When to leave the environment

A coupon stack is an environment with the vagueness taken out of it. Instead of the rest of the board is worth about this much, there is a pile of coupons falling in fixed steps, and a move is either a coupon off the top or a move on the board. An environment made of coupons builds it; two games in one environment puts two fights in one, and closed by naming the question the arrangement makes askable:

The coupon stack turns the same question into one about when to leave the environment, which is the form a Go player would recognise.

A Go player does not usually ask which of two endgame moves is bigger. They ask when to stop taking the small stuff and start on the big point, and the orthodox answer is a rule with a number in it: leave the environment when the coupon on top falls to the temperature of the hottest thing on the board.

The census #

The pool is nine fights: four plain switches, four with a follow-up, and one large symmetric switch. The mix is deliberate rather than convenient — a pool of plain switches alone would have found the orthodox rule exact everywhere and reported nothing, and a pool of nested fights alone would have found it inexact everywhere and reported the same nothing twice. Every ordered pair of them is put beside a stack of coupons falling in halves from five, and the whole thing is played out exactly — no strategy, no rule, just the full recursion over the sum.

The measurement is the coupon that would have been taken next when somebody first plays on the board. That number is the point at which optimal play judged the environment no longer worth staying in.

Sixty of the eighty-one leave exactly at the board’s temperature. Sixteen leave one coupon early, five one coupon late, and no departure exceeds one coupon — which the census asserts rather than reports.

Three quarters of the pairs following a rule exactly is a good score for a rule stated without qualification, and it is not the interesting number on this page. The interesting number is which quarter.

The exceptions are all sente #

Splitting the same eighty-one by whether either fight has a follow-up gives a much sharper picture.

Twenty-five of the eighty-one are pairs of plain switches, and all twenty-five leave on time. A plain switch is one whose options are numbers, so a move in it settles the fight and the opponent has nothing to answer.

All twenty-one departures are among the fifty-six pairs with a follow-up in them. The census asserts that too: a pair of plain switches leaving at the wrong coupon would stop the build.

Put the other way round, the fifty-six with a follow-up leave on time thirty-five times, which is 63 per cent against the plain pairs’ hundred. So having sente in the pair does not make the rule fail; it makes it fail sometimes, and the rest of this page is about which sometimes.

That is the same boundary every temperature result on this site turns out to have. The theory is stated for positions whose value is settled by the move that is made, and a position with an answer worth making is where the statement stops being exact — the same class that makes the sente crossover halve and makes the guarantee for playing the hottest stop one step down the stack.

Which way they are wrong #

Sixteen leave early and five late, and the asymmetry is worth reading.

Leaving early means playing on the board while the coupons are still worth more than the board’s temperature. That is what a fight with a follow-up buys: a move that will be answered costs the opponent a move as well, so it is worth more than its own temperature suggests, and taking it before the coupons have fallen that far is correct.

Leaving late means taking one more coupon than the rule says. Five pairs do it and three of them have a board temperature of three quarters, which is not a value the half-point coupons can stop at — the players take the coupon at a half because there is no coupon at three quarters, and the departure is an artefact of the grid rather than of the game.

So the honest split is sixteen genuine departures, all early, all caused by sente, and five that are mostly about the resolution of the stack.

Why the hottest is not always where they go #

Five pairs enter the colder fight first, and they are worth naming because they are the cleanest counterexample on the site to a rule most players treat as unconditional.

Each of the five has a fight with a follow-up in it and the follow-up is the reason. A move in a fight with an answer is worth more than the fight’s temperature, because it takes the opponent’s move as well as making one’s own; so a fight of temperature 1 with a follow-up can be worth entering before a fight of temperature 2 without one.

The thermograph does not say this on its face. The temperature is read at the top of the diagram and the follow-up is a feature at the bottom of one wall — which is exactly where the crossover factor lives too — so a player comparing two temperatures is comparing the wrong quantity on those five pairs.

What they should compare is the temperature plus what the answer is worth, which is not a quantity this site has a name for and is not a quantity a thermograph prints. That is a gap rather than a finding, and it is the same gap the early departures are made of: sente is worth something and nothing here measures how much.

What a Go player already knows #

The early departures are the recognisable half of this page, because they are a piece of Go folklore stated in the theory’s terms.

Take the sente first. A move that must be answered costs nothing in tempo — the player makes it, the opponent answers, and the player still has the move. So a sente move is worth taking before the position calls for it, which is exactly leave the environment early when the board has sente in it.

The census puts a size on the folklore. The move comes one coupon early, not two and not five. That is a smaller effect than the folklore’s confidence suggests and it is a real one, and it is measured over fifty-six pairs rather than argued.

And it explains why the rule looks exact to most players most of the time. An endgame is mostly gote — plain fights whose answers settle them — and on those the rule is exact. The sente points are the minority and the departure they cause is one step. A player following the orthodox rule strictly gives away one coupon on the positions where it does not apply.

Which fight, once they have left #

There is a second question the arrangement answers for free, and it is the one the rung below actually asked.

Seventy-six of the eighty-one enter the hotter fight first. Play in the hottest is right on 94 per cent of the pairs, and the five exceptions are again pairs with follow-ups.

That is worth stating as a comparison rather than as a number. Over a bare sum with no environment the same rule is exact far less often and its guarantee is loose by a factor of seven; here it is very nearly forced. The two measurements are of the same rule and they disagree because the surroundings differ, which is the standing lesson of every result in this field.

That is a much higher score than the same rule gets against optimal play in a bare sum, and the reason is the environment. A coupon stack gives both players something to do at every temperature, so a fight is entered when its temperature comes round rather than when it happens to be the largest thing left — and hottest first becomes nearly forced rather than nearly right.

So the environment makes the greedy rule better, not worse, which is the opposite of what a reader expects from a construction described as an adversary. An environment is not an adversary; it is a clock.

What the stack is standing in for #

Worth saying, because the whole construction is an idealisation and its idealisation is the thing that makes it useful.

A real board’s remaining moves are not a stack of coupons. They are positions with their own values, their own follow-ups and their own interactions, and modelling them as a monotone stack of independent one-move gains throws all of that away.

What it keeps is the one thing the question needs: a well-defined ambient temperature at every moment. The coupon on top is the ambient temperature, exactly, and it falls in known steps. That is what makes when to leave a question with a numerical answer rather than a matter of judgement.

The cost is that the answer is about the idealisation. On a real board the ambient is not monotone — a fight resolves and leaves a bigger one behind — and the departures this page measures at one coupon could be larger where the ambient jumps. That is the limitation to carry, and it is the same one the whole coupon apparatus carries.

The one number a player can carry #

Setting the three findings side by side gives a rule of thumb with its exception attached, which is the most a census of eighty-one pairs can honestly produce.

Leave the environment when the coupon falls to the hottest temperature on the board — and one coupon earlier if that fight has an answer worth making.

That is exact on all twenty-five plain pairs and on forty of the fifty-six with a follow-up, and it fixes fifteen of the sixteen early departures. What it does not fix is the five that leave late, three of which are the coupon grid and two of which are genuinely one coupon out in the other direction.

So a single amended rule reaches seventy-five of eighty-one, against sixty for the orthodox one. The amendment costs a reader one question — does the hottest fight have an answer worth making? — and that question is readable off the foot of a wall, so it costs nothing a player who draws thermographs does not already have.

The honest caveat is that this is a rule fitted to eighty-one pairs rather than derived. The mechanism behind it is sound and stated above; whether the amendment is one coupon on a pool with larger follow-ups is exactly what the rung above should measure, and the answer might well be that the amendment has a size rather than a value.

Two questions the stack answers and one it does not #

The environment supports three questions that are easy to run together, and the sweeps on this ladder answer them in different places, so it is worth separating them once.

When is a component entered? That is what a coupon stack measures directly: the value of the top coupon the first time optimal play moves in a given position. It is a reading about that position on that board.

When should a player leave the environment? That is a decision rather than a reading — stop taking coupons and start taking real fights — and it is the question a Go player has. It is answered by the same transcripts, but it is a statement about the board rather than about any component.

Which component should be entered first? That is a third question and the stack does not answer it at all. A transcript records an order and an order is a consequence of the whole board, so reading a ranking rule off it is reading a conclusion out of one instance.

The three come apart, and this ladder’s later rungs are the demonstration. The quantity that predicts when a component is entered — the larger of its two temperatures — turns out to be a bad ranking rule, worse than plain temperature and outside the guarantee. So an answer to the first question is not an answer to the third, however natural the step looks.

The habit that follows is to say which of the three a number is about whenever one is quoted. A departure coupon is about a board; an entry coupon is about a component on a board; and a ranking is about neither until it has been scored against optimal play on many boards.

What the census does not say #

Four limits.

Nine fights and two at a time. Eighty-one pairs from one pool. A board with five fights on it is a different measurement and the rule’s exceptions may compound; nothing here tests that.

Half-point coupons from five. The resolution of the stack sets the precision of every answer, and three of the five late departures are the grid rather than the game. A finer stack would remove those and would cost a much larger recursion.

Follow-up is a crude class. A fight is counted as having a follow-up when its Right option is not a number, which is one bit. How big the follow-up is does not enter, and the sixteen early departures are all one coupon regardless — so either the size does not matter or the pool does not have enough spread in it to show that it does.

And optimal play is not a player. Everything here is the full recursion over the sum, which is what a solver does and not what anybody does at a board. The value of the page for a player is the rule and its exception class, not the eighty-one numbers.

The convention, named #

Normal play. A coupon stack of n coupons with values falling by a fixed step is a game in which a move takes the top coupon and banks its value for the mover; the stack is built recursively and the banked total is a number at the bottom.

The ambient temperature at any moment is the value of the coupon on top. The board’s temperature is the largest temperature among the fights not yet resolved, computed from their thermographs, with a number given temperature −1.

Play is the full minimax over the sum of the fights and the stack, with Left moving first. The leaving coupon is the value that stood on top of the stack when the first move on the board was made; ties between a coupon and a board move are resolved in favour of the board move, which is the tie convention the play-out uses and is why the measured coupon sits at the temperature rather than one step below it.

Where the ladder goes next #

The coupons anchor has three rungs to here, and the four above it take the class this page identifies — positions with follow-ups — and turn it first into a formula and then into an attempted strategy that fails instructively.

How big the answer is grades the follow-ups by size, which this page’s pool could not do because all of its were about the same. With a position’s own temperature held at one and its follow-up’s swept from one to four, the departure coupon runs from 1 to 3.5, and across the whole grid the players leave at the larger of the two temperatures.

The two rungs after that try to use that quantity to order a board, and it is worth reading them in order because the first fails and the second succeeds by inverting it. The quantity that does not order a board promotes each component to the larger of its two temperatures and plays in the highest: over 220 three-component boards it plays exactly on 124 against playing-in-the-hottest’s 196, loses 85 of the 97 disagreements, and breaks Hotstrat’s guarantee outright on six. A quantity that predicts when a component is entered does not order the components.

A rule that beats the hottest then tries the reverse — discount a component by its answer’s temperature instead of promoting it — with an explicit prediction that it would not beat playing in the hottest. It does: 201 exact against 196, two thirds of the disagreements, inside the other rule’s guarantee, and the gap widens as the board grows.

The worst value in its own interval closes the anchor with a fact about ranking rules in general. Every discount weight strictly between nought and one scores the same and beats the choice of one at every board size — because a ranking rule’s score is a step function of its own coefficient, and one is exactly the value at which two components tie.