Combinatorial Game Theory
who wins, and by how much
Generator

Options handed to Left in 1 | −1

Options handed to Left in 1 | −1
Options handed to Left in 1 | −1. A position, and one candidate option after another added to it. Where the gift is one the player would never take the value does not move at all; where it is one they would, it does. The last column is the value of the enlarged form, computed by the same recursion as the original.

A position, and one candidate option after another added to it. Where the gift is one the player would never take the value does not move at all; where it is one they would, it does. The last column is the value of the enlarged form, computed by the same recursion as the original.

3 essays call gift-horse. 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

7 distinct positions, harvested by drawing the figure again at the options each essay asked for.

PositionWorth OutcomeDrawn in
{ | −1} −2 R An option nobody would take
{ | 0} −1 R An option nobody would take
{1 | −1} 1 | −1 N An option nobody would take · How hot a day gets · Nobody wants to move here · The operator chosen for one game · The operator that puts the star back · The values that are their own negatives
{1, {1 | −1} | −1} {1, {1 | −1} | −1} N An option nobody would take · Nobody wants to move here
{1, {2 | 0} | −1} {1, {2 | 0} | −1} N An option nobody would take
{1, 2 | −1} 2 | −1 N An option nobody would take
0 0 P A rule with no promise at all · A self-negative value costs a day · An option nobody would take · At least five hundred and seventy-one · Equal in every company · Fifty-two errors and seven sizes · How rare it is to be bigger · Nobody has to move · Nothing worth fighting over · The fight never runs backwards · The first theorem, and the winner it declines to name · The operator that puts the star back · The other way to move a row · The simplest game above both · The thirty that cancel themselves · The values that are their own negatives · Three players and no answer · Two people, four years apart, one theorem · What is left when the small change is thrown away · When a switch is not a switch · Where the order and the sum disagree · Who moves last · Start at the end and work backwards

Where it is called

Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.

Options handed to Left in 1 | −1. A position, and one candidate option after another added to it. Where the gift is one the player would never take the value does not move at all; where it is one they would, it does. The last column is the value of the enlarged form, computed by the same recursion as the original. Values

An option nobody would take

Every reduction of a form deletes. The gift horse principle adds: a move may be handed to a player for nothing, provided it is one they would never choose. Over all 484 additions to the values born by day two, 283 leave the value exactly where it was and the 201 that move it are precisely the ones the condition forbids — with the boundary at *not better*, which is a weaker demand than *worse*.

 What each move is worth to the player making it. For each position: every incentive, whether they are all strictly negative, whether the position is a number, and its temperature. The middle two columns are two different computations of the same fact. Values

Nobody wants to move here

A position is a number exactly when every move loses ground for the player making it. The test never mentions numbers, it disagrees with the ordinary one on none of the 1,474 values born by day three — and the reason a position fails it is not that somebody wants to move. It is that somebody cannot afford to wait.

 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. 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.

The whole library · The position index · The figures that play back