Combinatorial Game Theory
who wins, and by how much
Series

Loopy — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · limits
  2. 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.

    part 2 · limits
  3. 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.

    part 3 · limits
  4. 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.

    part 4 · limits
  5. 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.

    part 5 · limits
  6. 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.

    part 6 · limits
  7. 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.

    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.

    part 7 · limits

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