Optimal Reserve Price Research Articles

An information–Theoretic approach to partially identified auction models

We consider a situation in which we have data from ascending auctions with symmetric bidders, independent private values, and exogenous entry in which the bidders’ value distribution is partially identified. Focusing on the case in which the seller intends to use a second price auction, we discuss how to determine an optimal reserve price. We justify the use of maximum entropy, explore the properties of the estimand, determine the asymptotic properties of our maximum entropy estimator, evaluate its behavior in a simulation study, and demonstrate its use in a modest application. As an extension, we propose a maxmin decision rule with entropy regularization, which includes Aryal and Kim (2013) and the maximum entropy solution as extreme cases.

Multi-product pricing: A discrete choice model based on Markov Chain Choice Model combined with reservation prices

Finding the optimal selling prices for an assortment of multiple, substitutable products is a problem that occurs daily in many industries. To solve this problem, a new customer choice model is proposed, based on the Markov Chain Choice Model combined with reservation prices. A discrete version of the model is proposed and it is solved to optimality. This discrete model is instrumental in a simulation procedure to estimate the optimal reservation prices in the original continuous model. Important benefits of the proposed model are that the model is intuitive and close to reality, because it includes both the customers' willingness to pay and substitution behavior explicitly, and it can handle any type of correlation between products.

Balancing Attraction and Risk Revelation: The Optimal Reservation Price in Peer-to-Peer Loan Auctions

Balancing Attraction and Risk Revelation: The Optimal Reservation Price in Peer-to-Peer Loan Auctions

Optimal reserve prices for land auctions in Eastern Germany

ABSTRACT Privatization auctions are seen as culprit for rising prices in eastern Germany. This study aims to analyse whether privatization auctions could have earned even higher revenues by setting optimal reserve prices. For this purpose, a structural estimation approach is applied to determine reserve prices and to compare the resulting expected revenues with actual revenues. The data set includes land auctions in eastern Germany from 2005–2019. The empirical results illustrate that room for increasing revenues exists, but reserve prices would be required to be higher than actual land prices. A potential consequence of high reserve prices is the risk of delays in the privatization process.

Deposit Requirements in Auctions

We examine optimal auction design when buyers may receive future outside offers. The winning bidder may choose to default upon observing her outside offer. Under the optimal mechanism, the bidder with the highest value wins if and only if her value is above a cutoff, and the winner never defaults. The optimal auction takes the form of a second-price auction with a reserve price and a deposit by the winning bidder. Under regularity conditions, both the optimal reserve price and the deposit increase when the distribution of outside offers worsens. (JEL D44, D82)

Setting Reserve Prices in Second-Price Auctions with Unobserved Bids

In this work we consider a seller who sells an item via second-price auctions with a reserve price. By controlling the reserve price, the seller can influence the revenue from the auction, and in this paper, we propose a method for learning optimal reserve prices. We study a limited information setting where the probability distribution of the bids from bidders is unknown and the values of the bids are not revealed to the seller. Furthermore, we do not assume that the seller has access to a historical data set with bids. Our main contribution is a method that incorporates knowledge about the rules of second-price auctions into a multiarmed bandit framework for optimizing reserve prices in our limited information setting. The proposed method can be applied in both stationary and nonstationary environments. Experiments show that the proposed method outperforms state-of-the-art bandit algorithms. In stationary environments, our method outperforms these algorithms when the horizon is short and performs as good as they do for longer horizons. Our method is especially useful if there is a high number of potential reserve prices. In addition, our method adapts quickly to changing environments and outperforms state-of-the-art bandit algorithms designed for nonstationary environments. Summary of Contribution: A key challenge in online advertising is the pricing of advertisements in online auctions. The scope of our study is second-price auctions with a focus on the reserve price optimization problem from a seller’s point of view. This problem is motivated by the real-life practice of small and medium-sized web publishers. However, the proposed solution approach is applicable to any seller who sells an item via second-price auctions and wants to optimize its reserve price during these auctions. Our solution approach is based on techniques from machine learning and operations research, and it would be beneficial especially for sellers who start the selling process without any historical data and can collect the data on the outcomes of the auctions while making reserve price decisions over time. History: Accepted by RamRamesh, Area Editor for Data Science & Machine Learning. Supplemental Material: The supplementary material is available at https://doi.org/10.1287/ijoc.2022.1199 .

Bid signaling in first-price royalty auction

It is common that seller’s post-auction input affects winner’s valuation in the widely used royalty auctions. This study considers bidders’ signaling incentives and seller’s choice of reserve price in the first-price royalty auction. We first characterize the unique symmetric D1 PBE in which the quasi-separating bidding strategy can be decomposed into the equilibrium bid without signaling incentive and the signaling premium. The signaling premium increases in bidder’s type, indicating that a higher type has more incentive to raise bid to induce more effort from the seller. The optimal reserve price is lower than that under the post-auction complete information scenario, which leads to a more efficient allocation and a higher payoff for the seller. The optimal reserve price is non-monotonic in royalty rate while the seller’s equilibrium payoff increases in it, and full surplus extraction is obtained when the royalty rate approaches 100%.

The Generalized First- and Second-Price Auctions: Overbidding, Underbidding, and Optimal Reserve Price

The paper generalizes the models of the standard first- and second-price auctions (FPA and SPA) by using the Cobb-Douglas function to evaluate bidders’ two conflicting bidding preferences for the probability of winning the item (Cpwin) and the profit conditional on winning (Cprofit). Compared with the popular expected profit function, this function has the advantage of allowing the bidders to equally or differently value Cpwin and Cprofit from the point of view of multiple criteria decision making, and then can capture bidders’ bidding behavior in reality. Introduction of the Cobb-Douglas function into the FPA allows us to provide new interpretations for overbidding (underbidding) behavior and the dependence of the optimal reserve price on the number of bidders. Our new interpretations are all attributed to the bidders’ different preferences for Cpwin and Cprofit. In contrast, this introduction into the SPA does not matter. Our findings suggest that the seller should prefer the FPA (SPA) over the SPA (FPA) and set a lower (higher) optimal reserve price if he conjectures that the bidders have a stronger desire to win an auction (to obtain a profit). The above results help the seller to select between the two auctions and set a reserve price in practice.

Correlation-robust auction design

We study the design of auctions when the auctioneer has limited statistical information about the joint distribution of the bidders' valuations. More specifically, we consider an auctioneer who has an estimate of the marginal distribution of a generic bidder's valuation but does not have reliable information about the correlation structure. We analyze the performance of mechanisms in terms of the revenue guarantee, that is, the greatest lower bound of revenue across all joint distributions that are consistent with the marginals. A simple auction format, the second-price auction with no reserve price, is shown to be asymptotically optimal, as the number of bidders goes to infinity. For markets with a finite number of bidders, we (1) solve for the robustly optimal reserve price that generates the highest revenue guarantee among all second-price auctions with deterministic reserve prices, and (2) show that a second-price auction with a random reserve price generates the highest revenue guarantee among all standard dominant-strategy mechanisms.

In the basic auction model, the optimal reserve price may depend on the number of bidders

A widespread claim in the auction literature about the most well known auction setting with i.i.d. private values is that the optimal reserve price for a second price auction is independent of the number n of bidders. This is indeed the case if the virtual valuation function is increasing, but this result fails to hold if the virtual valuation is non-monotone. In such case the optimal reserve price is weakly increasing in n and, as n tends to infinity, it tends to the highest regular valuation with zero virtual value.

Projection of Private Values in Auctions

We explore how taste projection—the tendency to overestimate how similar others’ tastes are to one’s own—affects bidding in auctions. In first-price auctions with private values, taste projection leads bidders to exaggerate the intensity of competition and, consequently, to overbid—irrespective of whether values are independent, affiliated, or (a)symmetric. Moreover, the optimal reserve price is lower than the rational benchmark, and decreasing in the extent of projection and the number of bidders. With an uncertain common-value component, projecting bidders draw distorted inferences about others’ information. This misinference is stronger in second-price and English auctions, reducing their allocative efficiency compared to first-price auctions. (JEL D11, D44, D82, D83)

Semiparametric Quantile Models for Ascending Auctions With Asymmetric Bidders

The article proposes a parsimonious and flexible semiparametric quantile regression specification for asymmetric bidders within the independent private value framework. Asymmetry is parameterized using powers of a parent private value distribution, which is generated by a quantile regression specification. As noted in Cantillon, this covers and extends models used for efficient collusion, joint bidding and mergers among homogeneous bidders. The specification can be estimated for ascending auctions using the winning bids and the winner’s identity. The estimation is in two stage. The asymmetry parameters are estimated from the winner’s identity using a simple maximum likelihood procedure. The parent quantile regression specification can be estimated using simple modifications of Gimenes. Specification testing procedures are also considered. A timber application reveals that weaker bidders have 30% less chances to win the auction than stronger ones. It is also found that increasing participation in an asymmetric ascending auction may not be as beneficial as using an optimal reserve price as would have been expected from a result of Bulow and Klemperer valid under symmetry.

Scalable Optimal Online Auctions

This paper provides a set of tools to compute and implement optimal reserve prices for online auctions.

Economics of Strategic Network Infrastructure Sharing: A Backup Reservation Approach

In transitioning to 5G, the high infrastructure cost, the need for fast rollout of new services, and the frequent technology/system upgrades triggered wireless operators to consider adopting the cost-effective network infrastructure sharing (NIS), even among competitors, to gain technology and market access. NIS is a bargaining mechanism whose terms and conditions must be carefully determined based on mutual benefits in a market with uncertainties. In this work, we propose a strategic NIS framework for contractual backup reservation between a small/local network operator with limited resources and uncertain demands, and a more resourceful operator with excessive capacity. The backup reservation agreement requires the local operator (say, operator A) to reserve a certain amount of resources (e.g., spectrum) for future sharing from the resource-owning operator (say, operator B). In return, operator B guarantees availability of its reserved resources to meet the need of operator A. We characterize the bargaining between the operators in terms of the optimal reservation prices and quantities with and without consideration of their competitions in market share, respectively. The conditions under which competing operators have incentive to cooperate are explored. The impact of competition intensity and redundant capacity on performance under backup reservation are also investigated. Our study shows that NIS through backup reservation improves both resource utilization and profits of operators, with the potential to support higher target service levels for end users. We also find that, under certain conditions, operator B may still have the incentive to share its resources even at the risk of impinging on its own users.

A Theory of Auctions with Endogenous Valuations

We derive the revenue maximizing allocation of m units among n symmetric agents who have unit demand, and who take costly actions that in uence their values before participating in the mechanism. The allocation problem with costly actions can be represented by a reduced form model where agents have convex, non-expected utility preferences over the interim probability of receiving an object. Both the uniform m+1 price auction and the discriminatory pay-your-bid auction with reserve price constitute symmetric revenue maximizing mechanisms. Contrasting the case with exogenous valuations, the optimal reserve price reacts to both demand and supply. We also identify a condition under which the optimal mechanism is indeed symmetric, and illustrate the structure of the optimal asymmetric mechanism when the condition fails. The main tool in our analysis is an integral inequality, due to Fan and Lorentz (1954), involving majorization, super-modularity and convexity.

Sample Spacings for Identification: The Case of English Auctions With Absentee Bidding

This paper presents new nonparametric identification results for ascending auctions with independent private values. The standard identification approach is infeasible in the motivating setting, because absentee bidding conceals the number of bidders. I exploit insights from the statistics literature about the stochastic ordering of adjacent sample spacings. I show how to use such sample spacings to set-identify structural features, using an incomplete model and without knowing all highest bids or the number of bidders. Applying the sample spacing method to a small sample of wine auctions, I show that it identifies informative bounds on policy-relevant counterfactuals. It turns out that Sotheby's restricts full exploitation of the exclusion principle of optimal reserve prices. As a result, sellers set sub-optimally low reserve prices. They benefit up to 13% from adopting a common reserve price rule equal to 120% of the Wine Department's pre-auction value estimate.

Auctions with quantity externalities and endogenous supply

This paper studies the design of license auctions when the number of licenses allocated in the auction determines structure of the post-auction market. I first show that a sequence of conditional reserve prices that specify minimum acceptable bid at each supply level can be used to determine supply endogenously. Then I construct a static auction called multi-dimensional uniform-price auction and a dynamic auction called Walrasian clock auction that allow the auctioneer to condition reserve price on supply and allow bidders to condition bids on supply. I show that both proposed auctions can implement the efficient market structure that maximizes total surplus in the post-auction market in a dominant strategy equilibrium. I next characterize the optimal auction and show that the two proposed auctions can yield the optimal revenue under a sequence of optimal reserve prices.

Unimodal Bandits with Continuous Arms

We consider stochastic bandit problems with a continuous set of arms and where the expected reward is a continuous and unimodal function of the arm. For these problems, we propose the Stochastic Polychotomy (SP) algorithms, and derive finite-time upper bounds on its regret and optimization error. We show that, for a class of reward functions, the SP algorithm achieves a regret and an optimization error with optimal scalings, i.e., $O(\sqrtT )$ and $O(1/\sqrtT )$ (up to a logarithmic factor), respectively. SP constitutes the first order-optimal algorithm for non-smooth expected reward functions, as well as for smooth functions with unknown smoothness. The algorithm is based on sequential statistical tests used to successively trim an interval that contains the best arm with high probability. These tests exhibit a minimal sample complexity which confers to SP its adaptivity and optimality. Numerical experiments actually reveal that the algorithm even outperforms state-of-the-art algorithms that exploit the knowledge of the smoothness of the reward function. The performance of SP is further illustrated on the problem of setting optimal reserve prices in repeated second-price auctions; there, the algorithm is evaluated on real-world data.

Optimal reserve prices in sealed-bid auctions with reference effects

The reference effect and loss aversion are incorporated into the buyer’s utility in the symmetric independent private value models of sealed-bid auctions. The buyer’s equilibrium bidding strategy and the seller’s optimal reserve price are derived for the first-price and second-price sealed-bid auctions. In both auction mechanisms, the seller’s optimal reserve price and expected revenue are increasing in the reference point. We compare the seller’s expected revenues as well as the optimal reserve prices in the two auctions. The results show that the seller will set a higher optimal reserve price but obtain lower optimal expected revenue in the second-price auction compared to the first-price auction. Further, we extend the model to the gain-seeking case, and endogenize the reference point as the ex-ante expected price of the item in equilibrium. In contrast to the loss-averse case, the seller will set a lower optimal reserve price but obtain higher optimal expected revenue in the second-price auction compared to the first-price auction if the buyers are gain-seeking. With an endogenous reference point, similar results are obtained in terms of revenue comparison between the two auctions.

Asymptotically optimal prior-free clock auctions

Clock auctions have a number of properties that make them attractive for practical purposes. They are weakly group strategy-proof, make bidding truthfully an obviously dominant strategy, and preserve trading agents' privacy. However, optimal reserve prices and stopping rules depend on the details of underlying distributions, and so clock auctions have proved challenging to implement in a prior-free, asymptotically optimal way. In this paper, we develop a prior-free clock auction that is asymptotically optimal by exploiting a relationship between hazard rates and the spacings between order statistics. Extensions permit price discrimination among heterogeneous groups, minimum revenue thresholds, and quantity caps.