Close calls nothing resolves

Seventy-two of them were not silence split a stored Domineering strategy’s unanswered decisions in two: 202 where the rules say nothing, and 72 where the rules narrow to a single placement and it is the wrong one. It then fitted a panel of eleven tie-break rules to the 202 and found that indexing them by the value of the position covers 118 of them.

The price of taking the maximum named what had never been asked:

The rung above is the value on the seventy-two … Fitting the same value panel to them, one rule per value class, is the same computation on a smaller population and would say whether the residue is close calls the counts cannot resolve or close calls nothing can.

It is the second. The same procedure covers four of the seventy-two.

The panel cannot be appended #

The first thing the computation makes clear is that the two residues do not admit the same experiment, and the difference is structural rather than a matter of degree.

On the 202 the rules leave several candidates standing, so a tie-break has something to choose between and appending one costs nothing — it fires only where the rules were already silent, and everywhere else it never runs. That is what made the rung below’s experiment cheap to believe: a rule that cannot fire on a decision the strategy already answers cannot damage one.

On the 72 the rules have committed. They name one placement, it is the wrong one, and appending a twelfth rule to a list that has already produced a single answer changes nothing at all. The only way a panel rule can act is by replacing one of the two rules — and the natural candidate is the second, connectivity, which the rung below found responsible for about a third of the failures.

Replacing is not free, and that is the whole of what follows.

The choice of which rule to replace is also not arbitrary, and it is worth saying why connectivity rather than mobility. The rung below apportioned blame across the seventy-two and found about a third of them to be the second rule’s fault — cases where mobility had narrowed correctly and connectivity then picked the worse survivor. Those are the decisions a replacement has the clearest shot at, so connectivity is the rule with the best prospects and the smallest bill. Replacing mobility instead is the experiment with the larger target and it is left for the rung above, for reasons the last section gives.

Every substitution is a loss #

The best replacement — leave the position with the smaller birthday — fixes 14 of the 72 and breaks 404 of the 3,034 the original pair got right. Every rule in the panel is a net loss, from −52 to −498.

That is not a marginal trade. Connectivity is earning its place on the decisions nobody was complaining about, and the seventy-two are being bought with them at twenty-six to one at best.

It is worth reading the two ends of that table together. Leave the largest single piece breaks only 52 — the cheapest substitution there is — and fixes nothing at all, so it is nearly a null replacement for connectivity and buys nothing with the little it spends. Play at the cell with most neighbours breaks 504 and fixes six. Between them the panel spans an order of magnitude in cost and never once comes out ahead, which is a stronger statement than any single row of it.

And the breakage is not concentrated on hard cases. A substitution that broke only decisions the rules barely answered would be a substitution worth considering — trading a marginal 404 for a definite 14. The 404 are ordinary answered decisions: they are among the 3,034 where mobility and connectivity between them named a best placement, and replacing connectivity turns some of them into decisions where the survivor is worse. Nothing distinguishes them from the rest except that connectivity happened to be what settled them.

That is the sense in which connectivity is earning its place. Three rules and a tie-break established the rule list and measured what it answers; this page measures what the second rule is worth by removing it, and the answer is between 52 and 504 decisions depending on what replaces it.

And the value carries almost nothing #

The exchange rate would still be worth knowing if the value were informative on the seventy-two — a rule that predicted well and cost too much is a different situation from one that predicts nothing.

The best value rule fixes 14 and the scan-order control — take the first placement the scan reaches — fixes 12. Two decisions of seventy-two is the whole of what reading the value buys here.

And the control is not a weak one to be beaten. Scan order is an arbitrary rule with no content whatever, and it fixes twelve of the seventy-two — which says that a sixth of them are fixed by any change to the second rule, simply because the original pair’s answer was wrong and almost anything else would have been different. That is the baseline a rule has to clear, and the value clears it by two.

That is the sharpest form of the negative, and it is the same instrument the rung below used to establish the positive result one residue over. A panel that beats its control by a wide margin on 202 decisions and by two on 72 is a panel telling the truth in both places.

Why one residue and not the other #

The asymmetry has an explanation and it is worth stating, because it makes the negative useful rather than merely discouraging.

A silent decision is one where the rules have found several placements equally good by their own lights. The value is then a genuinely new piece of information about positions the rules cannot tell apart, and it distinguishes them — which is what seventy-two of them were not silence measured.

A wrong decision is one where the rules have found a difference and got its sign wrong. The mobility count and the connectivity count both fired, both preferred the same placement, and it is worse. Adding a value reading does not correct a preference; it produces a third preference, and there is no reason for a third opinion to agree with the truth when the first two agreed with each other and were wrong.

So the two residues are not two sizes of the same problem. One is a shortage of information and the other is misinformation, and a panel of rules is a remedy for the first only.

The mechanism also predicts the asymmetry’s direction, which is worth checking rather than admiring. A rule appended to a silent decision can only help: where the rules were silent, any choice is at least as good as the arbitrary one they would otherwise fall through to. A rule substituted into a wrong decision can help or hurt, because it displaces a rule that was right elsewhere. So the silent residue is a strictly favourable setting for a panel and the wrong residue is not, and one would expect a panel to do better on the first even if the value carried the same information about both.

What the sweep adds to that expectation is the size of the gap. The panel does not merely do better on the silent residue; it does well there and no better than chance here. Those are different claims, and only the second says the value has nothing to offer the seventy-two.

What this closes #

The anchor set out to account for a strategy’s failures and can now account for all of them.

3,034 decisions the rules answer, which is 94.5 per cent and is where three rules and a tie-break left it.

202 silent decisions, of which 118 are settled by a value-indexed rule. That residue is a shortage of information and it is largely repairable.

72 wrong decisions, of which 4 are. That residue is not repairable by any rule of this kind, at any price the rest of the strategy can afford.

Two per cent of the decisions, and rather less than that of the errors a player would notice — the price of taking the maximum found none of the seventy-two wrong by more than one reply, so the strategy’s irreparable failures are also its smallest ones. That is a comfortable place for a residue to sit and it is not an accident: a decision the two counts get badly wrong would be one where they disagreed with each other, and those are in the silent pile.

The honest headline is therefore that the strategy’s remaining errors are not a missing rule. What a strategy has to remember established the 3,308 decisions as the object, and the account of them is now complete: most are answered, a sixth of the residue is a shortage the value fixes, and the rest is a commitment the rules make wrongly and nothing available un-makes.

What a negative of this shape is worth #

Four of seventy-two is the kind of result that reads as a wasted rung, and it is worth saying why it is not.

It closes a question rather than opening one. The rung below offered two readings and had no way to choose between them — close calls the counts cannot resolve leaves a door open, and every subsequent rung on the anchor would have had to keep it open. The computation shuts it, and it shuts it with the anchor’s own panel rather than with a new instrument, so the result is comparable with everything the anchor has already measured.

It is measured against a control, twice. The best value rule beats the scan-order control by two decisions; every substitution’s cost is measured against the same population the original pair answers. Neither number would mean anything alone, and the pair of them is what makes the value carries almost nothing here a measurement rather than an impression.

It also leaves the anchor with a complete account rather than an open one. Three residues, three verdicts, and no category left saying this might yet be fixed by something — which is what a ladder is for and is rarer than it sounds.

And it prices connectivity, which nobody had. The rule list was assembled and scored as a list; removing the second rule and watching 52 to 504 decisions fail is the first measurement of what that rule is individually worth. That is a byproduct of the experiment and it is more useful than the experiment’s own answer — how many moves are worth making prices the first rule and nothing had priced the second.

What the solver computed, and how #

Every decision a stored Domineering strategy faces over regions of at most eight cells — 3,308 of them, which is the population what a strategy has to remember built and the anchor has used since.

For each, the rules run in order: narrow the candidates by the mobility count, and if more than one survives, narrow by connectivity. A decision is answered when every survivor is a best placement, wrong when exactly one survives and it is not, and silent otherwise. That is the rung below’s split and it reproduces its numbers: 3,034, 72 and 202.

Each panel rule is then run in place of connectivity: mobility first, then the rule, and the decision counts as fixed when every survivor is a best placement. The same substitution is run over the 3,034 already answered, and a decision that stops being answered counts as broken. Both counts come from the same procedure over the same population, so the exchange rate is a comparison rather than two measurements.

The value-indexed table is the rung below’s: group the decisions by the canonical value of the position they sit on, and count the classes where some single panel rule settles every decision in the class.

Two things are asserted rather than reported. Every substitution must have a negative net, since the page is written about none being worth making. And the value classes must cover a minority of the seventy-two while covering a majority of the silent decisions — the asymmetry is the finding, and a sweep where both were small or both large would make five of these figures wrong.

Where the model stops #

Regions of at most eight cells, and one strategy. The seventy-two are a small population and four of them being covered is a small number of a small number. What carries the conclusion is not the four but the exchange rate and the control: every rule costs more than it saves, and the best value rule beats a no-content rule by two.

Eleven rules, and they are the rung below’s eleven. A twelfth rule chosen with the seventy-two in view might do better, and it would be fitted to them — which is the mistake this anchor has avoided throughout by scoring every rule on a population it was not chosen for. So the honest claim is that this panel fails here, and that a panel assembled from what the seventy-two look like would need a fresh population to be believed on.

Only connectivity is replaced. Substituting for the mobility count instead is a different experiment and has not been run. The rung below’s finding that a third of the seventy-two are connectivity’s fault is what makes it the right one to try first, and the other two thirds are mobility’s — so a panel rule replacing mobility is the experiment with the larger target and the larger bill, since mobility answers more.

Normal play throughout, and best placement means one that keeps the position’s value where the exact evaluation says it should be, which is a normal-play comparison.

And the figures cannot show a wrong decision. Six tables of counts describe a strategy failing, and the object — a region with the placement the rules name and the better one they discarded, side by side — is a picture. The price of taking the maximum draws the margins those pairs differ by, and what neither page draws is one such pair with both placements marked.

Where the ladder goes next #

The tempo anchor has seven rungs: what a value leaves out, how many moves are worth making, what a strategy has to remember, three rules and a tie-break, what the residue is made of, how far wrong the wrong part of it is, and now what the value says about it.

The rung above is the mobility substitution. Two thirds of the seventy-two are the mobility count’s fault rather than connectivity’s, and this page has replaced only the second rule — so the experiment with the larger target has not been run. It is the same computation with one line changed, and it has a prediction attached that this page’s finding makes worth stating in advance: mobility answers far more of the 3,034 than connectivity does, so the exchange rate should be worse, not better, and a rule that fixed twice as many of the seventy-two would still be paying several times over for them. If that is what happens, the anchor can close: the strategy’s wrong decisions are not reachable by substitution at any price, and the only remaining move is to stop and search.

Two neighbours are worth the trip. Seventy-two of them were not silence is where the two residues were separated and where the panel first ran, and reading the two pages together is the whole of what the value buys a strategy. And when a real board falls apart says which of these regions a played game actually produces, which decides whether seventy-two unreachable errors is a small problem or an invisible one.