Make a market on the sum of three dice

Question

Three fair six-sided dice will be rolled. The contract settles at their sum. The interviewer will buy or sell against your quote before the roll.

Make a market, and explain how your width should differ for a one-shot bet versus a game played many times.

Accepted Answer

Fair value. Each die averages 3.5, and expectations add: E[S] = 3 	imes 3.5 = 10.5. Dispersion. One die has variance 	frac{35}{12}, and variances of independent dice add: \mathrm{Var}(S) = 3 	imes 	frac{35}{12} = 	frac{35}{4} = 8.75, \qquad \sigmaS \approx 2.96. The quote and its two regimes. As always with fair dice, everyone agrees fair is 10.5 and no one is better informed, so width buys protection against outcome variance only. What that variance is worth depends entirely on how often the game is played: - One shot for real money. A single roll has \sigma \approx 3, so selling at, sa…

Make a market on the sum of three dice

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