The coldest position has the biggest swing

The coin’s number is the average of two branches: Left’s chances if the flip names Left, and Left’s chances if it names Right. The average is the answer and the gap between the branches is a quantity of its own — it is how much the flip decides, which is to say how much having the move is worth.

The subject already has a number for that and it is the temperature. So there are two answers to how much is at stake, computed by completely unrelated routes, and they can be laid side by side.

The position that breaks the analogy immediately #

The empty position has a temperature of $-1$, which is the lowest the scale goes: it is a number, neither player wants to move in it, and a thermograph reads it as cold as anything can be.

Its swing is one. The largest there is.

Whichever player the coin names has no move and loses on the spot. One branch is nought, the other is one, and the flip decides everything there is to decide. Meanwhile the hottest value born on day two — a whole unit at stake, temperature 1 — has a swing of a half.

So the two quantities are not the same quantity measured twice, and the disagreement is not at the margins: the coldest position on the pool has twice the swing of the hottest.

What each is actually measuring #

The disagreement has a clean explanation and the explanation is worth more than the ordering.

The temperature measures what a player gains by moving. A thermograph is built from the two players’ options, cooled until the difference between moving and not moving vanishes, and the temperature is where that happens. It is a statement about the reward for taking a turn.

The swing measures what a player loses by being made to move. The coin’s two branches are Left moves and Right moves, and their gap covers both possibilities at once — a position can have a large gap because moving is wonderful or because being made to move is fatal.

The empty position is the second kind at full strength. There is no reward for moving; there is a total penalty. A thermograph has nothing to report, because cooling a position with no options changes nothing, and the swing is one because the penalty is everything.

That distinction has a name already. A position where being obliged to move is the whole problem is zugzwang, and the swing is the first quantity in this collection that measures it directly.

A swing is not a temperature and was never going to be #

Before the counts, it is worth saying why the two could not have matched even in principle, because that makes the near-match on the small pool the surprise rather than the mismatch.

A temperature is unbounded. A position with a hundred points at stake has a temperature of a hundred, and there is no ceiling. A swing is a gap between two probabilities, so it lives between nought and one and cannot exceed one however much is at stake. Whatever relationship holds between them, it cannot be a proportion: the swing must saturate and the temperature must not.

And they are scaled differently at the bottom too. A temperature of nought is a real place on the scale — it is where the all-small positions sit — and below it there is a further region reserved for numbers. A swing of nought is the bottom of its range with nothing beneath it. So the swing has one class where the temperature has two, which is why the comparison is drawn as a three-way split rather than as a curve.

What can be compared is the ordering and the sign, and those are what the counts below are about.

Which way the gap cuts #

The gap alone cannot tell the two kinds apart, and its direction can.

Left’s chances if Left moves, against Left’s chances if Right moves: when the first is larger, Left wants the move; when the second is larger, Left wants Right to have to move. Both are gaps of the same size and they mean opposite things.

On the day-two pool the direction lines up with the temperature perfectly. Every position whose temperature is $-1$ — every number — wants the other player to have to move. Every position with a positive temperature wants the move. Every position at temperature nought has a gap of nothing at all, so it wants neither.

Seven, seven and eight, and a three-way correspondence with no exceptions.

The correspondence, checked #

A correspondence that holds on twenty-two positions is a correspondence to check on fifteen hundred.

Two of the three survive exactly.

Every number wants the other player to have to move, on both pools, without exception. That is the zugzwang reading and it is what a number is: a supply of moves that a player spends and does not want to.

Every hot position wants the move, on both pools, without exception — all 1,122 of them on day three. That is what hot means, and the two machineries agree about it completely.

The third fails. On day two, all eight positions at temperature nought have a gap of nothing. On day three, 82 of them do and 255 of them want the move.

Three hundred and thirty-seven day-three values have temperature nought. Eighty-two of them have a flip that decides nothing and 255 of them want the move, and not one wants the other player to have it. So the third agreement does not merely fail; it fails entirely in one direction, which is a stronger statement than the count alone and one that needs an account.

Their swings are small and there are only two of them: an eighth or a quarter, and nothing else. So these are not positions where a great deal turns on the flip; they are positions where a little does, consistently, in the same direction, while a thermograph reports nothing at all.

What the eighty-two have that the others do not #

The surviving half of the third claim is worth stating, because it is the sharper direction and it holds everywhere.

A position whose flip decides nothing — whose two branches are equal — always has temperature nought. Eighty-two of day three’s values and eight of day two’s, and not one of them has any other temperature.

The converse fails. A position can have temperature nought and still care very much who moves, and 255 day-three values do. Temperature nought means the two players gain the same amount by moving, which is a statement about a difference; the flip deciding nothing means they arrive at the same chance, which is a statement about two absolute quantities. Equal differences do not make equal quantities, and the larger pool is where the distinction gets room to show.

The eighty-two are therefore a proper subclass of the all-small positions and not the whole of it, and nothing here had a name for them. They are the positions where the coin is genuinely blind: not merely unable to separate them from each other, but unable to see the position at all, because both of its futures look identical.

Where the swing meets the class #

One of the 255 is a position already met. The smallest of the seven values Left wins whoever moves and loses more often than not has temperature nought, a swing of an eighth, and a chance of seven sixteenths.

That is not a coincidence and it explains both findings at once. A position at temperature nought whose flip still cuts one way is a position where alternating play sees nothing to fight over and the coin sees a small persistent advantage to whoever is named. If the advantage runs against the player the outcome class favours, the class and the chance come apart — and that is exactly what the fourteen exceptional positions are.

So the two essays are measuring the same defect from two sides. Alternating play’s classes and the thermograph’s temperature both go blind in the same place, and the place is the all-small positions, which is the one part of the subject where a whole separate theory was needed to say anything at all.

The swing of a number #

Numbers are the one family where the swing has a formula, and it is short.

A lead of $n$ free moves has a swing of $2^{-n}$: the empty position swings by one, a lead of one by a half, a lead of two by a quarter, a lead of three by an eighth. The bigger the lead, the less the flip matters — which is the opposite of the temperature, flat at $-1$ for every number however large.

So on numbers the swing is doing the work the temperature explicitly declines to do. A thermograph says of every number that it is cold and says nothing about which; the swing says that the empty position is entirely at the mercy of the flip and a lead of five is almost immune to it, and that is a real distinction between two positions the temperature scale puts at the same place.

It is measuring the size of the cushion. A lead of $n$ survives $n$ unanswered turns and dies on the next, so the chance of dying is the chance of a run that long, and that is $2^{-n}$ whichever way it is derived. The temperature is measuring something else entirely and correctly reports nothing.

Why the small pool looked so convincing #

It is worth asking why twenty-two values produced a three-way correspondence with no exceptions, since the answer says what a small pool is good for.

Day two holds eight positions at temperature nought, and all eight of them are the simplest all-small values — star, up, down, and the handful a move away from them. Every one has options leading to the same place for both players, which is what makes the flip idle. The positions that break the correspondence are the ones with a small genuine asymmetry inside a zero temperature, and building one takes an extra day of depth: the smallest of them has an option that is itself a first-player win two moves deep.

So day two could not have contained a counterexample. The correspondence was not tested there; it was stated over a pool too shallow to hold the thing that refutes it.

That is a general shape and it is worth naming. A claim checked on the smallest complete pool is checked against every value of that depth and against nothing deeper, and the interesting failures in this subject are nearly all a question of depth — the thirty-four years between three complete solutions and any theory is the same lesson in a different currency.

The seventh figure, and what it is for #

A single free move is the smallest position where the direction of the gap can be read without any theory. If the coin names Right, Right has nothing to play and has lost — the best possible outcome for Left. If it names Left, Left spends the move and arrives at the empty position, which is an even flip.

So Left’s preference is unambiguous and it is for the other player to be named. That is the zugzwang direction, on the simplest number there is, and it is the pattern every number on both pools follows without exception.

It is also the clearest case of the two quantities measuring different things. A thermograph of a free move is a flat line at height one: nothing to fight over, no temperature, cold. The swing is a half, which is enormous, and it is a half because the position is one flip away from being over.

Why a number’s temperature is minus one here #

The temperature is the thermograph’s, computed by the site’s usual apparatus and reported as $-1$ for a number by the site’s usual convention. The swing is the gap between the two branches of the random-turn recursion, with a fair coin and normal play.

Two conventions of the comparison.

The swing is an absolute gap, not a rate. It runs from nought to one and is bounded by construction, while the temperature is unbounded above. So the two cannot be correlated directly and the comparison is between their orderings and their three-way split, which is why the figure prints the two branches rather than a scatter.

And a number’s temperature is $-1$ rather than nought here. That is the convention used here for distinguishing a position neither player wants to move in from one where moving is worth nothing, and the split above depends on it: with numbers at nought instead, the cold class and the idle class would be merged and the finding would disappear into a definition.

What a gap between two probabilities cannot be #

Two pools of constructed values. The correspondence that survives — numbers one way, hot positions the other — has an argument behind it in both cases, and the counts are over values the construction produces rather than positions any game reaches.

The swing is not additive. Neither is the temperature, and neither claim is made; but a reader used to the temperature of a sum being bounded by the temperatures of its parts should not expect the same here, since the coin’s number has no sum theory at all.

And nothing here measures a play. The swing says how much the flip decides and says nothing about what either player should do about it, which is the same gap between a number and a strategy that runs through this whole anchor.

What a player would do with it #

The swing has one reading that the temperature does not, and it is the practical one.

Playing the hottest component first is the strategy the temperature exists to justify: on a board of several parts, move where the most is at stake, because the opponent will take it otherwise. That rule is about a reward and it works because the two players are competing for the same gain.

Under random turns the corresponding rule would be about a risk: move where the flip going the wrong way would cost most. Those are different components on the same board, and the swing is the quantity that ranks them. A part with a large swing is a part where being passed over is expensive, and a part with none is a part that can wait however hot it is.

Nothing here tests that rule, and the reason is the one these essays keep arriving at: there is no sum theory under the coin, so a board of several parts has no number assembled from its parts’ numbers, and a rule for choosing between parts has nothing to be evaluated against. The swing is a quantity for one position and the strategy it suggests would be about many, which is the same wall the auction reading ran into from the other side.

Still open: the 255 #

The positions with temperature nought and a flip that decides are the ones with no account at all, and there are enough of them to look at.

Two things would say what they are. The first is whether they are the all-small positions that are not infinitesimally close to nought — an up-multiple or a tiny, where a player genuinely gains by moving despite the thermograph reporting no temperature. The second is the direction: all 255 want the move on day three and none wants the other player to have it, which is a one-sidedness that either has a reason or is an artefact of how the class was defined.

If it has a reason, the swing is measuring something inside the all-small world that the temperature scale bottoms out below — which would make it a candidate for the quantity the atomic weight is reaching for, arrived at from a completely different direction and with no theory attached to it yet.