Winner's curse at a charity art auction

Question

A painting has an unknown true resale value $V$ . Six appraisers each independently estimate it, and each estimate equals $V$ plus a noise term uniform on $[-12, +12]$ thousand dollars. The highest bidder wins and pays their bid.

If everyone bids their own estimate, by how much does the winner overpay on average, and how much should a rational appraiser shade below their estimate?

Accepted Answer

The curse. Each estimate is unbiased (E[	ext{estimate}] = V), but the winner is not a typical bidder, it is whoever's noise happened to be the largest. Conditioning on "I won" selects for having overestimated, so E[V \mid 	ext{I won}] < 	ext{my estimate}. Bidding your raw estimate therefore means systematically overpaying. Quantify it. With noise uniform on [-a, a] and n bidders, the winner's noise is the maximum of n such draws. The expected maximum of n i.i.d. Uniform[-a, a] variables is E[\max] = a\,\frac{n-1}{n+1}. For a = 12 and n = 6: E[\max] = 12 \cdot \frac{5}{7} = \frac{60}{7} \ap…

Winner's curse at a charity art auction

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