The purpose of this thesis is to construct symmetric mixed-strategy equilibria for auction games that model the "exposure problem," the problem faced by bidders in simultaneous auctions whose valuations are not additive. In these games identical bidders, identical objects and complete information are assumed. The objects are sold in simultaneous sealed-bid auctions.
Chapter 2 is concerned with the case of two objects and two bidders. There are two different cases: (i) the objects are complements, so the bidders' marginal valuations increase for the second object, and (ii) the objects are substitutes, so the bidders' marginal valuations decrease for the second object. The supports of the equilibria always consist of two curves. In the case of complements these curves are increasing, and in the case of substitutes the curves are decreasing.
In Chapter 3 the games all have two bidders and three objects. The bidders' marginal valuations increase for the second object and then decrease for the third. In all cases the support of the equilibrium mixture is two-dimensional, and it surrounds a three-dimensional set of best responses.
Chapter 4 analyzes majority auction games. These are auctions in which each bidder wants to win a specified fraction (more than half) of the objects. Each bidder receives no benefit from winning less than the specified fraction and no additional benefit from winning more than it. Symmetric equilibria having simple, intuitive forms are shown to exist in such games when the number of bidders is sufficiently large. This contrasts with the results in Chapter 3 for two bidders where the equilibria are more complicated.
In Chapter 5 a method is developed that transforms an equilibrium strategy profile from one auction to another. The method is constructive and does not require complicated computation. It enables an extension of the Revenue Equivalence Theorem to domains where it had not previously been known, in particular to simultaneous auctions with complete information and to auction environments having correlated private values and common values.
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