Two games in one environment

An environment of coupons turns two computed quantities into things that happen during a game. Beside the position sits a stack of coupons worth $t, t-\delta, \ldots, 0$; either player may take the top one instead of moving; and when the whole board is solved, the position’s contribution to the score is its mean and the coupon the players stop at is its temperature.

That is the theory’s own idealisation, and it has one position in it. A board has several. So the question this rung asks is which of the two measurements is a fact about the position and which is a fact about the board it was measured on.

What an environment is for #

The coupon stack is a laboratory instrument and it is worth remembering what it was built to replace.

A real board is a sum of local fights of different sizes, and a strong player’s whole skill is knowing which to take and when. The theory’s account of that is temperature and the hottest-first rule, both of which are statements about a position relative to its surroundings — and both of which are usually demonstrated on a position with no surroundings at all, which is the one case where the question does not arise.

Berlekamp’s answer is to build a surroundings with no surprises in it. A stack of coupons worth $t, t-\delta, \ldots, 0$ is a sum of switches of every temperature in turn, played hottest-first by construction, and the orthodox account is exact on it. Anything odd in a reading is therefore the position’s fault rather than the instrument’s.

That design has an assumption in it, and this rung is the assumption being tested. The instrument was designed for one position at a time, and a board has several.

The measurement that survives #

Start with the half that holds, because it is the reason the environment is worth building at all.

The contribution of the two positions together is the score of the whole board less the score of the stack alone, and it comes out at the sum of the two means, to within a coupon. That is not a small claim: it says the environment measures a sum the way it measures a part, and it holds on every pair tried.

It also has to hold, and the reason is one of this site’s standing results — means add even though temperatures do not. What the census does is confirm that the measuring instrument respects the theorem rather than merely that the theorem is true.

Why the means have to add and the temperatures do not #

The asymmetry the census reports is not an accident of the pool. It is a consequence of two facts this site has already measured, arriving in the environment’s own vocabulary.

The mean of a sum is the sum of the means, every time — that is the theorem, and it is why the contribution column survives company. Two positions beside a stack contribute what they would contribute separately, to within a coupon, because the quantity being measured is additive.

The temperature of a sum is not the sum of the temperatures. It is bounded by the hottest part and is often far below it, and two positions each worth fighting over can add to something neither player wants to touch. A reading that recovers a temperature is therefore reading a property of the whole board.

So “the coupon a fight is entered at” was never going to be additive. What the census establishes is that it is not even local: it moves for a position whose own thermograph has not changed at all, because the other game altered when entering it was worth more than taking a coupon.

The measurement that does not #

The stopping coupon is the environment’s reading of temperature, and temperature is exactly the quantity that does not add. So it should be no surprise that the reading moves — except that the reading is not about a sum. It is about one position, taken in company, and the company is not being measured at all.

Two ways it moves, and they run in opposite directions.

A hotter neighbour delays a fight. With $\{4 \mid 0\}$ on the board, the position $\{2 \mid \{1 \mid 0\}\}$ is not worth entering until the coupons have fallen to nothing: there is a bigger fight available and a fat coupon to take, and both are better. Alone it is entered at 1, which is the smallest coupon this grain has above its temperature of $\tfrac34$.

A hotter neighbour can also bring a fight forward. $\{2 \mid 0\}$ alone is entered at coupon 1. With $\{4 \mid 0\}$ beside it, it is entered at 2 — while a coupon worth 2 is still on the stack, which alone it would never do.

This is sente, from a direction that never mentions it #

The rung below this one found the same shape in a lone position: for a game whose follow-up is hotter than itself, the players stop at the follow-up’s temperature rather than at the position’s own. That was sente arriving out of a construction with no sente in it.

This is the other half of the same statement. Sente is a claim about a local fight and the rest of the board, and the coupon stack was built to be a rest-of-the-board with no surprises in it. Put a second real position on the board and the surprises come back — not because the stack is a bad instrument, but because “the coupon this fight is entered at” was never a local quantity.

The order of play, read off the transcript #

The transcripts are worth reading as transcripts rather than as evidence, because the line of play is the argument in its most concrete form.

In the hero figure the coupons go first. Both players take from the stack while the top coupon is worth more than either fight, which is hottest-first doing exactly what it is supposed to do: the stack is a set of fights of known size, and the two positions are two more.

Then $\{4 \mid 0\}$ is entered, at coupon 2 — its own temperature — because at that point it is the hottest thing on the board. Then more coupons. Then $\{2 \mid \{1 \mid 0\}\}$, at coupon 0, which is the bottom of the stack and a whole coupon later than the same position waits when nothing else is on the board.

What delayed it is not that the position got colder. It is that a better move existed for longer: with a second fight on the board, the moment when entering this one beats everything else available arrived later. The measurement is of the board’s schedule, and the position is one entry in it.

The control for that reading is the same position with the environment to itself, and it is worth drawing rather than quoting, because the difference between the two transcripts is the whole of the finding.

Two positions is already a board #

It is tempting to read this rung as a technicality — one extra position, a reading that wobbles — and the size of the effect argues otherwise.

$\{2 \mid \{1 \mid 0\}\}$ alone is entered at coupon 1. Beside $\{4 \mid 0\}$ it is entered at coupon 0. That is not a wobble of a fraction of a coupon: it is the whole remaining stack, and a player who had read the lone measurement as “this fight is worth entering when the board cools to 1” would have been wrong about every move between 1 and 0.

The other direction is as large. $\{2 \mid 0\}$ alone waits until coupon 1; beside a hotter fight it is entered at 2, a full coupon early, at a moment when the lone reading says the biggest thing available is the stack.

So the quantity is not approximately local. It is a schedule for a particular board, and the environment measures the schedule faithfully — which is the instrument working rather than failing. What the rung below could not see is that it was measuring a two-body problem with one body in it.

And two is the smallest number at which the problem exists at all, which is why the effect being this large at two is the finding rather than the caveat. A quantity that survived one neighbour and broke at six would be a quantity worth approximating; one that moves by the whole remaining stack on the first neighbour it meets is a quantity that was never about the position.

What the solver computed, and how #

One minimax over a sum of three parts: the first position, the second, and the coupon stack written out as a game.

The stack is a game rather than a rule. Coupons worth $t, t-\delta, \ldots, 0$ become a chain in which either player may take the top one, so “taking a coupon” is a move in a component like any other and nothing in the search knows the environment is special. The score is the sum of the components once every one of them is a number.

The stack itself is worth a sentence. Coupons of decreasing value are a sum of switches with distinct temperatures — $\{4 \mid -4\}$, $\{3 \mid -3\}$ and so on down — and a stack of $n$ of them is worth its alternating sum, which the rung below verifies against the recursion at five different grains. That is what makes it an instrument: its own reading is known exactly, so a difference in the whole board’s score is attributable to what was put beside it.

Three quantities come out of the line of play. Where each position was entered: the coupon on top the first time optimal play moved in it, which the search records rather than infers. The contribution: the whole board’s score less the stack’s own, both from the same first player. The error: the contribution against the sum of the two means, which is asserted to be within a coupon — the function throws if it is not, so the bound is a check rather than a claim.

The census runs every ordered pair from a pool of six, which is 36 boards, each a complete solution.

What a Go player would call this #

The vocabulary a strong Go player already has for the phenomenon is worth setting beside the measurement, because the two describe the same thing from opposite directions.

A player says a move is sente when the opponent has to answer it, gote when they do not, and — crucially — treats both as properties of the local shape. The rung below this one measures the first half of the correction: the same fight is answered while the rest of the board is quiet and ignored once it is busy, with the crossover at the follow-up’s own temperature.

This rung measures the other half. “When to take this fight” is not a local property either, and the coupon stack, which is the most orderly possible rest-of-the-board, still moves the answer as soon as a second real fight is on it.

What survives as local is the pair of numbers the thermograph computes: the mean and the temperature. Everything a player would call timing — which fight first, when to leave the big points, whether the opponent will answer — belongs to the board. That is not a defect in the theory; it is the theory’s own account of what a value is, arriving in the language of a game somebody actually plays.

Where the model stops #

Thirteen of thirty-six is the rate for this pool. Six positions taken two at a time, with coupons a whole point apart and a stack topping out at four. Thirteen of 36 is a rate for that pool and for those coupons, not a law; a pool of colder positions would move fewer readings and a pool of hotter ones more. What the census establishes is that the reading moves at all, which one pair also establishes; the count says it is common rather than pathological.

The coupon grid is coarse. With coupons a whole point apart, a reading can only move in whole points, and two positions whose fights differ by less than a coupon are indistinguishable to the instrument. Finer coupons cost an exponential search — the stack is a component with as many positions as coupons — which is why the rung below caps the stack at eighteen.

Both positions are hot. A pair including a number would have nothing to measure — a number is never entered at all, which the rung below reports as “never played in” — so the pool is chosen to make the question askable, and that is a choice rather than a sample.

And the environment is still idealised. A stack of coupons has no follow-ups, no threats and no interaction between its own parts. Two positions beside it is one step towards a board and not the same thing as one: a real endgame’s regions are hot, cold and connected by threats, and nothing here measures that.

What would settle it #

The measurement here is a difference between two runs, and there is a stronger experiment the machinery already supports: sweep the second position through a range of temperatures and watch the first position’s entry coupon move as a function of it. That is the shape the sente sweep has, with an ambient switch standing in for the rest of the board, and running it inside an environment rather than beside one would give a crossover with a number on it instead of a pair of readings that differ.

It is not run here because the search cost is the product of the two positions and the stack, and a sweep multiplies that by the number of temperatures. What the census establishes is the qualitative fact — the reading moves, in both directions, on a third of the pairs — and the quantitative version is a rung of its own. It has since been run, two rungs up, and it returns a number rather than a pair of readings: with a position’s own temperature held at one and its follow-up’s swept from one to four, the coupon the players leave at runs from 1 to 3.5, and across the whole grid they leave at the larger of the two temperatures.

Where the ladder goes next #

This rung establishes that the entry coupon is a fact about the board. The five rungs above turn that from a warning into a rule, and then find the rule is not the one anybody expected.

When to leave the environment asks the question a Go player actually has — not which fight to take but when to stop taking the small stuff. The orthodox answer is leave when the coupon falls to the hottest temperature on the board, and it is exact on sixty of eighty-one pairs. Every one of the twenty-one departures has a position with a follow-up in it, and every pair of plain switches leaves on time, which is the same split this page’s thirteen already hint at.

How big the answer is grades the follow-ups by size and gets a formula out of the class: the players leave at the larger of the position’s own temperature and its follow-up’s, across the whole grid. That is the sweep this page’s last section proposed and declined to run.

Then the ladder tries to use it, and the attempt fails instructively. The quantity that does not order a board promotes each component to that larger temperature and plays in the highest; over 220 three-component boards it plays exactly on 124 against hottest-first’s 196, loses 85 of the 97 disagreements, and breaks Hotstrat’s guarantee outright on six of them. A quantity that predicts when a component is entered does not order the components.

A rule that beats the hottest inverts it — discount a component by its answer’s temperature instead of promoting it — and does what nothing else on this site has: it beats playing in the hottest, 201 exact against 196, winning two thirds of the disagreements and widening its lead as the board grows. And the worst value in its own interval then finds that the coefficient chosen for that discount was the one bad choice in its own range, because a ranking rule’s score is a step function of its coefficient and one is precisely where two components tie.