pr.probability – Probability of fair value in a Die Game

You are allowed to roll a fair 6-sided die a maximum of 3 times. After any throw you can elect to “stop”. If you elect to stop after the ith throw then you will receive Xi dollars where Xi is the result of the ith throw, for i = 1, 2 or 3. For example, suppose you throw a 3 on your second throw and then elect to stop. You will then receive a payoff of $3 and will not proceed with the 3rd throw. You must stop after the 3rd throw if you have not elected to stop after the earlier throws.

(a) If you are risk-neutral what is the fair value of this game to you? Hint: Consider working backwards in time. e.g. If you have just one throw left what is the fair value of the game?

(b) If you are risk-averse and have log utility what is the fair value of this game to you?