### Consider private value auction with four bidders

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Consider a private value auction with four bidders and two identical goods. Supposeeach bidder demands just a single unit. The values for bidders 1, 2, 3, and 4 are v1 = 32, v2 =26, v3 = 45, and v4 = 22. The seller values each unit at v=20 (only seller knows his valuation).

a. In a Vickrey (second-price sealed bid) auction, which bidder(s) get the two units andhow much do they pay? Is the outcome Pareto efficient? (Hint: The auction will be hold twice to sell each unit).

b. Now suppose that the bidders have positive but diminishing values for the secondunit as well. In particular, the values are v1 = (32, 19), v2 = (26, 18), v3 = (45, 24),and v4 = (22, 14), where the first number is the value for the first unit and thesecond number is the value for the second. In a Vickrey auction, which bidder(s) getthe two units and how much do they pay? Is the outcome Pareto efficient? (Hint:The auction will be hold twice to sell each unit).

c. Suppose that the bidder’s valuations are still as described in part b above, and theseller actually knows bidders’ true valuations. Can the seller increase its profits byusing a different type of auction? If yes, prove your argument by giving a numericalexample. If no, explain why not.

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