The cheapest fight is not the yardstick

The cheap fights make the rule cheaper scored four candidate bounds on what it costs to move in the hottest component rather than play the whole board correctly, and one survived every line: twice the coolest temperature on the board. The essay then raised the objection to its own result. Every pool it swept had its coolest component at a quarter, three eighths or a half, and nearly every costly line cost a half. A cost of a half against a coolest part of a quarter is a ratio of two because the cost is a half. The test it proposed was a pool whose cheapest fight is worth much less — an eighth, with follow-ups to match — and a prediction for each outcome: if the cost stays near a half there, the ratio runs to four and the bound is not twice anything; if the cost falls with the coolest temperature, two is real.

Neither happens. The cost does not stay put and it does not fall. It rises, to seven eighths on a board whose coolest part is worth one eighth, and on a single board it can be watched rising as the cool part is made cheaper. The quantity that holds on every line is not about the cool end of the board at all.

Seven times the cheapest fight #

The pool the objection asked for takes the four cool components of the earlier copies pool and divides every number in them by four, so that each is a fight at temperature one eighth with a follow-up of the same temperature — $\{1/2 \mid \{3/8 \mid 1/8\}\}$ is one of them — and keeps the same three hotter parts: $\{2 \mid \{1 \mid 0\}\}$ at three quarters, $\{5 \mid \{4 \mid 0\}\}$ at one and $\{4 \mid 0\}$ at two. Two companions go beside it. One puts cool parts at four different temperatures, an eighth, a quarter, a quarter and a half, so the coolest part varies from board to board. The other drops two eighth-temperature fights into the pool whose temperatures are all halves, so that the only change from a pool already measured is the cheapest fight.

Over boards of three, four and five parts, every board played twice with each player moving first, the three new pools produce 4,536 lines, and twice the coolest temperature fails on 154 of them. On the eighths pool it fails on 56 lines of 65 that cost anything; the typical costly line costs seven eighths against a coolest part of an eighth. On the pool with cool parts at four temperatures the costs spread across seven values from an eighth to one, and 51 lines break the bound. On the halves pool with two eighths dropped in, the worst line costs a half, and the worst ratio is three — three eighths against an eighth.

The four earlier pools, measured the same way, break it nowhere. That is the whole of the earlier finding, and it is now visibly a finding about those four pools. A coolest component at a quarter or a half, a cost of a half or one: the ratio could not exceed two there because nothing on those boards was cheap enough for it to.

The new pools also dispose of a second observation that had hardened into a description. Over the earlier pools the cost took three values — nought, a half or one — on boards whose largest temperature ran to three, and the bound names the hottest part reported that shortness as a fact about the rule: whatever the board, the rule was either right or wrong by a half or a point. On the pool with cool parts at four temperatures the costs take seven values: an eighth, a quarter, three eighths, a half, three quarters, seven eighths and one. The quarter is the commonest, on 104 lines. So the short list was a property of pools whose values all had a half as their finest denominator, and the cost of the rule is as fine-grained as the temperatures on the board allow it to be.

That matters for how the earlier bound was read. A cost confined to three values on boards whose temperatures reach three looks like a cost governed by something small and fixed, and the coolest temperature was the small fixed thing in view. A cost that takes every eighth between an eighth and one, on boards whose hottest part is two, looks like what it is: a quantity set by the particular exchange the rule gets wrong, which can be anything the board’s own numbers make available.

The line the rule gets wrong #

A bound that fails is worth one failing line examined properly, and the eighths pool offers the same line over and over: a cool part, $\{5 \mid \{4 \mid 0\}\}$ and $\{4 \mid 0\}$, with Right to move.

The rule sees three temperatures — two, one and an eighth — and takes the two, exactly as playing the hottest states it. Right moves $\{4 \mid 0\}$ to nought. Left now has $\{5 \mid \{4 \mid 0\}\}$ to take, worth five to Left and still a fight if Right were to move there first, and takes it; Right collects the eighth; the board totals $4\tfrac{7}{8}$.

Best play moves in the part the rule ranked second. Right’s option from $\{5 \mid \{4 \mid 0\}\}$ is $\{4 \mid 0\}$ — a second copy of the switch already on the board — and two copies of $\{4 \mid 0\}$ are exactly the number 4. A switch and its twin cancel each other’s heat: whoever takes one, the other takes the other, and two hot fights add to a cold number. So Right’s move does not start a fight; it ends two. What remains is the number 4 and a fight of an eighth, Left takes the eighth, and the total is $4\tfrac{1}{8}$.

The difference is three quarters, six times the cool part’s temperature, on a board with a proved bound of two. And every part of that difference comes from the two hot components. The cool part only decides the change: in best play Left gets its eighth, in the rule’s line Right does, and a swing of a quarter is the cool part’s whole contribution to a cost of three quarters.

That is why the cost cannot be measured against the coolest temperature. The rule went wrong by ignoring a move that is a threat — Right’s move in $\{5 \mid \{4 \mid 0\}\}$ is the kind of move sente is a fact about the rest of the board describes, one whose value depends on what sits beside it — and what it lost was set by the temperature of that move, not by the cheap part the game ended on. A player sizing moves by what changes hands, as big is not the same as hot describes, would not have been saved either: by that count $\{4 \mid 0\}$ swings four points and $\{5 \mid \{4 \mid 0\}\}$ swings three. Both rules read one component at a time, and the point of the move the rule missed is what it does to a different component.

Watching the cost rise as the cheap part cheapens #

The hero figure makes the point in one picture. Hold the two hot parts fixed and let the cool part’s temperature run from a sixteenth to one.

For a plain switch $\pm t$ the cost to Right is exactly $1 - 2t$ until $t$ reaches a half, and nought beyond. The arithmetic is the line above with $t$ in place of an eighth: the rule’s line ends at $5 - t$, best play at $4 + t$, and the difference is $1 - 2t$. For a cool part with a follow-up, $\{4t \mid \{3t \mid t\}\}$, the cost is $1 - t$ all the way to one. Both are checked at twelve temperatures.

So on this board a warmer cool part makes the rule cheaper, and a colder one makes it dearer. A bound of the form some constant times the coolest temperature predicts the reverse — a cost that shrinks to nothing as the cheapest fight does. The earlier pools could not show this. Their cool parts sat at a quarter or a half, where $1 - 2t$ is a half or nought and $1 - t$ is three quarters or a half, and twice the coolest is a half or one. The two lines cross at $t = 1/4$ for the switch and $t = 1/3$ for the follow-up, and every earlier pool happened to sit on or beyond the crossing.

It also reframes the title of the essay before this one. The cheap fights did not make the rule cheaper in the sense of a bound falling with them. On the boards measured there, cheap fights were warm enough that the change they returned nearly paid for the threat the rule missed.

Who the opponent is #

There is a second finding, and it is a correction.

The cheap fights make the rule cheaper describes its cost as the difference between optimal play and the score “with the mover following the rule”. What its measurement actually computes is different: it plays the whole game with both players following the rule, which is the convention a rule that is never right and cannot be far wrong states openly and the bound names the hottest part argues for. The two sentences describe different quantities, and the difference matters.

With both players following the rule, the opponent sometimes blunders back. The measure then goes negative — the rule-user finishes ahead of optimal play because the other side also played by the rule — and on the earlier pools that happens on 548 lines, more than two thirds as many as the 761 lines that cost anything. A negative line was reported as costing nothing, so the share of free lines included every line on which one greedy player’s error was covered by the other’s. Let the opponent reply as well as it can instead, and no line can go negative, because the opponent is free to play exactly as optimal play would. Across all seven pools 611 lines that the first measure calls free cost the mover something under the second, 402 of them on the earlier pools.

The argument for the both-players convention is that it isolates what the rule costs its user from what a strong player can take off a weak one. That argument runs the wrong way. What a rule costs the person using it is precisely what an opponent who does not use it can take — a player who follows hottest-first will not meet a copy of themselves across the board. The both-players number is a well-defined quantity, a comparison of two regimes, and it is not the quantity the prose around it names. The earlier essay’s convention section now says what its measurement computed.

The finding that depended on the opponent #

The correction would be bookkeeping if no result moved. One does.

The earlier headline was that every line costing more than the coolest temperature has one or two components at that temperature, and that over 3,230 lines with three or more, not one does. With the opponent replying best, twelve lines with three copies exceed it. Four and five copies still produce none, and the worst ratio anywhere on those pools is still two — so the earlier pools do satisfy twice the coolest under either convention. What they no longer support is the story told about it: that several interchangeable cheap fights protect the rule. The protection was partly the opponent following the same rule and declining to steer.

That leaves the earlier essay with one sound measurement and one sound negative — the conjecture that the cost scales with the number of cheap parts is false under either convention — and a mechanism that was always offered as a story and now has less to explain.

The bound that was in the data all along #

With twice the coolest gone, the natural candidates are the proved bound and whatever lies between it and the cool end.

The second-largest temperature on the board holds on all 14,946 lines of the seven pools, under both conventions. It is strictly smaller than the proved bound on the 6,822 boards whose two hottest parts differ, and it is met exactly on 344 lines, so it cannot be lowered on these boards. The witness family approaches it from below: its second-largest temperature is one, and its cost, $1 - t$, tends to one as the cool part cools.

The witness also says why it is the natural candidate. The rule’s error there is to take the hottest part and leave the second-hottest for the opponent at a moment when the second-hottest carried a threat. What changes hands in that exchange is bounded by what the second-hottest part is worth fighting over, which is its temperature. That is a reason to expect the bound and not a proof of it, and what is at stake in a fight is exactly the thing a temperature measures only up to the fight’s own follow-ups.

The last table is the one that should have been drawn first. Of the 344 lines that meet the second-largest temperature exactly, 296 are on the four pools twice the coolest was read from — and on the pool whose temperatures are all halves, every one of its 91 costly lines costs exactly the second-largest temperature on its board. Only 54 of the 296 also cost exactly twice the coolest. The bound was being met, line after line, in the data that suggested the other one; the tables were sorted by the coolest part because the coolest part was the question being asked, and a table sorted by the coolest part shows a ratio to the coolest part.

What the measurement stands on #

Everything here is normal play and a cost in points. Each board is played twice, once with each player moving first. Optimal is a minimax over the whole sum, run to the end of every line. The rule is hottest-first as every account states it: take the component of largest temperature and, inside it, the option of best mean, with ties broken by position on the board. The cost is optimal play’s score less the rule’s, signed so that positive means the rule did worse.

Two conventions for the opponent are reported side by side and they are not interchangeable. The one this essay argues for lets the opponent choose every reply by minimax against a mover who follows the rule; it is never negative. The one the earlier measurements used has both sides following the rule. The totals quoted for the earlier pools under their own convention — 10,410 lines, 761 costing something, 48 exceeding the coolest — reproduce the earlier essay’s numbers exactly, which is the check that the two measurements differ only in the opponent.

What seven pools cannot settle #

The second-largest temperature is a conjecture, and a conjecture with a history. Twice the coolest held on 10,410 lines of four pools and failed at the first pool built to test it. This one holds on 14,946 lines of seven pools, three of them built to break its predecessor, and no pool here was built to break it. The pool that would test it is the witness turned round: a board where the hottest part is the threat and the second is a plain switch, or where three parts at nearly equal temperatures give the rule two wrong choices in succession.

Five parts is still not many, and every component here has at most one follow-up. A component whose follow-up has its own follow-up could make a threat several moves deep, and nothing here tests whether the second temperature bounds what such a threat costs.

And a bound is not a description. Half of the smaller temperature is the shape a description takes — a number read off the position that equals a cost — and the warming family has one: $1 - 2t$ and $1 - t$ on one board. That is a formula for a family of three components, not for the rule.

Still open: a board where the threat is the hottest part #

The witness has the threat in the second-hottest part and a plain switch on top. The case it does not cover is the reverse, and it is the one a bound in the second temperature has to survive: a board whose hottest part is itself a threat — a component like $\{5 \mid \{4 \mid 0\}\}$ scaled up — with a switch just below it and a threat of its own below that. The rule would then take the right part first and possibly the wrong part second, and a cost larger than the second temperature would have to come from an error made one move into the game rather than at its start. A pool of such components, swept at four and five parts with the opponent replying best, is the measurement. If the second temperature survives it, it is worth trying to prove; if it fails, the bound is a statement about where the first error is made, and a rule that beats the hottest is where a rule that looks further ahead is already on the page.