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second-price sealed bid auction-reserve price, true valuatio

1. Show that, in a second-price sealed bid auction with private values, bidders bid their true valuations of the object for auction.
2. In a second-price sealed bid auction, there are 2 bidders. The value to bidder i of the object for auction is Xi, and the realization of this value is information private to bidder i. The Xi's are uniformly and independently distributed over {0,1}. The object for auction is of no value to the seller. The seller's reserve price is r, and the seller and bidders are risk neutral. What is the seller's expected profit when r=0? What value for r is optimal for the seller, and what then is the seller's expected profit?

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See the attached file for complete help. The symbols and tables may not print here properly. Thanks.

Question
(a) Show that, in a second-price sealed bid auction with private values, bidders bid their true valuations of the object for auction.

In case of second price seal bid auctions we have following conditions:
1. bids are private information
2. bids are made simultaneously
3. highest bidder wins
4. winner pays second-highest bid

In this type of auction, even though no bidder knows any other bidder's true valuation of the item for sale. Yet, they bid their true valuation. To prove this, let us assume that there are two bidders with true valuations of v1 and v2. Now suppose their bids are bids are b1 and b2. Then
Expected gain to bidder 1 is
=(v1-b2)*P(win) + 0*P(Lose)
Where P(win) and P(loss) are probabilities of winning and losing the auctions respectively.
=(v1-b2)*P(win)
=(v1-b2)*P(b1>=b2)
The ...

Solution Summary

This post explains in simple words as well as mathematically that bidder in a second-price sealed bid auction bid their true value. Then it solves a numerical example to find out the optimal reserve price for the seller of the object to maximize profits. (approx 650 words)

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