Combinatorial Game Theory
who wins, and by how much
Concept

PSPACE — where it appears

The class of problems a machine can solve with a table that grows only as a polynomial, which is where deciding most games sits. The class is named after memory rather than time, and that is not an accident: a game is played by a machine that need only hold one line.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

Which questions are answerable. The theory is exact and much of it is expensive. Values are computable by definition; computing one for a position of any size is a different matter, and deciding the winner of a generalised board game is complete for PSPACE — as hard as anything solvable in polynomial space.

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.

complexity · Complexity A formula, drawn as a game. A token on a directed graph. A move slides it along an edge to a vertex not yet visited, and a player who cannot move loses. That is the whole game, and deciding who wins it is as hard as anything decidable in polynomial space — which is the strongest hardness claim anybody makes about a combinatorial game.

Hard, proved

A game is as hard as a logical formula when the formula can be drawn as the game. Here is the drawing — a quantified formula turned into a graph with a token on it — with every formula over three variables played both ways and required to agree.

complexity · Complexity What it costs to decide the winner. Three families of game, each at several sizes, with the number of distinct positions an exhaustive solver must evaluate beside the work its closed form does. The bars are logarithmic. Hardness is not about the size of the board or the length of the rules — Domineering has the shortest rule here — it is about whether anybody has found the shortcut.

The game with the shortest rule is the hard one

Deciding a generalised board game is PSPACE-complete, which is a statement about families and encodings rather than about size. Nim in the same subject is settled by one pass over the input at any size, and green Hackenbush by one pass over the edges — while Domineering, whose rules take a single line, has no shortcut anybody has found.

complexity · Complexity Every quantifier is a move. A quantified boolean formula with its quantifiers drawn as turns: an existential is a choice by the player to move, a universal a choice by the opponent. The same formula is put through the reduction to Generalized Geography and the two answers are checked against each other, so the prefix of quantifiers and the game beside it are one claim.

A puzzle asks once, a game asks alternately

Quantifier alternation is the whole difference between a puzzle and a game. One chooser is an existential and its answer is a witness somebody can check; two choosers taking turns is a prefix of alternating quantifiers, and the witness stops being an assignment and becomes a strategy.

complexity · Alternation A winning strategy on 3×3, drawn whole. The whole of one player's winning strategy on a small Domineering board: their own move at each of their turns, and every reply the opponent has at each of theirs. The strategy branches only where the loser chooses. Its size is what somebody would have to be handed to check the claim that this player wins, and it is far larger than the claim itself.

"Left wins" has no short proof

A complete solution of Nim on heaps of 7, 11 and 13 is 480 table entries. A winning strategy for the same position — one move of the winner's at each of their turns, and an answer to every reply — has 56,167,022 nodes in it. The answer is smaller than the proof by a factor of a hundred thousand.

complexity · Complexity One node per route, one node per position. For each board, the number of nodes in the recursion tree a solver with no memo table would walk, beside the number of distinct positions that tree contains, beside the longest run of moves in it. The first number is the cost of forgetting; the second is the size of the table that avoids it; the third is the stack, and it stays small however the other two grow.

The class is named after memory, and that is not an accident

A 4×4 Domineering board has 6,257,129 routes through it, 5,700 distinct positions, and a deepest line eight moves long. Those three numbers are three different resources, and the smallest of them is the one that gives games their complexity class.

complexity · Complexity Generalized Geography. A token on a directed graph. A move slides it along an edge to a vertex not yet visited, and a player who cannot move loses. That is the whole game, and deciding who wins it is as hard as anything decidable in polynomial space — which is the strongest hardness claim anybody makes about a combinatorial game.

A token on a graph

Geography is an impartial game whose position is a vertex and a history, so a ten-vertex graph has ten thousand states rather than ten. Take the arrows off and the same game is decided by a maximum matching — first player wins exactly when every maximum matching covers the start, verified on 41 vertices across eight graphs. One word in the rules separates a polynomial criterion from a PSPACE-complete problem.

impartial · Geography A bridge circuit, with a point on every link. The switching graph drawn as a bridge circuit, with a new point in the middle of every link in green and the original inner points in blue, already belonging to Short. Played as a game on the green points it gives the same verdict as the original game on links, because claiming a middle point is securing its link and deleting it is deleting the link.

A point with three neighbours

The switching game on links is settled by counting — enough links, arranged as two trees. Played on points instead, it is the game Hex belongs to, and the count is gone. The link game turns out to be the point game in which every contested point has exactly two neighbours; give one a third, and two graphs with the same points, the same links and the same number of separate routes can have opposite winners.

applied · Switching Using up edges instead of vertices. An undirected graph of 4 vertices and 4 edges, with the winner at every start of two games on it: vertex geography, where a move uses up the vertex it leaves, and edge geography, where it uses up the edge it crosses. The matching criterion decides the vertex game everywhere and is right about the edge game at 0 of 4 starts.

Using up the edges instead

Undirected geography is decided by a maximum matching when a move uses up the vertex it leaves. Use up the edge it crosses instead and the matching is exact on every tree — on a tree the two games are one game — and on nothing else. Over every connected graph on up to six vertices it names the winner at 480 of 745 starts once there is a cycle, it gets a four-cycle wrong from every start, and the more cycles a graph has, the more of its misses are wins that are really losses.

complexity · Geography One cycle, and how much of it the criterion misses. The matching criterion for undirected geography put to the game where a move uses up the edge it crosses, on every connected graph up to 6 vertices with exactly one cycle. It is right at 77 of 114 starts, and at none of the six on the graph whose cycle has six vertices.

Two graphs a rule cannot tell apart

The repair proposed for the matching criterion was to read the cycle as well. Over every connected graph with exactly one cycle up to six vertices — 21 graphs, 114 starts — eleven such rules reach at most 91, and the winner is not a function of the matching, the cycle's length, the start's distance from it, its degree and the edge count together: six cells of that table hold both verdicts, the smallest a pair of five-edge graphs.

complexity · Geography 6 turns in strict alternation, priced. One quantifier prefix taken apart into its blocks, with the size of a winning strategy computed a term at a time. A choice made after k of the opponent's turns has to be written down once for each of the 2^k lines the opponent can produce, so the total depends on where the opponent's turns sit and not merely on how many there are.

Twelve turns, and three different prices

The earlier essay prices a universal quantifier at a doubling and leaves it there. Twelve turns with six of them the opponent's cost 6, 63 or 384 decisions to write down, depending on nothing but the order the turns come in — and the cheap arrangements are cheap for only one of the two players. What a claim costs is the number of times the choosing changes hands.

complexity · Alternation

Named alongside it

The objects these essays reach for when they reach for this one.

ComplexityExhaustive searchIntractableCertificateGeneralized GeographyImpartialDomineeringNimAlternationExact evaluationPosition graphQuantified Boolean formula

All concepts