I was thinking of a game where a player pays one dollar to toss a coin. If the result is a head they get their $1$ dollar bet plus an additional $1$ dollar. If the outcome is a tails they lose their $1$ dollar bet and get nothing.
If $n = 1$ the fair price will be $0.5$ dollars to play. But what happens if we want to make it n tosses? It becomes complicated because at each iterative step the player has to decide whether the current pay off is greater or less then the expected outcome of tossing another coin.
I simulated this once and got a fair price to be $0.76$ dollars for $n = 20$ but I just cannot recall what my logic was.
Any suggestions will be appreciated.
The players pay off is the number of heads in $n$ tosses.