Tuesday, July 9, 2013

More on comparing zero probability sets

There are two devices, A and B, each of which generates an independent uniformly distributed number between 0 and 1. You have a choice between two games.

Game 1: You win if B generates the number 1/2.

Game 2: You win if B generates the number generated by A.

Perhaps you say: "I don't care. I have infinitesimal or zero probability of winning." To make you care, suppose the game is free but the payoff is infinite.

Intuitively, you're equally likely to win either game. There is no reason to choose one over the other, given that the two devices are independent. It's no easier or harder to get 1/2 than to get whatever number A generates.

But there is another way of seeing the situation. Graph the state space of the game with the x-coordinate corresponding to A and the y-coordinate corresponding to B. The state space is then the unit square. But on Game 1, the victory region is a horizontal line y=1/2 while on Game 2, it is the diagonal line y=x. But the diagonal line has the square root of two, approximately 1.414, as its length, while the horizontal line has unit length. So you should choose Game 2.

Really?  I still think it makes no difference.  (And that it makes no difference shows that rotation invariance does not apply to all cases of uniform distribution in the square, since the horizontal line when rotated becomes obviously shorter than the diagonal one.)