If the players are betting against each other, and by that I mean one wins what the other loses, then expectation must be zero (I am assuming that both currencies would appreciate by the same percentage). Quite simply, they'd be playing a zero sum game each expecting to win/lose half of the time.

What is confusing in this case is that when you win, you win more because your winnings appreciated. BUT you must realize that when Bob loses his $100, the $100 appreciated so he is actually losing more than the nominal values suggests.

In other words, when Bob wins he gets L100 (at time 1) which is equivalent to $(1+x)100 (at time 0) but when he loses, he loses $100 (at time 1) which is now worth L(1+x)100 (at time 0). (extremely important to realize i have put both values in terms of nominal terms at Time 0)

Since these payoffs are in nominal terms at Time zero, they are actually equal since at Time zero the $ and the L are 1:1. So he wins and loses an equal amount.