Combinatorial Game Theory
who wins, and by how much
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The thread: How much is at stake — page 3

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.
Two thousand values and no temperature. Every End-Nim row of up to six heaps, with its temperature. Not one of the 5,460 is hot, every one of them is worth an infinitesimal, and the 361 worth numbers are 361 rows worth nought. Particular games

A game with nothing at stake

The rung below explained a coincidence with a claim it did not compute: that End-Nim carries no hot self-negative values. The census says something stronger. Not one of 10,919 rows across three shapes of board is hot, every one of them is worth an infinitesimal, and the 361 rows the earlier census called numbers are 361 rows worth nought. The reason is one line of the rule, and 2,693 distinct values sit underneath it.

 The bend decides it. Whether the stop reading gives the mean and temperature, against whether either wall bends below the meeting point. Both off-diagonal cells are empty on all 138 values. Values

The bend is the condition

The rung below offered a description of the class its stop reading is exact on — neither wall bends below the meeting point — and a route to proving it: that a bend happens precisely when some option is neither a number nor an infinitesimal. The first is exact on all 138 values, both directions, no exception. The second is half right: every bent value has such an option and 49 unbent ones do too. And the eight apparent exceptions to the first turn out to be a bookkeeping convention.

 The same temperature, and four different departures. Positions with a temperature of one whose follow-ups are worth different amounts, with the coupon at which the players leave the environment. The departure tracks the follow-up. Temperature

How big the answer is

The rung below found every early departure from a coupon stack caused by a position with a follow-up, and could not say more: its follow-ups were all of a similar size, so the class it measured was one bit. A pool graded by follow-up size answers it. With the position's own temperature held at one, the departure runs from coupon 1 to coupon 3.5 as the follow-up's temperature runs from 1 to 4 — and over the whole grid the players leave at the larger of the two temperatures.

 A self-negative value costs a day. The values born by day three, by temperature, with the earliest self-negative one at each. The earliest is always the day after the temperature's own birthday, and the four temperatures with none are the four whose birthday is three. Sums and comparison

A self-negative value costs a day

The rung below placed the thirty values equal to their own negatives on the temperature scale and asked whether being self-negative forces anything about when a value can be born. It does, exactly: the earliest self-negative value of temperature t is born the day after t itself, which accounts for the four temperatures that carry one and the four that carry none. The guess it offered — that the first value of each temperature is a self-negative one — holds at four temperatures out of five and is not the shape of the answer.

 The stops stop working. Adding a switch to each of 120 day-three values, and asking whether the two stops of the sum are the sums of the two stops. They are on 330 of 720. Sums and comparison

What a fight does to a fight

A number added to a position shifts both its stops by itself; an infinitesimal moves neither. A hot game does neither: over 720 sums the two stops add on 330 and are wrong on the rest. What survives is the mean, which adds on every one of the 720 — and the failure has a bound, since no stop is ever out by more than twice the smaller of the two temperatures, a bound 222 of the sums attain exactly.

 A game is colder than its catalogue. The share of hot positions in the Domineering region catalogue against the share among the components a real game produces. Fifty-three per cent against sixteen. Temperature

What a game actually produces

Fifty-three per cent of the Domineering regions of at most eight squares are hot. Of the components eleven hundred random games actually produce, sixteen per cent are — and ten per cent once single squares are counted. The figure is the same on three sizes of board, so it is a property of play rather than of the board, and it says that every temperature census this site has taken over a catalogue overstates how hot the game is by a factor of three.

 The crossover by depth. How far the ambient temperature can rise with the move still answered, by how deep the fight goes. Where the answer settles the fight it is the follow-up's temperature; deeper it is that less a half. Temperature

A subtraction, not a factor

The crossover factor was a half on fights whose answer starts another fight, measured on a pool with two three-deep positions in it. A pool built to be deep gives twenty, and the factor does not survive them: the crossover is the follow-up's temperature less a half on eighteen of the twenty, and a factor of a half agrees with that only where the temperature is one — which nearly every position in the earlier pool had.

 The error is half the follow-up. Each bent value's true temperature, the temperature the stop reading gives it, the difference, and half the temperature of its hot option. The last two columns agree on every value. Values

Half a follow-up out

The rung below settled which values the stop reading is wrong about — the ones whose walls bend — and left the size of the error unmeasured. It is not bounded by anything readable off the diagram; it equals something readable off the diagram. On all thirty-two, the mean and the temperature are each out by exactly half the follow-up's temperature, and the temperature is always read too low.

 Two clauses, one formula. The law split by which of the two temperatures is the smaller. When the answer is colder the correction is half of it; when it is hotter the correction saturates at half the fight's own temperature. Temperature

Half of the smaller temperature

The correction to the sente crossover has been priced twice — first as a factor of a half, then as a subtraction of a half — each time on a pool whose answers were all about the same size. 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. Both earlier readings are regions of that one law.

 Three populations, three answers. How often something is worth fighting over, measured on the catalogue of shapes, on the pieces a played game produces, and on the whole board those pieces make up. Temperature

One fight makes a board a fight

The rung below found 16 per cent of the components a played game produces to be hot, against 53 per cent of the catalogue they are drawn from, and predicted that the share of hot boards would be much larger. Taking the same play-outs and tallying at the board gives 32 per cent — twice the piece figure and not ten times it, because a Domineering board carries only 1.68 pieces and the hot ones cluster on the same boards.

 The bound that needed no third number. The rung below's bound and the conjectured replacement, scored over 1,440 sums whose addend has a follow-up. The two-number bound holds everywhere and is attained; the three-number one fails. Sums and comparison

A bound with one number too many

The rung below bounded how far a hot addend can drag a stop — twice the smaller of the two temperatures — over sums whose addends were all plain switches, and conjectured that an addend with a follow-up would need twice the smaller of three numbers. Over 1,440 sums with bent addends the two-number bound holds everywhere and is attained 358 times, and the three-number version fails on 66.

 The ordering that does not order. Playing in the hottest component against playing by the larger of a component's two temperatures, over 220 boards. The proposed rule is exact far less often and its worst case is nine times as bad. Temperature

The quantity that does not order a board

The rung below found the players leaving an environment at the larger of a position's two temperatures, and proposed that a board should therefore be played in the order of that quantity. Over 220 boards of three components it plays exactly on 124 against playing-in-the-hottest's 196, loses 85 of the 97 disagreements, breaks Hotstrat's guarantee on six boards, and costs nine points on its worst one.

 The correction on eleven times the pool. The stop reading and the corrected stop reading scored against the true temperature over every non-number value born by day three. The correction was measured on thirty-two values and holds on three hundred and forty-eight. Values

A second level of stops

The rung below found the stop reading's error to be exactly half the follow-up's temperature and asked whether the correction survives a wider pool, survives two bends, and can be stated without a thermograph. It survives eleven times the pool, missing two values in 1,459. It needs no thermograph — the follow-up's temperature is half its own stop gap. And it does not survive two bends, because day three contains no value with two of them.

 Almost none of them is a number. Every three by three Amazons region holding one amazon of each colour, sorted by what kind of value it carries. Fifty-six of the two thousand are fractions and the great majority are hot positions. Particular games

The fractions that were not there

The rung below counted 1,452 fractions among the shared Amazons regions and asked which fractions they are. Fifty-six of them are fractions. The other 1,396 are hot positions with a fraction somewhere inside their options, counted by a regular expression looking for a slash — and the quantity the separation of the two amazons actually sets is not a denominator but a temperature.

 The rule that was supposed to lose. Five ordering rules on the same 220 boards. Playing where the temperature less the answer's is largest is exact more often than playing in the hottest component, which is what the rung below predicted it would not do. Temperature

A rule that beats the hottest

The rung below proposed the reverse of the rule that had just failed — discount a component by its answer's temperature rather than promoting it — and predicted, before the sweep, that it would not beat playing in the hottest component. It does. It plays exactly on 201 of 220 three-component boards against 196, wins two thirds of the boards where the two disagree, keeps inside a guarantee proved for the other rule, and the gap widens as the board grows.

 The board cools as it is played. Every position of a three by six Domineering board, grouped by how many dominoes are down. The share that are hot rises to four fifths and then falls to nothing. Temperature

The obstacle was the catalogue

The rung below could not measure the early game because its regions were too large for the catalogue, and asked for a bracket rather than a value. No bracket is needed: a twelve-square region evaluates in five milliseconds and an eighteen-square one in under a second. What was expensive was cataloguing every shape rather than sweeping the positions a board actually reaches — and the sweep says a board is hot four times in five three moves in, and cools when it breaks up.

 Which end, in four lines. The complete rule for where a sum's error lands, exact on every pair in the census. Three of its four cases are decided by the value being translated alone. Sums and comparison

Which end a sum lands at

The rung below found the errors in a translated stop clustered at the two ends of the range its bound allows — 660 at nought and 358 exactly on the bound — and asked for a rule saying which end a given pair lands at. There is one, in four lines, exact on all 1,440 sums. Three of the four cases are decided by the value being translated alone, and the property that decides them is the bend the switches ladder found for a different question entirely.

 The third temperature is not consulted. Positions grouped by their two top temperatures, with the third temperatures they hold between them. The crossover is the same for every position in a group however far apart their third temperatures are. Temperature

The two numbers at the top

The rung below found the crossover of a sente fight to be its temperature less half its answer's, and said a proof would settle the depth question with it. The depth question is settled without the proof, by construction: group the positions by their two top temperatures and the crossover is single-valued on every group, however far apart the third temperature is — and the formula survives a fourth level of fight, which the rung below never reached.

 The wall as an envelope. A thermograph with each option's contribution to its wall drawn over it, built from that option's two stops alone. The wall is the envelope of those contributions and it bends where the envelope has a corner. Values

The bend is in the stops

The rung below reduced the whole stop reading to one question — does this wall bend? — and asked whether that could be answered from the options' stops instead of from a diagram. It can, in four lines, and it gives more than the bend: on all 1,459 non-number values born by day three the options' stops determine the entire thermograph. One day deeper it breaks, and every failure is a value with a bent-walled option.

 The same distance, different room. Two shared Amazons regions with the amazons the same distance apart, differing in how many squares both of them can still reach. The one sharing more is the colder. Particular games

Room pulls two ways

The rung below found the distance between two amazons setting a shared region's temperature and asked for something finer — the squares each can reach, or the squares both can. Neither beats the distance on its own. Together they beat it by half as much again, and the reason is that they pull opposite ways: further apart is hotter, and sharing more reachable squares is colder.

 A plateau, not a point. The rule's score as the coefficient is varied on a fine grid. It is constant across the open unit interval and drops at exactly one. Temperature

The worst value in its own interval

The rung below scored a component by its temperature less its hottest answer's and asked what rate the answer should really be charged at. Every weight strictly between nought and one scores the same and beats the rung below's 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 where two components tie.

 The five hottest regions. Every eight-square Domineering region at the ceiling temperature, with which of the boards swept ever produces it. Temperature

Eight squares, and no hotter

The rung below found no Domineering position hotter than three halves on four boards and asked for the position that attains it. It is a region of eight squares, there are five of them up to symmetry, three are the hot core of an attaining board on every size swept — and the ceiling holds at nine and ten squares too, where the obvious extrapolation predicted seven quarters.

 One quantity, two currencies. Each bent value with how far its temperature falls short of its stop reading and how far its stop falls short of the translation bound. The second is exactly twice the first. Sums and comparison

The same number in two currencies

The rung below found bent-walled values falling strictly inside the translation bound and asked how far. The shortfall is the value's own hottest follow-up's temperature — exactly, on 400 of 408 pairs, and twice it on the other eight — which makes the whole error one expression. And it is the switches ladder's constant: a half there and a whole here, because a temperature is half a stop gap.

 Five premises, and the step. The claims an induction would need, with what checks each. The last row is the step and nothing here checks it. Temperature

The premises an induction would need

The rung below settled by a grouping test that a position's crossover depends on its own temperature and its answer's and on nothing below them, and asked for the induction. The four paragraphs are not written here; the checking they would rest on is. The law holds at five levels, survives translation, heating and cooling — and none of that is the step.

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