# Fight for a Dollar

**Logic**Level 5

Amy and Becca play a game in which they choose an amount to pay (\(A\) and \(B,\) respectively). These amounts determine each person's odds of winning the "prize pool" (i.e., Amy wins with odds \(A:B\) and Becca wins with odds \(B:A\)).

The "prize pool" of this game is the sum of **both of the payments plus an additional $1**. How big will the prize pool be if Amy and Becca play according to the pure-strategy Nash Equilibrium of this game?

Assume that they can pay any positive real number amount of money, and give your answer to 3 decimal places.