In 1738, J. Bernoulli investigated the St. Petersburg paradox, which works as follows. You have the opportunity to play a game in which a fair coin is tossed repeatedly until it conies up heads. If the first heads appears on the n.th toss, you win 2n dollars.

a. Show that the expected monetary value of this game is infinite.

b. How much would you, personally, pay to play the game?

c. Bernoulli resolved the apparent paradox by suggesting that the utility of money is measured on logarithmic scale (i.e., U (S n) = a log2 n + b, where S n is the state of having $n). What is the expected utility of the game under this assumption?

d. What is the maximum amount that it would be rational to pay to play the game, assuming that one’s initial wealth is $k?