Combinatorial Game Theory
who wins, and by how much
Series

Dead-ending — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Where running out of moves is permanent. Eleven rulesets, each walked position by position from three small boards, with every position at which a player has no move examined for whether any continuation gives them one back. Nothing here is evaluated: dead-ending is a property of the rules, and two boards worth the same value can differ on it. 9 of the 11 are dead-ending and 2 are not.

    Nobody comes back

    There is a class of games in which running out of moves is permanent, and it is the setting almost every modern misère result is stated in. Nine of this site's eleven rulesets belong to it across 5,334 positions; the two that do not are Toads and Frogs and Amazons, and Toads and Frogs loses the property to a single clause — delete the hop and it joins the list.

    part 1 · limits
  2. How often one position beats another. Misère comparison inside each ruleset's own universe. A quarter to a half of ordered pairs compare, and the ruleset that is not dead-ending is in the middle of the range.

    What the class does not buy

    Dead-ending is the hypothesis several modern misère results are stated under, and the rung below sorted this site's games into it without running the comparison those results are about. Running it: a quarter to a half of ordered pairs compare inside a ruleset's own universe, which is a great deal — and the ruleset that is not dead-ending sits in the middle of that range. Ten comparisons are lost when a universe is enlarged, and every one is lost to a dead-ending company.

    part 2 · limits
  3. One-sided, all three. The three option tests with their disagreements split by direction. None ever refuses a comparison that holds.

    Wrong in one direction only

    The rung below asked for the simplified comparison test the dead-ending hypothesis is supposed to license, and predicted it would agree with the quantifier on the dead-ending rulesets and not on Toads and Frogs. Written three ways and scored on 492 pairs, it agrees best on the ruleset that is not dead-ending — and never once refuses a comparison that holds, which makes it a sound filter and not a test.

    part 3 · limits
  4. Who gains, and how much. How much each test improves when dead-endedness is turned on, with the class-specific test beside the others.

    The clause that turns the class off

    Three rungs failed to find the dead-ending class doing measurable work, and each time the population was blamed. Toads and Frogs with and without the jump is the matched pair the anchor wanted — the same board with the class switched on and off — and on it the test the class licenses gains less from the class than a control that has never heard of it.

    part 4 · limits
  5. Neither quotient identifies anything. The number of misère-equivalence classes on each side of the matched pair, against the number of distinct positions.

    A quotient that identifies nothing

    The dead-ending class is famous for quotients rather than comparisons, so the matched pair was asked the question its own subject is about. Neither quotient identifies a single pair of positions, and both are separated by exactly five addends — because a quotient is small when its universe is poor, which is a choice of company and not a property of a class.

    part 5 · limits

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