Combinatorial Game Theory
who wins, and by how much
Concept

Depth — where it appears

How far the braces nest in a written form, which bounds the birthday of the value inside it and is attained by the canonical form. It counts levels where width counts options, and a form can be narrow and deep or wide and shallow.

Named by 6 essays across 3 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.

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 How old a form is, and how old its value is. Every one of the 256 forms born by day two, placed by the depth it is written at and by the birthday of the value it carries. Nothing sits above the diagonal, because a form cannot be younger than the value in it; the diagonal holds the forms written at exactly their value's birthday, and everything below it is a position written older than it needs to be. The count in each cell was obtained by canonicalising all 256 forms and measuring both depths.

How old a value is

A form's depth bounds the birthday of the value inside it, and reducing to canonical form attains the bound — for all 22 values born by day two, with no exception. Twenty-four of the 256 forms are older than what they are worth. The same reduction that makes the bound tight is what puts day three within reach: 98 option sets a side instead of four million, 9,604 forms, 1,474 values, a quarter of a second.

values · Numbers What each heap is worth. The value of a single heap of each size. Nothing here repeats: the forms grow deeper as the heap grows, which is what stops the impartial theory's periodic table from having an analogue.

Two players, two lists

Give each player their own list of how many counters they may take and the impartial theory stops applying. What survives is the outcome: it settles into a repeat, for every pair of lists, and that is a theorem. What does not survive is the value — on four of six pairs swept it has no repeat inside sixty heaps, and the birthdays are still climbing at the edge of the window.

positions · Partizan subtraction The reduction that puts options back. How the two reductions change the width of a form. Domination only ever removes an option. Bypassing a reversible option substitutes the answer's whole option list, so it can leave the form wider than it started — and the finished canonical form can be wider than the form it came from.

The reduction that puts options back

Canonical form is presented as simplification, and half of it is. Deleting a dominated option takes one away. Bypassing a reversible one substitutes the answer's whole option list, so it can leave the form wider than it started — and 60 of 32,428 forms end up with a canonical form wider than they are.

values · Reversibility 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

Named alongside it

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

BirthdayCanonical formExhaustive searchStar (∗)MemoisationUniquenessBorn on dayComparisonDay threeDominated optionEnumerationForm

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