Combinatorial Game Theory
who wins, and by how much
Series

Scoring — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A game where the last move decides nothing. Rows of coins taken from either end, with the exact score for each side moving first. Under the normal-play convention this family is settled entirely by the parity of the row — nobody is ever without a move until the coins run out — so normal-play theory returns the same answer for every row and it is not the answer anybody wants. The scoring answer depends on nothing but the numbers.

    Counting at the end changes everything

    Go is scored. So are Dots and Boxes, chess and almost everything anybody plays for money — and none of them is the kind of game this site's whole apparatus is built for. The simplest scoring game there is shows what that costs — the normal-play theory gives every position of it the same answer, and the answer is useless.

    part 1 · applied
  2. What a pass buys, and what it costs. Rows of coins solved with and without a pass. Milnor's mean-value theory needs a non-negative incentive to move, and rows containing a coin nobody wants break that condition — a player forced to take is a player who would rather have passed. Allow a pass and the condition is not merely satisfied but unbreakable, on every row in range. The price is that a player who may pass is never stuck, so the last-move convention has nothing to attach to and the game needs a separate rule to end at all.

    What a pass is worth to a theory

    The rung below finds fifteen of twenty-seven coin rows where having the move is a disadvantage, and those are exactly the rows Milnor's mean-value theory has to assume away. Allow a pass and the hypothesis stops being a hypothesis — nought violations, on every row in range. What it costs is the convention the rest of this site is built on.

    part 2 · applied
  3. The condition has to hold underneath, not on top. Pairs of coin rows sorted by where the incentive condition holds, with Milnor's bound checked on each pair. Rows that satisfy the condition at every subposition never break the bound. Rows that satisfy it only at the top break it on a counted fraction — and a reader who tested the row rather than the row's insides would have called those safe. The distinction is invisible from the position and decides whether the theorem applies to it.

    A hypothesis has to hold all the way down

    Milnor's bound is proved by induction over the play, so the condition it needs has to hold at every position the play can reach. Checked on the row instead, ninety-two pairs pass the test and twenty-four of them break the bound. Checked at every subposition, twenty-eight pairs pass and none breaks it.

    part 3 · applied
  4. A game beside its own mirror, and what is left over. Every coin row added to its own negative, played out exactly, with the resulting scores counted. Under the last-move convention every such sum is worth nothing, because the mirroring strategy guarantees the second player the last move. Here the same strategy is available and the score it produces is not nothing: the mirror of a coin conceded is another coin conceded. Gold is the sums that do come to nothing, which are a minority.

    Nothing to subtract with

    Comparison is defined by contexts and computed by subtraction, and the equivalence between the two is a theorem about groups. A scoring game is not one — sixty-six of eighty-one coin rows do not cancel against their own negatives — and the difference test then fails on a row compared with itself, which every context accepts and nothing certifies.

    part 4 · applied
  5. Four restrictions, and what each one buys. Four candidate classes of scoring game — every row, the incentive condition at the top, the same condition at every subposition, and the rows that cancel against their own negatives — scored on two families of coin rows for the mean-value bound, for comparison by subtraction, and for cancellation.

    The restriction that buys the most

    Four candidate classes of scoring game, scored on the same two families and the same three questions. The class everyone expects to be tiny — the rows that cancel against their own negatives — is empty on rows of three and the widest restriction on rows of four, where it holds fifteen rows against the hereditary class's twelve and gets all 225 of its comparisons right against 108 of 144. The trade everyone expected does not exist.

    part 5 · applied
  6. Cancelling is not pairing. For three sets of coin values and rows of two to seven coins, how many rows cancel against their own negatives, how many pair off as nested equal pairs, how many do both, and how many do one without the other.

    Cancelling is not pairing

    The coin rows that cancel against their own negatives looked like the rows whose coins pair off as nested equal pairs, and on rows of four they are exactly those. From six coins the description fails in both directions — twenty rows pair off perfectly and do not cancel, and one coin set has a hundred and thirty-six that cancel with no pairing at all — and a row of five coins cancels, though an odd row can never pair off. What does hold, on every row swept, is that the first player in a row plus its negative never finishes behind.

    part 6 · applied
  7. Cancelling rows add to cancelling rows. Every pair of cancelling coin rows of two, four and five coins from minus two, one and three, grouped by their lengths, with the number of pairs whose sum cancels against the sum of their negatives.

    A cancelling pair is a zero

    Two cancelling coin rows side by side cancel against their two negatives — all 190 pairs from rows of two, four and five coins, the odd row that cancels without pairing off included. And a cancelling row beside its negative is invisible next to any other row: in 741 tests against every row of one to three coins, neither score of the context moves. A non-cancelling pair moves a score in 452 of 780. Scoring games have no inverses in general; this class has them, and they behave as inverses must.

    part 7 · applied
  8. Even rows always reward the move. For four coin sets and rows of one to seven coins, the number of rows in which the player to move does at least as well as when the opponent moves first. Every even column is full.

    Even rows always reward the move

    Milnor's mean-value theory needs an incentive to move — the player to move must do at least as well as if the opponent moved first. On a coin row with an even number of coins that is not a hypothesis but a theorem: the first player can collect one whole parity class of coins, and one of the two classes holds at least half the total. So the condition excludes no even row whatever the coins, the class the earlier table called 'incentive at the top' was every row of four, and the hereditary condition is a condition on odd intervals alone.

    part 8 · applied

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