Combinatorial Game Theory
who wins, and by how much
Generator

{4 | 0} played out in a stack of 5 coupons

{4 | 0} played out in a stack of 5 coupons
{4 | 0} played out in a stack of 5 coupons. An idealised environment: coupons worth a fixed step less each, which either player may take instead of moving in the game. The rows are the line optimal play takes over the whole board, in order. What the game turned out to be worth is set beside its mean value, and the coupon the players stopped at beside its temperature — two quantities measured from the play, and two computed from the thermograph.

An idealised environment: coupons worth a fixed step less each, which either player may take instead of moving in the game. The rows are the line optimal play takes over the whole board, in order. What the game turned out to be worth is set beside its mean value, and the coupon the players stopped at beside its temperature — two quantities measured from the play, and two computed from the thermograph.

14 essays call coupon-stack. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

The positions it draws

14 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
{{5 | 3} | {2 | −4}} {{5 | 3} | {2 | −4}} L An environment made of coupons · What a move nobody makes is worth
{10 | {9 | 1}} {10 | {9 | 1}} L An environment made of coupons
{2 | {1 | 0}} {2 | {1 | 0}} L Two games in one environment
{2 | {1 | 0}}, beside 4 | 0 {2 | {1 | 0}} L Two games in one environment
{4 | {3 | −3}} {4 | {3 | −3}} L How big the answer is
{4 | {3 | 0}} {4 | {3 | 0}} L When to leave the environment
{5 | {4 | 0}} {5 | {4 | 0}} L An environment made of coupons
{5 | {4 | 0}}, beside 4 | 0 {5 | {4 | 0}} L The answer that starts another fight · Two games in one environment
2 | 0, beside 4 | 0 2 | 0 N Two games in one environment · When to leave the environment
3/2 3/2 L An environment made of coupons
4 | 0 4 | 0 N An environment made of coupons
4 | 0, beside {2 | {1 | 0}} 4 | 0 N Two games in one environment
4 | 0, beside {5 | {4 | 0}} 4 | 0 N The answer that starts another fight · Two games in one environment
4 | 0, beside 2 | 0 4 | 0 N Two games in one environment · When to leave the environment

Where it is called

Changing this generator changes every one of these figures.

{4 | 0} played out in a stack of 5 coupons. An idealised environment: coupons worth a fixed step less each, which either player may take instead of moving in the game. The rows are the line optimal play takes over the whole board, in order. What the game turned out to be worth is set beside its mean value, and the coupon the players stopped at beside its temperature — two quantities measured from the play, and two computed from the thermograph. Temperature

An environment made of coupons

Beside the game sits a stack of coupons worth 4, 3, 2, 1, 0, and a player may always take the top one instead of moving. Play the whole thing out and two quantities the theory computes are measured instead: {4 | 0} comes out worth exactly 2, its mean value, and the coupons stop at 2, its temperature. For {10 | {9 | 1}}, whose temperature is 1, they stop at 7/2 — because what the stopping coupon measures is the hottest temperature anywhere in the tree.

 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.

 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.

 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.

 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.

 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.

 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 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.

 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.

 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.

 Thirteen cells, thirteen scores. The rule scored in every cell of the unit interval on the designed pool, at three components. Temperature

A pool built to have an answer

The coefficient in the rule score a component by t − λa scored identically for every λ in the unit interval, because the rule reads an ordering and that pool's orderings changed at three places. A pool designed to have twelve crossings turns the interval into thirteen different rules, and all three board sizes agree on one cell: between a quarter and a third.

 Three orders, one of them right. The holder's rank above the crossover under three different orderings of the options. Temperature

Which top is the top

The crossover law's proof rests on the walls above the crossover being governed by the top two options, and the check was never run. Run on 23,586 heights it holds exactly — but only when the options are ranked by mean value. Ranked by the temperatures the law is stated in, it fails on a fifth of them.

 Two claims read as one. The proposed geometry separated into the claim the rung below established and the claim it needs but did not. Temperature

The four paragraphs prove something else

Three rungs earned the right to write the crossover law's proof as two straight walls meeting where the law says. The walls are straight — one of them everywhere, for a trivial reason, and the other only above the answer's own temperature. The crossover sits below that height, so the geometry holds nowhere the law is about, and where it does hold it proves the temperature instead.

 What a designed pool can say. Six statements about the coefficient, with which pool each rests on. Temperature

A second pool, designed differently

One designed pool put the rule's best coefficient between a quarter and a third, and all three board sizes agreed. A second pool, built by the identical greedy criterion from different material, has no cell that is best at every size — so the coefficient is a property of the pool and there is no number to find.

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