You win thirty-six over your die roll

Question

A game pays you \dfrac{36}{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: E\!\left[\frac{36}{X}ight] = 36 \cdot \frac{1 + 	frac12 + 	frac13 + 	frac14 + 	frac15 + 	frac16}{6} = 36 \cdot \frac{2.45}{6} = 36 	imes 0.4083 = \14.70. The naive trap is \dfrac{36}{E[X]} = \dfrac{36}{3.5} \approx \10.29. That understates the fair price because 1/X is convex on the positive reals, so by Jensen's inequality E[1/X] 1/E[X]. Intuitively, the low rolls pay huge: X=1 pays 36 and X=2 pays 18, and those big payouts pull the average up far more than the high rolls (which pay as little as 6) pull it down. Convex payoffs love volatility wh…

You win thirty-six over your die roll

Your answer

More Market Making questions