What a move is worth to the player making it

A player looking at a board asks a question the theory does not answer directly. Not what is this worth — that is the value, and it is computed — but what does moving here gain over moving there.

The gain has a name and a definition, and the definition is one subtraction. The incentive of Left’s move to an option G^L is G^L − G: where the move lands, minus where it started. Right’s incentive for G^R is G − G^R, the same subtraction with the sign that makes a good move for Right come out positive.

The subtraction is the ordinary one, so the answer is not a number. It is a game.

Why a game and not a number #

The temptation is to define the gain as a difference of scores, and the reason that cannot be done is the reason the subject exists.

Most positions are not worth a number. {5 | 1} is not 3; it is a fight whose settlement depends on who moves. Subtracting one such object from another cannot produce a number either, and the theory has no operation that would force it to — subtraction is defined on games, it produces games, and the difference is played rather than evaluated.

So an incentive is an object of exactly the same kind as the position it came from. It has an outcome class, a temperature, a mean, and options of its own. Left’s move in {5 | 1} has an incentive of {4 | 0}: another fight, worth two on average, with four points swinging on who takes it.

That is not a defect of the definition. It is the definition being honest about a fact players know: how much a move gains often depends on what the opponent does next, and a number cannot hold that dependency.

A worked incentive, and what it contains #

Take {5 | {4 | 0}} and compute Left’s incentive in full, because the answer is more interesting than the number that will be squeezed out of it.

Left’s option is 5. So the incentive is 5 − {5 | {4 | 0}}, which is 5 plus the negative of the position — and the negative of a position is the same position with the players exchanged, so the subtraction produces a game whose options mix Left’s 5 with Right’s reversed ones.

The recursion returns { {5 | 1} | 0}. That is not a number and not close to one: it is a fight with a follow-up, exactly like the position it came out of, and it has a temperature of its own.

Two things follow. The first is that “how much does this move gain” is a question whose answer needs the same machinery as the original question, so nothing has been simplified by asking it. The second is that the answer contains information a number would lose — the shape { {5 | 1} | 0} records that the gain itself is contested, which is the fact the rest of this essay turns on.

A player who wanted one number from that object would take its mean, which is 1. A player who wanted to know whether the move can be answered would look at its options, which is a different question with a different answer.

Moves are only partially ordered #

The consequence lands immediately and it is uncomfortable.

Games are partially ordered: given two of them, one may be greater, one may be smaller, they may be equal, or they may be confused — neither at least the other. Incentives are games. So of two moves available in a position, it can happen that neither incentive is at least the other.

When that happens there is no best move in the ordering sense, and a program that sorts the moves by their incentives is sorting objects with no total order on them. The usual response — take the mean of each incentive and sort by that — is a decision to throw away the part of the incentive that made the comparison fail, and it is worth knowing that is what is being done.

Comparing incentives is comparing options #

Before going further it is worth noticing that the comparison the last section performs is one this site already does constantly under another name, and that the two are literally the same subtraction.

Left’s incentives for two options are $G^{L_1} - G$ and $G^{L_2} - G$. Ask which is larger and the $-G$ cancels from both sides: the question is whether $G^{L_1} \geq G^{L_2}$, with no reference to $G$ at all.

So the order on Left’s incentives is the order on Left’s options, shifted by a constant. Two moves are confused as incentives exactly when the positions they lead to are confused, and “which move gains more” and “which option is better” are one question.

That is worth having because the second question has a name and a use. An option that is at least as good as another is said to dominate it, and the canonical form deletes every dominated option — nobody would take the worse of two moves when one is at least as good as the other, so the worse one carries no information about the position.

Which is why the universe is as large as it is #

Follow that one step and the partial order stops being an inconvenience and becomes the reason the subject has any content.

A canonical form has no dominated options, so its Left options are an antichain: a set of positions no two of which compare. If incentives were totally ordered there would be no antichains bigger than one — every set of options would collapse to its best member — and every canonical form would have at most one option on each side.

Count what that costs. The 22 values born by day two support 98 antichains, so a day-three form is one of $98 \times 98 = 9{,}604$ option-set pairs, which reduce to 1,474 distinct values. Were the 22 totally ordered, the antichains would be the 22 singletons plus the empty set — 23 of them — giving 529 forms and correspondingly fewer values.

So the confusion among incentives is what makes the universe big. Not by a little: it takes the number of option sets from 23 to 98 at the first step where it can act, and that number is squared to make the next day. Every essay on this site about how quickly the values proliferate is, underneath, an essay about moves that cannot be ranked.

The same fact, read as consolation and as cost #

That gives the partial order two readings, and both are worth carrying.

As a cost: a player choosing between two confused moves has no help from the ordering, and a program sorting moves by incentive is sorting objects that do not sort. The usual repair — take each incentive’s mean and sort on that — is throwing away exactly the part that made the comparison fail, and the section above says so.

As the reason there is anything to say: a game whose moves were totally ordered would have canonical forms with one option a side, which is to say it would be a game with no interesting positions in it. There would be no dominated options to remove because there would be nothing to remove them from, no thousand values at day three, and no theory beyond a count of how many moves each player has left.

The two readings are the same measurement. Where the theory is hard is exactly where it is not empty, and incentives are where that is easiest to see, because they are the objects a player is comparing when they say two moves look about as good as each other.

Nobody moves inside a number #

The clearest thing the incentive says is about the positions where nothing is at stake.

Take the number 1. Left’s only option is 0, so Left’s incentive is 0 − 1 = −1. Take 1/2: Left’s option is 0 and the incentive is −1/2. Take 2: the option is 1, the incentive is −1.

That is numbers avoid numbers restated in one line and derived rather than asserted. The theorem says a move inside a numeric component is never the right move when any other component has a move in it; in incentive terms it says the numeric component’s incentives are negative while the others’ are not, and negative is worse than not-negative.

The convention worth naming here is that this is a statement about components, not about moves in general. Moving to a number is frequently right; moving inside one is not.

Temperature is the largest incentive, priced #

There is one number a player usually wants: how much is this position worth moving in at all. That is the temperature, and it is what happens when the incentives are collapsed to a single figure.

Each incentive has a mean — the number it settles to when repeated — and the mean of an incentive is the natural candidate for “how much this move gains on average”. Take the largest of those means across all the options, and for most positions the answer is the temperature exactly.

The agreement is not decorative. It says the two ideas are one idea seen from two ends: temperature was defined by taxing moves until nobody wanted one, and an incentive is what a move gains, so the tax at which every incentive stops being worth paying is the tax at which the walls of the thermograph meet.

A switch is the easy case, because its options are numbers and the subtraction is forced to give the same fight on both sides. A position whose option is itself contested is the real test of the agreement.

Where the two come apart #

The agreement is not universal, and the exceptions are a recognisable class.

For {5 | {4 | 0}} the temperature is 1. Left’s incentive has a mean of 1, matching. Right’s incentive has a mean of 2 — twice the temperature of the position it was played in.

For {10 | {9 | 1}} the temperature is again 1, and Right’s incentive has a mean of 4.

What is happening is the thing Go players named centuries before anybody wrote a brace: the move is sente. Right can play it and Left must answer, so Right does not keep what the incentive appears to offer — the exchange happens, the board returns to something similar, and the accounting closes at one point rather than four.

The incentive is not wrong about the swing. It is answering “how far does this move move the position”, and the temperature is answering “how much of that is the mover going to keep”. Sente is a fact about the rest of the board is the essay about which of those two a player should be sorting on, and it turns on precisely these rows.

What a player should do with the pair #

Given a position with several components, the practical rule this all supports is short.

Move in the component with the largest temperature — that is playing the hottest, it has a proved bound, and it costs at most the largest temperature on the board against optimal play.

Do not sort by incentive means. Where the two orderings differ, the incentive is the one that overstates, and it always overstates in the same direction: it inflates sente moves, so a player following it plays them too early, every time.

The inversion is not a peculiarity of those two positions. Right’s move in {10 | {9 | 1}} moves the board a long way and keeps almost none of it, because Left answers at once — the incentive is measuring the distance and the temperature is measuring the keep. Any component with a follow-up of that shape can be made to overstate by as much as anybody likes, by pushing its options further apart while leaving the answer intact.

That is the same finding big is not the same as hot reports from the other direction, measured over 240 played-out lines rather than derived from a definition. Two routes to one conclusion is worth having, because the definition explains why the count overrates sente and the experiment says how often it matters.

Cooling, and the tax that kills every incentive #

The cleanest way to see temperature as a statement about incentives is to charge the tax and watch them die.

Cooling by t charges t for each move, which subtracts t from every incentive. Raise t far enough and every incentive is negative: nobody wants to move, the position freezes, and the number it freezes at is its mean. The height at which that happens is the temperature, and it is the least tax that makes every incentive unattractive.

So the three quantities this site keeps separate — incentive, temperature, mean — are one construction sliced three ways. The incentive is the gain from a move. The temperature is the tax at which every gain vanishes. The mean is what is left when it has.

The same shape in three other places #

The pattern — a quantity that is a game, collapsed to a number for practical use, with the collapse losing exactly one thing — is not confined to incentives, and it is worth naming because this site keeps meeting it.

A value collapsed to an outcome. The four outcome classes are what survives when a value is asked only who wins, and outcomes do not add is the report of what that collapse loses.

A thermograph collapsed to a temperature. The whole diagram is a function of the tax; the temperature is one height off it, and temperatures do not add is the report of what that collapse loses.

A position collapsed to its stops. Two numbers, exact about the points and blind to everything infinitesimal, which is where the fight stops.

In each case the collapsed quantity is the one a player can carry in their head, the uncollapsed one is the object the theorems are about, and the discrepancy is not noise. It is a specific, nameable class of positions: the sums for outcomes, the hot components for temperatures, the infinitesimals for stops, and the sente moves for incentives.

That is a useful habit to bring to any summary in this subject. Ask what the summary was computed from, and then ask which positions are in the kernel of the map — the ones the summary cannot distinguish. They are always the interesting ones, because they are exactly where the summary is about to mislead somebody.

Where the model stops #

Two limits worth stating.

Incentives do not add across components. The incentive of a move in one part of a sum is computed inside that part, and the sum’s own incentives are different objects. That is not a difficulty in practice — it is what makes local analysis possible at all — but it does mean that “the biggest incentive on the board” is a comparison of local quantities, and the theorem attached to it is about temperature rather than about the comparison.

An infinitesimal position has infinitesimal incentives, and they still decide games. In ↑ the incentives are ↓ and ↑∗, neither of which is any number at all. A position where every incentive is infinitesimal is a position where the temperature is zero and the game is decided by how many ups rather than by how many points — the machinery above says nothing there, and the machinery that does is a different one.

What the picture cannot show #

The figures draw incentives as values and cannot draw the thing that makes them awkward: that the ordering between two of them may not exist.

A row saying ‖ is drawn identically to a row saying > — a symbol in a column — and the difference between “this move is worse” and “these two moves are not comparable” is the whole content of the section on partial order. No arrangement of marks on a page distinguishes them, because incomparability is not a position on a scale; it is the absence of one.

The second invisible thing is the search. Each row of the first figure prints how many positions were walked to settle it, and those counts are two and three digits long for positions with three or four moves in them. The cost of asking what a move is worth grows with the position, and a figure that showed only the answers would suggest the answers are cheap.

The convention, named #

Normal play throughout, and short games throughout.

Both matter here. The definition of an incentive is a subtraction, subtraction is defined by negation, and negation works because G + (−G) = 0 — which is a normal-play theorem and fails under misère play completely. There is no such thing as the incentive of a move in a misère game, not because nobody has defined one, but because the subtraction it would be built from does not exist.

The other convention is quieter. Everything above measures a move by what it does to the value of the position, and a scoring game measures a move by the points it banks. The two agree where the normal-play skeleton of the scoring game agrees with it, and counting at the end changes everything is the essay about how far that is.