The restriction that buys the most

Four measurements here each establish one thing a scoring game lacks. Counting at the end changes everything finds the last-move theory returning one answer for every position. What a pass is worth to a theory finds a pass repairing Milnor’s hypothesis at the cost of the convention everything else is built on. A hypothesis has to hold all the way down finds the hypothesis needing to hold hereditarily. Nothing to subtract with finds comparison ceasing to be a subtraction.

Each is a measurement about the whole family, and between them they suggest a shape. A class small enough to have inverses would get the arithmetic and cover almost nothing; a class defined by the incentive condition covers a good deal and gets the bound and not the arithmetic; and nothing gets both. That is the trade this page was set up to price.

Priced on four classes, two families and three questions, the trade is not there. The class with inverses is the largest of the restrictions on one family and it buys all three, and on the other it is empty.

Four classes, three questions, one table #

The classes are the candidates those four measurements produce.

Every row is the baseline and the thing to beat. The incentive condition at the top is Milnor’s hypothesis as a reader would check it, on the position in front of them. The same condition all the way down is what an induction over the play can actually use. The rows that cancel against their own negatives is the smallest thing that could be called a group, and it is the class the arithmetic would need.

The questions are what each class is being asked to buy. Does the mean-value bound hold on every pair drawn from it? Does comparison by subtraction agree with comparison in every context, on every ordered pair from it? Does every member cancel against its own negative?

Asking the questions of pairs from the class rather than from the family is what makes this a comparison of classes rather than four readings of one sweep. A property that holds of the family restricted to a class is a property the class has; a property that holds of the family is a property nobody had to restrict for.

On rows of three, the expected answer #

The family of rows of three coins drawn from −2, 1 and 3 is where a hypothesis has to hold all the way down does its work, and on it everything behaves as expected.

The bound holds on 96 of the 378 pairs — a quarter — so a reader with no restriction at all has a mean-value theory that is wrong three times in four. Restricting to the fifteen rows satisfying the incentive condition at the top takes the pairs from 378 to 120 and the bound still fails on 24 of them, which is that essay’s finding and is why the hereditary reading exists. Restricting to the seven rows satisfying it everywhere takes the pairs to 28 and the bound holds on all of them.

So on this family the hereditary class buys exactly what it is advertised to buy: one property, at the price of keeping seven rows of twenty-seven.

The price is worth stating in the currency a reader would pay it in. Seven rows of twenty-seven is a quarter of the family, and the pairs fall from 378 to 28 — from three quarters of the positions a theory might be asked about to a fourteenth of them. A theorem covering a fourteenth of the cases is not useless, but it is a long way from the normal-play situation, where the mean value and the temperature are defined for everything without a hypothesis at all. Worth nothing, and worth fighting for develops those two quantities in their own setting, and the contrast is the whole reason a scoring theory is hard to state: the same quantities, defined the same way, needing a hypothesis that throws away nine tenths of the family before they can be used.

And the hypothesis is not one a reader can check by looking. Fifteen rows pass it at the top and seven pass it everywhere, so eight rows of twenty-seven look safe and are not — which is the same finding restated as a count of rows rather than of pairs.

It buys nothing else. Comparison by subtraction gets 24 of its 49 ordered pairs right — barely half — and not one of the seven rows cancels against its own negative. The cancelling class on this family is not small. It is empty.

On rows of four, it inverts #

Add a coin. The family of rows of four from the same three values has 81 rows, and every expectation the smaller family produced is wrong on it.

The bound holds on all 3,321 pairs, with no restriction at all. Milnor’s condition holds at the top of every one of the 81 rows, so the restriction that took an essay to distinguish from its hereditary version separates nothing here — the class of rows satisfying it is the family, row for row and pair for pair. A reader who arrived at this family first would have concluded that the mean-value theory needs no hypothesis.

And fifteen rows cancel against their own negatives. That is more than the twelve satisfying the incentive condition hereditarily. The class that was supposed to be the small expensive one is the larger of the two restrictions on this family.

What it buys is everything. The bound holds on all 120 of its pairs. Every row in it cancels, by construction. And comparison by subtraction agrees with comparison in every context on all 225 of its ordered pairs, where the hereditary class manages 108 of 144 and the unrestricted family 6,037 of 6,561.

So the answer to which restriction buys the most is the one with inverses, and it is not paying for the privilege in coverage. On rows of four it covers more than its rival and buys three properties to its rival’s one.

The fifteen are not a family a reader would have guessed at either. They include every constant row, the rows that repeat a pair, and the rows whose two halves mirror each other — and they exclude rows that differ from a member by one coin. Changing the last coin of −2, −2, 1, 1 from 1 to anything else takes the row out: −2, −2, −2, 1 added to its own negative scores 2 with Black moving first and −2 with White, which is a two-point advantage to whoever has the move on a position that is supposed to be nothing.

That is what a class defined by a test rather than by a construction looks like. The condition is checked on each row and the rows that pass it have no obvious shape in common, which is a reason to be careful about the word class and a reason the closure question at the end of this page is the one that matters.

Why coverage is not a property of the restriction #

Empty on one family and the widest restriction on the next is a strange thing for a class to be, and it is worth being clear about what it is not.

It is not a small-sample artefact: the rows of three are all 27 of them and the rows of four all 81, and both counts are exhaustive over their values. It is not the values changing, because they do not. It is a parity. A row of three coins is taken by three moves, so one player takes two coins and the other takes one, and a row added to its own negative is a six-coin position in which the mover’s advantage does not vanish. A row of four gives each player two, and the cancellation the mirror strategy would produce under normal play has somewhere to land.

That is the sense in which the coverage of the cancelling class is not a fact about the class. It is a fact about the family — and the same is true of the incentive condition, which separates fifteen rows of twenty-seven at length three and nothing at all at length four.

A restriction is a subset of a family, and every measurement of what it covers is a measurement of both. The expectation — a class with inverses would be very small — reads as a statement about scoring games and is a statement about whichever family somebody happened to check. The board falls apart, and the arithmetic changes is the same hazard one level down: a property that looks like the object’s turns out to be the sample’s.

The two ways a difference test is wrong #

The comparison column hides a distinction that decides how much any of this is worth.

A difference test can fail two ways. It can say yes where the contexts say no, which is unsound — a reader would replace a position by one that is worse in some context and lose a game they had won. Or it can say no where the contexts say yes, which is incomplete — a reader loses a simplification they were entitled to and nothing else.

On the unrestricted family of rows of three there are nine unsound pairs. Every restriction removes all of them: not one unsound pair survives the incentive condition, the hereditary condition or the cancelling condition, on either family.

What survives is incompleteness, everywhere. The hereditary class on rows of four refuses 36 of its 144 ordered pairs that the contexts accept; the unrestricted family of rows of four refuses 524 of 6,561. The cancelling class refuses none.

So the restrictions are worth more than the agreement counts suggest. A class whose difference test is merely incomplete is a class in which the test can be used — it proves what it claims and fails to prove some things that are true, which is what every sound-and-incomplete tool in mathematics does. Comparing positions sets out what the normal-play test is and why it is both; here the soundness is the thing a restriction buys first, and it is bought cheaply.

The counts also say something about where a scoring theory would have to be careful, and it is not where a reader would expect. The unsound pairs are all in the smaller family, where the bound is also at its worst — so the family on which the theory is hardest to state is the family on which a careless comparison is actually dangerous, and the family where the theory is easy carries no unsound pairs at all. A reader who checked the difference test on rows of four and concluded it was sound would be right about rows of four and wrong about the game.

What a class has to be before it is a class #

One thing none of the four candidates has is closure, and it is the property that would turn any of them into a theory rather than a test.

A class worth restricting to has to be closed under the sum: if two positions are in it, their sum should be too, or the arithmetic it licenses cannot be applied twice. Nothing above checks that, and the classes are defined by tests on a single row — this row satisfies the incentive condition, this row cancels — with no reason for the property to survive addition.

That is not a small omission and it is why this page compares what the classes buy rather than announcing one of them as the answer. The class with inverses on rows of four buys all three properties on the pairs measured, and whether the sum of two of its members is again one of its members is a question about rows of eight, which is outside the family and outside the search.

What the comparison cannot say #

Two families are not many. Rows of three and rows of four, from one set of three values, and the answer inverts between them. That is enough to establish that the coverage of a class is not a property of the class, and it is not enough to say what the classes look like on rows of five or six — where the parity argument above predicts the cancelling class reappears at every even length and vanishes at every odd one, and nothing here checks it.

The contexts are six. Comparison in every context means every context, and what is computed is comparison in six small ones. A pair the six accept might be separated by a seventh, so the contextual column is an upper bound on agreement and the unsound counts are lower bounds. The incompleteness counts are the reliable direction.

And closure is unchecked, as the section above says. Without it none of these is a class in the sense the theory would need, and the table is a comparison of tests rather than of theories.

The convention the classes are defined under #

Every row is played to the end: the coins are taken from the ends until none is left, and the score is the difference between what the two players took. There is no last-move convention anywhere on this page, which is the point of the whole question — counting at the end changes everything is what happens when one is imposed.

The negative of a row is the row with every coin’s value negated, which is the same position with the players exchanged, and it is what a negative has to be for cancellation to mean anything. The mean and temperature of a row are half the sum and half the difference of its two scores, which is Milnor’s definition and not an invention of this page.

The incentive condition is that the score with Left moving first is at least the score with Right moving first — having the move is not a disadvantage — and it is checked either at the row or at every subposition the play can reach, which is the distinction a hypothesis has to hold all the way down draws.

The surprise: the trade was a description of one family #

The expectation this page was written to test is a good one and it is stated in the right form: a stronger restriction covers less. Inverses are a strong thing to ask for, so the class with them should be small, and the arithmetic should be paid for in coverage.

Every part of that is true on rows of three, where the cancelling class is empty and the hereditary class is seven rows of twenty-seven. Every part of it is false on rows of four, where the cancelling class is fifteen rows and the hereditary class twelve, and the larger class is the stronger one.

What went wrong is that coverage was read as a property of the restriction. It is a property of the pair — restriction and family — and the two families here differ in a way that has nothing to do with scoring games at all. Four coins split evenly between two players and three do not.

That is worth carrying beyond scoring games, because the shape recurs wherever a class is proposed. Outcomes do not add is the same lesson about a summary: a quantity that describes a family perfectly can describe nothing about the objects in it. A class is a quantity of that kind. How much does this restriction cost is not a question with an answer until somebody says what it is restricting.

Still open: whether any of these is closed under addition #

The measurement this page most wants and cannot make is closure.

For each of the four classes, the question is whether the sum of two members behaves like a member — and behaves like has to be made precise, because a sum of two coin rows is a two-row position rather than a row and so is not in the family at all. The honest version is a test rather than a membership: for two members of a class, does the two-row position satisfy the class’s own condition, read as a condition on positions instead of on rows?

The incentive condition has an obvious reading on a sum, since a sum has two scores like anything else. Cancellation has one too: does the sum of two cancelling positions cancel? If it does, the cancelling class on rows of four is a group in the only sense a scoring theory could use it, and everything on this page about what it buys becomes a statement about an object rather than about a list. If it does not, then what was measured here is a set of rows with a shared property and not a class, and the right next question is what the smallest closed class containing them looks like.