Combinatorial Game Theory
who wins, and by how much
Ladder

Misere — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. normal playmisère playevery impartial position is a Nim heapno such reduction existsequal games can be swapped in any sumonly within a restricted universethe value is a single small integeran element of a quotient monoida canonical form exists and is uniquecanonical forms are enormousthe game is what mattersthe game is what mattersmisère quotients recover some of it, one game at a timeand there is no general theory, which after fifty years is a real result rather than a gap

    Misère play

    Change one word — the player who cannot move wins — and the games are identical, the strategies are not, and almost every theorem on this site stops being true. It is the cheapest possible modification and the most expensive.

    rung 1 · limits
  2. 6 classes under misère play4 under normal play — the Nim values 0, 1, 2, 3computed over all 28 positions with at most 6 heaps1{1}{2}{1,2}{2,2}{1,2,2}1{1}{2}{1,2}{2,2}{1,2,2}1{1}{2}{1,2}{2,2}{1,2,2}{1}1{1,2}{2}{1,2,2}{2,2}{2}{1,2}{2,2}{1,2,2}{2}{1,2}{1,2}{2}{1,2,2}{2,2}{1,2}{2}{2,2}{1,2,2}{2}{1,2}{2,2}{1,2,2}{1,2,2}{2,2}{1,2}{2}{1,2,2}{2,2}shaded — a lossfor whoever must movea dot is a sum thatleaves this universeeach class is named by the smallest position in it, and 1 is the empty positionthe table is a quotient of the universe drawn, not a proof about the whole game — which is a weaker claim, and the true one

    What survives misère play

    Misère play destroys the value theory, and something much smaller grows back. Fix one game, look only at sums of its own positions, and the classes that behave alike form a monoid — computed here, and larger than the normal-play answer every time.

    rung 2 · limits

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