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?