Combinatorial Game Theory
who wins, and by how much
Generator

A position that comes back

A position that comes back
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.

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.

4 essays call loopy. 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.

Where it is called

Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.

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. Where it stops

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.

 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. Where it stops

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.

 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. Where it stops

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.

 on: which positions play can return to. A position graph with the moves of both players drawn, and beside it the shortest sequence of moves that gets back to each position. A position play can return to is a position whose value is defined in terms of itself, so the recursion every value in this subject is built from has no base case there. A position with no way back is one the ordinary recursion terminates on. Where it stops

A stopper and how to find one

The class a value theory for loopy games would need is the ones with no infinite alternating run, and the qualifier does the work: seventy-nine of the two hundred and fifty-six two-node loopy games qualify and seventy-two of them have a cycle. Every one has a decided outcome, and under eight tests the seventy-nine collapse to six.

The whole library · The position index · The figures that play back