Collusion in Second‐Price Auctions under Minimax Regret Criterion

Collusion in auctions, with different assumptions on distributions of bidders' private valuation, has been studied extensively over the years. With the recent development of on‐line markets, auctions are becoming an increasingly popular procurement method. The emergence of Internet marketplaces makes auction participation much easier and more convenient, since no physical presence of bidders is required. In addition, bidders in on‐line auctions can easily switch their identities. Thus, it may very well happen that the bidders in an auction have very little, if any, prior knowledge about the distributions of other bidders' valuations. We are proposing an efficient distribution of collusive profit for second‐price sealed bid auctions in such an environment. Unlike some known mechanism, which balance the budget only in expectation, our approach (which we call Random k) balances the budget ex‐post. While truth‐telling is not a dominant strategy for Random k, it is a minimax regret equilibrium.     

Split‐Award Procurement Auctions—Can Bayesian Equilibrium Strategies Predict Human Bidding Behavior in Multi‐Object Auctions?

We analyze if and when symmetric Bayes Nash equilibrium predictions can explain human bidding behavior in multi‐object auctions. We focus on two sealed‐bid split‐award auctions with ex ante split decisions as they can be regularly found in procurement practice. These auction formats are straightforward multi‐object extensions of the first‐price sealed‐bid auction. We derive the risk‐neutral symmetric Bayes Nash equilibrium strategies and find that, although the two auction mechanisms yield the same expected costs to the buyer, other aspects of the two models, including the equilibrium bidding strategies, differ significantly. The strategic considerations in these auction formats are more involved than in single‐lot first‐price sealed‐bid auctions, and it is questionable whether expected utility maximization can explain human bidding behavior in such multi‐object auctions. Therefore, we analyzed the predictive accuracy of our equilibrium strategies in the laboratory. In human subject experiments we found underbidding, which is in line with earlier experiments on single‐lot first‐price sealed‐bid auctions. To control for regret, we organize experiments against computerized bidders, who play the equilibrium strategy. In computerized experiments where bid functions are only used in a single auction, we found significant underbidding on low‐cost draws. In experiments where the bid function is reused in 100 auctions, we could also control effectively for risk aversion, and there is no significant difference of the average bidding behavior and the risk‐neutral Bayes Nash equilibrium bid function. The results suggest that strategic complexity does not serve as an explanation for underbidding in split‐award procurement auctions, but risk aversion does have a significant impact.     

Optimal Nonparametric Estimation of First‐price Auctions

This paper proposes a general approach and a computationally convenient estimation procedure for the structural analysis of auction data. Considering first‐price sealed‐bid auction models within the independent private value paradigm, we show that the underlying distribution of bidders' private values is identified from observed bids and the number of actual bidders without any parametric assumptions. Using the theory of minimax, we establish the best rate of uniform convergence at which the latent density of private values can be estimated nonparametrically from available data. We then propose a two‐step kernel‐based estimator that converges at the optimal rate.     

Ranked Items Auctions and Online Advertisement

We study auctions for a set of commonly‐ranked items where each buyer has unit demand. This setting has promising applications in areas such as keyword auctions in the search engine advertising industry, the sale of quality‐ranked raw materials, etc. An auction mechanism suitable for this setting is the simultaneous pooled auction (SPA), where each bidder simultaneously submits a single bid and is allocated an object based on the rank of his bid among all the bids. We study how to improve the seller's expected revenue by enforcing a reserve price in an SPA. We find that the use of an appropriate reserve price may significantly increase the seller's revenue, especially when the number of items for sale is relatively large compared to the number of participating bidders. One inherent problem in the SPA is that some bidders may incur ex post losses; that is, they pay more than what they value the received objects. We propose a tailored VCG mechanism that generates the same expected revenue as the SPA does, while bidders do not incur any ex post loss. We also discuss the potential applications of this research to keyword auctions.     

First‐Price Auctions With General Information Structures: Implications for Bidding and Revenue

We explore the impact of private information in sealed‐bid first‐price auctions. For a given symmetric and arbitrarily correlated prior distribution over values, we characterize the lowest winning‐bid distribution that can arise across all information structures and equilibria. The information and equilibrium attaining this minimum leave bidders indifferent between their equilibrium bids and all higher bids. Our results provide lower bounds for bids and revenue with asymmetric distributions over values. We also report further characterizations of revenue and bidder surplus including upper bounds on revenue. Our work has implications for the identification of value distributions from data on winning bids and for the informationally robust comparison of alternative auction mechanisms.     

Collusion via Resale

The English auction is susceptible to tacit collusion when post‐auction interbidder resale is allowed. We show this by constructing equilibria where, with positive probability, one bidder wins the auction without any competition and divides the spoils by optimally reselling the good to the other bidders. These equilibria interim Pareto‐dominate (among bidders) the standard value‐bidding equilibrium without requiring the bidders to make any commitment on bidding behavior or postbidding spoil division.     

Speculation in Standard Auctions with Resale

In standard auctions resale creates a role for a speculator—a bidder who is commonly known to have no use value for the good on sale. We study this issue in environments with symmetric independent private‐value bidders. For second‐price and English auctions the efficient value‐bidding equilibrium coexists with a continuum of inefficient equilibria in which the speculator wins the auction and makes positive profits. First‐price and Dutch auctions have an essentially unique equilibrium, and whether or not the speculator wins the auction and distorts the final allocation depends on the number of bidders, the value distribution, and the discount factor. Speculators do not make profits in first‐price or Dutch auctions.     

Strategic Supply Function Competition With Private Information

A finite number of sellers (n) compete in schedules to supply an elastic demand. The cost of each seller is random, with common and private value components, and the seller receives a private signal about it. A Bayesian supply function equilibrium is characterized: The equilibrium is privately revealing and the incentives to rely on private signals are preserved. Supply functions are steeper with higher correlation among the cost parameters. For high (positive) correlation, supply functions are downward sloping, price is above the Cournot level, and as we approach the common value case, price tends to the collusive level. As correlation becomes maximally negative, we approach the competitive outcome. With positive correlation, private information coupled with strategic behavior induces additional distortionary market power above full information levels. Efficiency can be restored with appropriate subsidy schemes or with a precise enough public signal about the common value component. As the market grows large with the number of sellers, the equilibrium becomes price‐taking, bid shading is on the order of 1/n, and the order of magnitude of welfare losses is 1/n². The results extend to inelastic demand, demand uncertainty, and demand schedule competition. A range of applications in product and financial markets is presented.     

Nonparametric Identification of Risk Aversion in First‐Price Auctions Under Exclusion Restrictions

This paper studies the nonparametric identification of the first‐price auction model with risk averse bidders within the private value paradigm. First, we show that the benchmark model is nonindentified from observed bids. We also derive the restrictions imposed by the model on observables and show that these restrictions are weak. Second, we establish the nonparametric identification of the bidders' utility function under exclusion restrictions. Our primary exclusion restriction takes the form of an exogenous bidders' participation, leading to a latent distribution of private values that is independent of the number of bidders. The key idea is to exploit the property that the bid distribution varies with the number of bidders while the private value distribution does not. We then extend these results to endogenous bidders' participation when the exclusion restriction takes the form of instruments that do not affect the bidders' private value distribution. Though derived for a benchmark model, our results extend to more general cases such as a binding reserve price, affiliated private values, and asymmetric bidders. Last, possible estimation methods are proposed.     

Level‐k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner's Curse and Overbidding in Private‐Value Auctions?

This paper proposes a structural nonequilibrium model of initial responses to incomplete‐information games based on “level‐k” thinking, which describes behavior in many experiments with complete‐information games. We derive the model's implications in first‐ and second‐price auctions with general information structures, compare them to equilibrium and Eyster and Rabin's (2005) “cursed equilibrium,” and evaluate the model's potential to explain nonequilibrium bidding in auction experiments. The level‐k model generalizes many insights from equilibrium auction theory. It allows a unified explanation of the winner's curse in common‐value auctions and overbidding in those independent‐private‐value auctions without the uniform value distributions used in most experiments.     

Design of Auction Parameters to Reduce the Effect of Collusion

Government departments are increasingly turning to auctions to procure goods and services. Collusion among bidders, however, reduces competition and raises winning bid prices. Since conventional collusion control measures based on the redesign of auction mechanisms are less effective in government procurement auctions, there is a need to devise control measures that decrease the effect of collusion. This article demonstrates how the principles of design of experiments can be applied in a system dynamics model to find the auction parameter values that substantially reduce the effect of collusion in government procurement auctions. This research makes a number of contributions. First, it develops a feedback‐based dynamic mechanism of collusion in government procurement auctions. The mechanism proposes the winning bid price as being determined not by the total number of bidders but by the number of independent bidders. It defines each cartel as one independent bidder regardless of the number of bidders in the cartel. Second, the mechanism is tested by developing a system dynamics model to government auctions for procuring contracts for roadwork projects in India. Third, the principles of experimental design are applied to find the auction parameter values that ensure high bid participation and low winning price‐to‐reserve price ratios.     

Increasing Revenue by Decreasing Information in Procurement Auctions

We report on results of several laboratory experiments that investigate on‐line procurement auctions in which suppliers bid on price, but exogenous bidder quality affects winner determination. In procurement auctions, bidder quality may or may not be publicly known to all bidders, and the effect of this quality transparency on the auction outcome is one aspect of auction design that we examine. The second aspect of auction design that we examine is the effect of price visibility on the auction outcome, and the interaction between price visibility and quality transparency. In terms of price visibility, we consider two extreme cases: the sealed bid request for proposals (RFPs), and the open‐bid dynamic auction event. In terms of bidder quality transparency, we also consider two extreme cases: a setting in which bidder qualities are publicly known and the case in which they are private. We find that in our laboratory experiments, the RFP format is consistent in generating higher buyer surplus levels than does the open‐bid dynamic format. This advantage is independent of the quality transparency. In contrast, the open‐bid format is highly sensitive to quality transparency, generating significantly lower buyer surplus levels when the information about bidder quality is public.     

Keyword Auctions,Unit‐Price Contracts,and the Role of Commitment

Motivated by the enormous growth of keyword advertising, this paper explores the design of performance‐based unit‐price contract auctions, in which bidders bid their unit prices and the winner is chosen based on both their bids and performance levels. The previous literature on unit‐price contract auctions usually considers a static case where bidders' performance levels are fixed. This paper studies a dynamic setting in which bidders with a low performance level can improve their performance at a certain cost. We examine the effect of the performance‐based allocation on overall bidder performance, auction efficiency, and the auctioneer's revenue, and derive the revenue‐maximizing and efficient policies accordingly. Moreover, the possible upgrade in bidders' performance level gives the auctioneer an incentive to modify the auction rules over time, as is confirmed by the practice of Yahoo! and Google. We thus compare the auctioneer's revenue‐maximizing policies when she is fully committed to the auction rule and when she is not, and show that the auctioneer should give less preferential treatment to low‐performance bidders when she is fully committed.     

Bidding under auctioneer default risk

In a financially turbulent economy, participants of a procurement auction should consider in their bids the event of default of the auctioneer, which may result to substantial damages for the winning bidder. We examine a sealed bid auction, with private cost values and interdependence among the beliefs of the bidders about the auctioneer׳s default risk. The probability of payment of the bid price by the auctioneer is estimated by each bidder. For a first and a second price auction, we derive equilibrium bidding strategies, which address the risk of default and optimally adjust the bid price, introducing a risk premium in the form of an additional mark-up. A numerical illustration of the proposed strategies is provided. The effect of auctioneer׳s risk of default on the procurement project cost is examined. Financial arrangements that may be used to relax or eliminate the effect of the risk of default, such as early payment methods, third party guarantees or insurance programs are discussed and evaluated in comparison with the approach of risk premium on bid price.     

An Analysis of Scoring and Buyer‐Determined Procurement Auctions

In procurement auctions, the object for sale is a contract, bidders are suppliers, and the bid taker is a buyer. The suppliers bidding for the contract are usually the current supplier (the incumbent) and a group of potential new suppliers (the entrants). As the buyer has an ongoing relationship with the incumbent, he needs to adjust the bids of the entrants to include non‐price attributes, such as the switching costs. The buyer can run a scoring auction, in which suppliers compete on the adjusted bids or scores, or, he can run a buyer‐determined auction, in which suppliers compete on the price, and the buyer adjusts a certain number of the bids with the non‐price attributes after the auction to determine the winner. Unless the incumbent has a significant cost advantage over the entrants, I find that the scoring auction yields a lower average cost for the buyer, if the non‐price attributes are available. If the non‐price attributes are difficult or expensive to obtain, the buyer could run a buyer‐determined auction adjusting only the lowest price bid.     

The Effect of Timing on Jump Bidding in Ascending Auctions

We investigate the role of timing in ascending auctions under the premise that time is a valuable resource. Traditional models of the English auction ignore timing issues by assuming that the auction occurs instantaneously. However, when auctions are slow, as Internet auctions used for procurement often are, there are significant opportunity or monitoring costs to bidders, and the choice of the size of the jump bid becomes a strategic decision. We study the choice in the experimental laboratory by systematically varying the opportunity costs associated with fast bidding. When time is more valuable bidders respond by choosing larger jump bids. Surprisingly, the economic performance of the auction is not significantly affected. We develop a simple model of ascending auctions with impatient bidders that provides insights into the effect jump bids have on auction performance.     

Precautionary Bidding in Auctions

We analyze bidding behavior in auctions when risk‐averse buyers bid for a good whose value is risky. We show that when the risk in the valuations increases, DARA bidders will reduce their bids by more than the appropriate increase in the risk premium. Ceteris paribus, buyers will be better off bidding for a more risky object in first price, second price, and English auctions with affiliated common (interdependent) values. This “precautionary bidding” effect arises because the expected marginal utility of income increases with risk, so buyers are reluctant to bid so highly. We also show that precautionary bidding behavior can make DARA bidders prefer bidding in a common values setting to bidding in a private values one when risk‐neutral or CARA bidders would be indifferent. Thus the potential for a “winner's curse” can be a blessing for rational DARA bidders.     

Toward a Strategic Foundation for Rational Expectations Equilibrium

A step toward a strategic foundation for rational expectations equilibrium is taken by considering a double auction with n buyers and m sellers with interdependent values and affiliated private information. If there are sufficiently many buyers and sellers, and their bids are restricted to a sufficiently fine discrete set of prices, then, generically, there is an equilibrium in nondecreasing bidding functions that is arbitrarily close to the unique fully revealing rational expectations equilibrium of the limit market with unrestricted bids and a continuum of agents. In particular, the large double‐auction equilibrium is almost efficient and almost fully aggregates the agents' information.     

Reverse Auctions with Multiple Reinforcement Learning Agents*

Reverse auctions in business‐to‐business (B2B) exchanges provide numerous benefits to participants. Arguably the most notable benefit is that of lowered prices driven by increased competition in such auctions. The competition between sellers in reverse auctions has been analyzed using a game‐theoretic framework and equilibria have been established for several scenarios. One finding of note is that, in a setting in which sellers can meet total demand with the highest‐bidding seller being able to sell only a fraction of the total capacity, the sellers resort to a mixed‐strategy equilibrium. Although price randomization in industrial bidding is an accepted norm, one might argue that in reality managers do not utilize advanced game theory calculations in placing bids. More likely, managers adopt simple learning strategies. In this situation, it remains an open question as to whether the bid prices converge to the theoretical equilibrium over time. To address this question, we model reverse‐auction bidding behavior by artificial agents as both two‐player and n‐player games in a simulation environment. The agents begin the game with a minimal understanding of the environment but over time analyze wins and losses for use in determining future bids. To test for convergence, the agents explore the price space and exploit prices where profits are higher, given varying cost and capacity scenarios. In the two‐player case, the agents do indeed converge toward the theoretical equilibrium. The n‐player case provides results that reinforce our understanding of the theoretical equilibria. These results are promising enough to further consider the use of artificial learning mechanisms in reverse auctions and other electronic market transactions, especially as more sophisticated mechanisms are developed to tackle real‐life complexities. We also develop the analytical results when one agent does not behave strategically while the other agent does and show that our simulations for this environment also result in convergence toward the theoretical equilibrium. Because the nature of the best response in the new setting is very different (pure strategy as opposed to mixed), it indicates the robustness of the devised algorithm. The use of artificial agents can also overcome the limitations in rationality demonstrated by human managers. The results thus have interesting implications for designing artificial agents in automating bid responses for large numbers of bids where human intervention might not always be possible.     

ASSESSING SOCIAL COSTS OF INEFFICIENT PROCUREMENT DESIGN

This paper considers the social costs implied by inefficient allocation of contracts in a first‐price, sealed‐bid procurement auction with asymmetric bidders. We adopt a constrained (piecewise linear) strategy equilibrium concept and estimate the structural parameters of the bidders' distribution of costs. We estimate social costs defined as the predicted cost difference between the winning firm and the most efficient bidding firm. We also compare the expected procurement costs under two different auction formats. The data is collected from procurement auctions of road painting in Sweden during 1993–1999. The results indicate that the social costs of inefficient contract allocation is about 2% of total potential production cost and that an efficient second‐price auction would lower the expected procurement cost by 2.5%. (JEL: D44, H57, C15)