Fight for a Dollar
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.