Multiple-Object Auctions Around a Circle

This paper explores a multiple-object auction with the following three features: First, there is a capacity constraint, as each bidder can win at most one object from the auction. Second, the objects for sale and the bidders are located around the unit circle. A bidder's valuation of a certain object depends solely on the location of the object relative to the bidder; that is, on the distance between the object and the bidder. Third, instead of turning in separate bids on different objects, each bidder just states his location on the unit circle. His bids on different objects can then be derived from his stated location.The paper demonstrates that the Groves-Clarke logic applies to this particular setting, and it designs an auction mechanism that is ex-post efficient, under the capacity constraint, and incentive compatible. The paper also extends the Revenue Equivalence Theorem to this particular setting, using a proof different from Engelbrecht-Wiggans'(1988), by imposing less strict regularity conditions. Based on the circle setting, the paper shows that, with symmetric bidders, for any two different auction mechanisms applying the same allocation rule, the locations on the circle at which a bidder obtains the lowest expected payoff from either of these two mechanisms are the same. This location serves as the benchmark location to the circle version of the Revenue Equivalence Theorem in this paper.