Combinatorial Game Theory
who wins, and by how much
Series

Realisability — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The values the construction hands down, and the values games produce. The two lists counted against each other. The construction produces 1,474 values by day three; the eleven thousand positions swept here produce 1,193, and only 116 of those are on the construction's list. A value's birthday and a value's reachability have nothing to do with each other.

    The values nobody's game produces

    The construction hands down 1,474 values by day three. Seventeen rulesets on this site, swept to eleven thousand positions, produce 1,193 — and only 116 of those are on the construction's list. Two of the twenty-two values born by day two are produced by no position of any game here, and 1,077 of the values that are produced are born later than day three. A value's birthday and a value's reachability have almost nothing to do with each other.

    part 1 · values
  2. How old a value is, against how big a board it takes to show it. For each birthday, the range of sizes of the smallest position exhibiting a value of that age. The bars do not march rightwards: values born on the last day of the sweep are shown by six-piece positions, and values born on the fourth need up to fifteen.

    The cheapest way to show a value

    Eleven thousand positions from fifteen rulesets reach 1,193 values, and for each of them there is a smallest board that shows it. Set against the birthday the two measures agree hardly at all — until the numbers are taken out, at which point they agree rather well, and the whole apparent independence turns out to be a fact about integers.

    part 2 · values
  3. The birthday is a floor. How many more pieces a value's cheapest exhibit needs than the value has days. It is never fewer, on any of the 728 non-number values, and it is exactly none on 476 of them.

    The birthday is a floor

    The rung below measured a correlation of 0.73 between a value's birthday and the size of its cheapest exhibit, and asked which values are dearer than the birthday suggests. The relation is not a trend. Over all 728 non-number values the exhibit is never smaller than the birthday and is exactly the birthday on 476 of them, and the excess on the other 252 belongs to the game rather than to the value — the ruleset accounts for 40 per cent of its variance.

    part 3 · values
  4. The trend, running backwards. The excess grouped by the width of the value's canonical form. Values with wide option lists are exhibited nearer their birthdays than narrow ones, which is the opposite of what the proposal predicted.

    Wider costs less

    The rung below found the cheapest exhibit of a value never smaller than its birthday, exactly equal on two thirds, and the ruleset explaining 40 per cent of the rest. The variable it proposed for the remainder was the width of the form. Width and excess correlate at −0.39: the wider the value, the closer to its birthday it is exhibited, and inside a ruleset the relation cannot even agree on a sign.

    part 4 · values
  5. The excess is not a flat fee. The excess fitted against the birthday inside each ruleset with enough values to fit a line. A fee would have a slope of nought; every slope here but one is negative, so the excess is largest on the values born earliest.

    The entry fee was the cap

    Two rungs measured how much bigger a position has to be than the value it exhibits, and attributed what was left to the ruleset — Toads and Frogs paying 2.25 squares on everything, green Hackenbush paying nothing. Neither number is a property of the rules. Inside every ruleset the excess falls as the birthday rises, because the sweep's size cap censors exactly the values that would pay most — and three squares past the cap, Toads and Frogs exhibits values born later than the strip is long.

    part 5 · values
  6. Two orders of magnitude. The share of positions of a size whose value is one no smaller position exhibits. Every Hackenbush string is a new value and fewer than one Clobber row in a hundred is.

    The rate was the alphabet

    The rung below asked for a quantity a size cap cannot censor and proposed the rate: how many new values a ruleset produces per extra square. The rate is honest and it measures the notation — every ruleset grows at close to the number of symbols its positions are written in, and the seven span less than a factor of two. What separates them is the yield, which spans a hundred and nineteen.

    part 6 · values
  7. The yield, four sizes further. Toppling Dominoes rows to twelve, with the count of values not seen at any smaller size and the share of rows that is.

    The mirror was the floor

    Toppling Dominoes' share of genuinely new values had fallen from one to a half over eight sizes, and the rung below could not tell a floor from a slow fall. Four more sizes settle it: the distance above a half halves every two sizes. And the half is not a shortage of values but a symmetry — a row played from the other end is the same game, and the ruleset is as injective as that allows.

    part 7 · values
  8. Same group, three different yields. The rulesets with a trivial symmetry group, which the conjecture predicts must all have a yield of one.

    Three groups, and three yields

    The conjecture was that each ruleset's yield tends to the reciprocal of its symmetry group's order. Three rulesets have a trivial group and predicted yields of one; they measure 1.000, 0.531 and 0.204. And every colliding value in Push and Shove — all 175 of them — has two rows no symmetry relates.

    part 8 · values

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