Combinatorial Game Theory
who wins, and by how much
Theme

The thread: The theory runs out

Misère play, loops and computational hardness each break something essential. Knowing which of the three is biting is most of knowing where a game stands.
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 Where it stops

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.

 the boardworth 2 | −1/2outcome N{2 | −1/2}Left plays verticallyRight plays horizontally Particular games

Domineering

One player places dominoes vertically, the other horizontally, on a shared grid. The rules take one line, the values are a mess, and that mess is the point — this is what the theory looks like applied to a game nobody designed for it.

 ABCthe moves lead back to where they startedno base case, so the recursion never bottoms outa third outcome appears: neither player can force a winloopy game theory is a separate subject with separate machinery Where it stops

Loopy games

The whole theory assumes play stops. Allow a position to recur and the induction that every value rests on has nothing to stand on — and a fifth outcome appears that normal-play theory has no name for.

 001230412308123012123016123020123024heap size, and the value of a heap that bigperiod 4, from heap 0a heap of size n is worth ∗g(n) — and the whole game is the nim-sum of its heaps Impartial games

Grundy sequences, and where they stop being predictable

Computing one Grundy value is a mex. Computing all of them produces a sequence, and the sequences do something nobody has fully explained — most of them eventually repeat, some of them take thousands of terms to start, and for a few nobody knows whether they ever do.

 {6 | 0}t = 3a big fight{2 | 0}t = 1a smaller one{1 | 0}t = 1/2small change{0 | 1}no temperaturesettled — a numbercomponenthow much is at stakethe whole position is worth {{{19/2 | 17/2} | {15/2 | 13/2}} | {{7/2 | 5/2} | {3/2 | 1/2}}}and the first move goes in the hottest part, which is a theorem up to a small error rather than a rule of thumb Temperature

Playing the hottest

Given several independent fights, play in the one with most at stake. The rule is simple, it is what strong Go players do without being told, it is provably close to optimal — and it is provably not optimal, which is the interesting part.

 ∗N∗N0P∗Nblue toads move right · red frogs move leftevery value came out of the moves; none was chosen Particular games

Toads and Frogs

Toads shuffle right, frogs shuffle left, and either may jump over one of the other. A strip six cells long is worth exactly up. Another six-cell strip is worth exactly down. Nobody has a formula for which.

 the value of a Nim positioninstantthe Grundy value of a small subtraction gamelinearthe canonical form of a moderate positionexponential in theorywho wins a general Domineering boardno efficient methodwho wins a generalised board gamePSPACE-completecostthe definitions are constructive, so everything here is computable in principleand the practical range of an exact evaluator is a few dozen moves, which is the working constraint Where it stops

How hard is it

Every theorem on this site stays true at any size. The answers stop being reachable long before the games get interesting — deciding the winner of a generalised board game is PSPACE-complete, and an exact evaluator gives out after a few dozen moves.

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