Revenue-maximizing Auction Research Articles

Mechanism design with belief-dependent preferences

This paper studies mechanism design when agents have belief-dependent preferences, in that utilities depend on the agents' hierarchical posterior beliefs about types. For instance, agents may be subject to temptation, shame, image concerns, or privacy concerns. In this setting, the textbook revelation principle does not hold, since mechanisms can provide agents with information that affects posterior beliefs. This paper uses a psychological game framework suited for mechanism design, and provides a novel version of the revelation principle for belief-dependent preferences. The new revelation principle makes use of extended direct mechanisms that map each reported type into material outcomes and private suggestions of what posterior beliefs the agents should have. The paper shows that it suffices to use extended direct mechanisms that satisfy three conditions: Bayesian incentive compatibility, individual rationality, and a new condition called believability. The new revelation principle is used to find revenue-maximizing auctions when bidders have different types of image concerns. Moreover, it provides an alternate tool—distinct from Myerson's communication revelation principle—to study mechanism design with after-games.

Joint Pricing and Matching for City-Scale Ride-Pooling

Central to efficient ride-pooling are two challenges: (1) how to `price' customers' requests for rides, and (2) if the customer agrees to that price, how to best `match' these requests to drivers. While both of them are interdependent, each challenge's individual complexity has meant that, historically, they have been decoupled and studied individually. This paper creates a framework for batched pricing and matching in which pricing is seen as a meta-level optimisation over different possible matching decisions. Our key contributions are in developing a variant of the revenue-maximizing auction corresponding to the meta-level optimization problem, and then providing a scalable mechanism for computing posted prices. We test our algorithm on real-world data at city-scale and show that our algorithm reliably matches demand to supply across a range of parameters.

Approximately Revenue-Maximizing Auctions for Deliberative Agents

In many real-world auctions, a bidder does not know her exact value for an item, but can perform a costly deliberation to reduce her uncertainty. Relatively little is known about such deliberative environments, which are fundamentally different from classical auction environments. In this paper, we propose a new approach that allows us to leverage classical revenue-maximization results in deliberative environments. In particular, we use Myerson (1981) to construct the first non-trivial (i.e., dependent on deliberation costs) upper bound on revenue in deliberative auctions. This bound allows us to apply existing results in the classical environment to a deliberative environment. In addition, we show that in many deliberative environments the only optimal dominant-strategy mechanisms take the form of sequential posted-price auctions.

Revenue-Maximizing Auctions: A Bidder’s Standpoint

We address the problem of improving bidders’ strategies in prior-dependent revenue-maximizing auctions and introduce a simple and generic method to design novel bidding strategies if the seller uses past bids to optimize her mechanism. We propose a simple and agnostic strategy, independent of the distribution of the competition, that is robust to mechanism changes and local (as opposed to global) optimization of e.g. reserve prices by the seller. This strategy guarantees an increase in utility compared to the truthful strategy for any distribution of the competition. We generalize this result by showing that a best response, reaching the best tradeoff between beating the competition and reducing the reserve price in a large class of possible strategies, is a simple extension of this first strategy. Our new variational approach naturally yields itself to numerical optimization and algorithms for designing or improving the strategies in any given selling mechanisms. Our formulation enables to study some important robustness properties of the strategies, showing their impact even when the seller is using a data-driven approach to set the reserve prices. In this sample-size setting, we prove under what conditions, optimal bidding strategies can still improve the buyer’s utility. The gist of our approach is to see optimal auctions in practice as a Stackelberg game where the buyer is the leader, as he is the first one to move (here bid) when the seller is the follower as she has no prior information on the bidder.

Dynamic Mechanism Design: An Introduction

We provide an introduction to the recent developments of dynamic mechanism design, with a primary focus on the quasilinear case. First, we describe socially optimal (or efficient) dynamic mechanisms. These mechanisms extend the well-known Vickrey– Clark–Groves and D’Aspremont–Gérard–Varet mechanisms to a dynamic environment. Second, we discuss revenue optimal mechanisms. We cover models of sequential screening and revenue-maximizing auctions with dynamically changing bidder types. We also discuss models of information management where the mechanism designer can control (at least partially) the stochastic process governing the agents’ types. Third, we consider models with changing populations of agents over time. After discussing related models with risk-averse agents and limited liability, we conclude with a number of open questions and challenges that remain for the theory of dynamic mechanism design. ( JEL D44, D81, D82)

Optimal Auction Design for Flexible Consumers

We study the problem of designing revenue-maximizing auctions for allocating multiple goods to flexible consumers. In our model, each consumer is interested in a subset of goods known as its flexibility set and wants to consume one good from this set. A consumer's flexibility set and its utility from consuming a good from its flexibility set are its private information. We focus on the case of nested flexibility sets—each consumer's flexibility set can be one of the $k$ nested sets. We provide several examples where such nested flexibility sets may arise. We characterize the allocation rule for an incentive compatible, individually rational, and revenue-maximizing auction in terms of solutions to integer programs. The corresponding payment rule is described by an integral equation. We then leverage the nestedness of flexibility sets to simplify the optimal auction and provide a complete characterization of allocations and payments in terms of simple thresholds.

Robustly Optimal Auctions with Unknown Resale Opportunities

AbstractThe standard revenue-maximizing auction discriminates against a priori stronger bidders so as to reduce their information rents. We show that such discrimination is no longer optimal when the auction’s winner may resell to another bidder, and the auctioneer has non-Bayesian uncertainty about such resale opportunities. We identify a “worst-case” resale scenario, in which bidders’ values become publicly known after the auction and losing bidders compete Bertrand-style to buy the object from the winner. With this form of resale, misallocation no longer reduces the information rents of the high-value bidder, as he could still secure the same rents by buying the object in resale. Under regularity assumptions, we show that revenue is maximized by a version of the Vickrey auction with bidder-specific reserve prices, first proposed by Ausubel and Cramton (2004). The proof of optimality involves constructing Lagrange multipliers on a double continuum of binding non-local incentive constraints.

Conjugate information disclosure in an auction with learning

We consider a single-item, independent private value auction environment with two bidders: a leader, who knows his valuation, and a follower, who privately chooses how much to learn about his valuation. We show that, under some conditions, an ex-post efficient revenue-maximizing auction—which solicits bids sequentially—partially discloses the leader's bid to the follower, to influence his learning. The disclosure rule that emerges is novel; it may reveal to the follower only a pair of bids to which the leader's actual bid belongs. The identified disclosure rule, relative to the first-best, induces the follower to learn less when the leader's valuation is low and more when the leader's valuation is high.

Becoming the Neighbor Bidder: Endogenous Winner’s Curse in Dynamic Mechanisms

This paper addresses the problem of sequentially allocating time-sensitive goods, or one-period leases on a durable good, among agents who compete through time and learn about the common component of their valuation privately through experience. I show that efficiency is unattainable, and I identify simple variations of sequential second-price or English auctions that implement the second best and the revenue-maximizing auctions. When the units are divisible, I identify the corresponding auctions that allow for double sourcing. (JEL D44, D82)

Bounding the optimal revenue of selling multiple goods

Using duality theory techniques we derive simple, closed-form formulas for bounding the optimal revenue of a monopolist selling many heterogeneous goods, in the case where the buyer's valuations for the items come i.i.d. from a uniform distribution and in the case where they follow independent (but not necessarily identical) exponential distributions. We apply this in order to get in both these settings specific performance guarantees, as functions of the number of items m, for the simple deterministic selling mechanisms studied by Hart and Nisan [1], namely the one that sells the items separately and the one that offers them all in a single bundle.We also propose and study the performance of a natural randomized mechanism for exponential valuations, called Proportional. As an interesting corollary, for the special case where the exponential distributions are also identical, we can derive that offering the goods in a single full bundle is the optimal selling mechanism for any number of items. To our knowledge, this is the first result of its kind: finding a revenue-maximizing auction in an additive setting with arbitrarily many goods.

Inference of bidders’ risk attitudes in ascending auctions with endogenous entry

Bidders’ risk attitudes have key implications for the choices of revenue-maximizing auction formats. In ascending auctions, bid distributions do not provide information about risk preference. We infer risk attitudes using distributions of transaction prices and participation decisions in ascending auctions with entry costs. Nonparametric tests are proposed for two distinct scenarios: first, the expected entry cost can be consistently estimated from the data; second, the data does not report entry costs but contains exogenous variation in potential competition and auction characteristics. We also show the identification of risk attitudes in ascending auctions with selective entry, where bidders receive entry-stage signals correlated with their values.

Inference of Bidderss Risk Attitudes in Ascending Auctions with Endogenous Entry

Bidders’ risk attitudes have key implications for choices of revenue-maximizing auction formats. In ascending auctions, bid distributions do not provide information about risk preference. We infer risk attitudes using distributions of transaction prices and participation decisions in ascending auctions with entry costs. Nonparametric tests are proposed for two distinct scenarios: first, the expected entry cost can be consistently estimated from data; second, the data does not report entry costs but contains exogenous variations of potential competition and auction characteristics. In the first scenario, we exploit the fact that the risk premium required for entry - the difference between ex ante expected profits from entry and the certainty equivalent .is strictly positive if and only if bidders are risk averse. Our test is based on identification of bidders’ ex ante profits. In the second scenario, our test builds on the fact that risk attitudes affect how equilibrium entry probabilities vary with observed auction characteristics and potential competition. We also show identification of risk attitudes in a more general model of ascending auctions with selective entry, where bidders receive entry-stage signals that are correlated with private values.

Approximately optimal auctions for correlated bidders

We consider the design of dominant strategy incentive compatible, revenue-maximizing auctions for an indivisible good, when bidders' values are drawn from a correlated distribution. For independent distributions, Myerson showed that the optimal auction for risk-neutral bidders remains incentive compatible regardless of bidders' risk attitudes. We show that, for correlated distributions, the same is true when only two bidders are involved, whereas for more bidders, randomization can generate strictly more revenue. However, for risk-neutral bidders, we show this gain is never more than a 53-factor.This is a consequence of two results of independent interest: (1) a polynomial-time derandomization of auctions for two bidders; (2) a polynomial-time computable deterministic auction that 53-approximates the optimal revenue extractable from risk-neutral bidders.Moreover, we give a polynomial-time algorithm to compute optimal auctions for a constant number of bidders, and for any number of bidders we give polynomial-time algorithms with approximation factors arbitrarily close to 32.

How to approximate optimal auctions

Bayesian auction design investigates how to sell scarce resources to agents with private values drawn according to known distributions. A natural objective in this setting is revenue maximization. The seminal work of Roger Myerson presents a recipe for designing revenue-maximizing auctions in single-parameter settings. Myerson defines a function that maps each agent's value to a new number called the virtual value . The optimal auction then chooses the solution with maximum virtual surplus subject to the feasibility constraints. Often this underlying optimization problem is NP-hard and is complicated by the fact that virtual values may be negative. Prior work approximates the optimal auction by approximating this optimization problem. Such a solution can be interpreted as approximating the optimal auction point-wise, i.e., for each realization of values the revenue of the derived auction is close to that of the Myerson auction. In this letter, we suggest a different approach: Approximate the maximum virtual surplus on average with respect to the induced virtual value distributions . Our approach has the advantage that sometimes the point-wise optimization problem is not approximable to within any reasonable factor. However, by leveraging the fact that any distribution of virtual values has non-negative expectation, it is sometimes possible to get good average-case approximations. Furthermore, the optimal auction is itself an average-case guarantee: it maximizes the revenue on average with respect to the distributions, but it may lose substantial revenue on certain value profiles. Thus our average-case guarantee is without loss of generality. We showcase our approach using the problem of selling goods with positive network externalities. For this problem, the underlying optimization problem is inapproximable, yet with our approach we are able to prove that a natural greedy auction is a constant-factor approximation.

Strategic behavior modeling of multi-service overlay multicast networks based on auction mechanism design

Since the users of overlay multicast networks belong to different administrative domains, they are selfish in nature; resulting in degradation of performance. That is why strategic behavior modeling is a hot topic in the area of the overlay multicast networks. Mechanism design is the most versatile tool for strategic behavior modeling in microeconomics. In this paper, we model the strategic behavior of the selfish peers by leveraging the rich theory of mechanism design using the concept of economic auctions. By considering the bandwidth of services as the commodity, we design a revenue-maximizing auction mechanism. The sellers are either the origin servers or the peers who forward the digital multimedia content to their downstream peers. For each seller, the corresponding downstream peers play the role of buyers who are referred to as bidders. Each bidder submits a sealed bid to the corresponding seller. The highest bidder wins and pays its bid for the service. Also, we derive analytical closed-form expressions for upper bounds relevant to the performance metrics. The experimental validation proves the scalability and the efficiency of the proposed mechanism.

Optimal allocation mechanisms with single-dimensional private information

We study revenue-maximizing allocation mechanisms for multiple heterogeneous objects when buyers care about the entire allocation, and not just about the ones they obtain. Buyers’ payoff depends on their cost parameter and, possibly, on their competitors’ costs. Costs are independently distributed across buyers, and both the buyers and the seller are risk-neutral. The formulation allows for complements, substitutes and externalities. We identify a number of novel characteristics of revenue-maximizing mechanisms: First, we find that revenue-maximizing reserve prices depend on the bids of other buyers. Second, we find that when non-participation payoffs are type-dependent, revenue-maximizing auctions may sell too often, or they may even be ex-post efficient.

Auction design with opportunity cost

This paper studies revenue-maximizing auctions in an independent private value setting where potential bidders have known positive opportunity cost of bidding. The main findings are as follows. Firstly, there is no loss of generality in deriving the revenue-maximizing auctions within the class of threshold-entry mechanisms. Secondly, for any given set of entry thresholds, a second-price sealed-bid auction with properly set reserve prices and entry subsidy is revenue-maximizing. Thirdly, a variety of auctions are revenue-maximizing within the symmetric threshold-entry class. Two of them involve no entry subsidy (fee). Fourthly, we identify sufficient conditions under which it is in the seller’s interest to limit the number of potential bidders even if the revenue-maximizing symmetric threshold-entry auction is adopted. Lastly, the revenue-maximizing auction implements asymmetric entry across symmetric bidders in many cases.

Optimal quota allocation for a revenue-maximizing auction holder facing a random number of bidders

This paper is motivated by the following practical multi-unit auction. Before purchasing a vehicle in Singapore, a prospective buyer must bid for a certificate that entitles him to purchase the vehicle. Each month a certain number, say Q, of “certificates of entitlement” are made available and the highest Q bidders receive the right to purchase. Each successful bidder pays a “quota premium” equal to the bid submitted by the Qth highest bidder. We construct a stochastic optimization model using results from the theory of order statistics, where the objective is to determine the optimal number of units to be released in order to maximize the auction holders expected revenue. As the amounts bid and the number of bidders are both random, there may be wide – and, undesirable – fluctuations in the quota premia from one month to the next. We perform the optimization subject to a probabilistic constraint that limits the month-to-month fluctuations in quota premia. We show, assuming specific distributions for the bid amounts and the number of bidders, that in order to satisfy the probabilistic constraint one may have to reduce the available quota. For different bid random variables, the feasible set arising from the probabilistic constraint may assume different shapes. We also present a sufficient condition for comparing the size of the two feasible sets corresponding to different bid random variables. The results are illustrated by sensitivity analyses that examine the effect of varying parameter values on the optimal solution.

Implementation of the revenue-maximizing auction by an ignorant seller

In a general auction framework with independent private values, we propose a game, with a simple economic interpretation, that allows to implement the revenue-maximizing auction outcome when the seller ignores the distributions of the different bidders’ valuations. In this detail-free implementation procedure, an ascending-price contest is organized and the winner volunteers a payment to the seller; this payment can then be challenged by another bidder who knows the distribution of the winner’s valuation.

Optimal two-object auctions with synergies

We design the revenue-maximizing auction for two objects when each buyer has bi-dimensional private information and a superadditive utility function (i.e., a synergy is generated if a buyer wins both goods). In this setting the seller is likely to allocate the objects inefficiently with respect to an environment with no synergies (see Armstrong, RES (2000)). In particular, the objects may be bundled too rarely or a buyer may win the bundle even though her valuations for the goods are weakly dominated by the values of another buyer.