Fair price of a higher-die contract

EasyExpected Value Market Making & Betting Games~2m

You and a counterparty each roll one fair die. A contract pays you $12 if your die is strictly higher than theirs, and nothing on a tie or a loss. What is the fair value of this contract (its expected payout, in dollars)?

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By symmetry, $P(\text{you higher}) = P(\text{they higher})$ ; only the tie breaks the tie.
$P(\text{tie}) = \tfrac16$ , so $P(\text{win}) = \tfrac{1 - 1/6}{2} = \tfrac{5}{12}$ .

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Asked at: Options Market-Making, ETF Market-Making

Fair price of a higher-die contract

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