Combinatorial Game Theory
who wins, and by how much
Concept

Transposition — where it appears

A position reachable by more than one sequence of moves, which is one position rather than several. Merging them turns a tree into a graph and is the difference between a computation finishing and not.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

A position reached eleven ways is one position

A 4×4 Domineering board has 5,700 positions in it and 6,257,129 routes through them. Three heaps of 7, 11 and 13 have 480 positions and 7.6 × 10¹⁶ routes. The gap between those two numbers is not an optimisation — it is the difference between a search that finishes and one that does not.

complexity · Search Knowing who wins, and knowing what it is worth. Nine positions, each evaluated twice by an instrumented evaluator that starts with an empty cache. The third column counts what deciding the winner costs and the fourth counts what the canonical form costs, in the currency each question is actually paid in.

Knowing who wins, and knowing what it is worth

Deciding a winner expands positions. Computing a canonical form expands pairs of positions, because a comparison unfolds as a recursion over one subposition of each and the reduction makes many comparisons. Measured on the same nine positions by an evaluator that starts empty every time, the second costs between 1.3 and 279 times the first, and the ratio grows with the tree.

complexity · Value cost What a value costs to write down. Every one of the 1,474 values born by day three, grouped by the width of its canonical form, with the number of symbols the form takes when it is written out. Each count was obtained by walking the canonical form and counting its nodes, so a subposition appearing twice is counted twice — which is what writing it out does. The widest values of the day are not the longest to write.

What a value costs to write down

The canonical form is the smallest form of its value, and it is smallest in the one currency the reduction happens to spend: options. Counted in symbols it is nothing of the kind — the widest value born by day three is not the longest, the longest has six options rather than seven, and every canonical form on the day except the seven integers writes some position out twice.

values · Reversibility What a day of canonical forms costs, written out and written once. Three costs for the values born by each of the first three days: every node written every time it occurs, every distinct subposition of a single form, and every distinct subposition of any form of the day. The last is one node per value, and the gap between the first and the last widens as the construction goes on.

The same position, written once

Writing out the canonical forms of day three takes 24,940 nodes. Naming each distinct subposition once inside each form takes 10,102, and naming each distinct subposition once across the whole day takes exactly 1,474 — one per value, because nothing appears inside a canonical form that is not itself a value of the day.

values · Reversibility How much of the board a who-wins search walks. Positions a memoised who-wins search of 4 × 5 Domineering expands under six move orderings, drawn to scale against the 48,670 positions the board has. Leaving the opponent fewest replies expands 1,125; trying losing moves first expands 30,202; all six find the same winner.

The order a solver tries the moves in

A memoised search asking who wins 4 × 5 Domineering expands 1,125 positions when it tries first the move that leaves the opponent fewest replies, and 30,202 when it tries losing moves first — the same answer at twenty-seven times the price. The ordering that already knows which moves win is not the cheapest. A win needs one move and a loss needs all of them, so the price of an order is paid one level down, in the replies it leaves.

complexity · Search

Named alongside it

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

Game treeMemoisationCanonical formExhaustive searchPosition graphDomineeringBirthdayComplexityDay threeDepthFormInteger

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