Combinatorial Game Theory
who wins, and by how much
Theme

The thread: How much is at stake — page 5

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.
The more cheap fights there are, the cheaper the rule is. Hottest-first play against optimal play over 10,410 lines, grouped by how many components share the lowest temperature on the board. Lines costing more than that temperature occur only where one or two components share it; over the 3,230 lines with three or more, none does. What it costs

The cheap fights make the rule cheaper

A conjecture stands that playing the hottest part costs at most the coolest temperature times the number of parts sharing it — proposed on a range where that number never exceeds two. Swept to five-part boards over 10,410 lines it is false, and false the other way round: every line costing more than the coolest part has one or two parts at that temperature, and over the 3,230 lines with three or more, not one does.

 What the flip is worth, and what is at stake. Left's chances if Left moves and if Right moves, with the gap between them beside the position's temperature. The two are answers to the same question computed by different routes, and they do not order the positions the same way. Where it stops

The coldest position has the biggest swing

How much a flip is worth is the gap between the coin's two branches, and it is a rival to the temperature — both answer how much is at stake. They disagree at once: the empty position has the lowest temperature there is and a swing of one, twice the hottest thing born on day two. On the small pool the two quantities look like a perfect three-way correspondence, and 255 of day three's values break it.

 What control is worth, to a box. The margin the player who does not have to open nets, beside the formula that predicts it: the total less four boxes for every long chain after the first. Computed against the solver on every endgame of long chains in range. Out in the world

Four boxes for every chain after the first

Nimstring answers who is forced to open and says nothing about the score. The margin has a formula: the controller nets the total less four boxes for every long chain after the first — two surrendered and two not taken, each time control is kept. Checked against the solver on 175 endgames it is exact on 172, never too generous, and exact wherever it promises the controller anything at all. The three it misses are the three where it promises nothing.

 The cost falls as the cool part warms. Cost of hottest-first play against the temperature t of the coolest part, on one three-part board with Right to move. For a plain switch the cost is 1 − 2t and vanishes at a half; for a cool part with a follow-up it is 1 − t. Twice the coolest temperature, drawn dashed, lies below both curves for small t. What it costs

The cheapest fight is not the yardstick

Hottest-first play was found to cost at most twice the coolest temperature on the board, on pools whose coolest parts sat at a quarter or a half. Put a fight worth an eighth on the board and the rule costs seven times it. On one board the cost falls as the cool part warms, which is the opposite of a bound in the coolest. The number that holds on all 14,946 lines of seven pools is the second-largest temperature.

 The residue of a sum is the hotter part's. Every hot day-two value added to every hot day-three value, 7,854 sums in all, sorted by what happens to the residues: when the parts differ in temperature, on all 4,712 such pairs, the sum's residue is the hotter part's alone; when they are equally hot the residues add on 880, the sum grows colder on 2,237 and cancels to a number on 25. Temperature

The hotter residue survives

Charge a hot position its own temperature and subtract its mean, and an infinitesimal is left — its residue. Add two hot positions and the means add, the temperatures do not, and the residues do something stranger than either: on every one of 4,712 pairs whose parts differ in temperature, the residue of the sum is the hotter part's residue and nothing of the colder's. Cooling is additive, and a part cooled past its own temperature is a bare number. Residues add only when two parts are exactly as hot, and even then the fights usually cancel into something colder. The summary forgets; the game does not.

 The second temperature, with the threat on top. Hottest-first play against the best reply on four pools whose hottest component is a threat of the form {10 | {8 | 0}}, with a switch just below it and a smaller threat below that. Over 2,254 lines the cost never exceeds the second-largest temperature on the board and meets it exactly on 97. What it costs

The price of a tie

The second-largest temperature bounded hottest-first play on 14,946 lines, and the pool that should have broken it puts a threat on top. It does not break: over 2,254 more lines the cost never passes the second temperature and meets it on 97. What the pool finds instead is where the rule goes wrong — one move into the game about as often as at its start — and why. Almost every line that meets the bound is a tie: two parts equally hot and the rule taking the first on the board. Break ties well and the lines meeting it fall from 636 to 51 on the earlier pools, and to none on any board with the threat on top.

 Answering locally wins back half. Six cheap tie-breaks for hottest-first play and the searched one, scored over 17,200 lines of eleven pools by the share of board order's excess cost over the search each recovers: first on the board 0%, prefer to deny the opponent one -28%, prefer a follow-up of one's own 21%, answer where the opponent just played 50%, answer locally, else deny 32%, answer locally, else a follow-up 71%, locally, a follow-up, then an urgent denial 52%, the best, by search 100%. What it costs

Answer where the opponent played

Most of what hottest-first play costs is the choice between parts equally hot, and a search could make that choice well. Six choices a player could make without searching were scored on all 17,200 lines of eleven pools. Answering where the opponent has just played wins back half of what board order loses, and 95 per cent of it on the pools with a threat on top; preferring a part with a follow-up of one's own wins back a fifth; the two together, 71 per cent. Preferring to deny the opponent a follow-up is worse than taking the first part on the board, and denying only an urgent one takes the combination back down to 52. What is left is mostly at the opening move, where there is nothing yet to answer.

 Colder exactly when the residues cancel. Equally hot pairs of hot values, 3142 from days two and three and 3000 sampled from day three, sorted by whether the sum stays as hot, grows colder or is a number, and by whether the two residues cancel. Every sum that stays as hot has residues that do not cancel, and every sum that grows colder or is a number has residues that do. Temperature

Colder exactly when the residues cancel

Add two fights of one temperature and the sum either stays that hot or grows colder, and which one happens is decided by the smallest thing either fight carries. The sum grows colder exactly when the two residues are negatives of each other — on all 3,142 equally hot pairs from days two and three and 3,000 sampled from day three, and by a two-line argument from cooling's additivity. What survives the cancellation is another matter. Group the colder sums by their parts' thermographs and the survivor differs in every group that holds more than one pair: {1 | 0} and {1 | 0, ∗} draw the same diagram and leave different infinitesimals behind. The diagrams fix only the survivor's temperature: nought, or exactly the lowest bend of either part.

 Four rows of lamps and three winning cuts. A Nim board of four rows of up to seven lamps, lit 7, 5, 3 and 2, with each row's binary beside it. The columns add without carrying to 011, a nim-sum of 3; the three rows carrying the middle bit each have a winning cut, to 4, to 0 and to 1, and the lamps each cut would put out are shaded. How it was found

A machine that knew one theorem

In 1940 a machine of relays played Nim against the public at the New York World's Fair, on four rows of up to seven lamps, and is reported to have won about nine games in ten. Everything it knew was Bouton's theorem, which is three column parities and one test per row. The record is the interesting number, and it is a measurement of the visitors rather than of the machine: against a visitor choosing at random the machine wins 98.5 per cent, and nine in ten is what a visitor earns by playing perfectly once seven lamps or fewer are lit. On the board the game is usually set up on, a visitor handed the first move cannot win at all.

 A price in stones per turn. Small boards with the exact number of stones Breaker needs per turn to stop a Maker placing one, the rate the biased Erdős–Selfridge potential guarantees, and the number Maker needs per turn to beat a Breaker placing one. Out in the world

A win priced in stones

Change the rate of play and a yes-or-no question becomes a number. Against a Maker placing one stone a turn, Breaker needs two stones to stop three in a row on every board up to 4 × 4 and six by three, and three on 5 × 4. Against a Breaker placing one, a Maker needs two stones on most boards drawn at one, and three to make four in a row on 4 × 3. The extra stone is enough to erase noughts and crosses altogether, and the biased sum is exact there and high everywhere else.

All themes