Auction Design Problem Research Articles

Auction design with heterogeneous priors

We consider an auction design problem with private values, where the seller and bidders may enjoy heterogeneous priors about their (possibly correlated) valuations. Each bidder forms an (interim) belief about the others based on his own prior updated by observing his own value. If the seller faces uncertainty about the bidders' priors, even if he knows that the bidders' priors are within any given distance from his, he may find it worst-case optimal to propose a dominant-strategy auction mechanism.

Diffusion auction design

This paper studies an auction design problem for a seller to sell a single commodity in a social network, where each individual (the seller or a buyer) can only communicate with her neighbors. The challenge is to design a mechanism to incentivize the buyers, who are aware of the auction, to further propagate the information to their neighbors, so that more buyers can participate in the auction and hence, the seller will be able to make a higher revenue and a higher welfare. We build a general framework for this new scenario and propose several novel diffusion auctions, which not only incentivize the buyers to report their valuations on the commodity truthfully, but also to propagate the auction information to all their neighbors. Particularly, the direct extension of the well-known Vickrey-Clarke-Groves (VCG) mechanism on social networks can have the incentives, but it will decrease the seller's revenue or even lead to a deficit. We also show that in the social network setting all efficient mechanisms that are individually rational and incentive compatible can lead to a deficit. The goal in this article is to increase the seller's revenue by attracting more buyers, so we give up welfare maximization and propose a class of mechanisms called critical diffusion mechanisms. It is proved that both the seller's revenue and the social welfare achieved in critical diffusion mechanisms are not less than that given in the VCG mechanism before attracting new buyers. The intuition behind the proposed mechanisms is that buyers who join the mechanism earlier have higher priorities to buy the commodity. If a buyer does not win the commodity because of her propagation, then she will be compensated. The formalization of the problem has not been well-studied in the literature of mechanism design, and there are many open problems worth further investigation. The study of this problem will provide insights for the emerging market based on the participants' recommendations via their social networks.

Auction Design with Ambiguity: Optimality of the First-Price and All-Pay Auctions

We study the optimal auction design problem when bidders' preferences follow the maxmin expected utility model. We suppose that each bidder's set of priors consists of beliefs close to the seller's belief, where closeness is defined by a divergence. For a given allocation rule, we identify a class of optimal transfer candidates, named the win-lose dependent transfers, with the following property: each type of bidder's transfer conditional on winning or losing is independent of the competitor's type report. Our result reduces the infinite-dimensional optimal transfer problem to a two-dimensional optimization problem. By solving the reduced problem, we find that: (i) among efficient mechanisms with no premiums for losers, the first-price auction is optimal; and, (ii) among efficient winner-favored mechanisms where each bidder pays smaller amounts when she wins than loses: the all-pay auction is optimal. Under a simplifying assumption, these two auctions remain optimal under the endogenous allocation rule.

Regret Minimization and Separation in Multi-Bidder Multi-Item Auctions

We study a robust auction design problem with a minimax regret objective, where a seller seeks a mechanism for selling multiple items to multiple bidders with additive values. The seller knows that the bidders' values range over a box uncertainty set but has no information on their probability distribution. We propose a mechanism that sells each item separately via a second price auction with a random reserve price and characterize an upper bound on its optimality gap. We show that this upper bound vanishes and that the proposed auction becomes optimal if the bidders are symmetric. We then interpret the auction design problem as a zero-sum game between the seller, who chooses a mechanism, and a fictitious adversary or 'nature,' who chooses the bidders' values from within the uncertainty set with the aim to maximize the seller's regret. We characterize the Nash equilibrium of this game analytically when the bidders are symmetric. The Nash strategy of the seller coincides with the optimal separable second price auction, whereas the Nash strategy of nature is mixed and constitutes a probability distribution on the uncertainty set under which each bidder's values for the items are comonotonic. We also study a restricted auction design problem over deterministic mechanisms. In this setting, we characterize the suboptimality of a separable second price auction with deterministic reserve prices and show that this auction becomes optimal if the bidders are symmetric.

Robust auction design under multiple priors by linear and integer programming

It is commonly assumed in the optimal auction design literature that valuations of buyers are independently drawn from a unique distribution. In this paper we study auctions under ambiguity, that is, in an environment where valuation distribution is uncertain itself, and present a linear programming approach to robust auction design problem with a discrete type space. We develop an algorithm that gives the optimal solution to the problem under certain assumptions when the seller is ambiguity averse with a finite prior set $${\mathcal {P}}$$ and the buyers are ambiguity neutral with a prior $$f\in {\mathcal {P}}$$ . We also consider the case where all parties, the buyers and the seller, are ambiguity averse, and formulate this problem as a mixed integer programming problem. Then, we propose a hybrid algorithm that enables to compute an optimal solution for the problem in reduced time.

Mechanism Design in Social Networks

This paper studies an auction design problem for a seller to sell a commodity in a social network, where each individual (the seller or a buyer) can only communicate with her neighbors. The challenge to the seller is to design a mechanism to incentivize the buyers, who are aware of the auction, to further propagate the information to their neighbors so that more buyers will participate in the auction and hence, the seller will be able to make a higher revenue. We propose a novel auction mechanism, called information diffusion mechanism (IDM), which incentivizes the buyers to not only truthfully report their valuations on the commodity to the seller, but also further propagate the auction information to all their neighbors. In comparison, the direct extension of the well-known Vickrey-Clarke-Groves (VCG) mechanism in social networks can also incentivize the information diffusion, but it will decrease the seller's revenue or even lead to a deficit sometimes. The formalization of the problem has not yet been addressed in the literature of mechanism design and our solution is very significant in the presence of large-scale online social networks.

Auction design without quasilinear preferences

I study the canonical private value auction model for a single good without the quasilinearity restriction. I assume only that bidders are risk averse and the indivisible good for sale is a normal good. I show that removing quasilinearity leads to qualitatively different solutions to the auction design problem. Expected revenue is no longer maximized using standard auctions that allocate the good to the highest bidder. Instead, the auctioneer better exploits bidder preferences by using a mechanism that allocates the good to one of many different bidders, each with strictly positive probability. I introduce a probability demand mechanism that treats probabilities of winning the indivisible good like a divisible good in net supply 1. With enough bidders, it has greater expected revenues than any standard auction, and under complete information, it implements a Pareto efficient allocation.

Auction optimization using regression trees and linear models as integer programs

In a sequential auction with multiple bidding agents, the problem of determining the ordering of the items to sell in order to maximize the expected revenue is highly challenging. The challenge is largely due to the fact that the autonomy and private information of the agents heavily influence the outcome of the auction.The main contribution of this paper is two-fold. First, we demonstrate how to apply machine learning techniques to solve the optimal ordering problem in sequential auctions. We learn regression models from historical auctions, which are subsequently used to predict the expected value of orderings for new auctions. Given the learned models, we propose two types of optimization methods: a black-box best-first search approach, and a novel white-box approach that maps learned regression models to integer linear programs (ILP), which can then be solved by any ILP-solver. Although the studied auction design problem is hard, our proposed optimization methods obtain good orderings with high revenues.Our second main contribution is the insight that the internal structure of regression models can be efficiently evaluated inside an ILP solver for optimization purposes. To this end, we provide efficient encodings of regression trees and linear regression models as ILP constraints. This new way of using learned models for optimization is promising. As the experimental results show, it significantly outperforms the black-box best-first search in nearly all settings.

Optimal minimum bids and inventory scrapping in sequential, single-unit, Vickrey auctions with demand learning

Retailers often conduct sequential, single-unit auctions and need to decide the minimum bid in each auction. To reduce inventory costs, it may be optimal to scrap some of the inventory rather than holding it until it is auctioned off. In some auctions, the seller may be uncertain about the market response and hence may want to dynamically learn the demand by observing the number of posted bids. We formulate a Markov decision process (MDP) to study this dynamic auction-design problem under the Vickrey mechanism.We first develop a clairvoyant model where the seller knows the demand distribution. We prove that it is optimal to scrap all inventory above a certain threshold and then auction the remaining units. We derive a first order necessary condition whereby the bidders’ virtual value at an optimal minimum bid equals the seller’s marginal profit. This is a generalization of Riley and Samuelson’s result from the one, single-unit auction case. When the virtual value is strictly increasing, this necessary condition is also sufficient and leads to a structured value iteration algorithm.We then assume that the number of bidders is Poisson distributed but the seller does not know its mean. The seller uses a mixture-of-Gamma prior on this mean and updates this belief over several auctions. This results in a high-dimensional Bayesian MDP whose exact solution is intractable. We therefore propose and compare two approximation methods called certainty equivalent control (CEC) and Q-function approximation. Numerical experiments suggest that Q-function approximation can attain higher revenues than CEC.

Optimal Design for Multi-Item Auctions: A Robust Optimization Approach

In this paper, we revisit the auction design problem for multi-item auctions with budget constrained buyers by introducing a robust optimization approach to model (a) concepts such as incentive compatibility and individual rationality that are naturally expressed in the language of robust optimization and (b) the auctioneer’s beliefs on the buyers' valuations of the items. Rather than using probability distributions (the classical probabilistic approach) or an adversarial model to model valuations, we introduce an uncertainty set based model for these valuations. We construct these uncertainty sets to incorporate historical information available to the auctioneer in a way that is consistent with limit theorems of probability theory or knowledge of the probability distribution. In this setting, we formulate the auction design problem as a robust optimization problem and provide a characterization of the optimal solution as an auction with reservation prices, thus extending the work of Myerson [Myerson RB (1981) Optimal auction design. Math. Oper. Res. 6(1):58–73] from single item without budget constraints to multiple items with budgets, potentially correlated valuations, and uncertain budgets. Unlike the Myerson auction where the reservation prices do not depend on the item, the reservation prices in our approach are a function of both the bidder and the item. We propose an algorithm for calculating the reservation prices by solving a bilinear optimization problem that, although theoretically difficult in general, is numerically tractable for the polyhedral uncertainty sets we consider. Moreover, this bilinear optimization problem reduces to a linear optimization problem for auctions without budget constraints and the auction becomes the classical second price auction. We report computational evidence that suggests the proposed approach (a) is numerically tractable for large scale auction design problems with the polyhedral uncertainty sets we consider, (b) leads to improved revenue compared to the classical probabilistic approach when the true distributions are different from the assumed ones, and (c) leads to higher revenue when correlations in the buyers' valuations are explicitly modeled.

Dynamic Auctions with Demand Learning

The minimum bid is a key design variable in retail auctions as it affects the number of posted bids and the clearing price. Retailers often conduct sequential, single-unit auctions online as a revenue generation and inventory clearing tool, and need to dynamically decide the minimum bid in each auction depending on the inventory level at that time. To reduce inventory costs, it may in fact be optimal to scrap some of the inventory rather than holding it until it is auctioned off. In auctions of fashion goods, seasonal products, new products, and where the seller is new to the market, the seller may be uncertain about the market response. Such a seller may want to dynamically learn the demand by observing the number of posted bids. This introduces an additional, exploration versus exploitation trade-off. We formulate a Markov decision process (MDP) to study this dynamic auction-design problem. We consider a Vickrey mechanism, and a random number of bidders with private, independent, and identically distributed valuations. We first develop a clairvoyant model where the seller knows the demand distribution. We prove that it is optimal to scrap all inventory above a certain threshold and then auction the remaining units. We derive a first order necessary condition, which states that, in any inventory level, the bidders' virtual value (as defined by Myerson) at an optimal minimum bid should equal the seller's marginal profit. This is an intuitive generalization of Riley and Samuelson's well-known result from the one single-unit auction case. Under the standard regularity condition that the virtual value is strictly increasing, our first order necessary condition is also sufficient and leads to a structured value iteration algorithm for efficiently computing optimal policies. In that case, we prove that optimal minimum bids (i) decrease with increasing inventory levels, and (ii) are higher than optimal minimum bids in one single-unit auction. We also prove that optimal scrapping thresholds and optimal minimum bids both decrease with increasing inventory costs.We illustrate our results and perform sensitivity analyses using examples with a Poisson-distributed number of bidders and Beta-distributed private valuations.In the second half of the paper, we assume that the number of bidders is Poisson distributed but the seller does not know its mean. The seller uses a mixture-of-Gamma prior on this mean and updates this belief over the sequence of auctions while simultaneously deciding minimum bids and scrapping quantities. This results in a high-dimensional Bayesian MDP whose exact solution is intractable. We therefore implement a certainty equivalent control (CEC) heuristic instead. We extend our clairvoyant structural result to CEC and numerically investigate the effect of initial inventory and mean demand on CEC performance.

Price of Fairness on Networked Auctions

We consider an auction design problem under network flow constraints. We focus on pricing mechanisms that provide fair solutions, where fairness is defined in absolute and relative terms. The absolute fairness is equivalent to “no individual losses” assumption. The relative fairness can be verbalized as follows: no agent can be treated worse than any other in similar circumstances. Ensuring the fairness conditions makes only part of the social welfare available in the auction to be distributed on pure market rules. The rest of welfare must be distributed without market rules and constitutes the so-calledprice of fairness. We prove that there exists the minimum ofprice of fairnessand that it is achieved when uniform unconstrained market price is used as the base price. Theprice of fairnesstakes into account costs of forced offers and compensations for lost profits. The final payments can be different than locational marginal pricing. That means that the widely applied locational marginal pricing mechanism does not in general minimize theprice of fairness.

Auction Design Without Quasilinear Preferences

I analyze private value auction design and assume only that bidders are risk averse and have positive wealth effects (i.e. the good is normal). I show removing the standard quasilinearity restriction leads to qualitatively different solutions to the auction design problem with respect to both efficiency or revenue maximization. On efficiency, I show that probabilistic allocations of the good can Pareto dominate the second price auction; and there is no dominant-strategy mechanism that is both Pareto efficient and individually rationality. On revenue, I construct a probability demand mechanism with greater expected revenues than standard auctions when there are sufficiently many bidders. In addition, I take a new approach to studying bid behavior when types are multidimensional. Instead of characterizing bidder's interim incentive constraints, I place bounds on their bids, and I show that these bounds are sufficient for obtaining revenue comparisons.

An optimal auction for capacity constrained bidders: a network perspective

This paper examines the problem of a seller with limited supply selling to a group of agents whose private information is two-dimensional. Each agent has a constant marginal value for the good up to some capacity, thereafter it is zero. Both the marginal value and the capacity are private information. We describe the revenue maximizing Bayesian incentive compatible auction for this environment. A novel feature of the analysis is an interpretation of an optimal auction design problem in terms of a linear program that is an instance of a parametric shortest path problem on a lattice.

Mutually Destructive Bidding: The FCC Auction Design Problem

In general, synergies across license valuations complicate the auction design process. Theory suggests that a “simple” (i.e., non-combinatorial) auction will have difficulty in assigning licenses efficiently in such an environment. This difficulty increases with increases in “fitting complexity.” In some environments, bidding may become “mutually destructive.” Experiments indicate that a properly designed combinatorial auction is superior to a simple auction in terms of economic efficiency and revenue generation in bidding environments with a low amount of fitting complexity. Concerns that a combinatorial auction will cause a “threshold” problem are not borne out when bidders for small packages can communicate.