Auctions with Limited Commitment

Abstract

We study auction design in the standard symmetric independent private values environment, where the seller lacks the commitment power to withhold an unsold object off the market. 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 which is small for many distributions, 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 combines insights from bargaining, auctions, and mechanism design. * We wish to thank

Reputational Bargaining with Unknown Values

“…Making low offers is incentive compatible when the buyer is unsure about the seller's cost, but not when the buyer is unsure about the seller's discount rate. 3 Our work also contributes to the literature on the efficiency of bargaining under incomplete information pioneered by Stokey (1981) and Gul, Sonnenschein, and Wilson (1986) and has been revisited recently by Strulovici (2017) and Liu, Mierendorff, Shi, and Zhong (2019). The driving forces behind the inefficient equilibria in our model differ from those identified in existing works, such as the gains from trade can be arbitrarily close to zero (Ausubel and Deneckere, 1989), players face higher order uncertainty (Feinberg and Skrzypacz, 2005), players' values are interdependent (Deneckere and Liang, 2006), new players arrive over time (Fuchs and Skrzypacz, 2010), the seller has stochastic time-varying costs (Ortner, 2017), and players face uncertainty about the future costs of delay (Fanning, 2018).…”

Sequential Search Auctions With a Deadline

“…Reynolds and Wooders (2009) show that, when bidders are risk-averse, a buy-price then auction mechanism can generate higher revenue than an auction mechanism, as the presence of a buy-price option reduces bidders' risk-premium. In a similar model, Liu et al (2017) investigate and justify the online buy-price auctions as a valid selling mechanism. In a recent paper, Zhang (2017) studies the optimal sequence of posted-price and auction in a sequential mechanism, where running an auction is more costly for a seller.…”

“…Said (2011Said ( , 2012 studies sequential auctions of multi-unit product with changing population, yet in a different environment to our model. Liu et al (2017) study sequential auctions in the case of limited commitment. Other recent literature on revenue management includes Board and Skrzypacz (2016) with forward-looking buyers, in the case of full commitment, and Dilme and Li (2017), who study revenue management with the arrivals of strategic buyers in the case of no commitment.…”

Credibility in Second-Price Auctions: An Experimental Test

“…In particular, Baliga et al (1997) and Bester and Strausz (2001) study environments where the designer has agency to determine the outcome of the mechanism, which is a feature that is reflected prominently in our experiment. Liu et al (2019) consider multi-period auctions with limited commitment across periods: in particular, the seller cannot rule out lowering the reserve price in future periods. In more general allocation problems, Hakimov and Raghavan (2021) introduce the related concepts of "verifiability" (i.e., mechanisms that allow participants to check if their assignments are correct) and "transparency" (i.e., mechanisms in which the designer cannot cheat without being detected).…”