Mathematics of Backgammon 

Position 1. Should the player on roll redouble? 
 The player who enters first loses a lot if he hasn't doubled, since he is very likely to gammon his opponent.
 Possession of the cube has little importance; whoever comes in first will simply blast his opponent off the board. The cube is only likely to come to life if a shot is hit in the bearoff.
So the advantage of rolling first seemed to outweigh the value of owning the cube, and I offered to play this as a proposition, where I would redouble and my opponent wouldn't, alternating first rolls. However an analysis by Bob Floyd (see "Riding the Tiger") shows that, even though the redoubling side should win that proposition, it is not necessarily right to recube.
Bob is a professor of computer science at Stanford with an international reputation in the design and analysis of algorithms, and a strong interest in mathematical aspects of backgammon. After seeing Bob's analysis, I went back to the drawing board and came up with this:

Position 2. Should the player on roll redouble? 
Nevertheless, the great advantage of the player on roll outweighs the value of owning the cube, and the roller must redouble (if he can afford it).
In this case, the infinite series discussed by Bob will add up, because each term decreases. Each successive term is multiplied by 2 and by 16/36 (the chance of a miss).
The idea of the cube rising to some astronomical level seems crazy — but consider that every time you play backgammon for money there is no limit on the cube. You need to be at least half crazy to play backgammon in the first place.