You win two dollars raised to your die roll

Question

A game pays you 2^X dollars, where X is one roll of a fair six-sided die. What is the fair (zero-EV) price to play? Would you actually pay that much, and how would you make a market on the game?

Accepted Answer

The fair price is the expected payout, averaging 2^X over the six equally likely faces: E[2^X] = \frac{2 + 4 + 8 + 16 + 32 + 64}{6} = \frac{126}{6} = \21.00. The tempting wrong answer is 2^{E[X]} = 2^{3.5} \approx \11.31, barely half. By Jensen's inequality, an exponential payoff is convex, so E[2^X] 2^{E[X]}, and here the gap is enormous because 2^X rockets from 2 to 64 across the range. The single X=6 face pays 64 and, at weight 1/6, contributes more than 10 to the fair price by itself: convex payoffs are dominated by their tail. Would you pay 21? At the fair price the game is zero EV but w…

You win two dollars raised to your die roll

Your answer

More Market Making questions