The winner's curse in a jar-of-coins auction

Question

A jar of coins has an unknown true value $V$ . Ten bidders each independently estimate its value, say each estimate is $V$ plus a noise term uniform on $[-10, +10]$ , and the highest bidder wins and pays their bid.

If everyone bids their own estimate, what happens to the winner? By how much should a rational bidder shade below their estimate?

Accepted Answer

The curse. Each bidder's estimate is unbiased: E[	ext{estimate}] = V. But the winner is not an average bidder, the winner is the person whose noise term was the highest. Conditioning on "I won" selects for having overestimated. So: E[V \mid 	ext{I won}] < 	ext{my estimate}. If everyone naively bids their estimate, the winner systematically overpays, this is the winner's curse. It is a selection effect: winning is bad news about your own estimate. Quantify the overpayment. With noise uniform on [-10, 10] and n bidders, the winner's noise is the maximum of n such draws. The expected maximum …

The winner's curse in a jar-of-coins auction

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