Alice is selling two identical watches by auction. There are three bidders, Bob,

Question

Alice is selling two identical watches by auction. There are three bidders, Bob, Charlie and Daphne, and each will buy at most one watch. Bidders’ values are independent and private. Bidders always have the option not to participate, which is the same thing as bidding $0. Bob’s value for the watch is $400. He doesn’t know what the other bidders will do, but his best guess is that Charlie’s bid will be uniformly distributed on the interval [0,500] and Daphne’s on the interval [500,1000].

(a) Suppose Alice runs a first-price auction: All three bidders submit bids simultaneously, and the top two bidders each pay the highest bid and receive a watch. What would you advise Bob to bid? Why?

(b) Suppose Alice runs a second-price auction: All three bidders submit bids simultaneously, and the top two bidders each pay the second-highest bid and receive a watch. What would you advise Bob to bid? Why?

(c) Suppose Alice runs a third-price auction: All three bidders submit bids simultaneously, and the top two bidders each pay the thirdhighest bid and receive a watch. What would you advise Bob to bid? Why?

Need C most of all

Explanation / Answer

A) Bid 0. Daphne defintly wins one price, since her smallest bid is 500 according to Bob. Hence bob will have to pay 500 if he comes second, making his utility -100. Therefore his optimal choice is to bid 0.

B) Bid 200. Daphne defintly wins one price, since her smallest bid is 500 according to Bob. Hence BobCharlie will have to pay the amount he bids if he comes second. This makes a first price auction betwee Bob and Charlie. In case of uniformly distributed belief structure, for an N player first price auction optimal bidding strategy is v-v/N, where v is the valuation of player.

C)Bid 400. Daphne defintly wins one price, since her smallest bid is 500 according to Bob. Hence BobCharlie will have to pay the amount the other person bids if he comes second. This makes a second price auction betwee Bob and Charlie.

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