# Theory of Menu Auction and Applications

## Abstract

My doctoral dissertation contains three essays on menu auction and its related applications. The first chapter is a theoretical generalization of classical menu auction model, and the second and the third chapters are applications to a resource allocation problem and an industrial organization problem. Menu auction (Bernheim and Whinston, 1986) is a first-price package auction with complete information. They show that every Nash equilibrium under some refinements always leads to an efficient outcome. Therefore, this becomes a natural efficiency benchmark for package auction designs (e.g., Ausubel and Milgrom 2002). Menu auction can also be viewed as a model of economic influence where the auctioneer is going to choose an action which affects bidders' payoff so that each bidder tries to influence the outcome by monetary transfer to the auctioneer. This framework is widely adopted in political lobbying models where the special interest groups lobbying the government over trade policies (e.g., Grossman and Helpman 1994). However, the applicability is limited by quasi-linear preferences and the absence of budget constraints. In my first chapter, ``Menu Auctions with Non-Transferable Utilities and Budget Constraints'', I extends Bernheim and Whinston's (1986) menu auction model under transferable utilities to a framework with non-transferable utilities and budget constraints. Under appropriate definitions of equilibria consistent with subgame perfection, it is shown that every truthful Nash equilibrium (TNE) is a coalition-proof Nash equilibrium (CPNE) and that the set of TNE payoffs and the set of CPNE payoffs are equivalent, as in a transferable utility framework. The existence of a CPNE is assured in contrast with the possible non-existence of Nash equilibrium under the definition by Dixit, Grossman, and Helpman (1997). Moreover, the set of CPNE payoffs is equivalent to the bidder-optimal weak core. The second chapter relates menu auction to a resource allocation problem. Kelso and Crawford (1982) propose a wage-adjustment mechanism resulting in a stable matching between heterogeneous firms and workers. Instead of a benevolent social planner, in ``Profit-Maximizing Matchmaker'' (w. Hideo Konishi), we consider a profit-maximizing auctioneer to solve this many-to-one assignment problem. If firms can only use individualized price, then the auctioneer can only earn zero profit in every Nash equilibrium and the sets of stable assignments and strong Nash equilibria are equivalent. Otherwise, the auctioneer might earn positive profit even in a coalition-proof Nash equilibrium. This reinforces Milgrom's (2010) argument on the benefit of using simplified message spaces that it not only reduces information requirement but also improves resource allocation. The third chapter applies menu auction in an industrial organization problem. In ``Choosing a Licensee from Heterogeneous Rivals'' (w. Hideo Konishi and Anthony Creane), we consider a firm licensing its production technology to rivals when firms with heterogeneous in production costs competing in a Cournot market. While Katz and Shapiro (1986) show that a complete transfer in duopoly can be joint-profit reducing, we show that it is always joint-profit improving provided that at least three firms remain in the industry after transfer. While transfers between similarly efficient firms may reduce welfare, the social welfare must increase if the licensor is the most efficient in the industry, contrast with Katz and Shapiro (1985) in the duopoly environment. This has an important implication in competition regulation. Then we investigate relative efficiency of the licensee under different licensing auction mechanisms. With natural refinement of equilibria, we show that a menu auction licensee, a standard first-price auction licensee, and a joint-profit maximizing licensee are in (weakly) descending order of efficiency.