The paradox is that, if both players know their doubling cube strategy, doubles are accepted more often than not. It's easiest to explain why in the context of a money game; there are additional considerations in a tournament match that we can discuss later.

When playing strictly for money, each game stands on its own. The players agree in advance how much money they will wager on each point. Say they play for $2 a point. Then a simple game is worth $2 to the winner. A gammon is worth $4, as is a doubled game. A doubled gammon is worth $8, and so on. (This can get expensive!)

OK now, suppose we are playing for $2 a point. After a while you think your position is so good that you have a 70% chance of winning the game (nothing in backgammon is certain!). You properly turn the cube to 2 and offer it to me.

Now, here is what I am thinking. "If I drop the cube right now, I will lose 1 point, or $2. Dropping will cost me $2, so dropping is worth minus $2 to me. On the other hand, if I take the cube, I will have a 70% chance of losing $4. However, I will also have a 30% chance of winning $4! The value to me of accepting the cube is therefore 0.70 x (-$4) + 0.30 x (+4) = -$2.80 + $1.20 = -$1.60. Since minus $1.60 is better than minus $2, I should take the cube even though I only have a 30% chance of winning!"

Now, you might ask how the game can have a value of minus $1.60, when I know for sure I will either win $4 or lose $4. Think of the minus $1.60 as the average result that you would obtain if you played many games against this same opponent and always accepted the cube when it was offered to you when the winning odds are 70-30. If you encountered this situation 100 times, you would expect to win about 30 times and lose about 70. Your net loss would be ($120 - $280) or $160. A loss of $160 over 100 games works out to $1.60 per game.

Note that if you dropped the cube every time, you would lose $200 or $2.00 per game.

Of course, if you and your opponent are evenly matched you should also encounter the reverse situation 100 times, where you get ahead, double at 70-30 and he accepts.