Combinatorial Game Theory
who wins, and by how much
Concept

On, the game that never stops — where it appears

The game whose only move, for either player, is back to itself, so it never ends and drags every sum containing it to a draw. It is a stopper in one direction and never ends in the other, and any sum containing it inherits the problem.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

A position that comes back. Three positions whose moves lead round in a circle. Every value in this subject is defined by recursion on the options, and that recursion assumes play ends — here it need not, so the definition has nothing to stand on and the outcome may be a draw, which normal-play theory has no name for.

Loopy games

The whole theory assumes play stops. Allow a position to recur and the induction that every value rests on has nothing to stand on — and a fifth outcome appears that normal-play theory has no name for.

limits · Loopy a loop with a way out: what the backward analysis settles. A position graph in which the moves can lead back to where they started. The labels are the order in which a backward analysis settles each position, starting from the ones where a player has already run out of moves. Positions the analysis never reaches are drawn — and there is no test for that; being unreachable is what a draw is.

Start at the end and work backwards

When play can return to where it started there is no bottom for the recursion to stand on. What replaces it begins at the positions where somebody has already lost and propagates outwards — and the positions it never reaches are exactly the draws. There is no test for a draw, and there does not need to be.

limits · Loopy a cycle of three: what the backward analysis settles. A position graph in which the moves can lead back to where they started. The labels are the order in which a backward analysis settles each position, starting from the ones where a player has already run out of moves. Positions the analysis never reaches are drawn — and there is no test for that; being unreachable is what a draw is.

An outcome with no value behind it

Retrograde analysis labels positions in rounds, outward from the ones already lost. Whatever is still blank when nothing more can be deduced is a draw — and there is no separate test for a draw, because a draw is exactly the residue the method never reaches.

limits · Loopy on + off: what the backward analysis settles. A position graph in which the moves can lead back to where they started. The labels are the order in which a backward analysis settles each position, starting from the ones where a player has already run out of moves. Positions the analysis never reaches are drawn — and there is no test for that; being unreachable is what a draw is.

One part that never ends

The game called `on` has one move and it is back to itself. Add anything to it — a star, a point, its own mirror image — and the whole board is drawn. So `off` is exactly the negative of `on` and their sum is not zero, which is the group law failing for a reason that has nothing to do with who is winning.

limits · Loopy a loop with a way out under three rules for never ending. One graph, one labelling, and three ways of reading the residue the labelling never reaches. A draw is not a computed outcome here — it is what is left over — so declaring infinite play a win for one side is a legal alternative that costs no extra search and changes who wins.

When never ending is a win

Retrograde analysis labels a position a win when somebody can force the opponent to be stuck, and leaves everything else blank. Calling the blanks draws is a rule from outside the game — and two other rules are available. The labelling does not change under any of them; only the residue does, and on a three-cycle that residue is every position on the board.

limits · Loopy What the outcome of a loopy sum can be. One row and one column per loopy outcome class, and each cell lists every outcome a sum of two such positions was found to have. Most cells hold several. The cell where both parts are drawn holds one.

The one outcome that adds

Finite outcomes do not add: two first-player wins can sum to anything. Loopy play has seven outcome classes instead of four and adds even less — of the 28 cells in the table, eleven hold several answers. Two do not, and they are the two worth having: a second-player win added to anything leaves the outcome alone, and a draw added to a draw is a draw. A draw added to anything else is not.

limits · Loopy Cancellation, by exhaustion. The law checked on every triple of values born by day two, and then put to work: two Domineering regions compared directly and compared again inside a larger board. The comparison never changes, which is the licence every decomposition on this site is drawn under.

What can be struck out

From G + X = H + X it follows that G = H, in one line, by adding −X to both sides. It is the shortest theorem here and the most used: it is what makes comparing two boards region by region legitimate. Over 10,648 triples the hypothesis fires 484 times and the conclusion holds 484 times — and the licence expires in three separate directions, each of which loses the same axiom in a different way.

sums · Negation Every region of three positions, counted. The 262,144 graphs on three positions reduced to the regions that are genuinely three positions with a cycle in them, and then split by whether the two-position vocabulary has a name for both of their sides.

Four thousand nine hundred regions with no name

Two positions give 256 regions and ten names cover every side of all of them. Three positions give 262,144 graphs, 110,934 genuine loopy regions — and 4,931 of those have a side that no name in the two-position vocabulary reproduces, with 3,990 of them named on one side and blank on the other. The count the earlier essay left open comes back in the affirmative.

history · Notation The guess, and what it covered. Two attempts to name the leftover sides out of the old vocabulary: every pair of the six stoppers, and every two-position region that is a stopper, each with small finite games added. Both cover nothing, and the count of distinct leftovers is what remains.

The names are not built out of the old ones

The guess was that a three-position region's missing names would be sums of two loopy ones — on plus over, and that family. Built and tried, every pair of the six stoppers covers none of the 4,931 regions that need one, and so does every two-position stopper there is, all seventy-nine of them with small games added. Thirteen names have to be invented, and forty-eight cover the whole census against ten at two positions.

history · Notation

Named alongside it

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

DrawLoopyExhaustive searchOutcome classTerminationNormal playRetrograde analysisDisjunctive sumEnumerationFixed pointMisère playPosition graph

All concepts