Essays on Auction Theory and Application

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## Abstract

This dissertation contributes to auction theory with application of the theory to the analysis of some real-life problem.

In Chapter 1, I study the problem of competition between contest designers where they offer differentiated prizes to a group of contestants with some minimal effort requirements. The equilibrium among contestants is either a separating equilibrium, where strong contestants participating in high-prize contest and weak contestants in low-prize contest, or a mixing equilibrium, where strong players participate in high-prize contest with probability 1, middle-type players randomize between the two contests, and weak players go to low-prize contest with certainty. I then solve an equilibrium of contest designers where one designer's choice of minimal effort level is assumed to be non-strategic. Finally, I provide conditions such that the assumed non-strategic choice of minimal effort level is optimal and thus characterize at least part of the equilibrium set, which expands the knowledge on competing auctions.

In Chapter 2, I apply auction theory to analyze the eﬀect of a merger on ﬁrms’ research and development (R&D) investment. There is a substantial literature on the eﬀects of mergers on product prices, but the eﬀects of mergers on other outcomes, such as R&D investment spending, are less studied. I develop a model for evaluating the likely eﬀects of a merger (or joint research venture) on the R&D eﬀorts of competing ﬁrms. The R&D process is modeled as an all-pay contest (auction) among ﬁrms, with the payoﬀ from investment going to the ﬁrm that invests the largest amount. I provide an explicit characterization of the equilibrium in a multi-player asymmetric all-pay contest model. The equilibrium solution then is applied through simulation to calibrate the eﬀects of mergers on ﬁrms’ R&D eﬀorts and eﬃciency as well as on social welfare. I ﬁnd that each ﬁrm is expected to exert more eﬀorts after a merger, but if there are only few ﬁrms premerger, a merger reduces total R&D eﬀort. A merger may also cause ineﬃciency, but the loss in eﬃciency is low. My results also show that net surplus increases after a merger if the number of ﬁrms is small.

In Chapter 3, I study a problem of sequential auctions and extend the standard model of sequential second-price auctions to a dynamic game with an inﬁnite horizon with one new buyer entering the auction every period. I ﬁrst derive properties of the symmetric and stationary equilibrium, where buyers bid according to their private valuation less a pivotal continuation value, and I also show that the price path in such equilibrium is weakly decreasing. Imposing preconsistent beliefs, I give the conditions under which a stationary equilibrium exists.