Problem for Discussion Section

Suppose that you are given a choice between playing two coin-flipping games. Each one costs $100 to play.

In the first game, if you get 3 heads in a row, you get $1000. Otherwise, you lose. If you lose, the value of the game is -$100, and if you win the value is $1000 - $100 = $900.

In the second game, you also need to get 3 heads in a row to win. However, if you get 3 heads in a row, you continue flipping until you get tails. Your payoff is equal to 2

^{n+1}, where n is the number of heads in a row. So, if you win, the least you can win is 2^{4}, or $16. If you get to 10 heads in a row, you win 2^{11}, or $2048. If you get top 14 heads in a row, you win $32,768. As in the previous game, you subtract your $100 entry fee. So, if you lose, the value of the game is -$100. If you get 14 heads in a row, the net value of the game is $31,768.

Which game has the higher expected value? Which game would you rather play? Can you make a case for why it might be rational to prefer the game with lower expected value?