Combinatorial Game Theory
who wins, and by how much
Concept

Greedy play — where it appears

Taking whatever is available now, which is the rule most games are taught by and the rule most solvers are measured against. It is the control every strategy on this site is measured against, and it is not always the one that loses.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

A 2 × 3 board of boxes, 6 still on the table. A Dots and Boxes position drawn as dots and lines, and — where the figure asks for it — the same position as a strings-and-coins graph: one coin per box, one string per line, and the border lines running to the ground. Lines already played are solid, lines still available are dashed, and a box with no strings left has been pocketed. The footer carries the exact net score the solver computes from here and the normal-play verdict on the same position.

The game in every exercise book

Dots and Boxes is played by more people than every game in this collection put together, and everybody is taught the same rule — take every box available. The rule is wrong. Establishing that takes a solver rather than an opinion, and the solver says how wrong, on which boards, and by how many boxes.

applied · Dots and Boxes Move where it is hottest. four independent components of one position, ordered by temperature. The temperature is how much a player loses by moving somewhere else instead, so the hottest component is the one to take — and a component that is already a number has no temperature at all, because nobody gains by moving in it.

Playing the hottest

Given several independent fights, play in the one with most at stake. The rule is simple, it is what strong Go players do without being told, it is provably close to optimal — and it is provably not optimal, which is the interesting part.

temperature · Temperature How much changes hands, against how much is at stake. Two ways of choosing where to move, run against optimal play over every board from a pool of three components. Biggest-first takes the component where the most changes hands, which is the count in every endgame book; hottest-first takes the one with the highest temperature. They disagree on most of these boards, the count costs points more often, and — the difference that matters — the count sometimes loses more than the largest temperature on the board, which is the bound the theory's rule is guaranteed to keep.

Big is not the same as hot

A player sizes a move by how much changes hands when it is played, which is the number in every endgame book. The theory sizes it by temperature. On a plain switch the two agree exactly, so nothing shows; on a move with a follow-up they come apart, and the count gives up more than the guarantee the theory's rule carries.

temperature · Temperature 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.

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.

temperature · Strategy 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.

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.

temperature · Strategy 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.

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.

temperature · Strategy What the opponent's choosing is worth. Every position answered twice: against an opponent who searches, and against one following a fixed rule with no search in it. Only a loss can change, so the share is taken over the losses. The spread between games is the measurement — in one of them nearly every loss is recovered and in another none is.

The opponent stops choosing

Replace one player by a rule with no search in it and the question has one chooser left, which is a puzzle rather than a game. Nim recovers five of its six lost positions that way, and six of seven on three heaps of five. Domineering recovers six of a hundred and twenty-two while the fixed rule throws away a winning move eighty-eight times, and one Clobber board recovers none at all — because on that board no rule can misplay.

complexity · Alternation 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.

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.

applied · Dots and Boxes

Named alongside it

The objects these essays reach for when they reach for this one.

HeuristicExhaustive searchMove selectionSenteTemperatureDisjunctive sumMean valueStrategyAmbient temperatureComponentEndgameHot game

All concepts