# how to divide money fairly before the game is played?

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.

• Why wouldn't you simply give them each their bets back? Otherwise, I suppose I'd just give them their expected winnings. – lulu Dec 22 '16 at 17:09
• I also don't understand the payout rules for $C$. Say $AA$ has bet $100$ and $BB$ has bet $20$. If $C$ happens, who gets what? – lulu Dec 22 '16 at 17:11
• thank you for asking. The situation is that we don't have a history of how much each of them are betting. but we know together that would be $X$ dollars. if $C$ happened, then $AA$ would win 50 and $BB$ would lose 10. – kensaii Dec 22 '16 at 17:19
• If we don't know how much they each bet, how do you know that $AA$ would win $50$? Anyway, how can $AA$ win $50$? There's only $120$ in the pot... – lulu Dec 22 '16 at 17:22
• Regardless of the rules...assuming that you know how to calculate the three payouts, just give each player their expected gain/loss. – lulu Dec 22 '16 at 17:23