The same position, reduced two ways
A position with several reductions available at once, taken in two different orders. Every step deletes an option nobody would play or bypasses one that backfires, and the two trails end at the same form — which is what uniqueness of the canonical form actually claims, and it is a statement about the process rather than about the answer.
6 essays call
reduction-orders. The drawing above is what it returns with no arguments at all; every
call below passes it something, because a placement that passes nothing draws whichever
member of the family the generator happens to default to rather than the one its essay argues
about.
The positions it draws
5 distinct positions, harvested by running this generator again at the options each essay passed it.
| Position | Worth | Outcome | Drawn in |
|---|---|---|---|
{−1, 0, ∗ | 1} |
1/2 |
L | The reduction that always shrinks |
{0, ∗ | ∗} |
↑ |
L | The reduction that always shrinks |
{0, ↑ | 0, ∗2, ↓} |
∗ |
N | How much a list of options can lose · How wide a form can get |
{0, ↑ | 0, ∗2, ↓} |
∗ |
N | Two hundred and fifty-six ways to write twenty-two things · The reduction that puts options back |
{0, ∗ | 0, ∗2, ↓} |
{0, ∗ | ↓} |
N | The reduction that always shrinks |
Where it is called
Changing this generator changes every one of these figures.
Values
Two hundred and fifty-six ways to write twenty-two things
Every game whose options come from the four born on day one — there are 256 of them, and between them they carry 22 values. The reduction that collapses one to the other has choices in it at every step, and uniqueness is the claim that none of the choices matters.
Values
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
The reduction that always shrinks
Canonical form is two reductions and they are not the same kind of operation. Deleting a dominated option removes one option and can do nothing else; bypassing a reversible one substitutes a whole option list. Over the 256 forms born by day two, deleting alone finishes 225 of them and accounts for 480 of the 520 options that come off — and the 31 it cannot finish are almost all the ones with a star in them.
Values
How much a list of options can lose
Deleting a dominated option is the reduction with no surprises, and how many options it takes is decided by the shape of the order rather than by the values in it: the survivors are the maximal elements, and the count is the length of the list less the number of them. The essay separating the two reductions closed by predicting that the longest chain would give the number. It is a lower bound, exact on 3,859 of the 7,315 four-option lists and wrong on the rest.
Values
How wide a form can get
Bypassing a reversible option replaces it with a whole option list, so a form grows in the middle of its own reduction. Whether it can come out wider than it went in is the question that leaves standing, and over 64,515 forms built from day-two options the answer is no, not once — the growth is real, it is transient, and the widest canonical form reached is exactly as wide as the widest form that reaches it.
Values
A factor, and not an overhead
Deciding who wins a form searches the form's own tree. Reducing it to canonical form searches a difference game for every comparison, and a difference game is a sum. Over 256 forms the reduction expands 5.46 times as many positions — and the ratio runs from 0.58 at one option to 9.80 at eight.
The whole library · The position index · The figures that play back