Where the fight stops

A fight ends. Two players take turns grabbing at a contested position, and at some point neither wants it any more — every option left is a number, nobody is in a hurry, and the score is whatever has been collected on the way.

The number reached that way is a stop, and every position has two of them: the one Left reaches by moving first, and the one Right reaches by moving first. They are the coarsest honest summary a fight admits, they are computed by the same recursion as everything else here, and they have been sitting in plain sight on every thermograph this site has drawn.

What playing on means #

The definition is a mutual recursion and it is two lines.

A number stops. If a position is a number — 3, or −1/2, or 0 — then both its stops are that number. Nobody moves in a number if there is anything else to do, which is what numbers avoid numbers says, so the fight is over and the score is the number.

Anything else keeps going. For a position that is not a number, Left’s stop is the best Right stop among Left’s options, and Right’s stop is the worst Left stop among Right’s. The switch of players in that sentence is not a slip: Left moves, and then it is Right’s turn to move first from wherever Left left it.

Written out, LS(G) = max over Left's options of RS, and RS(G) = min over Right's options of LS. The recursion terminates for the same reason every recursion here terminates — the options are simpler than the position, and eventually they are numbers.

The result is a pair of numbers rather than one, and the pair is not symmetric. For {5 | 1} the stops are 5 and 1: Left moving first gets 5, Right moving first gets 1, and the four points between them are what the two are fighting over.

The recursion, run once by hand #

It is worth running the two lines on a position where they do something, because the arithmetic is short and the answer is not the one the notation suggests.

Take {5 | {4 | 0}}. Left’s only option is 5, a number, so RS(5) is 5 and therefore LS of the whole position is 5 — Left moves, the fight is over, Left has five points.

Right’s only option is {4 | 0}, which is not a number, so the recursion continues. LS({4 | 0}) is the best RS among Left’s options there, and Left’s only option is the number 4, so it is 4. Right’s stop for the whole position is the worst LS among Right’s options, and there is only one: 4.

So the pair is (5, 4). Right moving first ends up at 4, one point below where Left moving first ends up, and the second line of the recursion is where the whole reversal happened: Right’s move looked like a move to 0 and was a move to a position Left immediately converts into 4.

That is a two-step calculation and it is the general shape of every stop calculation on this site. The recursion alternates the two players’ extrema, and each alternation is one exchange of the fight being played out in advance.

The stops are the feet of the walls #

Nothing new had to be built to compute them, which is worth dwelling on, because it is the sort of coincidence that turns out to be a definition.

A thermograph plots what each player gets when a tax of t is charged on every move. Charge nothing — read the diagram across at t = 0 — and what is left is what each player gets from ordinary play with no tax at all. That is the stop.

So the left wall’s foot is LS and the right wall’s foot is RS, on every thermograph in this collection, drawn since the first one and never labelled until now. This collection has been drawing the stops since its first thermograph while talking about the point where the walls meet.

That figure is the reason stops are worth a name of their own. The written position {5 | {4 | 0}} looks like a five-point fight — an option at 5 against an option that starts at 0 — and it is a one-point fight, because Right’s move invites an answer that takes most of it back. The stops are 5 and 4 and they say so; nothing about the notation does.

Reading the pair #

Two numbers, and there are exactly two things to do with a pair: subtract them and average them.

The difference is what the fight is worth. For a plain switch the temperature is exactly half of it, which is where the factor of two in every endgame book comes from. For a position with a follow-up it is not — {5 | {4 | 0}} has stops one apart and a temperature of 1 rather than 1/2 — because temperature is a statement about a tax and the stops are a statement about untaxed play.

The average is not the mean value. For {5 | 1} the stops average to 3 and the mean value is 3, so nothing shows. For {5 | {4 | 0}} they average to 4.5 and the mean is 4. For {2 | {1 | 0}} they average to 1.5 and the mean is 1.25. The mean sits between the stops, always, and lands at their midpoint only when the fight is plain.

Both of the last two are positions with a follow-up, and it is worth drawing one of each kind, because which side the follow-up is on decides which way the mean slides off the midpoint.

Now the same position with the follow-up on the other side, which is not the mirror image of it: Left’s option is contested instead, and the quantity that moves is the left stop.

Two summaries, and neither contains the other #

The stops are one pair of numbers and the mean-and-temperature are another, and the essay so far has compared them one position at a time. Over a whole day the comparison is sharper, and the answer is that they are genuinely rival summaries rather than two spellings of one.

Take the 1,474 values born by day three. They produce 36 distinct pairs of stops and 43 distinct mean-and-temperature pairs, and neither is a function of the other:

17 of the 36 stop-pairs cover more than one mean-and-temperature. Knowing where a fight starts and finishes does not say what it settles at or how urgent it is.

9 of the 43 mean-and-temperature pairs cover more than one stop-pair. Knowing the settled value and the stake does not say where untaxed play ends up.

So a reader holding either pair is holding two numbers that the other pair cannot reconstruct, in both directions. That is a stronger statement than the average is not the mean value: it says the two summaries are measuring different things and that neither is the more informative.

And the mean really is between them #

One relation does hold, everywhere. The mean lies between the two stops — at or above the right stop, at or below the left — on all 1,474, with no exceptions.

That is worth having as more than a check, because it says what the two summaries share. Both are bracketing the same position on the same number line: the stops from the outside, by playing on until nobody wants to move, and the mean from the inside, by taxing until nobody wants to move. Taxing and playing on are two ways of exhausting the same fight, and they arrive at points in the same order.

Where they differ is what happens between. The stops record the two endpoints of untaxed play and say nothing about how much of the interval is contested; the mean and temperature record a centre and a width and say nothing about where play actually lands. On a plain switch the interval is symmetric and the two accounts coincide — which is why the essay’s first example shows nothing.

And it explains which of the four numbers survives addition. Means add, exactly, across any sum. Stops do not: a sum’s stops are not the sums of its parts’ stops, because playing on in a sum means choosing which component to play on in. So the summary that composes is the one built by taxing, and the summary that describes actual play is the one that does not — which is an uncomfortable division and is the reason both are kept.

The plain case, where nothing is hidden #

It is worth being explicit about the case where the stops say nothing new, because it is the case most positions in most textbooks are in.

A switch — a position both of whose options are numbers — has its stops handed to it: LS is the left option and RS is the right one, because both options stop immediately. So for {5 | 1} the stops are 5 and 1, the difference is 4, the temperature is 2, the mean is 3, and every one of those numbers can be read straight off the position.

Run three of them and nothing is ever computed. {5 | 1} has stops 5 and 1, mean 3, temperature 2; {2 | 0} has stops 2 and 0, mean 1, temperature 1; {3 | −1} has stops 3 and −1, mean 1, temperature 2. Every quantity in every one of those rows is a rearrangement of the two numbers the position was written with, and the machinery that draws switches is doing arithmetic a reader could do faster.

That is why a fight with a follow-up is the interesting case and a switch is not, and it is why the switch a player is imagining is a different essay: the summary and the position coincide here and separate everywhere else.

What the pair cannot see #

Now the failure, and it is total rather than partial.

A position whose two stops are equal has a stop pair that says nothing at all. Zero has stops 0 and 0. So does ∗. So does ↑, and ⇑, and ↑∗, and ∗2, and every other infinitesimal — every position smaller than every positive number and larger than every negative one, which is to say every position with no points in it whatever.

Those positions are not the same as each other. Their outcomes are P, N, L, L, N and N respectively: four different answers to “who wins”, behind one pair of numbers.

The diagram of one of them says the whole thing, and it is the emptiest picture in this collection.

The reason is structural rather than accidental. The stops are built out of numbers by taking maxima and minima of numbers, so the whole apparatus lives on the number line, and the infinitesimals are precisely the part of the theory that does not. Asking a pair of numbers about ↑ is asking a ruler how loud something is.

The ordering the stops cannot report is real and is established the only way it can be, by playing the difference. Up is greater than nought; star is confused with it; two ups are greater than one up; down is less than nought. Four facts, four positions, and every one of the four with the same pair of stops and the same picture above.

Stops do not add #

The other limitation is the one that matters in a real position, because real positions are sums.

Take two fights and put them side by side. The mean of the sum is the sum of the means — that is a theorem and it is what makes orthodox accounting possible. The stops do nothing of the kind.

{5 | 1} has a left stop of 5 and {2 | 0} has a left stop of 2. The sum has a left stop of 5, not 7, because Left cannot move first in both components: taking one of them hands the other to Right.

This is the same fact as outcomes do not add wearing a different coat, and it has the same consequence: a summary of a component that does not survive addition is a summary of that component alone. Stops describe a fight; means describe a fight’s contribution to a board.

That division of labour is why the theory carries both. It is also why temperature exists: temperature is the quantity that says which component to fight in first, and it is derived from how the stops behave under a tax rather than from the stops themselves.

The one thing the pair does decide #

Set against two failures that large, it is worth saying what the stops are actually good for, because the answer is sharp.

When the two stops straddle zero, the position is a fight and whoever moves first wins it. {5 | 1} has both stops positive, and it is a win for Left whoever starts — Left has points banked there whatever Right does. {1 | −1} has stops 1 and −1, and it is a first-player win: each player, moving first, ends up on their own side of zero.

The second of those is worth drawing beside the first figure on this page, because the two are the same shape and the verdict is different, and the only thing that changed is where the pair of feet sits relative to nought.

That is a genuine decision procedure and it covers every position whose stops differ, which is every position with any points in it at all. What it does not cover is the case where the two stops are equal, and the case where they are equal is exactly the case the previous section was about.

So the honest statement of what a pair of stops is worth is: it settles the outcome of every position that is not infinitesimally close to a number, and it settles nothing about the rest. The subject’s whole second half — up, star, atomic weights, all-small games, the values that decide Clobber and Domineering endgames — lives in the residue.

That division is not a defect of the stops. It is the reason the theory has two halves at all: one arithmetic of numbers for positions with points in them, and one order-theoretic apparatus for positions without, meeting at the place where the stops give up.

What many copies do to them #

One more measurement, because it makes the difference between the two summaries concrete.

The mean is a rate and the stops are a single outcome. Piling up copies exposes which is which, and the mean is the one that survives, which is exactly why the mean is defined by the pile rather than by the position.

Where the model stops #

Three limits, stated rather than implied.

Stops are defined for short games only. The recursion above needs the options to be simpler and eventually to be numbers. A loopy position has no such bottom, its play need never stop, and it has no stops in this sense — the word would be a bad joke there.

They assume the normal-play convention. The definition ends with “somebody is left facing a number and the score is that number”, which is a statement about a game whose end is reached by running out of moves worth making. Under misère play the last move is a loss rather than a formality, and the whole account of stopping has to be rebuilt.

They say nothing about how the fight was conducted. A stop is where play arrives, not what the moves were, and two positions with identical stops can require completely different play to reach them. That is the same gap a strategy is not a certificate is about, in miniature.

What the picture cannot show #

The figures here draw one position at a time, and two of the claims are about all positions.

That the mean always lies between the stops, and that a switch always has its options as its stops, are theorems rather than observations, and no finite collection of thermographs establishes them. The figures show the claims holding in the cases drawn; the site’s own recursion is where the claims are actually true, and the check that the drawn values agree with it is what the site’s gate does.

The other invisible thing is more interesting. Every figure above draws the stops as marks on a line, and a mark on a line looks like a position on the number line, which is precisely what an infinitesimal is not. A reader who takes the drawing literally will conclude that ↑ sits at zero, and ↑ does not sit anywhere: it is above zero and below every positive number, a place the drawing has no room for. Drawing it as a mark at zero is a compromise made knowingly, and the figures that separate the infinitesimals do it by comparison rather than by position for that reason.

The convention, named #

Everything above is normal play, and everything above assumes the score is what has been collected when the fighting stops.

That second assumption is stronger than it sounds and it is a convention rather than a fact about games. A scoring game — Go, Dots and Boxes, anything played for points — pays out on a count at the end rather than on who is left unable to move, and counting at the end changes everything. The stops are the point at which the normal-play machinery gets closest to a score, and the resemblance is what makes them useful for Go endgames; the resemblance is not an identity, and the essay on scoring is the one that says where it breaks.

Within normal play the stops are exact, computed rather than estimated, and blind in exactly one direction. Everything they miss is smaller than every positive number, and everything they miss is what decides a game with no points left in it.