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.