Two players are going to take turns trying to guess a number selected randomly from 1 to 15. After each guess if the number was not chosen they will both be told whether the actual number is higher or lower. A player must make a reasonable guess among the possible numbers remaining on his turn.
What is sought in each of the following cases is the best strategy for each player and the chance they will win.
Case 1: Unlimited (up to 15 if needed) guesses. The winner gets $30.
Case 2: Only 4 guesses allowed total. If the number is guessed the winner gets $20 and the loser gets $10. If the number is not guessed neither gets anything.
Well, very interesting problem Jer, at least for Case 2.
Case 1 seems straightforward. Neither player wants to give the other player (who goes next) any more information than needed, so each player in his turn should pick either the lowest or highest of the numbers which are still reasonable (since they are all equally likely to be correct). Player 1 gets a maximum of 8 guesses, and wins with probability 8/15. Player 2 gets a maximum of 7 guesses, and wins with probability 7/15.
I'll start my Case 2 analysis in a separate posting.