What a move nobody makes is worth

Sente is a fact about the rest of the board established the shape of the idea. A move is sente when the opponent has to answer it locally rather than spending the move elsewhere, and whether they have to is decided by the ambient temperature — by what is available on the rest of the board, not by anything in the position itself.

Turn that round. If Right’s move here is sente over some band of ambient temperatures, then Left’s move here is reverse sente over that band: it prevents an exchange Right was going to get for nothing, and it is the move a player is told, in every Go book ever written, to take before the opponent does.

The awkward question is what it is worth.

They are not the same band, and the gap is the essay.

The phrase itself is worth pinning down before anything is measured. Reverse sente is a Go term and it names a move by its effect on the opponent rather than by anything it achieves: the player making it gains little or nothing directly, and what they buy is the removal of a free exchange from the opponent’s account. Every account of it in the literature prices it by a rule of thumb, and the rules of thumb disagree with each other.

Two prices a reader arrives with #

The first is the local temperature. Every other move on the board is priced at the temperature of the region it is in, and a player choosing where to move compares those numbers. If reverse sente is a move like any other, it costs what its region is worth.

The second is the swing of the exchange it prevents. Right’s sente move here gains something and then has to be answered; the size of that gain — twice the temperature of the follow-up — is what the reverse sente denies. If reverse sente is a defensive move, it is worth what it saves.

Over the sixty measurements in the sweep, the gain matches the local temperature once and matches the swing twice. Neither is the price.

How the measurement is made #

Put the local position beside an environment of temperature $t$ — a stack of moves each worth $t$, which is what an ambient temperature is when made concrete. Then compare two boards, both with Left to play:

• Left takes the local move, and the whole board is played out exactly from there;
• Left plays in the environment instead, leaving the fight standing, and the whole board is played out exactly from there.

Both boards are finished by an evaluator that computes rather than estimates, so the two final scores are exact. The gap between them is what the local move was worth relative to the best alternative available, which is the only sense in which a move has a price at all.

That is a number about a pair of boards. Change the environment and it changes, which is the same lesson the rung below teaches about sente itself and is why this anchor exists.

The two bands come apart #

$\{5 \mid \{4 \mid 0\}\}$ is the sharpest case. Right’s move in it is answered at every ambient temperature from a quarter to two — a wide band, and the position is a textbook sente. Left’s move in it is worth taking at a quarter and at nothing above.

So the fight is sente over a band eight times as wide as the band over which reversing it is the right thing to do. A player who reads this is sente for the opponent, so take it first has a rule that is correct at the bottom of the range and wrong through most of it.

$\{2 \mid \{1 \mid 0\}\}$ is the same shape at a smaller scale: sente over a quarter and a half, worth reversing at a quarter alone.

$\{\,\{6 \mid 2\} \mid \{1 \mid -3\}\}$ is the case where the two bands coincide — sente over a quarter to two, worth taking over a quarter to two — and it is the only one of the six where they do.

Six fights, and what each one does #

Reading the table row by row is worth the time, because the six behave in four different ways.

$\{\,\{5 \mid 3\} \mid \{2 \mid -4\}\}$ has a follow-up on both sides and a temperature of two. Right’s move is answered up to an ambient of three; Left’s is worth taking up to one. The gap is the widest in the sweep and the position is the one the double sente essay is built on, so the two bands are already known to be complicated before the reverse is asked about.

$\{5 \mid \{4 \mid 0\}\}$ is the plain sente: a single follow-up, of temperature two, sitting under a fight of temperature one. Answered up to two, worth reversing at a quarter.

$\{2 \mid \{1 \mid 0\}\}$ is the same shape, smaller, and behaves the same way — which is the check that the first result is about the shape rather than about the numbers.

$\{\,\{6 \mid 2\} \mid \{1 \mid -3\}\}$ is the case where the bands coincide, and what makes it different is that its two follow-ups are nearly equally hot. Neither player can afford to leave it, so the fight is live for both over the same range.

And the two with numbers underneath — $\{4 \mid 0\}$ and $\{\,\{4 \mid 2\} \mid 0\}$ — have no sente band at all and a wide taking band, which is the control.

Why the gain is largest at the bottom #

Every one of the six has its largest gain at the lowest ambient temperature swept, and that is not an artefact.

The gain is a comparison against the best alternative, and the alternative is worth the ambient temperature. Lower the ambient and the alternative gets worse while the local move stays the same size, so the difference widens. Raise it and the difference narrows until it crosses zero, which is the crossover the table reports.

Read that way, the taking band has a simple description that the crossover height does not: a reverse sente is worth taking exactly while it beats the biggest move elsewhere, which is a tautology until the two quantities are computed, and the point of computing them is that neither is the number the player has to hand. The biggest move elsewhere is easy to see on a board. What the local move is worth is not the local temperature, and that is the finding.

The rule that prices every move at the temperature of its region — play in the hottest component — is the rule this comparison is measured against, and where it loses is exactly where a move is worth something other than the temperature of the region it is in. The taking band is the answer to when does that happen, and it is narrow.

Why the bands differ #

The reason is that a reverse sente is not defensive in the way the phrase suggests. It is a move, and it costs a move, and the alternative to making it is making a different move somewhere else that is worth the ambient temperature.

At a low ambient temperature the alternative is worth little, so almost anything beats it and the reverse sente is taken. As the ambient rises the alternative gets better while the reverse sente stays the same size, and at some height the elsewhere move wins. The exchange the reverse sente was preventing does then happen — and it does not matter, because the player who allowed it took a bigger move in exchange.

That is exactly the argument for not answering a sente move at a high ambient temperature, run one ply earlier. The two are the same phenomenon and the two crossovers are at different heights, because one compares the follow-up against the environment and the other compares the whole local exchange against it.

The follow-up’s temperature is what decides whether Right’s move has to be answered; the whole fight is what decides whether Left’s move is worth making. Those are two different heights on two different walls of the same thermograph, so the two bands they generate have no reason to agree, and the finding is that on five of six they do not.

The two fights that are not sente at all #

Two of the six locals have a number as the follow-up: $\{4 \mid 0\}$ and $\{\,\{4 \mid 2\} \mid 0\}$. Right’s move in them leads nowhere — the position after it is settled — so Right’s move is never sente, at any ambient temperature, and the sente band is empty.

They are in the sweep on purpose. Left’s move in each is worth taking over a wide band, up to an ambient of one and a half, and its gain at a low ambient is the largest in the whole sweep: three and a half points for $\{4 \mid 0\}$.

That is what an ordinary big gote move looks like when it is measured this way, and it establishes the control. A large gain at a low ambient temperature is not evidence of reverse sente; it is evidence that the fight is large. The word reverse sente only means anything when there is a sente to reverse, and the two columns of the sweep are what separate the two cases.

The one time the temperature is right #

The gain equals the local temperature exactly once in sixty measurements, and the coincidence is worth naming rather than hiding, because a rule that is right once in sixty is a rule somebody will believe on the strength of the one.

The reason the temperature usually fails is structural. Temperature is the height at which nobody wants to move any more — a property of the position alone. What a move is worth is a comparison with an alternative, which involves the rest of the board. The two agree when the alternative is exactly a move of that temperature and the local exchange settles in one ply, and they part company as soon as either clause fails.

The swing does better — twice in sixty — and fails for the opposite reason. The swing is what the exchange is worth if it happens; the reverse sente is worth the difference between the exchange happening and something else happening instead, which is smaller whenever the something else is worth anything.

The same question, on fights that do not settle #

Six positions is a small pool and five of the six have an answer that ends the exchange. That is the easy half of the family, and the whole of the anchor above this one is about what changes when the answer starts another fight instead — so the census is run again on a pool built the other way.

That last row is the widest gap this site has measured — a factor of sixteen between the band over which a move must be answered and the band over which preventing it is worth a move. It is also the position rung one uses to show that a fight can be worth almost nothing and its answer a great deal, arriving here as the same fact priced from the other side.

The two three-level fights, $\{6 \mid \{4 \mid \{3 \mid 1\}\}\}$ and $\{8 \mid \{6 \mid \{5 \mid 3\}\}\}$, are the pair worth noticing. Both are sente only up to an ambient of a half — well under their follow-ups’ temperature of 1 — and both are worth reversing up to 3/4, which is above their sente band rather than at the bottom of it. So on this pool the taking band is not a sub-band of the sente band at all, and the ordering the first census suggested does not hold.

What the sweep is really measuring #

There is a temptation to read the gain column as the value of the move, and it is worth resisting.

The number is the difference between two boards, so it depends on the environment as much as on the position — and the environment here is a stack of equal coupons, which is a mathematical convenience rather than a board anybody plays. On a real board the alternative move is a specific move in a specific region with its own follow-ups, and the comparison is with that rather than with an abstract temperature.

What survives the abstraction is the shape of the answer: a band, with a crossover, whose position depends on the local diagram in a way neither of the two obvious numbers captures. That shape is the finding, and the numbers are its evidence.

The move that is never made #

There is something peculiar about the object being priced, and it is what makes the anchor worth a third rung.

If the reverse sente is correctly taken, the exchange it prevents never happens. If it is correctly not taken, the exchange happens and the reverse sente is never played. Either way, one of the two moves is a move that appears in no game record — and the whole of its value is the difference it makes to which of the two records occurs.

That is not a paradox, but it is a genuine difficulty in learning to see these moves. The evidence for a reverse sente being worth taking is a game that was not played, and a player who takes them correctly for years never sees the disaster avoided.

The measurement above is a way of seeing the counterfactual: both boards are played out, both scores computed, and the difference between the record that happened and the record that did not is a number on a table.

What this shares with the rung below #

Double sente is not a property of the position found that the phrase names a relationship between a position and a band of ambient temperatures, not a feature of the position, and that a position can be sente for one player, then for both, then for neither, as the ambient falls.

Reverse sente inherits all of that and adds a second band on top. So a full description of one local fight needs four numbers rather than one: the temperature at which Right’s move stops being sente, the temperature at which Left’s stops being sente, the temperature at which Left’s move stops being worth taking, and the temperature at which Right’s does. Nothing in the position announces any of them, and the last two are what this rung computes.

That is also why the orthodox endgame account cannot price these moves. It summarises each region by its mean and takes the stakes in turn from hottest down, which is a procedure over one number per region; a reverse sente is a comparison between two whole boards and has no number to contribute to such a sum. The account gets the move order right anyway whenever the taking band and the temperature order happen to agree, and this page is the measurement of how often that is.

Why a move nobody makes has a value at all #

The object being priced here is unusual enough to be worth defending before it is measured, because a reader can reasonably ask what it means to value something that does not happen.

A threat is not a move. It is the availability of a move, and availability is a property of a position rather than of a line of play. So the question is not what the move earns when it is played — a threat that gets answered earns nothing directly — but how the position differs from one in which the move is absent, which is a comparison between two games and is exactly the sort of thing this subject can compute.

That reframing is what makes the quantity finite and definite. A move nobody makes has no consequences to add up, and a position with a threat in it has a value that differs from the position without it. The difference is a game, the comparison is a search, and neither mentions whether anybody ever plays it.

It also explains why the value is not the size of the move. A threat worth ten points that will certainly be answered contributes nothing like ten points: what it contributes is the opponent’s obligation to spend a move answering, and an obligation to move is worth whatever a tempo is worth on that board. So a large threat and a small one can be worth the same, and a threat can be worth more than the move that carries it out.

Which is the sense in which this is a fact about the rest of the board rather than about the move. The threat is the same object in every position; what changes is what the opponent has to give up to answer it, and that is a quantity the local shape cannot carry.

The convention, and the limits #

Normal play, disjunctive sums, and an environment of equal coupons. The evaluator plays both boards out exactly, so no estimate enters at any point, and the scores compared are the scores of finished games.

Three things the sweep cannot say.

It cannot say what happens when there are two local fights, which is the situation a reverse sente actually arises in: taking one leaves the other, and the comparison is three-way rather than two-way. Every number above is a single fight against an environment.

It cannot say anything about ko, and Go players will notice the omission. A ko changes what “answering” means, because the answer may be a threat elsewhere, and the rule that makes Go finite is doing work the abstraction here quietly assumes away.

And it cannot say what the crossover height is as a formula. Eleven positions across the two censuses produce eleven crossovers, and where they fall is clearly a function of the diagram; what function is not settled here, and both pools are small enough that they should not be guessed from. The second pool does establish one negative that the first could not: the taking band is not always inside the sente band, so the two are genuinely independent readings rather than one nested in the other.

Where the ladder goes next #

The sente anchor has three rungs to here, and the six above are one crossover being measured, corrected three times, and finally proved to depend on nothing below the top of the position.

The answer that starts another fight finds the rule this anchor has carried from the start — a move is answered while the ambient temperature is below the follow-up’s — to be exact only when the answer ends the fight. When the answer starts another one the crossover moves, and what the halving is a function of locates the distinction in a feature of one wall: whether the follow-up’s right wall rises straight before it leans.

Then the correction itself is corrected twice. A subtraction not a factor builds a deliberately deep pool and finds the crossover to be the follow-up’s temperature less a half rather than half of it, with the two readings agreeing only where that temperature is one. Half of the smaller temperature then shows both to be regions of one law: over 128 fights with answers from a number up to a temperature of six, the correction is half the answer’s temperature, saturating at half the fight’s own.

The last two rungs settle how deep the dependence goes. The two numbers at the top groups positions by their two highest temperatures and finds the crossover single-valued on every group, however far apart the third is — so the answer depends on two numbers and nothing below them, established by construction rather than by proof. And the premises an induction would need says exactly what is still missing: the law holds at five levels and survives translation, heating and cooling, and none of that is the inductive step.

So the quantity this page prices is real and the number it is priced at moved three times, each time because a pool was narrower than the law it was being asked about.