Which option the reduction keeps

Domination is the half of the reduction to canonical form that only ever deletes. Left’s option A goes when some other Left option B satisfies B ≥ A, because a player offered both would never take the worse one, and what is left when the deleting stops is the set of options nothing else beats — the maximal elements of a partial order.

How much a list of options can lose counted that over every option list of two, three and four day-two values, and closed by naming the question it had not asked:

Which options are deleted, rather than how many. The reduction keeps the maximal elements, and the question of what a maximal element of a real game’s option list looks like is a question about positions rather than about orders. In Domineering, is the maximal option the one that takes the most space, or the one that leaves the opponent least?

Two descriptions, one question, and a game to ask it on. The answer is nine to one and it is the second.

What is being counted #

The pool is every Domineering board of six shapes — 2 × 2, 2 × 3, 2 × 4, 3 × 3, 2 × 5 and 3 × 4 — with every possible arrangement of covered squares on it, and from each of those the list of moves Left has. Lists with fewer than two options are dropped, because a list of one has nothing to delete.

That leaves 1,586 option lists carrying 4,434 options between them, of which 2,463 survive domination. So the reduction throws away 44 per cent of everything Left is offered, which is a much larger share than the two-, three- and four-element lists of the rung below suggested and which is worth reading before anything else on this page: most moves on a Domineering board are dominated by another move on the same board.

More than half the lists — 842 of the 1,586 — come down to a single surviving option. On those positions the reduction has not merely narrowed Left’s choice; it has made it, and the value of the position is the value of one particular move.

The two readings, stated precisely #

Each reading is a rule for picking options out of a list without evaluating anything, and each has to be scored fairly.

Taking the most room picks the option after which Left has the most further moves. It is the reading a player has in their head when they talk about keeping the position open: play the domino that leaves the most places to put the next one.

Leaving the opponent least picks the option after which Right has the fewest replies. It is the reading behind taking away his moves, and in Domineering the two players’ moves are perpendicular, so a vertical domino that breaks a horizontal run is doing this and nothing else.

A reading is counted right on a list when every option it names survives the reduction. That is deliberately the weaker of the two possible tests. A reading naming one of three survivors has said something true; a reading naming a deleted option has said something false. Scoring the other way — demanding that a reading name all the survivors — would penalise a correct description for being less than exhaustive, and neither reading claims to be exhaustive.

Under that test the scores are 754 of 1,586 for room and 1,425 of 1,586 for replies.

Why forty-seven per cent is worse than it sounds #

The room reading scores just under half, and half is roughly what a rule with no content scores here.

The average list has 2.8 options and 1.55 survivors, so a rule that picked an option at random would name a survivor rather often. The right comparison is not against nought; it is against chance, and 47.5 per cent is close enough to chance that the reading cannot be said to be describing the reduction at all.

That is the finding worth carrying, and it is a mild surprise. Keeping one’s options open is sound general advice in games and it is the intuition most players would bring to a Domineering board. It does not describe what the reduction does.

Why the other reading wins #

The mechanism is in the definition of the order, and once seen it makes the ninety per cent look inevitable rather than lucky.

B ≥ A means B − A ≥ 0, which means Left wins B − A moving second. What Left needs in that difference game is for her position in B to answer whatever Right does in −A. The quantity that governs whether she can is not how many moves she has; it is how many moves Right has, because Right moving in the copy of −A is what she has to answer.

So the order on Left’s options is sensitive to Right’s mobility in a way it is not sensitive to Left’s own. An option that leaves Right nothing is above almost everything, whatever it leaves Left; an option that leaves Left a great deal but leaves Right a great deal too is comparable with very little.

That is the same asymmetry the incentives measure from the other side, and the same one comparison itself is built out of, and it is worth putting the two together. The incentive of a move is what the move is worth to the player making it; the order on the moves is decided by what they are worth to the other player. A position’s value is built from both and a player’s intuition usually tracks only the first.

What the reading gets wrong #

One hundred and sixty-one lists have the replies reading naming a deleted option, and they are not scattered.

They are lists where two moves leave Right the same number of replies and the two are not comparable — the reading names both, one survives and the other does not. That is the reading failing for the reason a count fails generally: a number cannot separate two positions that a game distinguishes. Right having three replies on one board and three on another says nothing about whether those replies are the same replies or lead to the same places.

So the reading is right about the direction and blind to structure, which is a fair description of every quantity a player carries in their head while playing. Its ninety per cent is high enough to be a rule of thumb and its failures are exactly where a rule of thumb should be expected to fail: between two moves that look the same by the count.

When the reduction makes the move #

Eight hundred and forty-two of the 1,586 lists have exactly one survivor, and it is worth stopping on what that means before the census moves on.

A position whose Left option list reduces to one option is a position where Left’s choice has been settled by the order rather than by a search. Every other move she has is at least matched by that one, so playing it can never be worse — and the reduction found that out by comparing options with each other, which is much less work than evaluating the position.

That is a practical fact and not a theoretical one. A solver that reduces before it recurses does 4,434 comparisons here and then recurses on 2,463 options rather than 4,434, and on 842 boards it recurses on one. The saving a decomposition buys is the other large economy in this subject and it is a different kind: decomposition splits the position, and domination narrows the choice inside it.

The two do not compose, and the census says by how much they fail to. A board that has fallen into separate regions has its Left options split between them, and two options in different regions are almost never comparable — each leaves the other’s region untouched — so decomposition reduces what domination can do.

On boards still in one piece the reduction deletes 52 per cent of Left’s options. On boards already broken up it deletes 37. Both figures are over roughly seven hundred and nine hundred lists respectively, so the gap is not a small-sample effect, and it is the sort of interaction a solver author would want to know about before assuming the two savings multiply.

A day and a board are different populations #

Worth saying plainly, because the two rungs of this anchor measure the same operation on different things.

The rung below’s lists are subsets of day two, chosen because they are every option list of two, three and four values. That is the right population for a question about the shape of an order, and it has one property a board never has: any set of values can be an option list, so the orders that arise are every order that fits.

A board’s option lists are not arbitrary. They are the moves of one position, they share almost all of their structure — two options on a 3 × 4 board differ by two squares out of twelve — and the resulting orders are correspondingly special. One thousand and eighty-four of the 1,586 boards’ lists — 68 per cent — contain a comparable pair, where an arbitrary list of values usually does not, and that is why the deletion rate here is so much higher than the rung below’s. A list with no comparable pair in it is an antichain and loses nothing at all; a third of the boards’ lists are in that position and two thirds are not.

So the two rungs are not the same measurement at two scales. One says what domination does to orders and the other says which orders a game produces, and the second is the one a player is standing in front of.

What a player should take from it #

The census has a reading for somebody sitting in front of a board, and it is narrower than the ninety per cent suggests.

It is not a rule for finding the best move. A surviving option is one nothing else beats; there may be three of them and the position may be a loss whichever is taken. Domination narrows the choice and does not make it, and the value of the position is still what decides who wins. Knowing who wins and knowing what it is worth are already two different questions, and which move is a third.

What it is is a rule for discarding. A move that leaves the opponent more replies than another available move is very likely dominated, and a player who never plays such a move loses nothing on nine boards in ten. That is the same shape of advice a rule with a guarantee gives about temperature, and it comes with the same caveat: a bound on how much can be lost is not a promise of playing well. That is a rule about what not to consider, which is the useful shape for a person and the useful shape for a solver.

And it says the intuition to distrust is the other one. Keeping one’s own options open is the reading that scores at chance. A player following it is not making a mistake in any given position — she is following a description of the reduction that turns out not to describe it, and the positions where the two readings disagree are exactly the ones where the choice matters.

The sharpest form of that is the 3 × 4 board, where the most-room reading is right on 592 lists of 1,239 and the replies reading on 1,083. Nearly five hundred positions separate them, and every one is a board where a player playing to keep her options open plays a move the reduction had already discarded.

Why the winning description is the counter-intuitive one #

Takes the most space scores 47 per cent and leaves the opponent fewest replies scores 90, and the gap is worth explaining, because the loser is the description a player would offer and the winner is not.

Space is a quantity about the position. How much of the region a placement occupies is a fact about the board and about nothing else, and it is what a reader sees when looking at a drawing.

Replies are a quantity about the opponent. How many placements remain for the other player after this one is a fact about the pair — the shape left behind and what the opponent can do with it — and it is not visible in the same glance.

Under normal play the second is the one the theory is about. A move is a resource spent, and the game is decided by who runs out; so the value of a placement is what it does to the opponent’s supply rather than to the mover’s own footprint. A description in terms of space is measuring the wrong side of the exchange.

That also explains why the loser scores as well as 47 per cent rather than at chance. Space and replies are correlated — a placement occupying an awkward part of a region usually removes options as well — so the wrong quantity inherits some of the right one’s accuracy, which is exactly the confound that appears whenever two related quantities are scored separately.

The useful instruction is to describe a move by what it denies rather than by what it takes, which is the same reading the incentive gives of why moving is always bad.

What the census does not say #

Four limits, and the third is the one that bounds the claim.

Six rectangles is not Domineering. The largest board here is twelve squares, and a real game is played on eight by eight. Nothing says the ninety per cent survives a board with room for genuinely long-range interaction; what it does say is that the reading is right on every one of the six sizes measured and that the gap between the readings widens with the board rather than narrowing.

Positions rather than play. Every arrangement of covered squares with an even count is included, and some of those are not reachable in a game from an empty rectangle. They are legitimate positions of the same game starting from a different board, and excluding them would mean enumerating play rather than positions — a different and much larger computation.

Only Left’s options. Domineering is symmetric under exchanging the players and rotating the board, so Right’s lists are Left’s lists on the transposed boards and counting both would double every number without adding a fact. On a game without that symmetry the two sides would have to be counted separately and might not agree.

And a reading that names a survivor has not been shown to be the mechanism. Ninety per cent is a score, not an explanation. The argument in the section above is an argument about the difference game and it is a sketch; turning it into a statement means showing that Right’s mobility bounds the comparison, which is a claim about all positions and not about 1,586.

The convention, named #

Normal play. Left places vertical dominoes and Right horizontal ones, which is this collection’s standing convention and the one that makes a tall region good for Left.

An option is dominated when another option of the same player is greater than or equal to it in the partial order on values, with equality excluded so that two equal options both count as surviving; the comparison is ge on canonical forms, computed by the recursion. Survival is a property of one round of deletion rather than of the whole reduction — reversibility is not applied here, because bypassing a reversible option changes the list rather than shortening it and the question is about what deleting keeps.

Both readings count moves available, not squares covered. A move is a legal placement, so a 2 × 3 board with nothing on it offers Left two moves and Right four.

Where the ladder goes next #

The dominance anchor has three rungs: what deleting removes, what decides how much, and now what the survivor looks like on a board.

The rung above is the mechanism rather than the score. The argument that Right’s mobility governs the order is a paragraph about the difference game; making it a statement means bounding the comparison by a count of the opponent’s moves, and the 161 failures are the data that would say what the bound has to be weakened to. That is a claim about positions in general, and this page has 1,586 of one game.

Two neighbours are worth the trip. The simplest game above both reads the same order for a different purpose — not what the reduction removes but what structure the order carries — and finds far more of it than a partial order is entitled to. And which shapes are worth fighting over is the same game’s regions asked what kind of value they carry, where the answer also turns on which orientations fit and where the two players’ moves stop interfering.