One fight makes a board a fight

What a game actually produces played eleven hundred Domineering games out at random and measured the temperature of every component they produced. Sixteen per cent were hot, against 53 per cent of the catalogue of shapes those components are drawn from — so a real game produces the cold shapes and mostly leaves the hot ones alone.

That page closed on the quantity a player actually faces:

The rung above is the board rather than the component. A board is a sum and its temperature is the largest of its parts’, so a board with one hot piece is hot however many cold ones surround it — and the share of boards that are hot is a different and probably much larger number than the 16 per cent here.

It is larger. It is 32 per cent, which is twice the piece figure rather than the several-fold jump the sentence expects, and the reason for the modest size is a count nobody had taken.

What a board’s temperature is #

A board in play is a disjunctive sum of its connected regions, and the temperature of a sum is the largest of its parts’ — that much does compose, even though the temperatures themselves do not add. So a board is hot exactly when at least one of its pieces is, and a single hot region makes the whole board worth fighting over however many dead regions surround it.

That is the arithmetic the prediction was built on, and it is right. What it leaves out is how many pieces there are.

There is a second way to see the same thing, and it is the one that makes the modest size feel inevitable. A Domineering board stays connected for a long time: a domino placed in the middle of an open board does not cut it, and it takes several placements before a region is pinched off. So for most of the game the board is one piece, and asking whether the board is hot is asking whether that piece is.

A board carries 1.68 regions on average. If hotness were independent across pieces, a board would be hot roughly 1.68 times as often as a piece — and it is hot 1.96 times as often. The extra is clustering: a hot board carries 2.55 pieces against the 1.68 of an average one, so the boards with fights on them are the boards that have broken into several parts.

So the answer to how much larger is: as much larger as there are pieces, plus a little. The prediction’s mechanism was sound and its magnitude was not, and the missing ingredient is that Domineering boards do not break into many pieces until very late.

The measurement, and what it cannot see #

Eight thousand eight hundred and forty-eight board positions arise in these play-outs and 4,525 of them are settled — every piece small enough for the evaluator. The others contain a region of more than eight squares and their temperature is unknown, so they are excluded rather than guessed at.

That exclusion is not neutral and the honest reading has to state which way it leans. The unsettled positions are the early ones, where the board is still mostly one region, and a large region is more likely to be hot than a small one. So 32 per cent is a share among boards that have already broken up, and the true share over all positions is probably higher.

The floor is easy to state: 16 per cent of all 8,848 positions are known to be hot, since a board is only counted hot when it is shown to be. Between that floor and the settled figure of 32 per cent is where the answer lies, and closing the gap needs an evaluator that reaches regions of a dozen squares.

One more thing the exclusion does is worth naming, because it is a general hazard rather than a detail of this sweep. The positions an evaluator cannot reach are not a random sample of positions. They are the early ones, the large ones and — in a game whose pieces shrink as it goes — the ones most likely to differ from the rest. Every census on this site that stops at a size has the same shape of blind spot, and the only honest treatments are to report a floor, to say which way the missing cases lean, or to reach further.

Through the game #

The curve has a clear shape once enough positions are settled to read it. From the eighth move onward — where nearly every position is settled — the hot share runs 22, 21, 25, 28, 23, 15, 5, 1 and 0 per cent.

The fight arrives in the middle game and leaves before the end. That matches how the game feels and it also explains itself: early, the board is one large region and there is no separation to fight over; late, the pieces are two and three squares and a piece that small has nothing at stake. In between, the board has broken into pieces of four to seven squares, which is exactly the size range the catalogue’s hot shapes occupy.

The last three moves are the sharpest part of it. By move fourteen only 5 per cent of boards are hot, and by sixteen none is. A Domineering endgame is a counting exercise: who has more moves left, played out with no decisions worth making. That is the reverse of the usual picture, in which the endgame is where the subtlety lives — here the subtlety is over before the endgame starts.

How hot, when hot #

The temperatures a real board takes are not spread out at all. Of the hot boards, 64 per cent are at exactly $1$, 21 per cent at $1\tfrac14$, and the rest at a quarter, a half or three quarters. Nothing reaches $1\tfrac12$ more than a handful of times and nothing reaches $2$.

A temperature of one means the move is worth one domino: taking the fight now rather than later gains a move’s worth of board. That is the commonest thing at stake in Domineering, and it puts a limit on how much a player’s judgement can be worth — the whole difference between playing the hottest component and playing at random, on a single decision, is bounded by the size of the fights available, and those fights are all about one size.

That is also why the greedy rule does so well here. When the fights available are all worth about the same, the cost of picking the wrong one is small, and a rule that gets the ranking roughly right loses very little to one that gets it exactly right.

What this changes about the rung below #

The rung below reported the piece figure as evidence that a played game avoids the hot shapes. That reading survives — the catalogue is 53 per cent hot and the pieces produced are 16 — but the conclusion a reader would draw from it does not.

A player does not meet pieces, they meet boards, and a third of the boards a game passes through have something at stake. So the apparatus of temperature is engaged for a third of the game rather than for a sixth of it, and engaged precisely in the middle third where the decisions are.

The corrected sentence is worth stating carefully, because it is the sort that gets repeated: Domineering produces cold pieces and warm boards. The pieces are cold because a played board breaks into small regions and small regions have nothing at stake; the board is warmer because it takes only one exception among its pieces to make the whole thing a fight.

What a player should take from it #

Three things, and the first is the one that changes a habit.

A cold piece is not a cold board. A player looking at a region and finding nothing at stake in it has established a fact about that region and nothing about the position. The board’s temperature is a maximum, so it is decided by the hottest piece, and the piece worth checking is the one a player is least inclined to look at again.

The middle game is where judgement is worth having. A third of the settled boards are hot and nearly all of them are in the middle third of the game. Before that the board is one region and there is nothing to choose between; after it the pieces are too small to hold a fight. So the reading that pays is concentrated, which is worth knowing for a player deciding when to think.

And the fights are all one size. With 85 per cent of the hot boards at a temperature of one or one and a quarter, the cost of getting a comparison wrong is bounded by about a quarter of a move. That is the quantitative version of a familiar feeling — that Domineering rewards counting far more than it rewards judgement — and it is why the count of dominoes each side can place does as well as it does.

Why 1.68 pieces is the number that decides it #

The prediction this page refutes rests on an implicit model, and writing the model down says exactly which of its assumptions was wrong.

The prediction was: if a piece is hot one time in six, and a board has several pieces, then a board is hot far more often than one time in six — because a board is hot when any of its pieces is, and any is a much weaker demand than each. Formally, if a board has $k$ pieces each hot independently with probability $p$, the board is hot with probability $1 - (1-p)^k$, which for $p = 0.16$ rises to 58 per cent at five pieces and 80 at ten.

Two things in that model are wrong and they pull the same way.

$k$ is 1.68, not five. A Domineering board that has broken up has usually broken into two pieces, and often one of them is a single square with nothing in it. So the any is being taken over fewer than two things most of the time, and $1 - (1-p)^{1.68}$ is 26 per cent rather than 58.

And the pieces are not independent. The hot ones cluster: a board that has a hot region tends to have another, because both come from the same middling stage of the game where regions are large enough to fight in and small enough to have broken off. Positive correlation between the events makes any closer to each than independence would, which pushes the figure down again rather than up.

So the gap between 26 and 32 is what the correlation costs, and the gap between 32 and 80 is what the piece count costs — and the second is much the larger. The prediction failed on an assumption nobody stated, which is that a decomposed board has many pieces. It has two.

What this does not say #

Four limits.

Random play, not good play. Every game here is played at random by both sides, which is the only way to sample positions without the sampler’s own strategy deciding what it sees. A game between players who know the theory would produce a different distribution, and probably a colder one late — a good player takes the hot piece early, which removes it.

Three board sizes. 4 × 5, 5 × 5 and 6 × 6, and the hot share rises with the board — 30, 29 and 37 per cent — so a 9 × 9 board would be hotter still and by an unknown amount. The rise is what the piece-count mechanism predicts, since a larger board breaks into more pieces.

Eight squares is the evaluator’s reach here, and everything above that is unsettled. The by-move curve is unreadable before the eighth move for exactly this reason, and the early game is where a genuinely large fight would live if the game had one.

And a hot board is not a board where anything is easy. The temperature says something is worth fighting over; it does not say which piece, and which part to move in is a separate computation over the values. What a board’s temperature buys a player is the knowledge that the choice matters at all.

Who measured this, and why it is unusual #

The temperature of a position class is a standard object and the temperature of a population is not. Conway’s temperature is defined for one game; Berlekamp’s endgame practice applies it to the components of a real board; nobody in either tradition asks what share of the positions a game passes through are hot, because the question needs a solver and a sampler rather than a theorem.

That is what makes the pair of rungs here worth having. The catalogue’s 53 per cent is a fact about the shapes anybody could write down; the pieces’ 16 per cent is a fact about what the rules actually produce; and the board’s 32 per cent is a fact about what a player sees. Three different numbers for one question, and the difference between them is entirely in what is being counted.

The general lesson is a warning about enumerated catalogues, and this site now has it twice. A catalogue is a population somebody chose, and its statistics describe the choosing. What a game actually produces made the point for shapes; this page makes it for the sums those shapes sit in, and the correction goes in the opposite direction — the play-out cools the pieces and the summing warms the board.

The convention, named #

Normal play, Domineering, Left placing vertically.

A board position is the whole board after some dominoes have been placed, and a piece is one connected region of free squares. A region of a single square is not a piece here, since neither player can use it and it can never be hot.

A board is hot when the largest temperature among its pieces is above nought, and settled when every piece is at most eight squares, which is what the evaluator reaches. A board with a larger piece is neither hot nor cold in these tables; it is unknown, and it is counted in the denominator only of the floor.

A number is given temperature $-1$ by convention on this site, so the cold boards are those whose every piece is a number or an infinitesimal.

One sentence is worth keeping. A board is hot when any piece of it is, so a player who has checked every piece but one has checked nothing — and the piece most often skipped is the small one, which is exactly where this game keeps its fights.

Where the ladder goes next #

The cold anchor reaches five rungs to here, and the two above overturn this page’s headline in one direction and close the anchor in the other.

The obstacle was the catalogue removes the limitation this page treats as unavoidable — that the early game cannot be measured because its regions are too large — and finds it was never a limitation of evaluation. A twelve-square region evaluates in five milliseconds and an eighteen-square one in under a second. What was expensive was building a catalogue of every shape up to a size, which grows as the shapes do; sweeping the positions a board actually reaches is cheap, because a board reaches very few of them.

With the early game measurable the picture inverts. A board is hot four times in five three moves in, and it cools as it breaks up — so the 32 per cent this page reports is an average over a game that spends its opening hot and its ending cold, and the average describes neither half.

That is worth carrying back to the piece figure as well. Sixteen per cent of components hot is a statement about the components a whole game produces, and a game produces most of its components late. Any census weighted by position rather than by phase will report the endgame’s numbers, because the endgame is where the positions are.

Eight squares and no hotter then supplies the ceiling this anchor has been circling. No Domineering position anywhere in the sweep is hotter than three halves; the attaining region has eight squares, there are five of them up to symmetry, three are the hot core of an attaining board at every size — and the ceiling holds at nine and ten squares, where the obvious extrapolation from the smaller boards predicted seven quarters.