The quantity that does not order a board

How big the answer is put fights of graded depth into a coupon environment and found when the players leave it: at the larger of the position’s two temperatures — its own, and its answer’s. A fight at temperature 1 whose answer sits at 4 is taken at coupon 3½, not at coupon 1.

That page closed on the consequence it could not test:

If the departure is set by the larger of a position’s two temperatures, then a board of several components should be played in the order of that quantity rather than of the temperature — and whether a rule that says so beats playing in the hottest component is a measurement of exactly the shape this site already makes for strategies.

It does not beat it. It loses, badly, and the reason says something about what the two questions are.

The rule, in one line #

Both rules pick a component and then play the best move inside it, and they differ only in the picking:

• hottest takes the component with the largest temperature;
• the ordering rule takes the component with the largest departure quantity — the larger of its own temperature and its hottest answer’s.

Everything else is identical: the same pool of ten hot components, the same sums, the same optimal opponent, and the same choice of move once the component is chosen. So any difference in score is a difference in ordering and nothing else.

What it costs #

The pool is the site’s standing one: ten hot components including plain switches, fights whose answers start further fights, and two positions whose only move is into a fight. It was built for the strategy sweep precisely so that a rule ignoring follow-ups would be punished, which makes it the right pool for a rule that attends to nothing else.

On 123 of the 220 boards the two rules end with the same score — usually because they pick the same component. On the 97 where they differ, the hottest rule is better 85 times.

Where they agree, they usually agree because the hottest component is also the one with the largest departure quantity — a plain switch has no answer, so its two numbers coincide, and a pool with only plain switches in it would show no difference at all. The 97 disagreements are exactly the boards containing a component whose answer is hotter than it is.

That ratio is what makes this a refutation. A rule that lost narrowly might be worth keeping for the positions it handles better; a rule that loses seven disagreements in eight is choosing wrongly and being rescued by the positions where the choice does not matter.

The worst board is worth reading. It is $\{10 \mid \{9 \mid 1\}\}$ beside two copies of $\{0 \mid \{0 \mid -10\}\}$. The two copies have temperature nought — nothing is at stake in them — but each has an answer at temperature 5, so the ordering rule ranks them above a component worth ten and spends its first two moves in them. It scores 1 against optimal play’s 10.

And it breaks the guarantee #

The hottest rule is not merely a good rule; it is the one rule in this subject with a theorem behind it. Hotstrat guarantees a score of at least the board’s mean less its temperature, whatever the opponent does, and the sweep confirms it on all 220 boards.

The ordering rule breaks that bound on six of them. A heuristic that loses on average is a heuristic; one that breaks the only proved bound available is not a candidate at all, because what a bound buys is precisely the promise that the loss is limited. Losing that promise is a different kind of failure from losing points.

Why the same number answers one question and not the other #

The mechanism is one sentence and it is worth stating carefully, because the conjecture was reasonable.

Leaving an environment is a decision about when. A fight with a large answer is worth taking early, because whoever takes it collects the answer as well — the exchange and its follow-up come as a pair. So the departure is set by the larger of the two temperatures, which is what the rung below measured.

Ordering a board is a decision about where. Moving into a fight with a large answer hands the answer to the opponent: the mover takes the exchange and the opponent takes the follow-up. So the very property that pulls a player out of an environment should push them away from the component when there is somewhere else to move.

The two are not in tension once said out loud. An environment is a supply of moves of known size that nobody else wants first; a board of components is a supply of moves the opponent is competing for. A follow-up is an asset against a coupon stack and a liability against an opponent.

That also explains the twelve boards the ordering rule wins. They are the boards where the large answer is unanswerable — where the component’s follow-up is one the opponent cannot take profitably — and there the rule is right by accident rather than by reason.

There is a second reading of the worst board that makes the point without any theory. The two components worth nought are worth nought because nobody wants to move in them: whoever moves hands the opponent a fight worth five. The ordering rule looks at them and sees a five; a player looks at them and sees a position to stay out of. The departure quantity measures how big the fight inside a component is, and a board asks whose fight it will be.

The gap grows with the board #

On boards of two components the two rules disagree 14 times and the hottest rule is better on all 14. On boards of three they disagree 97 times and the hottest rule is better on 85.

More components mean more opportunities to be sent into the wrong fight, so a bad ordering compounds — each wrong choice hands the opponent an exchange, and the opponent has more places to take one. That is the ordinary behaviour of a per-move rule and it is worth checking rather than assuming, because a rule that failed only on small boards would be a different finding — one about the pool rather than about the rule.

What a player should do instead #

Two readings for somebody at a board, and they are simpler than the rule they replace.

Play in the hottest component, and use the answer as a tie-break the other way. When two components are equally hot, the one whose answer is smaller is the safer one to open, because opening the other hands the opponent more. That is not measured here and it is the natural shape of a repair.

And treat a cold component with a hot answer as a place not to move. The worst board above is made of two of them, and both rules would have done better leaving them alone entirely — which is what optimal play does. A component nobody should move in is a component whose ordering does not matter, and any rule that ranks it highly is spending moves on it.

The general lesson is worth stating in the site’s own terms. A quantity measured in an environment describes a position against a supply of moves; a quantity used to order a board describes a position against an opponent. This site has now measured both for the same number, and they point opposite ways.

What this does not refute #

Three things survive, and it is worth separating them from the rule that does not.

The departure measurement stands. Players do leave an environment at the larger of the two temperatures; that is the rung below’s finding and nothing here touches it. What fails is the inference from it.

The follow-up is still a real quantity. Half of the smaller temperature finds it entering the sente crossover exactly, and a bound with one number too many finds it staying out of a bound on the stops. Three pages here have now asked the same question of the same number and got three different answers, which is the honest state of it: the answer’s temperature matters where the timing of an exchange is at stake and not where a value or an ordering is.

And a better ordering rule may exist. What is refuted is use the departure quantity, not the temperature is the last word: nothing here proves the temperature optimal, and the hottest rule itself is exact on 89 per cent of these boards rather than on all of them. A rule that discounted a component by its answer — the reverse of this one — is the natural thing to try next, and it is one line away.

What it cost to find out #

The measurement is a hundred lines and about a second, which is worth recording because the conjecture had stood since the rung below was written.

Both rules are one line each inside the same strategy driver; the sums are the pool’s own triples; the optimal opponent is the exact evaluator this site uses everywhere. So the whole experiment is a new branch in a switch statement and a sweep that was already written — which is the shape a well-posed conjecture should have, and the reason the rung below could name it as a piece of work rather than as a hope.

A conjecture that costs a second to refute and an essay to state is a good conjecture, and this ladder has now produced two of them.

Predicting a threshold is not ranking #

The failure here has a general shape and it is worth stating, because the reasoning that produced the rule is sound at every step and the conclusion is still wrong.

The rung below establishes when a component is entered: the players leave the environment at the larger of a position’s two temperatures. That is a statement about one component against a falling ambient temperature — a threshold, measured on a board with the position and a stack of coupons and nothing else.

The rule proposed here reads that threshold as a priority: enter the component with the highest threshold first. And the step from one to the other is the error, because a threshold says at what ambient level a component becomes worth entering, not what is lost by entering another one instead.

Two components can have thresholds in one order and be worth taking in the other. A component with a high threshold and a small immediate gain is entered early against coupons because nothing better is on the stack; put a genuine fight beside it and the fight is worth more now, whatever either threshold says. The stack has no alternatives in it, and a ranking rule is entirely about alternatives.

That is why the rule loses 85 of its 97 disagreements rather than a scattered few. It is not a good rule with a blind spot; it is a quantity from a one-body problem applied to a many-body one, and the disagreements are exactly the boards where the difference between the two problems shows.

The lesson is worth carrying past this ladder. A measurement taken against an environment is a measurement of a component in isolation, and turning it into a decision between components needs a second argument that nothing in the environment supplies.

What the census does not say #

Four limits.

The pool is hot by construction. Every component in it has something at stake, so the boards are all fights and none is the quiet endgame where an ordering rule matters least. That makes the comparison sharp and it also makes it a comparison about hot boards only.

Ten components and sums of three. The pool is this site’s standing strategy pool, built so that a rule ignoring follow-ups would be punished. It is small, and a board of six components is untested.

One opponent. The strategy is measured against optimal play, which is the hardest test and not the commonest situation. Against a fallible opponent the ordering rule might lose less, and nothing here says by how much.

The move inside the component is fixed. Both rules take the option maximising what the mover keeps, so the comparison is of orderings and not of strategies in full. A different move rule could change both columns.

And the guarantee is Hotstrat’s, not a general bound. Mean less temperature is proved for the hottest rule; the six boards where the ordering rule falls below it are boards where a rule with no theorem behind it does what rules with no theorems do.

Who proved the rule this one had to beat #

Hotstrat is Berlekamp’s, and the bound it carries — a score of at least the mean less the temperature — is the one general promise this subject makes about playing a board without evaluating it. The proof is short and it is about the largest temperature: moving where the stake is highest means the opponent’s best reply cannot cost more than that stake, and the loss telescopes.

That argument is worth having in mind when reading any proposed replacement, because it says exactly which quantity the guarantee comes from. A rule that orders by something else is not a variant of Hotstrat with a better tie-break; it is a rule with no proof, and the six broken bounds above are what that costs.

The environment is Berlekamp’s too, and the pairing of the two is the reason the conjecture was tempting: the coupon stack was built to make temperature measurable, and it measures the departure quantity just as readily. Getting a number out of an instrument is not the same as knowing what the number governs, and this page is one measurement of that gap.

The convention, named #

Normal play throughout. A board is a disjunctive sum of components, each a hot position; a component’s temperature is read off its own thermograph, and a number has none.

The departure quantity is the larger of a component’s temperature and the temperature of its hottest option — the quantity the rung below found the players leaving an environment at.

A rule plays exactly on a board when the score it reaches against optimal play equals the score optimal play reaches against itself. Hotstrat’s guarantee is the board’s mean less its temperature, which the hottest rule is proved never to fall below.

Scores are from Left’s side throughout, with the rule playing Left and the optimal player playing Right.

The whole page reduces to one sentence worth carrying. A number measured against an environment describes a position facing a supply of moves, and a number used to order a board describes a position facing an opponent — and here the same quantity points opposite ways in the two settings.

Where the ladder goes next #

The coupons anchor has five rungs to here, and this one has just refuted the rule it proposed. The two above invert it and then find its coefficient was the one bad choice available.

A rule that beats the hottest takes the reverse of what failed here — discount a component by its answer’s temperature rather than promoting it — and does what nothing else on this site does: it beats playing in the hottest component. 201 exact on 220 three-component boards against 196, two thirds of the boards where the two disagree, inside the guarantee proved for the other rule, and the gap widening as the board grows. The prediction made before that sweep was that it would fail, which is worth knowing before reading it.

The worst value in its own interval then finds the discount’s coefficient of one to be exactly the wrong choice, and for a reason that applies to any rule of this shape. A ranking rule’s score is a step function of its own coefficient: what the rule does depends only on the order the scores put the components in, and that order changes only where two scores cross. Every weight strictly between nought and one gives identical play — and one is precisely a crossing, where two components tie and the ranking is settled by a tie-break rather than by the rule.

Read together with this page they make one argument in three parts. Promoting by the answer’s temperature loses badly, discounting by it wins, and the winning coefficient is any interior value rather than the round number anybody would pick. The direction of the correction and the size of it are two separate findings, and this page establishes the first by getting the sign wrong.