Combinatorial Game Theory
who wins, and by how much
Series

Infinitesimals — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Smaller than every positive number, and not zero. Values that sit between zero and every positive number, each compared with zero and with 1/1024. Every relation drawn was computed by playing the difference, and one of them is confusion — neither greater, smaller nor equal. None of these is a number, and in a close game they are the entire margin.

    Infinitesimals

    Some positions are positive — Left wins them whoever moves first — and smaller than every positive number, including a millionth and a millionth of that. They are the values that decide close games, and the smallest of them is a single move's worth of nothing.

    part 1 · values
  2. How many ups, bracketed. Every position here is all-small, so no number says anything about it and the yardstick has to be ↑ instead. Each bar spans the multiples of ↑ the position lies between: the largest it is at least, and the smallest it is at most. Four of the seven are pinned to a single multiple of ↑; the rest keep a band that comparison cannot narrow, the widest being ∗ at four ups of slack.

    How many ups

    When every component of a position is smaller than every positive number, no number can decide it. What decides it is a count of ups — and comparison can pin that count down exactly, except when a star is present, when it cannot.

    part 2 · sums
  3. Tiny and miny: infinitesimals with a scale. Positions that are greater than zero and smaller than every positive number, and which are nevertheless strictly ordered among themselves — the larger the subscript, the smaller the value. Being smaller than everything positive is not one size of thing; it is a whole scale, and up sits above all of it.

    Tiny, miny, and the sizes below every size

    An empty two-by-four Domineering board is worth less than nothing and more than every negative number. It is not up, not down and not a fraction — it is a miny, and the minies come in sizes, strictly ordered among themselves below a floor no number reaches.

    part 3 · values
  4. Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal.

    The class where nobody runs out first

    Three stones in a row — blue, blue, red — and the position is worth exactly up. Clobber cannot produce anything else, because adjacency is symmetric — a player has a move precisely when the opponent does, and a game with that shape can never be worth a whole move to anybody.

    part 4 · values
  5. The bracket of a sum, against the sum of the brackets. Two all-small positions, the interval of multiples of ↑ each lies between, those two intervals added coordinatewise, and the interval the sum actually lies between. The added one always contains the computed one — greater-than survives addition — so the bracket never widens under a sum. Where it narrows, the parts were each too vague to pin down and the sum is not.

    When the ups add

    Atomic weight brackets do not add over a sum — they bound it. Over all 120 pairs from a fifteen-game family the sum's bracket came out exactly the sum of the parts' brackets 56 times, strictly narrower 64 times, and wider never; and the rule separating the two is one line long, because every one of the 54 pairs with a pinned part is exact and only 2 of the other 66 are.

    part 6 · sums

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