: I have a question concerning the liveliness of the cube. In a money game, assuming a
: dead cube the takepoint is 25%; assuming a completely alive cube the takepoint becomes
: 20%. Common wisdom says that the real takepoint is something like 21.5% (corresponds to
: an equity of -0.57). This would mean that one considers the cube to be alive to 70%. How
: can we compute mathematically this number?

No, we can't.  It depends on the type of positions.  Two examples:

a) A long race.  Here the trailer is likely to have a reasonably
efficient recube if things go his way, so the cube is relatively alive.

b) A holding game -- way behind in the race (so must hit a shot), but the
trailer having a perfect prime which he will be able to maintain for
several rolls.  In this case, if the trailer does hit his shot he will
suddenly become a huge favorite and lose his market by a mile.  Thus, the
cube is relatively dead.

The above are examples of positions where gammons are impossible or very
unlikely.  If gammons are in the air, things get more complex still.  For
example, consider a blitz position.  Assuming the trailer has a close
pass/take decision, this means that his win percentage is way above 25%
(to compensate for the many gammons he loses).  Thus, he will be able to
get in even more recubes than normal.  Also, this type of position tends
to lead to very efficient recubes when the game turns around, since
usually the trailer makes his improvements slowly.  Therefore, the cube
is even more alive than normal.

The 21.5% figure is the generally agreed figure for the "average"
position.  However, the actual number definitely depends on the position
in question.

: Also, in some scores for example 3-away 4-away the live takepoint is 19% but the dead
: takepoint is around 34%, what is considered to be the real takepoint? Instinctively, I
: would guess that the recube of the trailer is more alive than usual due to the very high
: takepoint of 40% of the redouble. Again, I would be very interested in how one could
: compute this number.

Yes, the cube is more alive than usual due to the takepoint of the
redouble.  And, once again, for the reasons above, there is no way to
compute this number.