Winner's curse with bell-curve estimation error

Question

A private company has an unknown fair value $V$ . Ten analysts each estimate it as $V$ plus independent normal noise with standard deviation $\sigma = \$ 4$ million. The highest bidder wins and pays their bid.

By how much does the winner overpay on average, and how does that overpayment grow as more analysts compete?

Accepted Answer

The curse, again. The winner is whoever drew the largest positive noise, so conditional on winning your estimate is biased high. You overpay by the expected maximum of n noise terms. Expected max of normals. Unlike the uniform case, the expected maximum of n i.i.d. standard normals has no tidy closed form, but its values are well known. For n = 10 the expected maximum is about 1.54\,\sigma. With \sigma = 4: E[\max] \approx 1.54 	imes 4 \approx \6.16 	ext{ million}. So the winning analyst's estimate sits about \6.16 million above true value on average, and bidding it raw locks in roughly tha…

Winner's curse with bell-curve estimation error

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