Combinatorial Game Theory
who wins, and by how much
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The thread: How much is at stake — page 2

A position is worth something on average and worth something more to move in first, and the second number is the one a player feels. Temperature is that number, and most of what it measures is not where it is expected.
What a deeper position does to the shape. Thermographs side by side, two of them, with temperature running up each panel and value across it: {{6 | 2} | {1 | −3}}, with a bend where an option's own fight cools out; {4 | −1}, straight-walled. A wall that runs straight has nothing changing hands below the meeting point; a bend is an option's own fight cooling out at a lower temperature than this position's, and it is where a decision passes from one player to the other. The two marks on each base line are the stops — what each player gets by moving first with no tax charged. Values

The switch a player is imagining

Every account of a hot position ends up as "worth about m, and worth t to move in", which is the switch {m+t | m−t}. For a plain fight that summary is the position exactly. For a fight with anything behind it the leftover is not a rounding error — on one position here it is a whole second fight of temperature two.

 4 | 0 and {2 | {1 | 0}} in the same environment. Two positions and one coupon stack, solved as a single board. The rows are the line optimal play takes; the coupon on top when each position is first entered is compared with the coupon it was entered at when it had the environment to itself. The mean contributions still add and the entry coupons need not agree. Temperature

Two games in one environment

A coupon stack measures a position: play the whole board out and the coupon the players stop at is the temperature, the score is the mean. Put a second position beside the first and one of the two measurements stops working. Over 36 ordered pairs the contributions still add to the means every time, and the coupon a fight is entered at moves on 13 of them — without either position changing.

 What the rule costs. Every sum of three components from a fixed pool, played out twice: once with one side following the rule "move where the stake is largest" and once with both sides evaluating exactly. The rule is not optimal, the gap is bounded, and the bound is the largest temperature on the board. Temperature

A rule with a guarantee

Evaluating a sum of a dozen fights is impossible; following a rule is not. Move where the stake is largest, and over 220 sums of three hot components the rule scores exactly what perfect play scores in 196 of them, is never more than one point behind, and never ends more than the largest single stake below the mean. The rule that is supposed to be different — answer the threat — chose differently in none of the 220.

 Every day-three value, moved by every quarter. Four claims counted over 3,000 translations — 120 values, each moved by every quarter from −3 to 3: that adding a number leaves the temperature alone, that it shifts both stops by exactly itself, that the interval between the negated stops is where the two players are confused, and that the only failures of the last are on its endpoints. Sums and comparison

What a number does to a fight

Adding a number to a position moves everything and changes nothing: over three thousand translations the temperature never once shifted and both stops moved by exactly the number added, every time. What the number decides is whether the fight is worth having — and the interval where the two players are confused is exactly the open interval between the negated stops, right in all 2,890 cases away from its endpoints and wrong in 110 that are all on them.

 Where a switch stops being a switch. The same Left option with the Right one raised past it. While the Left option is above the Right one both players want to move, the bar spans a real fight and the temperature is half the gap. Where the two meet the position is the number plus a star — no longer a switch, and not a number either. Above that the simplicity rule takes over: the value is the simplest number strictly between the options, and there is nothing to fight about. Values

When a switch is not a switch

A position {a | b} with numbers on both sides is a fight only while a is above b. Sweeping the boundary with a fixed at 2 and b climbing from −2 to 3 turns up three regimes rather than the two the definition suggests: eight fights whose temperature is exactly half the gap, one position at a = b that is 2∗ and is not a number, and beyond that numbers chosen by the simplicity rule — which on 8 of 12 sampled cases is not the midpoint.

 The fight never runs backwards. Each of the 22 values born by day two, drawn from its right stop to its left stop — what Right gets moving first, and what Left gets moving first, once the fight has been played out to a number. Every bar runs the same way. The left stop is never below the right one, which is what "both players are trying to improve their own position" amounts to, and the cold rows, where the two coincide, are drawn as a single point. Values

The fight never runs backwards

Left's stop is never below Right's — in every one of 1,780 distinct values, computed twice by two independently written routes, with nothing that disagreed anywhere. The inequality is what makes a mean value well defined and a fight a fight; and where it collapses to equality, 433 of the 460 cold positions turn out not to be numbers at all.

 Shove strips, and what each is worth. A shelf of positions with the value the recursion returns beside each. Every one is a number: Shove has no hot positions at all, which is unusual for a partizan game and is the first of the essay's three claims. Particular games

Nothing worth fighting over

Shove is a strip of coins beside a cliff, and both players have completely different moves. Every one of its 728 positions is worth a number, so nobody ever wants to move; the winner is the owner of the coin furthest from the cliff, in all 728; and the number the board is worth is not the sum of its coins — that reading is exact on 126 strips and wrong on 588 of the other 602.

 The whole Domineering catalogue, chilled. Every Domineering board this site evaluates, with its value, its temperature, and what cooling by exactly one does to it. Thirteen of the fifteen become cold — a number, or a number plus a star — and the two that do not are the two whose temperature was above the tax. Temperature

The operator chosen for one game

Chilling is cooling by exactly one, and the one is not derived from anything. It is chosen because Domineering mostly runs at that temperature — and measured against this site's whole Domineering catalogue it turns thirteen of fifteen boards into numbers or numbers plus a star, and warms thirteen of the fifteen back exactly. They are not the same thirteen: eleven boards do both, two freeze too far to be recovered, and two stay hot and come back on the nose.

 Where 1,474 values sit on the scale. The temperature of every value in the pool, counted. The floor is −1 and only numbers are on it; the next rung up is 0, and everything there is a number with an infinitesimal added. Above that the scale is continuous and the counts thin out. Temperature

Below zero

Temperature is described as urgency and urgency has no obvious bottom, but the scale stops at −1 and only the numbers are on it. The rung above is exactly zero, and the 337 values born by day three that sit there are the ones a number cannot be told from — flat thermograph, nothing at stake, and an infinitesimal that no number can see. Two independent computations agree on the classification for all 1,474.

 Four rules over 220 sums. Each rule plays every sum against an opponent evaluating exactly. Two of the rules come with a bound and two do not; the coldest rule is the control, and it violates the bound often enough to show that being inside it is a real constraint rather than a description of the pool. Temperature

A rule with no promise at all

Playing in a hottest component comes with a bound: never more than the largest single temperature below the mean of the board. Over 220 sums the bound holds 220 times — and so does the bound for a rule with nothing behind it, which scores exactly what perfect play scores on 205 sums against the hottest rule's 196. The control that shows the bound is doing work is the rule that plays the coldest component, which breaks it 74 times and loses up to eleven points.

 Where a sum's temperature actually lands. Every pair drawn from 45 hot positions, with the temperature of the sum set against the larger of the two temperatures. The bound is never broken and it is almost never used: 864 of the 1035 sums sit exactly at the maximum and 160 are frozen. Temperature

How cold a sum of hot games can be

The temperature of a sum is at most the largest temperature in it, and the bound leaves the whole interval below it open. Over 1,035 pairs the sums do not use that interval: 864 sit exactly at the maximum, 160 are frozen outright, and eleven land anywhere in between — every one of them with a component whose wall bends.

 The move that stops the other one. Six local fights, each placed beside an environment of known temperature. The last two columns are the band over which taking the local move beats spending the move outside, and the band over which the opponent's move in the same fight has to be answered. They are not the same band. Temperature

What a move nobody makes is worth

If Right's move in a fight has to be answered, then Left's move in the same fight prevents an exchange Right was going to get for nothing. That is a reverse sente, and pricing it is the awkward case: over sixty measurements the gain matches the local temperature once and the follow-up's swing twice, and the band over which the move is worth taking is not the band over which the move it reverses is sente.

 What is left when the copies pair off. For each position, the difference between n copies and n times the mean, reduced to canonical form. The first two are drawn and the last column says what the sequence does after them: half of these settle into a short cycle and the rest produce a new leftover every time, all of them the same bounded size. Temperature

What is left when the copies pair off

A pile of n copies stays within a bounded distance of n times the mean, and the distance never grows. The difference is a game rather than a number, and what it actually is has a much better answer: for a plain switch it alternates between one fight and nothing at all, and for a fight with a follow-up it is different every time — bounded in size and unbounded in complexity.

 Adding two thermographs. Every pair drawn from the 15 values born by day two that are not numbers — 120 sums — with the two walls added pointwise and compared against the true diagram of the sum. The added walls are always an outer bound and the means always add; the whole diagram is right for 92 of the 120, and the 28 it is wrong for are exactly the pairs in which both components are hot. Temperature

When two thermographs can be added

The temperature of a sum is not the sum of the temperatures, and the natural repair is to add the whole diagrams instead. Over every pair of hot values born by day two the added walls always bound the true ones and the mast always comes out right — and the whole diagram is right exactly when at most one of the two components is hot, which is precisely the case a reader has no use for.

 How hot a day gets. The hottest value born by each of the first three days, with every temperature that occurs on it. Day one tops out at nought, day two at one, day three at two — a day buys exactly one degree — and the value attaining the maximum is unique each time. Each temperature was computed as the height at which that value's two thermograph walls meet. The temperatures of day three are exactly the half-gaps between the numbers born by day two, which is what puts a hole in the scale at 7/4. Temperature

How hot a day gets

A day of construction buys exactly one degree of temperature — nought, then one, then two — and the value attaining the maximum is unique on every day: ∗, then {1 | −1}, then {2 | −2}. The distribution underneath is not tidy at all: it peaks at a half, leans to the right of the peak, and has a hole in it at one and three quarters where nothing is born.

 Five rules over 120 sums built to punish greed. Each rule plays every sum against an opponent evaluating exactly, on a pool whose components are traps: a large immediate gain that hands the opponent a larger follow-up. The pool was built to punish the greedy rule and does not — that rule scores a move by the stop it leaves, and a stop already contains the follow-up. What the traps catch is the rule below it, which scores a move by the territory it takes and loses up to 16. Temperature

A pool built to punish greed

The rung below found a rule with no theorem behind it beating the rule with one, and predicted that a pool of deliberate traps would reverse the result. It does not. The traps miss, because the rule called greedy scores a move by the stop it leaves and a stop already contains the follow-up — and the rule the traps do catch, losing sixteen points where the guaranteed rule loses five, had to be written to make the point.

 The switch formulas, off the hypothesis they were stated for. Values born by day three with exactly one option a side, split by whether both options are numbers. On the twenty-one that satisfy the textbook hypothesis the midpoint and half-gap formulas are exact; on the 146 that do not, the same formulas read off the two stops instead hold about three quarters of the time. Values

A fight with no midpoint

The mean of {a | b} is the midpoint and the temperature is half the gap — on the twenty-one values of day three where a and b are numbers. One hundred and forty-six others have the same shape and not the hypothesis, and the repair that suggests itself, reading the two stops instead of the two options, holds on about three quarters of them and no more.

 The same population counted twice. The temperature scale over the positions this site has enumerated, once with every position counted and once with every distinct value counted. The two disagree about how much of the subject is hot, about what the commonest hot temperature is, and about whether a number is the usual thing for a position to be worth. Temperature

How hot a real position is

Counted one value at a time, a tenth of the subject is hot. Counted one position at a time — every board this site has enumerated, all 11,397 of them — it is a twentieth, two thirds of the positions are worth numbers outright, and ten of the seventeen rulesets never produce a hot position at all.

 Where sente stops, by what the answer costs. The largest ambient temperature at which a local move is still answered, sorted by how deep the fight below the answer runs. When the answer ends the fight the crossover is the follow-up's temperature; when the answer starts another fight it is exactly half of it, and a third level does not halve it again. Temperature

The answer that starts another fight

A local move is answered while the ambient temperature stays below the follow-up's — and that rule, which this site has carried since the anchor opened, is exact only when the answer ends the fight. When the answer starts another one the crossover is exactly half the follow-up's temperature, on every position tested, and a third level of fight does not halve it again.

 Where the thirty sit on the scale. How many of the values equal to their own negatives carry each temperature. Fifteen sit at nought, fourteen are hot, and one is a number — so the subgroup runs the whole length of the scale rather than living at the cold end of it. Sums and comparison

The thirty that cancel themselves

Thirty values born by day three are equal to their own negatives, and every one of them has a mean of exactly nought and two stops that are exact opposites. Neither property comes close to picking them out — 496 values of the day have a mean of nought — and half of the thirty are hot, one of them the hottest value the day produces.

 How far down the stack the guarantee reaches. The temperatures of a sum's components, sorted largest first, with each position asked whether playing in the hottest component can lose more than the temperature sitting there. The first two never fail; the third fails on 681 sums. Temperature

A schedule instead of a number

Playing in the hottest component loses at most the largest temperature on the board — the classical guarantee, stated against one number. Sorting the temperatures and reading the guarantee one step further down gives a promise 47 per cent smaller that is never breached over 1,734 sums. Two steps down it fails 120 times, so the schedule has exactly one step of slack in it.

 Two feet, and the one that halves the crossover. The thermographs of two follow-ups, drawn to the same scale. The first has a right wall that leans in from the axis and its move is answered up to the follow-up's full temperature; the second has a wall rising vertically first and its move is answered only to half of it. Temperature

What the halving is a function of

A move is answered while the board is cooler than the follow-up's temperature — or half of it, depending which of two classes the position is in, and the classes were stated in terms of forms. They are a feature of one wall: whether the follow-up's right wall rises straight up before it leans. Fifty-five positions, no exception, and the crossover becomes something a reader can see.

 When the players stop taking coupons. Every pair of fights from a pool of nine, played beside a coupon stack, with the coupon standing when somebody first plays on the board. Sixty of the eighty-one leave exactly when the coupon falls to the board's temperature. Temperature

When to leave the environment

A Go player's question is not which fight to take but when to stop taking the small stuff. Put two fights beside a stack of coupons and the orthodox answer — leave when the coupon falls to the hottest temperature on the board — is exact on sixty of eighty-one pairs. All twenty-one departures have a fight with a follow-up in them, and every pair of plain switches leaves on time.

 What the correction buys. The packing count as a point against the packing count as an interval, on the regions worth numbers. The point is right on 141 of 315; the interval contains the value on 209, at a mean width of about half a move. Particular games

The moves a player can be talked out of

The difference of the two players' largest domino packings is the value of a Domineering region on 141 of the 315 worth numbers. The count is optimistic for its owner and pessimistic for the other, and one number cannot be both — so it becomes an interval, from what a player can be reduced to against what the opponent can achieve. The interval contains the value on 209, is a single point on 505 of the 1,042 regions, and never exceeds two moves wide.

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