Just made up a problem. Have ideas, but no proof yet – came here for professional help.
Tonight an important match of Metagame (box/football/baseball/chess) takes place. Everyone is hyped. Everyone is making bets.
The result of the game will be represented with a real number between $0$ and $1$.
We have intel on $2$ (two) mathematicians. Mathematician $A$ believes that the probability density distribution of the game result $x$ goes as $F_a(x) = 2-2x$. At the same time, mathematician $B$ believes that game result is distributed as $F_b(x) = 2x$.
Both matematicians take bets rationally: if their expected winning is more than zero (read: equals zero, you stingy fox) they will take the bet. At the same time, the maximum that each of them will bet is 10 dollars.
You need to make two (different) bets with $A$ and $B$, so that no matter what game results are, you get profit. The money you can bet is technically non-limited, however, risking more than 10 dollars for a single bet is obviously a bad step – if you lose this bet, winning the other will barely compensate for your loss.
Make a strategy with maximal guaranteed profit, and a proof of its perfection.