Auctions with Limited Commitment∗

We study auction design with limited commitment in a standard auction environment. The seller has a single object and can conduct an infinite sequence of standard auctions with reserve prices to maximize her expected profit. In each period, the seller can commit to a reserve price for the current period but cannot commit to future reserve prices. We analyze the problem with limited commitment through an auxiliary mechanism design problem with full commitment, in which an additional constraint reflects the sequential rationality of the seller. We characterize the maximal profit achievable in any perfect Bayesian equilibrium in the limit as the period length vanishes. The static full commitment profit is not achievable but the seller can always guarantee the profit of an efficient auction. If the number of buyers exceeds a cutoff, the efficient auction is optimal. Otherwise, the efficient auction is not optimal, and we give conditions under which the optimal solution consists of an initial auction with a non-trivial reserve price followed by a continuously decreasing price path. The solution is described by a simple ordinary differential equation. Our analysis draws insights from bargaining, auctions, and mechanism design. ∗We wish to thank Jeremy Bulow, Yeon-Koo Che, Jacob Goeree, Faruk Gul, Philippe Jehiel, Navin Kartik, Alessandro Lizzeri, Steven Matthews, Benny Moldovanu, Bernard Salanié, Yuliy Sannikov, Vasiliki Skreta, Andrzej Skrzypacz, Philipp Strack, Alexander Wolitzky, and various seminar and conference audiences for helpful discussions and comments. Parts of this paper were written while some of the authors were visiting the University of Bonn, Columbia University, ESSET at the Study Center Gerzensee, Princeton University, and the University of Zurich. We are grateful for the hospitality of the respective institutions. Mierendorff gratefully acknowledges financial support from the Swiss National Science Foundation and the European Research Council (ESEI-249433). Shi gratefully acknowledges financial support by the Social Sciences and Humanities Research Council of Canada. †Columbia University, [email protected] ‡Columbia University and ESEI–University of Zurich, [email protected] §University of Toronto, [email protected]