A shortlist with nothing at the top

The pairing removes moves it cannot name turns a statement about positions into a statement about moves. A Sylver Coinage move handing the opponent a position whose gaps pair off around its own largest gap hands them a win, so it can be struck off by one pass along a strip of numbers with no search anywhere in it. It removes a quarter of a full search and never removes a winning move.

It leaves a list. On the 583 positions whose own gaps pair that way, 9.58 moves of 12.27 survive, and the essay ends by asking the obvious question about them: whether the survivors have anything in common beyond surviving. If the winner is always the largest of them, or always the one closing the fewest gaps, that is a rule and it is one more scan away.

It is not. Seven quantities were computed for every survivor of every position and each was scored two ways, and the best of them beats chance by two and a half points on one measure and loses to chance on the other.

Seven things a scan can see #

A rule that names a move has to be built out of things a scan can see. A Sylver Coinage position is a set of unnameable numbers — its gaps — and a move is one of them; what a pass over the strip can produce about a move is the number itself, how many gaps naming it closes, and what the position it reaches looks like. Seven such quantities exhaust the plausible list:

the largest survivor and the smallest; the survivor closing the fewest gaps and the one closing the most; the survivor leaving the largest gap behind and the one leaving the smallest; and the survivor nearest the position’s own largest gap, which is the quantity the pairing argument itself is built out of.

Each was read as an order over that position’s survivors and scored on the positions the mover wins. Two scores, because they are different questions. How often the order’s first choice is a winning move is what a player following the rule would care about. Where in the order the first winning move sits is what a search ordering its moves by the rule would pay.

The second needs a baseline and the baseline is not one half. A position with 9.58 survivors and 1.68 winners among them puts the first winner, on average over random orders, at 4.28 — because more winners pull the first one forward. Anything worse than 4.28 is an ordering a search would be better off ignoring.

Every one of the seven is worse than 4.28. The best, at 4.34, is barely distinguishable from it; the worst is 5.08.

Twenty-four per cent, and twenty-two by chance #

The first score is the one that looks, at first, like there is something there.

The best ordering — take the survivor leaving the smallest gap behind — puts a winning move first on 145 of 583 positions, which is 24.9%. Three others reach 140, or 24.0%. That is a quarter, and a quarter sounds like a rule that works one time in four.

The baseline is 22.3%. A position with 1.68 winners among 9.58 survivors gives a scan choosing at random a winner 22.3% of the time, averaged over the 583, and the best rule in the table beats that by 2.6 points. Four of the seven land within two points of chance in one direction or the other, and one — take the survivor leaving the largest gap — is seven points below it, at 15.3%.

That is the shape of a measurement with nothing in it. A rule with real content would show as a gap of tens of points on a list this short, the way the parity of what a winning move closes does: a parity with a first exception finds a tendency that nearly decides the game, at better than nine cases in ten. What the seven show instead is that the winning move is where it is.

The two scores also disagree slightly, and the disagreement is worth a sentence rather than a section. Four of the orders are a little better than chance at putting a winner first and all seven are a little worse than chance at where the winner sits overall. A rule can have a whisker of signal at the top of its order and anti-signal below it, and that is what a rule with no content looks like when it is scored twice.

What is actually left #

The lists are not short in the way shortlist suggests. Twenty-two of the 583 come down to one move — and that one move is then forced, so the test has solved those positions outright — but 223 of them keep twelve, thirteen or fourteen, which is nearly everything that was legal.

The shape of that distribution is the reason no order helps. A test that cut the typical position from twelve moves to three would leave an object small enough for a rule to sort; one that cuts twelve to nine and a half leaves the same problem with a slightly shorter list.

It also puts the saving from the pairing removes moves it cannot name in its place. A quarter of a search is a real quarter and it is a constant factor, and the reason a constant factor is all it is shows here: the branching has fallen from 12.27 to 9.58, which multiplies over the depth of a search and never changes its character. A strategy is not a certificate is the general form of that — a saving that keeps the shape of the object cannot turn an intractable question into a tractable one.

Four positions, and where the winner is in each #

The table above is a summary and the summary is easy to argue with, so it is worth watching four positions.

Three of the four have gaps running 1 to 7 with an eighth gap above, at 14 or 15, and the test leaves the same six moves each time: 3, 4, 5, 6, 7 and the large one. On two of them the winner is the large one. On the third — gaps 1, 2, 3, 4, 5, 7, 8 and 14 — the survivors are 3, 4, 5, 7, 8 and 14, and the winner is 4.

The three positions differ by one gap. Two of them are won by the move the pairing argument would reach for and one is won by a small number in the middle of the list, and a scan comparing the two positions has nothing to compare: the same six survivors, the same largest gap, the same count of unnameable numbers.

That is why the shares in the first table sit where they do. The largest survivor wins a quarter of the time not because largeness matters but because a list of nine or ten with one or two winners in it will hand any fixed choice about a quarter of the positions.

A sieve instead of an order #

An order that cannot be trusted at the top might still be trusted at the bottom. A rule that cannot say which survivor wins might still say which cannot, which is exactly the shape the pairing test itself has — and a second pass of that kind would compose with the first.

Read that way, the seven are worse. Take the largest survivor and nothing else: the list falls from 9.58 moves to one, and on 443 of 583 positions the move discarded includes every winner the position had. Three of the seven cut that hard and all three are wrong on better than three positions in four.

The gentlest of them — keep the survivors leaving the largest gap behind, which ties often and so keeps several — halves the list to 4.89 and is still wrong on 264 positions, which is 45%. There is no setting among the seven where the cut is worth the risk: the ones that cut are wrong, the ones that are less wrong barely cut, and every one of them is wrong often enough that a search using it would have to search again.

That is the difference between this and the pairing test, and it is the whole of it. The order the moves are tried in prices what a good ordering is worth to a search, and the answer there is a great deal — which is exactly why the absence of one here is a result rather than a shrug. The pairing test is not a heuristic. It is a proof applied to a child position, so a move it strikes off is a move that provably loses, and it can be run before a search because it is never wrong. Nothing among these seven is a proof of anything, and a filter that is right 55% of the time is not a filter.

Twenty-two positions the test actually finishes #

There is one place where the shortlist is the answer, and it is small and worth separating from the rest.

Twenty-two of the 583 positions come out of the test with a single move left. On those, the move is forced: every other legal move has been shown to hand the opponent a position the opponent wins, and a position known to be a win for its mover with one move unaccounted for is a position whose one move wins. The test has solved them, and the cost was one pass along a strip of numbers.

That is the only unconditional thing on this page, and the shape of it is worth naming because it is the shape every useful pruning rule has. The test does not know which move wins. It knows which moves lose, and when it knows that about all but one of them the remaining move is named by elimination rather than by description. A rule that identified the winner directly would be a stronger object; a rule that eliminates hard enough gets there anyway on the positions where the elimination happens to be nearly total.

It happens on 3.8% of the positions in range, and there is no sign of the share growing. The twenty-two are spread across the genera rather than concentrated at one end, and the distribution above shows why: the lists get longer as the positions get larger, because the number of gaps grows faster than the share the test can strike. A position with fourteen survivors is further from elimination than a position with six, and the large positions are the ones a player actually needs help with.

So the honest summary of the test’s value is two numbers rather than one. It settles 3.8% of these positions outright, and on the remaining 96.2% it saves a constant factor on a search that still has to run. How much a list can lose measures the same trade in a different game, and the lesson transfers: a pruning rule’s worth is what it does to the positions it does not finish, because those are nearly all of them.

Where the moves that do win turn out to be #

The reason all of this is worth doing is that the target is usually a single number. Most of the 583 positions have exactly one winning move; the average is 1.68, and the distribution is concentrated at the bottom.

A position with one winner among ten survivors is a position where a rule is worth a great deal and a rule that is right a quarter of the time is worth almost nothing, because the cost of being wrong is the search the rule was supposed to replace. That asymmetry is why the two scores were both computed: an ordering that is right a quarter of the time and puts the winner second the rest of the time would be worth having in a search, and none of these does that either — the mean rank is 4.34 at best, against 4.28 for knowing nothing.

What the census cannot say #

Seven quantities are not all quantities. Everything scored here is a single number read off a move or its child, and a rule could be a combination — the largest survivor unless it closes an even number of gaps, say, or a comparison between two children. Nothing here rules that out, and the reason the seven were chosen is that they are the ones a one-pass scan produces, which is the cost the question was asked at.

Sixteen unnameable numbers is where the census stops. There are 583 paired positions in that range and the shares above are shares of those 583. A rule that only bites on larger positions would be invisible here, and the number of semigroups grows fast enough that the next few genera would need a different enumeration.

And only the paired positions are scored. The pairing test applies at every position in the game, but the positions counted here are the ones strategy stealing settles, because those are the ones known to be wins and therefore the ones with a winning move to find. What the survivors look like at a position nobody has settled is a different census with no answer column to score against.

The convention the verdicts are computed under #

Normal play: the player who cannot move loses, which in Sylver Coinage is the player who names 1, since after 1 every number is nameable and nobody can move at all. That is the convention the game that is a number system sets out and every account of the game here uses, and 1 is excluded from every move list above for the same reason it is excluded there — naming it is a resignation rather than a move.

The winner of each position is computed by the recursion over semigroups rather than by any of the rules being scored, so the answer column is independent of everything being scored against it. The pairing test’s soundness is checked separately and holds: no move it strikes off is ever a winner, over every position in the census.

The surprise: the pruning was the result, and it was the whole result #

Every move closes the largest gap proves a class of positions won and names no move. The pairing removes moves it cannot name turns that into a test on moves, which removes a quarter of a search and still names no move. The natural reading of those two together is that the naming is one step away — that a property this sharp about which moves lose must be close to a property about which move wins.

The two are not close and the measurement says how far apart. The test that says this move loses is a theorem, and it transfers to a child position exactly. Every attempt here to say this move wins is a guess scored against the answer, and the best guess available beats guessing by two and a half points.

The reason is visible in what each object is. A losing move can be recognised from the position it reaches, because the position it reaches is settled by an argument. A winning move cannot be recognised from the position it reaches, because what makes a position lost for its mover is that every move from it leads to a win — which is a statement about the whole subtree and not about the position. The pairing test is a certificate of loss that happens to be short; a certificate of win is a strategy, and a strategy is not a certificate prices those.

So the asymmetry this page measures is not a fact about Sylver Coinage. It is the same asymmetry that makes how hard is it a question with the answer it has, arriving in a game where one side of it has a one-line test and the other has none.

Still open: whether a rule reading two moves at once does better #

Every quantity here is read off one move at a time, which is what makes a scan a scan. The obvious thing it cannot do is compare.

A rule reading two survivors at once could ask which of them leaves the opponent with the shorter shortlist — the pairing test applied one level down, counting rather than striking — and prefer the move that leaves the fewest survivors. That is a quadratic scan rather than a linear one, it is still enormously cheaper than a search, and nothing here scores it.

The measurement that would settle it is the same table with that ordering added: the share of positions whose winner it puts first, against the 22.3% chance gives, and the mean rank against 4.28. If it lands where the seven did, the shortlist is as far as any scan reaches and the remaining work is the search this game has never been able to avoid. If it does substantially better, then what a scan cannot see about a move is visible in what the move leaves, and the next question is how many levels of that are worth paying for.