How many moves are worth making

What a value leaves out opens the tempo anchor with the observation that two positions worth the same value can differ in everything else, and names the quantities the equivalence throws away:

Beyond length are the others: which move achieves the value, how much of the position a strategy has to remember, and how the position looked before it was reduced. Each is invisible to the equivalence and each matters to somebody.

The first of those is countable, and counting it on a real game turns a remark into a measurement.

What a best move is here #

A best move is one that leaves the most behind. For each of Left’s placements, the region left over is evaluated and its right stop taken — what Left ends with if Right replies first and the fight is played out to the end — and the placements attaining the largest of those are the best ones.

That is not the definition a reader might expect, and the reason is worth a paragraph.

The obvious definition is the move that achieves the left stop: $\mathrm{LS}(G) = \max_{G^L} \mathrm{RS}(G^L)$, so the moves attaining the maximum are the ones that realise what the position promises. That identity holds when $G$ is not a number and fails when it is — a vertical domino is worth 1, its only move leaves nothing, and nothing has a right stop of nought rather than of one.

The failure is not an accident of bookkeeping. It is the content of numbers avoiding numbers: in a number-valued position neither player wants to move at all, so no move achieves the value and the definition has nothing to pick out. Half this catalogue is numbers, so the definition that survives is the relative one — the best of what is available, which is defined whether or not anybody wants to play it.

What the value does not say #

One thousand and thirty-four regions of at most eight squares give Left at least one placement, and between them they carry 125 distinct values.

Sixty-three of the 125 have two regions that disagree about how many best moves there are. Half the values in the catalogue.

Fifty-two disagree about whether there is a choice at all — one region with a single best placement and another, worth exactly the same, with several.

The two regions in that figure are equal in the strongest sense this subject has: equal in every company, interchangeable in any sum, indistinguishable by any game-theoretic test. And a player at the board is in a different situation in front of each — one has a decision to make and the other does not.

Why this is not a defect #

It would be easy to read that as the value failing at something, and it is worth saying plainly that it is not.

The value answers one question completely: in any company, who wins and by how much. It answers it so completely that two positions with the same value can be swapped in any sum with no effect whatever, which is the theorem the whole subject stands on.

How many moves are worth making is a different question, and there is no reason a complete answer to the first should carry an answer to the second. Two positions with the same value have the same canonical form — the form is determined by the value — and the canonical form is precisely the position with the redundancy taken out. A dominated option is deleted because nobody would choose it, and the count of moves a player is choosing between is exactly what deletion destroys.

So the gap this page measures is not a limitation of the value; it is the shadow of the reduction. The value is a fact about the equivalence class and the move count is a fact about the representative, and the reduction is what stands between them.

What grows and what does not #

The more interesting half of the census is what happens as the regions get bigger.

A region of four squares offers Left 1.75 placements on average and 1.50 of them are best ones. A region of eight offers 3.69 and 1.58 are best.

The number of moves more than doubles and the number worth making does not move. The share falls from 86 per cent to 43, and the count of regions with a unique best move rises with the count of regions.

That is the practical shape of the finding. A larger region is not a richer position with more good options in it; it is the same small number of good options buried in more bad ones. A player is choosing from a longer list for the same answer, which is what makes a big board hard rather than merely long.

Where the good moves are #

The regions themselves say something the counts do not, and it is visible once a few are drawn.

A best placement is nearly always one that keeps Left’s remaining dominoes from being broken up — a move into a column that is about to be cut, or one that leaves the surviving squares in a shape Right cannot easily spoil. That is the same phenomenon the packing bracket prices from the other side: a placement that wastes a future placement is a placement worth avoiding, and a region where several placements avoid it equally is a region with a free choice.

What the census cannot do is turn that into a rule. Every attempt at one on this ladder has run into the same wall: the properties visible on the drawing — squares, runs, orientations — are counts, and which placement is best depends on what happens after it. The move census is the same fact, counted on the moves instead of on the values.

The two ends of the distribution #

Five hundred and eighty-four of the 1,034 regions have exactly one best move, and 167 have every move a best one. The rest are in between.

The 167 are worth a moment, because they are the case a player would most like to recognise. A region in which every placement is equally good is a region where the decision is free — put the domino anywhere and the value is unchanged — and they are 16 per cent of the catalogue, concentrated in the small sizes: eleven of the twenty five-square regions, and only 84 of the 729 eight-square ones.

And the value does not identify them either. They are spread across values that also contain regions with a unique best move, which is the same finding as the section above stated at the extreme.

The 584 with a unique best move are the other extreme and the more common one. In more than half the catalogue there is exactly one placement worth making, and knowing what the region is worth does not say which.

The two counts move in opposite directions as the regions grow, which is the same trend seen once more. At five squares eleven of twenty regions give a free choice and ten have a unique answer; at eight squares 84 of 729 are free and 405 have a single answer. A small region is usually a position with nothing to decide, and a large one is usually a position with exactly one right decision — and neither of those facts is legible from the value.

That is also the honest reason a player finds Domineering hard. It is not that the good moves are numerous and finely balanced; it is that there is generally one, it is not marked, and the quantity the theory computes is silent about which.

A gap that does not close with better notation #

One reading of all this would be that the value is simply the wrong summary and a richer one would carry the move count too. It would not, and the reason is worth stating because it is a theorem rather than a difficulty.

Any summary that is preserved by the disjunctive sum — any quantity a player can compute per component and combine — is a function of the value. That is what equality in every company means: two positions with the same value are interchangeable in every sum, so a quantity distinguishing them cannot survive the substitution.

So the move count cannot be added to the value. Not because nobody has found the right notation, but because a quantity distinguishing the two regions in the figure above would distinguish positions that no sum can tell apart, and would therefore not compose.

That puts the gap this page measures in the same family as the other things this site keeps finding on the far side of the equivalence — the length of the game, the memory a strategy needs, the position before it was reduced. They are not missing from the value. They are the price of it composing.

What this suggests about play #

A practical reading, and it is the reason which part to move in sits on a different anchor from this one.

A player facing a board made of several regions has two questions: which region and which square in it. The first is answered by the values — that is what the whole apparatus of sums and incentives is for, and this site has an anchor about it. The second is not answered by anything the values carry, and this census is the measurement of how much is left over: on more than half the regions there is a single right answer and the value does not know it.

That division of labour is worth naming, because the two questions get very different amounts of attention in the literature. The first has a theory. The second has a search.

The catalogue this is measured on #

The regions are the ones the domineering anchor has been using for four rungs, and it is worth saying what they are, because the census’s shape depends on it.

Every connected shape of at most eight squares, taken up to the reflections, with its value computed exactly by the recursion — 1,042 of them, of which 1,034 give Left somewhere to place. The catalogue exists because a board that falls apart is easier, and a real Domineering position is a sum of these rather than one of them.

The distribution of sizes is why the last figure’s trend is trustworthy and its endpoint is not. There is one region of two squares and 729 of eight, so the eight-square row is nearly three quarters of the census and the small rows are anecdotes. What the trend shows is a quantity that does not move across a population growing by a factor of ninety, which is a stronger statement than six equally-weighted points would be.

Counting good moves is not counting moves #

The quantity here is easy to confuse with a simpler one, and the difference is the whole reason the count stays near one and a half.

How many moves a player has is a property of the drawing: count the placements. It grows with the region, roughly linearly in the free squares, and a large region has many.

How many moves achieve the value is a property of the game tree. A placement achieves the value when the position it leaves is worth what the whole position is worth to that player — which is a comparison, decided by play, and has nothing to do with how many placements exist.

So the second does not grow with the first, and there is no reason it should. A region twice as large offers twice as many placements and the same handful of good ones, because being good is a comparison against the best available rather than a share of what is available. The count staying near one and a half however large the region gets is that statement measured.

That has a consequence for how a strategy scales, and it is the encouraging half of this anchor. The table of choices grows with the number of positions and not with the number of moves per position, so a bigger board multiplies the rows and leaves the width alone. A player facing a large region does not have a larger decision in front of them; they have the same small decision, in a position that took longer to reach.

It also says why which move is a harder question than how many. A count near one and a half means the choice is usually forced, and a forced choice is easy to describe and hard to locate — which is exactly the shape the rungs above run into.

What the census does not say #

Four limits.

The best-move definition is greedy. Maximising the right stop of the remainder is a one-ply reading, and a move that is best by that test need not be best in a longer sense. It is the natural reading and it is the one a player at a board applies, and a definition looking two moves ahead would give different counts.

One game, one player, one player’s orientation. Everything here is Left’s vertical placements on a Domineering region. Right’s horizontal ones are the reflection and give the same numbers by symmetry, and a second game would say whether the flat count of best moves is about Domineering or about regions.

A best move is a best option, not a winning move. The definition maximises the right stop of what is left, which is the natural greedy reading and is not the same as a move in an optimal strategy for a whole board. A region in a sum is played according to the sum, and which part to move in is where that difference is measured.

Eight squares. The trend in the last figure is six points and they are consistent, and six points are not a proof that the count of best moves stays flat for ever. A region of twelve squares is where the claim would be worth checking and where the evaluation gets expensive.

And ties are exact ties. Two placements count as equally best when the right stops agree to within a billionth, and the stops here are dyadic rationals computed exactly, so the tolerance is doing no work. A definition of nearly best would give different and probably larger counts, and would need a scale to be nearly best on.

The convention, named #

Normal play, Left placing vertical dominoes and Right horizontal, and a region is a set of orthogonally connected empty squares taken up to the reflections.

The value is the canonical form, computed by the recursion over the whole region and compared by name. Two regions are said to have the same value when the recursion returns the same canonical form, never when they look alike.

A best move for Left is a placement maximising the right stop of the region left behind. The right stop is what Left is left holding when Right moves first in the remainder and both play the fight out with no tax charged, and it is read off the foot of the right wall of the remainder’s thermograph.

Where the ladder goes next #

The tempo anchor has two rungs to here: the quantities a value discards, and now the first of them counted.

What a strategy has to remember sizes the whole gap rather than one corner of it. A player who wants to win rather than to predict has to store 3,308 choices across 4,269 positions carrying 128 values between them — twenty-six entries for every number the theory supplies, which is the omission stated as a table.

Three rules and a tie-break then compresses that table almost completely. Three geometric rules applied in order — leave the opponent fewest replies, then keep the region whole, then take whichever placement comes first — answer 94.5 per cent of it, and the fourth and fifth rules anybody would add answer not one line more.

Seventy-two of them were not silence takes the residue apart and finds it is not one thing. Seventy-two of the 181 unanswered decisions are the rules speaking and being wrong, which is a worse failure than falling silent: a rule that declines can be backed by a search and a rule that answers confidently cannot. On the 109 that really are silence, a rule chosen per value answers more than half, and the star class is settled outright by leaving the younger position.

So the anchor’s arc is a large omission, an exhaustive table of it, a three-sentence compression of the table, and a residue that turns out to be several small situations rather than one hard one.