Answer where the opponent played

Hottest-first play is a rule a person can follow at a board: find the component with the largest temperature and move in it. A rule with a guarantee proves it never costs more than that largest temperature, and the cheapest fight is not the yardstick found, over thousands of lines, that it never cost more than the second-largest. The price of a tie then took the rule apart. The rule leaves one decision open — which part to take when several share the hottest temperature — and the original sweeps settled it by taking whichever comes first on the board. Breaking those ties with minimax instead took the costly lines of eleven pools from 2,859 to 966. Most of what the rule costs, in other words, is not the rule. It is the tie.

That measurement used a search to break the ties, which says how much there is to win back and nothing about how a player could win it. The essay ended by naming two tie-breaks a player could actually use, and this one scores them.

Choices a player could make without searching #

The two named candidates are reflexes every Go player has. Answer locally: if one of the tied parts is the one the opponent has just moved in, take it. Prefer a follow-up: if one of the tied parts is such that the mover’s own move there leaves something still hot in that part — a move that threatens another — take it. The second is close to what a rule that beats the hottest does with its discount, applied only where temperature cannot decide.

A third is the mirror of the second and deserves testing because it sounds just as sensible. Deny a follow-up: prefer the tied part in which the opponent’s move would leave the opponent something hot — take away the opponent’s threat before the opponent can use it. And the local rule says nothing when the opponent’s last move was not into one of the tied parts, so it combines naturally with either of the others as a fallback: answer locally, and otherwise prefer a follow-up of one’s own, or otherwise deny one.

None of them searches. Each reads the board as a player sees it: the parts, their temperatures, the option the rule would take in each, and where the last move was made. When none of a rule’s preferences applies it falls back on board order, so every cheap rule is a refinement of the old one rather than a replacement.

Everything else about hottest-first stays fixed. The rule-follower moves in a hottest part, and inside that part takes the option of best mean; the opponent replies with the move that does the rule-follower most harm. That is the convention under which a cost cannot be negative, and it is the convention under which the second-largest temperature was measured, so the new costs are comparable with the old ones line for line.

Every line, every tie-break #

The eleven pools are the seven of the cheapest fight and the four with a threat on top from the price of a tie. Every board of three, four and five parts is played twice, once with each player first: 17,200 lines, each played seven times.

The first two rows are the old results, reproduced: 2,859 costly lines by board order, 966 by search. The others fall between, and the ordering is not the one a reader would guess from the names.

Answering locally is the strong one. It takes the costly lines from 2,859 to 1,741 and recovers half of board order’s excess in total value. On the four threat pools it is very nearly the search: 95 per cent of the gap, with no line made worse. A follow-up of one’s own is the weak one. It recovers a fifth overall and almost nothing on the threat pools, where the parts that tie are a threat and the switch another threat has become, and neither the threat nor the switch leaves the mover anything hot. The two together recover seventy-one per cent, more than either, because — as the worked lines below show — they succeed on different boards.

And denying the opponent a follow-up loses ground. It makes 849 lines dearer than board order and only 262 cheaper, and in total it costs more than taking the first part on the board would. Combined with answering locally it still recovers something, because the local answer does most of the work, but less than the local answer does alone.

One number the table checks and does not print, because it is nought for every row: no tie-break takes any line past the second-largest temperature. That is what the price of a tie observed of every tie-break move — none gives up more than the temperature the tied parts share — and it survives every way of choosing among them. The bound belongs to hottest-first whichever tie-break it uses.

Answering where the opponent played #

The line that motivated the local rule is the largest cost in the threat pools.

Under board order, Left takes one of the big threats and Right answers in the small one, turning it into a switch at temperature four — exactly as hot as the big threat still standing. The rule takes the first of the two equally hot parts, which is the threat, and Right takes the switch; the four points in it go with the tie.

Under the local rule the second move would have been the switch Right had just made, so the exploitation stops working, and the figure shows something the counts alone would hide: the best reply changes. Right no longer answers in the small threat. It plays in the second big threat instead, Left answers there, and only then does Right turn to the small threat, which Left also answers; the game finishes level with best play. The cost of board order was never the rule-follower’s move in isolation. It was a reply that manufactured a tie away from the last move, where board order would resolve it the wrong way, and a rule that looks at the last move removes the reason to make that reply.

That is also why the local rule is nearly perfect on the threat pools. A threat hands its opponent a switch when the opponent plays in it — that is what makes it a threat — and on boards built of scaled threats the switch so made ties with the threats still standing. The tie is always created by the last move, so it is always resolved correctly by answering it.

A follow-up where there is nothing to answer #

The local rule is silent when the tie is not where the last move was, and there the follow-up earns its place.

After Right takes the threat and Left answers the switch it made, the board holds a plain switch at an eighth and a fight at an eighth with a follow-up: Right’s move in the fight leaves $\{\frac38 \mid \frac18\}$, still hot, to be played later. The tie is not at the last move, so the local rule has nothing to say and board order takes the plain switch. Preferring a follow-up takes the fight, and the line ends where best play ends.

So the two cheap rules are not competitors. One reads the last move and the other reads the shape of the tied parts, and the boards on which each is wrong are mostly boards on which the other has nothing to say. That is why putting the local answer first and the follow-up second recovers seventy-one per cent when neither alone recovers more than half.

Where each rule earns its share #

The aggregate hides large differences between pools.

On the pool of four fights at a half, on the eighths and on the pool of cool parts at four temperatures, answering locally recovers between ninety and ninety-nine per cent by itself; these are pools where most ties are created by the opponent’s last move. On the pool where every temperature is a half, three eighths or a quarter — the pool on which the price of a tie found every costly line to be a tie — neither rule alone does much, and the combination recovers about half. And on the whole-number pool with follow-ups, the largest pool, the combination recovers under half of a large gap.

The pools the local rule dominates are the ones built of copies or of threats: parts that come in families, so that a move in one member makes a tie with another. The pools where it struggles have parts of many different shapes at the same temperature, and there the question of which to take is about the parts rather than about the last move.

The price of a cheap rule #

A search never does worse than board order, by construction. A cheap rule can, and each of these does.

Here Right’s move makes $\{1 \mid 0\}$ out of one of the fights, at the same temperature as the untouched fight whose follow-up is $\{\frac32 \mid \frac12\}$. Answering locally takes the switch Right made; the fight should have gone first, and the rule gives up a half. Board order happened to take the fight. The local rule makes six lines of the whole census dearer than board order, and the combination sixty-two, against 1,138 and 1,556 made cheaper. Those are small prices, and they are the price of not searching: each cheap rule is a guess about what a tie means, and on some boards the guess is wrong where the arbitrary choice was lucky.

The denying rule’s record — 849 lines worse, 262 better — shows what a guess costs when it is the wrong guess on most boards. Taking away the opponent’s follow-up sounds prudent. On these pools it is usually a tempo spent on a part the opponent was not going to profit from, while the part with the follow-up for the mover goes begging.

What the cheap rules leave #

The combined rule is still dearer than the search on 417 lines, and their location is the most useful thing the sweep finds.

Every one of the 417 loses value at a tie, so the residue is still entirely the tie-break’s; the rule’s temperatures are right. And on 264 of them the first loss is the opening move, where there is no previous move for the local rule to read. The smallest example is $\{5 \mid 1\} + \{5 \mid 1\} + \{6 \mid \{3 \mid 1\}\}$ with Left to move. All three parts sit at temperature two. Neither $\{5 \mid 1\}$ nor Left’s move in $\{6 \mid \{3 \mid 1\}\}$ leaves Left a follow-up, so the combined rule falls back on board order and takes a $\{5 \mid 1\}$; the right opening is $\{6 \mid \{3 \mid 1\}\}$, because Right’s move there would have left Right the follow-up $\{3 \mid 1\}$. That is the denying rule, which is right on this board and wrong on balance across the census. Over half the residue sits on the whole-number pool, where boards of this shape are common.

So the gap between the cheap rules and the search has a location and a character. It is concentrated at the opening move, before the game has given the player anything to read, and it is concentrated where the right choice is the one no single cheap preference makes reliably. Once play is under way the last move carries most of the information a tie-break needs.

A denial that knows when it pays #

The residue points at a refinement, and it is worth testing before calling the opening tie a search. Plain denial fails because it spends a tempo taking away follow-ups the opponent was never going to use. A denial that pays is one that removes a follow-up which would keep the initiative — the kind of move sente is a fact about the rest of the board describes, where the opponent’s move leaves something hotter than anything else on the board, so the answer is forced and the move comes back. That is still a reading a player can make by looking: for each tied part, ask what the opponent’s move there would leave, and whether it would be the hottest thing left.

So a seventh tie-break goes after the combined rule: answer locally; else prefer a follow-up of one’s own; else deny the opponent a follow-up, but only an urgent one — one hotter than every other part on the board.

It does not even fire on the smallest residual board. Right’s move in $\{6 \mid \{3 \mid 1\}\}$ would leave $\{3 \mid 1\}$, at temperature one, and the two copies of $\{5 \mid 1\}$ beside it are at two; by the only measure of urgency a player can read off the parts, the follow-up is not urgent, and yet denying it is the whole difference between the rule and best play. Plain denial gets this board right, for the wrong reason, and gets 849 others wrong.

Over the census the urgent denial loses ground. It repairs 53 of the 417 residual lines and breaks 289 lines the combination had right, so the combination’s seventy-one per cent falls to fifty-two; the lines made dearer than board order rise from 62 to 351; and on the whole-number pool, where the residue is concentrated, the share recovered falls from 43 per cent to 11. On the threat pools it changes nothing, because the local answer has already decided every tie there.

That settles the dichotomy the price of a tie closed on. It said that if a cheap tie-break recovered most of the gap, hottest-first had a complete statement that was still cheap; and if none did, the tie-break was a search in disguise. The answer is both, divided cleanly by the stage of the game. Once play is under way, answering the last move and preferring one’s own follow-up recover most of what a search would, and on the pools built from threats nearly all of it. At the opening tie, before anything has happened, the right choice depends on how the parts will interact over the rest of the game, and three different readings of the parts — one’s own follow-ups, the opponent’s, and the opponent’s urgent ones — each get it right on some boards and wrong on more.

The convention every cost depends on #

The rule-follower plays hottest-first throughout, with the tie-break under test, and moves in a part to its option of best mean. The opponent is not following any rule: at each turn it plays the move that minimises what the rule-follower goes on to get, knowing the rule. A different opponent — one who also plays hottest-first, as the very first sweeps assumed — would credit the rule-follower with the opponent’s own mistakes and make every tie-break look better. Temperatures are the thermograph temperatures of the components, and “the second-largest temperature” is read off the board before the first move.

What the pools cannot show #

The pools are built, not sampled. Eleven pools of five to ten components, chosen by earlier essays to probe particular shapes, are not a population of real endgames; a pool of parts in families flatters the local rule, and the proportions here would move on a different pool.

The cheap rules are the ones named, and not the best cheap rules. A rule reading two moves back, or preferring the part whose follow-up is hottest, might do better; nothing here searches the space of tie-breaks. And each rule falls back on board order, so part of every figure is board order’s luck.

And the bound is observed. No tie-break pushes a line past the second-largest temperature on these 17,200 lines, which is evidence for the conjecture and not a proof of it.

Still open: whether the opening tie has a price #

The opening tie resisted every cheap reading tried, which makes it the natural next object: not a rule for it, but its size. On the 264 lines whose first loss is an opening tie, what the search wins over board order is a number per line, and the question is whether that number is bounded by something smaller than the second-largest temperature — the difference between the tied parts’ follow-ups, say, or the temperature of the hottest thing any of them leaves. If it is, then hottest-first with the combined tie-break has a sharper guarantee than the one a rule with a guarantee proves, stated as the largest temperature after the opening; and a player who knew it would know exactly how much the one decision they cannot make cheaply is worth. If it is not, the opening tie is where hottest-first stops being a rule at all, and a schedule instead of a number is the other place in these essays where a rule’s cheapness turned out to rest on a decision it did not state.