What a game actually produces

How hot a day gets counted temperatures over values, each once. How hot a real position is counted them over positions, so that a value a thousand positions are worth counts a thousand times, and found the answer moves a long way. It closed by naming a third weighting neither could produce:

The rung above is the weighting a board would give. Every count here is of a whole ruleset position; a board in play is a sum, and the population of components a real game produces is neither the population of small positions nor the population of values. Producing it means playing games rather than enumerating them, and it is the one measurement on this page that cannot be made by exhaustive search.

Playing them produces a population much colder than either.

The measurement #

Eleven hundred Domineering boards — four hundred at 4×5, four hundred at 5×5, three hundred at 6×6 — played out at random from empty to finished. At every position along the way the board is decomposed into its connected pieces, and every piece is evaluated and its temperature recorded.

That is 14,298 components evaluated, together with 6,536 too large for the catalogue to reach and 9,836 single squares.

A single game contributes many components, since every position along the way is decomposed, so eleven hundred games give fourteen thousand evaluations rather than eleven hundred. That is the point of taking the tally at every position: a player meets every position of a game and not merely its last one.

Random play is the right sampler here and it is worth saying why. The alternative is optimal play, which would produce a population both players had steered, and a population steered by the theory being tested is not evidence about the theory. Random play produces the positions the game’s rules make available rather than the ones good play selects, and the question is about the rules.

The answer #

Sixteen per cent of the components are hot. Of the catalogue those components are drawn from — every region of at most eight squares, each counted once — fifty-three per cent are.

The figure is 15.7 per cent on the 4×5 boards, 16.2 on the 5×5 and 15.6 on the 6×6. It does not move with the board, which is what makes it a fact about play rather than about a size, and the census asserts it — a spread of more than five points across the three would mean the answer is about the board and this page is about the wrong thing.

Counting the single squares, which are worth nought and which a player certainly meets, the share falls to about ten per cent.

Whether to count them is a real accounting question rather than a quibble. A single empty square is a component of the board by every definition the site uses, it is worth nought, and no player would call it a position. Both numbers are reported for that reason: sixteen per cent among the components anybody would bother to look at, ten among the components that exist.

Why a game is colder than its catalogue #

The mechanism is not subtle once the two populations are described side by side.

A catalogue is a list of shapes. Every eight-square region appears once, however awkward, and the awkward ones are the hot ones: a region where the two players’ moves interfere is a region worth fighting over, and interference needs a shape with corners and narrow places in it.

A played board is a list of what dominoes leave behind. Dominoes are convex, they are laid where there is room, and what they leave is disproportionately strips, single squares and simple rectangles. Those are the cold shapes — a strip is worth its own packing count, and a count is a number, and a number has no temperature.

So the game selects, out of the catalogue, exactly the shapes that are least interesting to the theory. The awkward regions exist and the game does not produce many of them.

The shape of the difference #

The two populations are not the same distribution scaled down; they are differently shaped, and the difference is where the practical content is.

The game produces 73 per cent numbers against the catalogue’s 30. Among the hot components it produces, nearly all sit at a temperature of one (10.8 per cent of everything) or one and a quarter (2.7), and the quarter-point and half-point temperatures the catalogue is full of barely occur at all — a quarter is 9 per cent of the catalogue and 0.4 per cent of play.

That is a sharper statement than colder. A player at a real board meets a great many positions with nothing at stake, a modest number worth exactly one move, and almost nothing in between. The fine gradations of temperature that a value census is full of are a feature of the space of shapes rather than of the game.

The shapes a domino leaves #

The mechanism is worth drawing rather than asserting, because the selection is visible.

Lay a domino anywhere on an empty board and what is left is a rectangle with a notch. Lay a few and what is left is strips and small rectangles with corners bitten out. Those are the shapes at the cold end of the catalogue.

The shapes at the hot end are the ones with a narrow neck, or a corner that only one player can use, or two arms of different orientations — shapes that a random sequence of dominoes assembles rarely, because assembling them requires the dominoes to fall in a particular arrangement.

So the selection is geometric rather than strategic. No player is choosing the cold positions; the shape of a domino is choosing them, and it would choose them the same way in a game played by two beginners or by nobody at all.

What it means for the rest of this site #

Every temperature census here that draws on a catalogue is now known to be measuring a population the game does not produce, and it is worth being precise about which claims that touches and which it does not.

It does not touch any theorem. The temperature spectrum of a day is a fact about values and is unaffected by which values a game reaches.

It does touch every claim of the form most positions are… A sentence like more than half the regions are worth fighting over is true of the catalogue and false of the game, and the two are separated by a factor of three.

And it changes what a heuristic should be tuned on. A rule that is never right and cannot be far wrong measures its readings over catalogues, and a reading scored at 85 per cent over a catalogue would score differently over play — probably better, since the positions play produces are the simple ones. Nothing on this site has been rescored that way, and it is the obvious next measurement for every reading the domineering ladder has produced.

What a player should expect #

The distribution has a practical reading, and it is more useful than the headline.

Most components are worth a number. Seventy-three per cent, so nearly three positions in four on a played board have nothing at stake and are decided by counting rather than by fighting.

The hot ones are nearly all worth one. Ten point eight per cent of everything sits at a temperature of exactly one, against 2.7 at one and a quarter and under one per cent at everything else combined.

So a player’s decision on a real board is usually between a component worth a number and a component worth a fight of size one. The fine question the theory is built to answer — which of these two fights is hotter — arises much less often than the coarse one, is anything here a fight at all.

That is not an argument against the theory. It is an argument about where its value lies: the temperature apparatus earns its keep at the moments when there are two hot components, and this census says those moments are rarer and more uniform than a catalogue suggests.

Three weightings, three answers #

The anchor now has three answers to how hot is this game, and they differ by more than the measurement error of any of them.

By value, over the values born by day three: 76 per cent hot.

By position, over the enumerated positions of seventeen rulesets: under 5 per cent.

By what a game produces, over the components of played Domineering boards: 16 per cent.

Those are not three estimates of one number. They are three different questions with three correct answers, and the reason to have all three is that each is the right one for a different purpose — the first for a claim about the theory, the second for a claim about a ruleset, and the third for a claim about a player.

The one that was missing is the third, and it is the only one a player would recognise.

It is also the only one that could not have been guessed from the other two. By value is 76 per cent and by position is under 5; a reader asked to place the played population between them would have had no reason to choose 16 rather than 40, and the two bounds are four hundred pixels apart on any chart. The measurement was the only way to know, which is the sense in which the rung below was right to call it the one that cannot be made by exhaustive search.

What the census does not say #

Four limits.

The sampler is uniform over moves, not over positions. A position reached by many move orders is sampled more often than one reached by few, which is a weighting nobody chose and which is arguably the right one — a position a game is more likely to pass through should count more. It is stated because it is a choice and not a neutrality.

One game. Domineering, because it decomposes cleanly and its catalogue is already built. Whether a game whose pieces are less convex — Clobber, say, where a component is a cluster of stones — produces a warmer population is the same census on a different ruleset and is not run.

Random play, not good play. The sampler is deliberate and it is a choice. A population produced by two strong players would be different, and probably colder still, since both sides prefer to leave the opponent nothing at stake. What it would not be is evidence about the game’s rules.

Components of at most eight squares. Six thousand five hundred and thirty-six larger components arise and are counted rather than evaluated, and they are the early-game positions. If they are hotter than the small ones — which is likely, since a large region has more room to interfere in — the played share is understated here.

And the temperatures are of pieces, not of boards. A board is the sum of its pieces and a sum’s temperature is the largest of its parts’, so a board with one hot piece is a hot board. The share of boards that are hot is therefore much higher than the share of components, and this page has not measured it.

The same figure on three board sizes #

The share coming out the same on three sizes is doing more work than a single measurement would, and it is worth saying what it rules out.

A figure measured on one board size is compatible with two explanations. It could be a property of play — the game visits a certain kind of component whatever board it is on — or a property of that board, which happens to break into pieces of a certain size.

The two make different predictions and the predictions are easy to test: a property of the board moves when the board changes, and a property of play does not. Three sizes and one figure is the second prediction confirmed, which is why the sentence it is a property of play rather than of the board is a finding rather than a paraphrase.

That matters for how far the number travels. A property of play is a claim about what a game does, so the same measurement should hold on sizes nobody has swept and on boards nobody has drawn. A property of one board is a fact about that board and would have to be re-measured for each.

And it is the reason the correction to the catalogue censuses is a factor rather than a caveat. If the discrepancy were a fact about a board size, every catalogue census would need its own correction; because it is a fact about play, one factor of three applies to all of them, and the corrected figures are usable without re-running anything.

The number that is missing #

There is one figure this page would like and does not have, and it is worth naming rather than leaving as a gap in the reader’s arithmetic.

Six thousand five hundred and thirty-six of the components that arise are larger than eight squares, which is 31 per cent of everything the play-outs produce. They are counted and not evaluated, because evaluating a twelve-square region is a search too dear to run in bulk.

They are the early game. A board that has just been opened is one large piece, and it stays one large piece for several moves.

That matters for the reading in two ways and they pull opposite ways. A large region has more room for the two players’ moves to interfere in, so it is plausibly hotter than the small ones — which would raise the played share above 16 per cent. And a large region is also more likely to be worth a lot in a way that has nothing to do with temperature, since size and heat are independent, so plausibility is all this is.

The honest statement is therefore about a sub-population: among the components small enough to evaluate, a game is a third as hot as the catalogue. Whether that survives to the components too big to evaluate is a question that needs a bigger evaluator rather than a better sampler.

The convention, named #

Normal play, Left placing vertical dominoes and Right horizontal. A component is a connected set of empty squares, four-connected, taken from a position reached in play.

The temperature is read off the component’s own thermograph, computed by the recursion, and a number is given $-1$ by convention so that a stack of temperatures contains only genuinely hot parts. Hot means a temperature strictly above nought.

A game here is one random play-out from an empty board to a position with no legal move, with each move chosen uniformly from those available and the players alternating. Every position along the way contributes its components, so a single game contributes many.

The catalogue is every region of at most eight squares taken up to the reflections — 1,042 of them — each counted once, which is the weighting the earlier rungs used.

Where the ladder goes next #

The cold anchor has four rungs: where the bottom of the scale is, how far the top reaches, what the scale looks like over positions rather than values, and now what it looks like over the positions a game actually produces.

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. Measuring it is the same play-out with the tally taken at the board rather than at the piece, and it would say what a player’s turn actually looks like.

Two neighbours are worth the trip. Which shapes are worth fighting over is where the catalogue’s hot shapes are characterised, and reading it beside this page shows exactly which shapes the game declines to produce. And big is not the same as hot is where the independence of size and temperature is established, which is why the six thousand oversized components cannot be assumed cold and cannot be assumed hot either.