Nobody wants to move here

There are two ways to find out whether a position is worth a number, and they have nothing in common.

The first walks the tree. A position is a number when every Left option is strictly below it, every Right option strictly above it, and the same holds all the way down — which is a recursion over the whole form and a definition that mentions numbers in its own statement. The second never mentions a number at all. It asks, of each move available, whether taking it makes things worse for the player who takes it; and if the answer is yes every time, the position is a number.

The claim is that the two columns always agree. Stated as a theorem it is short — a game is a number if and only if all of its incentives are negative — and stated that way it is easy to nod at and hard to feel. What follows is the census that makes it a fact about this site rather than a quotation, together with a second finding the census produced and nobody was looking for.

What an incentive is, and why it is a game #

The word is borrowed from economics and the object is not a number. The incentive of a move is the difference between the position after the move and the position before it, computed from the mover’s own side of the board:

$$\text{Left's incentive for } G^L = G^L - G, \qquad \text{Right's incentive for } G^R = G - G^R$$

Both are games, obtained by subtracting one position from another, which means adding the mirror image and reducing. A difference of two games is exactly the object comparison is built out of, and the four things it can be — positive, negative, zero, or confused with zero — are the four things the comparison can report.

Negative means the move costs the mover something. That is the ordinary case and it is worth dwelling on, because it sounds wrong. A player about to move in a position worth $2$ has one move, to $1$, and it takes a whole point off what the board is worth to them. Under normal play the last player to move wins, so a move is a resource spent, and a position where every move is a loss is a position both players would rather leave alone.

That is what a number is. Not a quantity: a standoff.

The test, run over a whole day #

Five rows are an illustration. The theorem is a claim about every position there is, so the check runs over every value born by day three — the whole set of values reachable from nothing in three moves’ worth of construction, 1,474 of them, enumerated rather than sampled.

The agreement is the theorem and is the less interesting half. The classification is the finding.

A position fails the test when some incentive is not strictly negative, and not-negative is three different things. It could be positive: the mover gains by moving. It could be zero: the move is free, gaining nothing and costing nothing. Or it could be confused with zero: the difference is neither above nor below, which is the fourth relation and the one with no analogue among numbers.

Over the whole of day three, it is the third every time. Not one position born by day three has a move that gains its mover; not one has a move that is exactly free. Every single position that is not a number is not a number because a move is confused with doing nothing.

What that actually says #

“Confused with zero” means the difference $G^L - G$ is a first-player win. Left moving to $G^L$ neither gains nor loses in any absolute sense; what it does is seize the initiative, and whether that was worth doing depends on who moves next.

So the picture a reader is likely to have — that a hot position is one where a move is profitable, and that players fight over it because there is something to win — turns out to be wrong in a precise way. Over an entire day of values there is no profitable move anywhere. Moving is never good. What happens in a hot position is that moving is not clearly bad either, and so the player who is forced to wait is the one who suffers.

The vocabulary the subject uses — hot, at stake, worth fighting over — is all about how urgent it is to move rather than how much moving gains. That is not a loose way of speaking. It is exactly what the census says.

Zero passes for a reason worth stating #

One of the 1,474 positions passes the test with no incentives at all. The position with no moves has no options, so “every incentive is strictly negative” is a claim about an empty collection, and a claim about an empty collection is true.

That is not a technicality to be waved through. The whole theory is built by recursion from that position, and every number on the site descends from it. The base case of the number test is a quantifier over nothing — which is the same reason the position with no moves is a second-player win, and the same reason the recursion terminates at all. A definition that had to make an exception for the empty position would be a definition with a hole at the bottom of it.

The count is reported separately in the census for the same reason. A perfect agreement that rested on vacuous cases would be a perfect agreement about nothing; here it rests on one, and the other 1,473 do real work.

Temperature is the same measurement, read louder #

If a position is not a number, at least one incentive fails to be negative. How badly it fails is a number, and that number has a name.

The temperature of a position is how much tax has to be charged on every move before the position freezes into a number. Cooling by $t$ subtracts $t$ from every Left option and adds $t$ to every Right one — it makes moving more expensive — and the temperature is the tax at which the last non-negative incentive finally goes negative.

So the number test and the temperature are the same question at two resolutions. The test asks whether any incentive fails to be negative, and answers yes or no. The temperature asks by how much the worst one fails, and answers with a number. A position of temperature zero has no incentive above zero and is cold; a position with a strictly positive temperature has one, and the temperature is how far above.

That is a satisfying arrangement and it has one wrinkle, which is worth saying out loud because it is the case a reader will meet first.

The infinitesimals, which are cold and not numbers #

$\uparrow$ is not a number. Its temperature is zero. Both statements are true and they look like a contradiction.

The resolution is that “temperature zero” and “every incentive negative” are not the same condition after all. $\uparrow$ is $\{0 \mid \ast\}$, and Left’s incentive for moving to $0$ is $0 - \uparrow = \downarrow$, which is negative. Right’s incentive for moving to $\ast$ is $\uparrow - \ast = \uparrow\!\ast$, which is neither positive nor negative nor zero: it is confused with zero. So $\uparrow$ fails the incentive test — correctly, since it is not a number — while its thermograph is a vertical line, because the failure is by an amount no number measures.

This is the standing pattern of the subject and it appears here in a compact form. The coarse measurement — the temperature — is a number and is enormously useful. The fine measurement — the incentive — is a game, is harder to read, and is the one that decides. Small things decide is not a slogan about edge cases; it is a statement about which of two measurements is complete.

The three claims that follow, all of them checkable #

Three familiar theorems fall out of the test once it is stated in this form, and each of them reads better as a statement about incentives than in its usual dress.

Numbers avoid numbers. The theorem says that in a sum containing a number and something hot, a player should never move in the number. In incentive language it is almost a tautology: every incentive in the number is strictly negative, and there is an incentive elsewhere on the board that is not, so moving in the number is strictly worse than moving where the fight is.

A number is where a fight stops. The stops are defined by playing on until somebody faces a number, and now the stopping condition has a reason rather than a convention behind it: a number is exactly the position at which both players have run out of moves they can afford.

Adding a number cannot start a fight. Adding $x$ to $G$ shifts every incentive by nothing at all — the $x$ appears on both sides of $G^L - G$ and cancels — so translation leaves the temperature alone. A number can decide who wins and cannot decide whether there is anything to win.

The case the census could not produce #

A theorem is worth more when the reader can see what its failure would look like. Two of the three ways to fail the incentive test came back empty over day three, which raises the obvious question: can they happen at all?

A positive incentive would be a move that improves the mover’s position outright — $G^L > G$, so Left is better off having moved than having the position with the move still available. Under normal play that is impossible, and the reason is one line: $G^L - G > 0$ would say Left wins $G^L - G$ moving second, and Right moving second in that game has the mirror strategy of answering in the copy the opponent moved in, which never loses. So the first empty column is empty everywhere and not merely on day three.

An incentive of exactly zero looks like a different matter and is not. It would be a move that changes nothing at all: $G^L = G$, a Left option worth exactly what the position it sits in is worth. The same theorem forbids it, because the theorem is not about positivity — it says $G^L \ntriangleright G$, that no Left option is greater than or equal to its own position, and equality is one of the two things that rules out.

The proof is the one above, stated for the relation rather than for the strict inequality. Play $G^L - G$ with Right to move. Right answers in the second copy, moving $-G$ to $-G^L$, and the position is $G^L - G^L$ with Left to move — a second-player win, so Left loses. Right moving first therefore wins $G^L - G$, which is exactly the statement that $G^L - G$ is not greater than or equal to nought. It is strictly negative or it is confused, and there is no third possibility.

So the two empty columns are not two facts of different kinds. They are one theorem read twice, and the census’s classification is not a fact about day three at all: every incentive of every short game is either strictly negative or confused with zero, on day four and on a board with forty pieces on it, for the same three lines of play.

The check that would have told the two apart, had they been different, is to run the census on forms rather than on values — since a fact about canonical forms fails on unreduced ones and a theorem does not. Run over all 256 forms built from day-one options it comes back the same: no positive incentive and no zero one. Run over the 5,584 options carried by the 1,474 values of day three, no option is greater than or equal to the position it belongs to, not once.

So this is the shape a good theorem has from the inside. The census was designed to separate two counts and found them both nought; the argument then explained both with one sentence; and the sentence is the sentence an option nobody would take is built on, arriving here from the opposite direction. That page adds options that fail $H \ntriangleright G$; this one observes that a position’s own options already fail it, so every option a position has was already a gift it could have been handed.

What the solver computed, and how #

Each of the 1,474 values is a canonical form held as a small tree. For each one the incentives are built by taking the difference with each option — construct the mirror image, add, reduce — and then compared with zero by the ordinary comparison, which itself plays the difference game and reads off its outcome. Nothing is quoted from a table: the incentive of $\{2 \mid 0\}$'s Left option is computed as $2 - \{2 \mid 0\}$, canonicalised, and asked which of the four outcome classes it lands in.

The classification of failures is the same computation read one step further. A failing incentive is tested against zero for being greater, then for being equal, then for being confused, and the three counts are kept apart. Two of the three came back empty, which is the finding.

The check that could have gone wrong and did not is the agreement itself: the two tests use no shared code beyond the comparison, and a disagreement on any of the 1,474 would have been reported as a disagreement rather than reconciled. There were none.

Where the model stops #

The census is a day, and a day is finite. Day three holds 1,474 values, day four holds a number nobody has written down, and the theorem is about all of them. What has been established here is that the equivalence holds everywhere it has been checked and that the classification of failures holds everywhere it has been checked; the second of those is the more fragile claim, since it is a statement about how positions fail rather than about whether, and one day further out is one day this site cannot reach.

There is a second limit, and it is the one that matters more. Everything above is normal play. Under misère play the last player to move loses, so a move is no longer a resource spent, and the sentence “moving is always bad” — the sentence the whole picture rests on — is inverted at the bottom of the tree and mangled everywhere above it. Incentives can still be computed; they no longer classify anything, because misère play has no negatives and the difference of two positions is not a difference.

Where the ladder goes next #

The rung below this one asks what a single move is worth and answers with a game. This one asks what happens when every move is worth less than nothing, and finds the numbers. The rung above asks the question the census refused to answer: what the incentives look like one day further out, where the classification could still come apart.

What the theorem in the section above does to that plan is worth saying, because it removes the interesting half of it. The classification cannot come apart one day further out: every incentive is strictly negative or confused with zero is proved rather than measured, so a day-four census would find the same two empty columns for the same reason and would be measuring the arithmetic rather than the day.

What a day further out could still change is the agreement — the claim that the incentive test and the number test never disagree — and that is a theorem too. So the honest statement of what is left is narrower and more interesting than check the next day: it is that this page’s two findings have quite different standing, one of them a census that confirms a theorem and the other a census that was superseded by one, and the way to tell which is to try to prove it before running it.

Two neighbours are worth following instead. What a number does to a fight takes the cancellation noticed above and measures it over ten thousand translations. And the reduction that puts options back is about the other half of the canonical form — the half where an option nobody would take is not deleted but replaced, and the form gets bigger.