The switch a player is imagining

Ask a strong player what a contested corner is worth and the answer comes in two numbers: it is worth about so much, and it is worth about so much to move there first. Ask the theory and the answer is a mean and a temperature — the same two numbers, computed.

Two numbers with those meanings name a position on their own. A mean of m and a temperature of t describe the switch {m+t | m−t}: a fight whose settlement is m and where moving first is worth t. So every summary of a hot position is implicitly a claim that the position is that switch.

The claim can be tested by subtraction, and it fails.

What the summary is, exactly #

The construction has a name in the literature — it is heating a number — and it is the inverse of the operation that produced the summary in the first place.

Cooling a position by t charges a tax on every move; raise the tax to the position’s temperature and the position freezes into its mean. Heating the number m by t runs the reverse: it pays a bonus of t for moving, turning the number back into a fight. The result is {m+t | m−t}, and it is what a two-number summary describes.

For the plain case the round trip is exact.

Where it is exact #

A switch — a position whose two options are both numbers — is its own summary.

{5 | 1} has a mean of 3 and a temperature of 2, and {3+2 | 3−2} is {5 | 1} again. Subtract the two and the answer is 0, not approximately zero: a second-player win in the strongest sense, interchangeable with nothing at all inside any sum.

That is worth stating plainly because it explains why the summary is trusted. Most positions in most textbooks are plain switches, the summary loses nothing on any of them, and a habit formed on those positions carries into the ones where it does lose something.

Where it is not #

Now the measurement, and it is starker than “approximately right”.

Take { {6 | 2} | {1 | −3} } — a fight where both options are themselves fights. Its mean is 3/2 and its temperature is 5/2, so its summary is {4 | −1}. Subtract:

{ {6 | 2} | {1 | −3} } − {4 | −1} = {2 | −2}.

The leftover is not small. It is a switch of temperature 2 — a whole second fight, worth as much to move in as most positions on this page — sitting inside the gap between a position and the two numbers that were supposed to describe it.

The same subtraction on the other positions in this site’s stock of examples gives leftovers of temperature 1 for {5 | {4 | 0} }, temperature 1 for { {4 | 2} | 0}, temperature 1/2 for {2 | {1 | 0} } and temperature 1 for {10 | {9 | 1} }. Every one of them is a fight rather than a discrepancy, and none of them is zero.

Each of those four is worth putting beside its own summary, because what the four have in common is not what a reader would guess from the worked example.

So a bend is not what goes wrong here, and a whole point of Right’s stop is.

Why the leftover is a fight rather than a number #

The reason is visible in the walls, and it is not the reason the worked example makes it look like.

A switch’s thermograph is two straight lines meeting at a point: Left’s wall runs down from the left option at slope one, Right’s runs up from the right option, and they cross at the temperature. That is all a switch can do, because its options are numbers and a number’s wall is vertical. It also means a switch’s feet are forced: run two lines of slope one down from a meeting point at height $t$ over value $m$ and they reach the ground at $m+t$ and $m-t$, and nowhere else.

The summary inherits that. Whatever position it was built from, the summary stops at $m+t$ and $m-t$ — and a position’s own stops are under no such obligation. On all five of the worked examples they are somewhere else: 2 and 1 against 4 and −1, 5 and 4 against 5 and 3, 2 and 0 against 3 and 0, 2 and 1 against 2 and 1/2, 10 and 9 against 10 and 8.

So what the summary loses is the pair of stops, and it loses them whether or not anything bends. Three of the five have a visible bend and two are straight-walled throughout, and all five leave a fight behind. A reader who took the worked example’s bent walls for the mechanism would predict that $\{5 \mid \{4 \mid 0\}\}$ and $\{10 \mid \{9 \mid 1\}\}$ are their own summaries, and both of them are not.

The second visible bend is on the other wall, which is worth seeing once so that the bend does not get attached to one side of the diagram.

Where a wall does bend, the bend cannot be matched by a summary that produces straight ones: the mismatch is a whole function rather than an offset, and subtracting the two leaves an object with structure in it. Where no wall bends, the stops have moved anyway.

The two-number summary is therefore not an approximation that could be improved by better numbers. It is a projection onto a two-parameter family, and everything the family cannot represent survives the projection intact.

What a leftover of temperature two actually costs #

An error that is itself a fight is a different kind of error from an error that is a fraction of a point, and the difference shows up as soon as the summarised position is put beside something else.

Suppose a player summarises { {6 | 2} | {1 | −3} } as {4 | −1} and then reasons about a board holding that region and one other. The reasoning is now being done about a board that differs from the real one by {2 | −2} — a component nobody has drawn, which neither player can see, and which is hotter than most of the regions being counted.

If the difference were a number the reasoning would still be sound about the ordering of the other components: adding a constant to a board changes every line by the same amount. A fight is not a constant. It changes different lines by different amounts, because whether its two points are collected depends on who ends up with the move, and that depends on the parity of everything else.

So the practical damage of the summary is not that the count comes out two points wrong. It is that the count comes out wrong by an amount that depends on the rest of the board, which is precisely the dependency a per-region summary was adopted to avoid.

That is the general reason this site keeps insisting on the difference between a value and an estimate of one. An estimate whose error is a number can be carried through the arithmetic. An estimate whose error is a game has to be re-examined every time the board changes.

Two numbers, and the family they name #

It is worth counting the parameters, because the failure is a counting argument and nothing more.

A summary made of a mean and a temperature has two degrees of freedom. The switches {m+t | m−t} form a two-parameter family, and there are exactly as many summaries as there are switches — the map from one to the other is a bijection.

The short games born by day three already number in the thousands, and the ones born by day four cannot be counted on this site’s evaluator at all. Whatever the exact figures, the target of the map is two-dimensional and its source is not, so the map is enormously many-to-one and every fibre of it holds positions that no mean and no temperature can tell apart.

The essay could stop there, and the reason it does not is that the counting argument says nothing about which positions get confused or how badly. That is what the subtraction is for: it names the fibre — every position with the same mean and temperature — and it measures the spread inside it by computing the difference between a member and the family’s own representative.

Making the summary is that operation run backwards, one step at a time. Cooling $\{4 \mid -1\}$ raises the tax on moving until the fight freezes at its mean of 3/2, and the height it freezes at is the temperature of 5/2; heating the number 3/2 by 5/2 walks the same strip in the other direction and arrives back at $\{4 \mid -1\}$. That round trip is why a summary is always a switch, and it is the reason the target of the map has exactly two parameters in it.

Counting the fibres #

The section above says the map is enormously many-to-one and declines to say how many-to-one. It is worth saying, because the number is not the one a reader would guess and it changes what the summary is being accused of.

Take the 1,474 values born by day three and compute each one’s mean and temperature. There are 43 distinct pairs.

Not forty-three hundred; forty-three. A summary made of two numbers, applied to a day of the construction holding one and a half thousand values, produces forty-three different answers — so the average fibre holds thirty-four values, and twenty of the forty-three pairs are the ones that name a single value each.

The day before is the same statement in miniature: 22 values, 13 pairs, and the largest fibre holding six.

That is a much stronger claim than the counting argument makes. Two parameters against a large set guarantees collisions in principle; what the count shows is that the collisions are the normal case, and that a mean and a temperature are closer to a coarse classification than to a description with error bars.

The fibre that holds a fifth of the day #

One fibre holds 291 values, and it is the pair $(0, 0)$ — mean nought, temperature nought.

Those are the positions with nothing to fight over and nothing to settle at: $\ast$, $\uparrow$, $\downarrow$, $\uparrow\!\ast$, $2\!\cdot\!\downarrow\!\ast$, $\{\ast 2 \mid 0\}$, $\{\uparrow\!\ast \mid \downarrow\}$ and two hundred and eighty-four more. Every infinitesimal born by day three is in it, and the summary assigns all of them the same two numbers, which are both zero.

So the summary’s worst case is not a hot position with a bent wall. It is the whole infinitesimal world, collapsed to a point. The essay’s worked example loses a fight of temperature two, which is dramatic; this loses the distinction between a position worth nothing to anybody and one that is a first-player win, and between $\uparrow$ and $\downarrow$, which are on opposite sides of zero.

Three of them side by side is three copies of one picture, which is the whole of the complaint.

That has a reading in play. A player summarising every region by a mean and a temperature has, by construction, written down zero and zero for every part of the board where the points have run out — which is exactly the position at the end of an endgame, and exactly where a tempo decides the game. The two-number account is at its most complete when the fighting is at its hottest and least informative when the game is closest.

What the count does not say #

Two cautions, because the number is striking enough to be over-read.

Forty-three is a count for one day. Day four has more values and, presumably, more pairs; whether the ratio grows, shrinks or holds is not established here, and a single day is one data point about a sequence.

And a large fibre is not the same as a bad summary. The $(0,0)$ fibre is large because every infinitesimal is in it, and a player who has correctly determined that a region is worth nothing and has nothing at stake has learned something true and useful. The complaint is not that the summary lies; it is that it stops there, and the section below on what it gets right is not a concession but the other half of the same measurement.

What the summary is right about #

It would be wrong to leave the impression that the summary is worthless, because it is the single most useful thing a player carries.

It is exactly right about the mean. The mean of the position and the mean of its summary are equal by construction, and the mean is the quantity that adds across a board and that many copies converge to. A player counting up a whole board is adding means, and the summary is faithful to every one of them.

It is exactly right about the temperature. The summary was built with the position’s temperature, so it agrees about which component is hottest, and the rule that says to move in the hottest is unaffected by everything the summary loses.

Rank the position, its summary and a cooler third component by what is at stake and the first two come out tied at 5/2, with $\{2 \mid 0\}$ behind them at 1. That tie is not a coincidence and not an approximation: the summary was built with the position’s own temperature, so it can never disagree about which component is hottest. The ranking is one of the two things the projection preserves exactly.

So a player using a mean and a temperature to decide where to move is on solid ground, and a player using them to decide what the position is is not.

The account a Go player writes #

The place this matters in practice is the endgame count, where the summaries are added up and the temperatures alternated.

The orthodox account takes three regions summarised as switches — $\{4 \mid -1\}$, $\{3 \mid 1\}$, $\{2 \mid 0\}$ — adds their means, and alternates their temperatures in falling order to price the moves. Set beside the exact value computed from the same three regions, the two agree, and the agreement is not luck: every region there is a plain fight, so every summary is the region.

The account is exact when every region is a plain switch, and it is exact for the same reason the subtraction above came back zero: on those regions there is nothing else there. On a board with follow-ups the account is working with summaries that have each thrown away a fight, and the first time temperature told somebody something is the history of what it took to notice.

Playing by the summary, and what it costs #

The clean way to measure a summary is to build a player out of it and score it.

Build the pool out of three positions and the three summaries they reduce to — $\{5 \mid 1\}$ and itself, $\{\{6\mid2\}\mid\{1\mid-3\}\}$ and $\{4 \mid -1\}$, $\{5 \mid \{4\mid0\}\}$ and $\{5 \mid 3\}$ — and play whole boards out of them under each rule. Where the two rules pick differently they are picking between a position and something carrying the same two numbers, which is the disagreement this page exists to show is possible.

The experiment is the same one a bound instead of an answer runs, and its conclusion carries over: playing by temperature has a proved bound, which is what makes the summary usable despite everything it drops. A rule that is wrong about the position and right about the ordering is a good rule, provided nobody forgets which half is which.

The habit this is an instance of #

Three of the four measurements collected here have the same shape, and it is worth naming once rather than three times.

A practical rule computes a number. The position’s answer is not a number. The rule is therefore exactly right on the class of positions where the answer happens to be one, and unbounded elsewhere.

For the greedy Dots and Boxes rule the class is boards with no chain long enough to be worth declining. For the chess tempo count it is endings where both sides have the same options. For the two-number summary here it is positions whose options are numbers.

What makes the pattern worth stating as a test rather than as an observation is that the class is always identifiable in advance. Given a rule of thumb about a game, the question to ask is: on which positions is the value a number? — and the honest scope of the rule is that set and nothing larger.

The corollary is the useful half. A player who knows the class knows when to stop trusting the summary, and the boundary is not subtle: it is the moment a component acquires an option that is itself worth fighting over. That is a thing anybody can see on a board, without computing a single value.

Where the model stops #

The leftover is not always a switch. {2 | −2} came out of one particular subtraction; the others leave objects with nested braces and no short name. The claim is that the leftover is a game with a temperature, not that it is anything tidy.

Nothing here says the summary is the best two-number description. A different pair of numbers — the two stops, for instance — describes a different two-parameter family, and it fails on a different set of positions. The mean and the temperature are the pair with the theorems attached; being the best summary in some other sense is not a claim anybody has made or measured.

And the exactness result is about switches only. Every position whose options are numbers is its own summary; positions with one numeric option and one contested one are not, even when the contested side looks harmless.

What the picture cannot show #

Every figure here draws the position and the summary as two thermographs side by side, and the eye is very good at seeing that two curves differ.

What it is bad at is judging how much they differ, because the leftover is not the gap between the curves. It is the value of a subtraction, computed by a recursion on both trees, and its temperature of 2 is not readable from the picture at all: the two thermographs on this page differ visibly by about a point at their widest, and the game between them is worth two.

The other thing not shown is the family. The claim is about a projection onto every switch, and the figures show five positions and their images under it. That the projection loses something in general is a statement about all hot positions, and it is established by the argument about where a switch’s feet have to be rather than by the drawings.

The convention, named #

Normal play, short games, and one further convention that is easy to miss.

The summary {m+t | m−t} uses a mean and a temperature that were both computed with a tax charged uniformly on every move, which is what cooling is. That is a modelling decision, and it is the one that makes temperature a single number: a tax that varied by component would produce a different theory and a different summary.

Where the tax cannot be charged uniformly — where one component’s moves are cheap and another’s are not, as in a scoring game with a clock or a ko threat — the mean and the temperature stop being the right two numbers, and nothing above applies. That boundary is the subject of counting at the end changes everything, and it is a boundary rather than a caveat.