Coupons — the series · Combinatorial Game Theory

{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 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.

part 1 · temperature

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.

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.

part 2 · temperature

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.

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.

part 3 · temperature

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.

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.

part 4 · temperature

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.

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.

part 5 · 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.

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.

part 6 · temperature

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.

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.

part 7 · temperature

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

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.

part 8 · temperature

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

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.

part 9 · temperature