## Are there undefined equities in backgammon?

Backgammon bots presuppose that every backgammon position
has a well-defined *equity*, which in the case of a money game
means the expected payoff if both sides employ “perfect play”
(or more accurately, *Nash equilibrium* play).
However, if the cube value is truly unlimited,
then it is not clear that the equity of an arbitrary position
in backgammon is always finite.

It has been recognized for a long time that there are some
positions in backgammon whose equity is probably undefined,
although to the best of my knowledge, this has never been
mathematically proven. My contribution to this topic has been
to exhibit a backgammon position for which one can rigorously
prove that the equity is either undefined or zero.
Anyone with some backgammon experience can see that the
equity is “obviously” not zero,
so this comes very close to a rigorous proof that
the equity is undefined. Perhaps someone reading this
can close the gap by proving rigorously that the equity is not zero.
For details, see
this post and
this post
that I made to the BGOnline forums.
(You may need to set your browser's text encoding to "Western"
to get some of the characters to display properly.)

One catch with my position is that one can show,
via retrograde analysis, that it cannot be reached from the
initial position in backgammon. However, very similar positions
can be reached.