On the Existence of Pure Strategy Monotone Equilibria in Asymmetric First‐Price Auctions

We establish the existence of pure strategy equilibria in monotone bidding functions in first‐price auctions with asymmetric bidders, interdependent values, and affiliated one‐dimensional signals. By extending a monotonicity result due to Milgrom and Weber (1982), we show that single crossing can fail only when ties occur at winning bids or when bids are individually irrational. We avoid these problems by considering limits of ever finer finite bid sets such that no two bidders have a common serious bid, and by recalling that single crossing is needed only at individually rational bids. Two examples suggest that our results cannot be extended to multidimensional signals or to second‐price auctions.     

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.     

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.     

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.     

Behavior in Multi‐Unit Demand Auctions: Experiments with Uniform Price and Dynamic Vickrey Auctions

We experimentally investigate the sensitivity of bidders demanding multiple units of a homogeneous commodity to the demand reduction incentives inherent in uniform price auctions. There is substantial demand reduction in both sealed bid and ascending price clock auctions with feedback regarding rivals' drop‐out prices. Although both auctions have the same normal form representation, bidding is much closer to equilibrium in the ascending price auctions. We explore the behavioral process underlying these differences along with dynamic Vickrey auctions designed to eliminate the inefficiencies resulting from demand reduction in the uniform price auctions.     

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.     

Do Auction Parameters Affect Buyer Surplus in E‐Auctions for Procurement?

Although the initial euphoria about Internet‐enabled reverse auctions has given way to a cautious but widespread use of reverse auctions in business‐to‐business (B2B) procurement, there is a limited understanding of the effect of auction design parameters on buyer surplus. In this paper, we study the effect of bidding competition, information asymmetry, reserve price, bid decrement, auction duration, and bidder type on buyer surplus. We collected field data on more than 700 online procurement auctions conducted by a leading auctioneer and involving procurement items worth millions of dollars. Consistent with the predictions of auction theory, the results indicate that bidding competition, reserve price, and information sharing affect buyer surplus. Unlike previous findings in the consumer‐to‐consumer context, we find that bid decrement and auction duration have no effect in B2B procurement auctions. Our results suggest that use of the rank‐bidding format increases buyer surplus when incumbent suppliers participate in the auction. We discuss the theoretical and managerial implications of these findings for future research and for optimal design of online procurement auctions.     

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.     

Market‐Based Allocation with Indivisible Bids

We study multiunit double auctions accepting bids with indivisibility constraints. Modeling the auction problem as a Multiple Choice Knapsack Problem and using dynamic programming, we show that incremental computations during bid processing can speed the handling of key auction operations such as clearing and quoting. We propose different price‐quote policies and study their influence on the efficiency of market‐based allocation. Using a reconfigurable manufacturing scenario where agents trade large quantities of multiple goods, we demonstrate potential benefits of supporting indivisibility constraints in bidding. These benefits are highly sensitive to the form of price quote provided, indicating interesting tradeoffs in communication and allocation efficiency.     

Reverse Auctions to Innovate Procurement Processes: Effects of Bid Information Presentation Design on a Supplier's Bidding Outcome

Information technologies (ITs) are being used to innovate various procurement processes. This research study focuses on the supplier‐side effects of IT design choices to conduct reverse auctions, which are increasingly used to procure a wide range of products and services. IT–enabled reverse auctions enhance supplier participation across geographical boundaries, leading to more efficient pricing. However, there are growing concerns about the adverse effects of IT–enabled reverse auctions on a supplier's performance. Supplier‐side issues are gaining prominence in the reverse auction literature and are critical for the long‐term success of reverse auctions. Therefore, we focus on suppliers’ bidding outcomes and assess how the design of an IT–enabled reverse auction facilitates the auction bidding outcomes of participating suppliers. Specifically, we examine the effects of two types of bid information presentation design—full price visibility and partial price visibility—on supplier's auction bidding outcomes, across auctions with different cost certainty and suppliers bargaining power vis‐à‐vis the buyer. The results of this study contribute new knowledge about the ways to use IT for creating effective auction designs and innovating procurement through auctions to enhance both the buyer's and suppliers’ performance. We present the detailed theoretical contributions of our study and discuss the managerial implications for designers of reverse auctions.      

Clocks and Trees: Isomorphic Dutch Auctions and Centipede Games

We report an experiment on effects of varying institutional format and dynamic structure of centipede games and Dutch auctions. Centipede games with a clock format unravel, as predicted by theory but not reported in previous literature on two‐player tree‐format centipede games. Dutch auctions with a tree format produce bids close to risk neutral Nash equilibrium bids, unlike previous literature on clock‐format Dutch auctions. Our data provide a new, expanded set of stylized facts which may provide a foundation for unified modeling of play in a class of games that includes centipede games and Dutch auctions.     

On E‐Auctions for Procurement Operations

Electronic auctions have revolutionized procurement in the last decade. In many situations, they have replaced negotiations for supplier selection and price setting. While they have often greatly reduced transaction costs and increased competition, they have also encountered problems and resistance from suppliers resenting their intrusion on cooperative supplier/buyer relationships. In response to these issues, procurement auctions have evolved in radical new directions. Buyers use business rules to limit adverse changes. Some procurement auctions allow bidders to offer variants in the specifications of products to be supplied. Most important, some suppliers are allowing bidders to bid on packages of items, not just individual items. This tends to change procurement auctions from zero‐sum fights over supplier profit margins to win‐win searches for synergies. These changes have opened up many new research areas. Researchers are trying to improve how to deal with the computational issues involved in package auctions and to analyze the new auctions forms that are evolving. In general, equilibrium incentives are not known, and dealing with ties in package auctions is an issue. Computer scientists are analyzing the use of computerized bidding agents. Mechanisms that combine auctions with fixed buy prices or with negotiations need to be analyzed.     

Contractors’ and Agency Decisions and Policy Implications in A+B Bidding

The focus of this study is on the A+B transportation procurement mechanism, which uses the proposed cost (A component) and the proposed time (B component) to score contractors’ bids. Empirical studies have shown that this mechanism shortens project durations. We use normative models to study the effect of certain discretionary parameters set by state transportation agencies on contractors’ equilibrium bidding strategies, winner selection, and actual completion times. We model the bidding environment in detail including multi‐dimensional bids, contractors’ uncertainty about completion times, and reputation cost. The latter refers to a private penalty that accrues to tardy contractors from increased cost of posting bonds and reduced prospects of winning future projects. Our model explains why contractors may skew line‐item bids and why winners frequently finish earlier than bid. It has several policy implications as well. For example, we recommend that agencies set the daily incentive, disincentive, and road user cost to be equal and not cap incentives. This is a departure from current practice, where incentives are often capped and weaker than penalties. Furthermore, we show that agencies may be justified in setting daily road user cost strictly smaller than the true cost of traffic disruption during construction.     

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.     

Common Value Auctions with Insider Information

Bidding is studied in first-price common value auctions where an insider is better informed than other bidders (outsiders) about the value of the item. With inexperienced bidders, having an insider does not materially reduce the severity of the winner's curse compared to auctions with a symmetric information structure (SIS). In contrast, super-experienced bidders, who have largely overcome the winner's curse, satisfy the comparative static predictions of equilibrium bidding theory: (i) average seller's revenue is larger with an insider than in SIS auctions, (ii) insiders make substantially greater profits, conditional on winning, than outsiders, and (iii) insiders increase their bids in response to more rivals. Further, changes in insiders' bids are consistent with directional learning theory (Selten and Buchta (1994)).     

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.     

Competing Mechanisms in a Common Value Environment

Consider strategic risk‐neutral traders competing in schedules to supply liquidity to a risk‐averse agent who is privately informed about the value of the asset and his hedging needs. Imperfect competition in this common value environment is analyzed as a multi‐principal game in which liquidity suppliers offer trading mechanisms in a decentralized way. Each liquidity supplier behaves as a monopolist facing a residual demand curve resulting from the maximizing behavior of the informed agent and the trading mechanisms offered by his competitors. There exists a unique equilibrium in convex schedules. It is symmetric and differentiable and exhibits typical features of market‐power: Equilibrium trading volume is lower than ex ante efficiency would require. Liquidity suppliers charge positive mark‐ups and make positive expected profits, but these profits decrease with the number of competitors. In the limit, as this number goes to infinity, ask (resp. bid) prices converge towards the upper (resp. lower) tail expectations obtained in Glosten (1994) and expected profits are zero.     

Competition for a Majority

We define the class of two‐player zero‐sum games with payoffs having mild discontinuities, which in applications typically stem from how ties are resolved. For such games, we establish sufficient conditions for existence of a value of the game, maximin and minimax strategies for the players, and a Nash equilibrium. If all discontinuities favor one player, then a value exists and that player has a maximin strategy. A property called payoff approachability implies existence of an equilibrium, and that the resulting value is invariant: games with the same payoffs at points of continuity have the same value and ɛ‐equilibria. For voting games in which two candidates propose policies and a candidate wins election if a weighted majority of voters prefer his proposed policy, we provide tie‐breaking rules and assumptions about voters' preferences sufficient to imply payoff approachability. These assumptions are satisfied by generic preferences if the dimension of the space of policies exceeds the number of voters; or with no dimensional restriction, if the electorate is sufficiently large. Each Colonel Blotto game is a special case in which each candidate allocates a resource among several constituencies and a candidate gets votes from those allocated more than his opponent offers; in this case, for simple‐majority rule we prove existence of an equilibrium with zero probability of ties.     

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.     

网上拍卖销售与逆向拍卖采购下的库存管理

以电子商务为背景研究网上拍卖销售与逆向拍卖采购下的库存管理。在该问题中,零售商一方面利用网上拍卖销售产品,另一方面利用逆向拍卖进行采购。对于单阶段情形,证明网上拍卖下零售商的期望收益函数是采购量的严格递增的凹函数,从而得到零售商的最优采购策略和供应商的最优投标策略。对于多阶段情形,利用马尔可夫决策过程理论建立该问题的最优控制模型,得到零售商的最优采购策略和供应商的贝叶斯-纳什均衡投标策略均与基本库存策略相类似。