Market on the Middle Die

Question

I roll three fair dice and pay you the middle value (the median of the three) in dollars. Make me a market: theo first, then a bid and offer you'd honor both ways.

Accepted Answer

Theo by symmetry. Relabel every face x mapsto 7 - x. Each die's distribution is unchanged, and the middle value of the three maps to 7 minus the middle value. So the median's distribution is symmetric about 3.5, and mathbbE[textmedian] = mathbf3.5. One sentence, no casework. Compare the slog: enumerating 216 outcomes to discover what symmetry hands you for free. Width. The exact distribution is P(textmed = k) = frac16, 40, 52, 52, 40, 16216 for k = 1,dots,6, which gives sigma approx 1.37. Note that's tighter than a single die (sigma approx 1.71): the middle order statistic ignores the extremes, so it concentrates. A market like 3.2 / 3.8 is honest; quoting one-die width (say 2.9 / 4.1) leaves edge on the table, and quoting 3.45 / 3.55 on a one-shot payout is showing off, not pricing. Why interviewers like it: max-of-three (theo approx 4.96) and min-of-three (approx 2.04) both require real computation; the median quietly doesn't. Candidates who notice which statistic they've been handed, and reach for symmetry before arithmetic, read as traders rather than calculators. The trap: assuming every order statistic needs the CDF grind, or doubting the symmetry and hedging with a lopsided

Market on the Middle Die

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