AbigailII: Interesting discussion. In your situation, I would actually do the opposite. Consider the two extremes:

If both you and your opponent keep redoubling, the entire match will be decided in just one game. That means that you're just one mistake away from losing the match. More specifically, you would lose 26% of the time in this case (not considering draws).

In the other extreme case, if nobody doubles and each game is only for one point, your chances of winning a long match are significantly higher. I'm not going to do the math, but the more games you play, the smaller the chance is that your opponent will win more games than you. Your risk of losing the match will be significantly lower than those 26%.

In general, I would say that the stronger player should be conservative in using the cube, while the weaker player should be eager to use it. I even have a feeling that this is true in backgammon as well, but I wonder what stronger bg players would think about this?