There are 2 people betting against each other, called $AA$ and $BB$, and they make a pool of bet money of $X$ dollars, and we don't know the history who bet how much, e.g. $AA$ may bet with more confidence as \$2 for \$1. The ratio is obviously not 1 to 1.
There are 3 outcomes: $A,B,C$ of the game. Assume the probability associated with the event $A$ is $a$, $B$ is $b$, and $C$ is $c$.
If $A$ is the outcome, then $AA$ takes all the money.
If $B$ is the outcome, then $BB$ takes all the money.
If $C$ is the outcome, then $AA$ win half of the money $AA$ bet, and $BB$ lose half of the money $BB$ bet.
At the end, the game is canceled. How should we divide the money to $AA$ and $BB$ according to their probabilities that would seem fair?
It is a zero-sum game, in the sense that expected gain/loss of $AA$ + expected gain/loss of $BB$ = 0. (thanks to @lulu's comment)
The third constraint $c$ can be used to deduce another equation, IMHO.