headline. A reading with 100 per cent accuracy on 2 per cent of positions is nearly useless and looks perfect; a reading with 90 per cent accuracy on everything is more useful and looks worse.

The design question is therefore where to put the silence, and it has a right answer here. A bracket declines exactly when its two ends disagree about the comparison being asked, which is the case where any single-number reading would have had to guess. So the silence lands on the positions the reading has no information about rather than on positions chosen to flatter the accuracy — and that is what makes the two numbers honest rather than tunable.

Which is the property to look for in any approximation of this shape. Silence chosen by the method is a measurement; silence chosen by the author is a hit rate with the failures removed, and the two are indistinguishable from the outside unless the rule for declining is stated first.

What this says about approximations generally #

The pattern here is worth abstracting, because it recurs.

An approximation that is a bound — that says the answer lies in this range and is never wrong about the range — has a hit rate and a silence rate, and the useful question is what fraction of its silence is expensive. Forty-three per cent of the decidable comparisons recovered sounds poor; two thirds of the silences being free changes the reading completely.

A rule that is never right and cannot be far wrong is the same shape of object in the temperature theory, and it is measured the same way: against what the truth would have said. A bound that has never been compared with the truth is a claim about a method rather than about a game.

What the sweep cannot say #

Rows of six, and 126 of them. The bracket’s behaviour on longer rows is not measured here and there is no reason to expect the 43 per cent to hold: longer rows have wider brackets, wider brackets overlap more, and the recovered fraction should fall.

Nothing here is about two-dimensional Clobber, which is where the game is actually played and where the values run out of reach at 4×4.

And nothing here computes an atomic weight. The bracket is the evidence for one; the calculus that produces the number itself is a different object, and this essay measures the evidence rather than the conclusion — deliberately, because the evidence is what a reader can check.

Where the ladder goes next #

clobber has three rungs: a game where nobody can be ahead in moves, one row of it evaluated, and now the approximation that makes those values usable measured against the truth.

The rung above is the calculus. The atomic weight is a number and the bracket brackets it; computing the number, and checking that it always lies inside the bracket, would close the gap this essay measures from the other side. The interesting count would be how often the number is strictly inside — which is to say, how often comparison alone genuinely cannot reach it.

Two neighbours are worth the trip. Atomic weight is where the bracket is defined and where its two ends are shown to be comparisons rather than estimates. And the class where nobody runs out first is the family Clobber belongs to by the shape of its rule, where the same measure applies to every member and the same difficulty arrives.