Why doesn't this bet have a positive expectation for both players?

Assume that the $ and the £ are 1:1.

Bob and Joe make a bet. Bob puts up $100 and Joe puts up £100. Bob wins if the £ appreciates in value relative to the $, and Joe wins if the $ appreciates in value relative to the £. Assume the probability is 50/50 of either player winning.

Why doesn't this bet have a positive expectation for both players? Because if the £ appreciates in value relative to the $, then Bob wins £100, and those £100 are worth more than the $100 he had to wager. And if the $ appreciates in value relative to the £, then Joe wins $100, and those $100 are worth more than the £100 he had to wager.

Common sense tells me that no bet can have a positive expectation for both players. Plus, if this example really worked, then you'd have investors all over the world just making bets with each other all the time and getting filthy rich off it :). But I'm struggling to explain/prove why. Both players are risking X to win Y where Y is greater than X, and each has a 50/50 chance of winning.

Can someone help me understand the reasoning?

Thank you.

Re: Why doesn't this bet have a positive expectation for both players?

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.