Equilibrium Bidding Research Articles

Empirical implementation of nonparametric first-price auction models

Nonparametric estimators provide a flexible means of uncovering salient features of auction data. Although these estimators are popular in the literature, many key features necessary for proper implementation have yet to be uncovered. Here we provide several suggestions for nonparametric estimation of first-price auction models. Specifically, we show how to impose monotonicity of the equilibrium bidding strategy; a key property of structural auction models not guaranteed in standard nonparametric estimation. We further develop methods for automatic bandwidth selection. Finally, we discuss how to impose monotonicity in auctions with differing numbers of bidders, reserve prices, and auction-specific characteristics. Finite sample performance is examined using simulated data as well as experimental auction data.

Cyclical Bid Adjustments in Search-Engine Advertising

Keyword advertising, or sponsored search, is one of the most successful advertising models on the Internet. One distinctive feature of keyword auctions is that they enable advertisers to adjust their bids and rankings dynamically, and the payoffs are realized in real time. We capture this unique feature with a dynamic model and identify an equilibrium bidding strategy. We find that under certain conditions, advertisers may engage in cyclical bid adjustments, and equilibrium bidding prices may follow a cyclical pattern: price-escalating phases interrupted by price-collapsing phases, similar to an “Edgeworth cycle” in the context of dynamic price competitions. Such cyclical bidding patterns can take place in both first- and second-price auctions. We obtain two data sets containing detailed bidding records of all advertisers for a sample of keywords in two leading search engines. Our empirical framework, based on a Markov switching regression model, suggests the existence of such cyclical bidding strategies. The cyclical bid-updating behavior we find cannot be easily explained with static models. This paper emphasizes the importance of adopting a dynamic perspective in studying equilibrium outcomes of keyword auctions. This paper was accepted by Pradeep Chintagunta and Preyas Desai, special issue editors. This paper was accepted by Pradeep Chintagunta and Preyas Desai, special issue editors.

Management Insights

Management Insights

Position Auctions with Consumer Search

This article examines a model in which advertisers bid for “sponsored-link” positions on a search engine. The value advertisers derive from each position is endogenized as coming from sales to a population of consumers who make rational inferences about firm qualities and search optimally. Consumer search strategies, equilibrium bidding, and the welfare benefits of position auctions are analyzed. Implications for reserve prices and a number of other auction design questions are discussed.

Out‐of‐Equilibrium Bids in First‐Price Auctions: Wrong Expectations or Wrong Bids

Deviations from risk‐neutral equilibrium bids in auctions can be related to inconsistent expectations with correct best replies or correct expectations but deviant best replies (e.g. because of risk aversion, regret, quantal‐response mistakes). To distinguish between these two explanations, we use a novel experimental procedure and study expectations together with best replies in symmetric and asymmetric auctions. We extensively test the internal validity of this setup. We find that deviations from equilibrium bids do not seem to be because of to wrong expectations but because of deviations from a risk‐neutral best reply.

Bayesian Inference in Structural Second-Price Common Value Auctions

Structural econometric auction models with explicit game-theoretic modeling of bidding strategies have been quite a challenge from a methodological perspective, especially within the common value framework. We develop a Bayesian analysis of the hierarchical Gaussian common value model with stochastic entry introduced by Bajari and Hortaçsu. A key component of our approach is an accurate and easily interpretable analytical approximation of the equilibrium bid function, resulting in a fast and numerically stable evaluation of the likelihood function. We extend the analysis to situations with positive valuations using a hierarchical gamma model. We use a Bayesian variable selection algorithm that simultaneously samples the posterior distribution of the model parameters and does inference on the choice of covariates. The methodology is applied to simulated data and to a newly collected dataset from eBay with bids and covariates from 1000 coin auctions. We demonstrate that the Bayesian algorithm is very efficient and that the approximation error in the bid function has virtually no effect on the model inference. Both models fit the data well, but the Gaussian model outperforms the gamma model in an out-of-sample forecasting evaluation of auction prices. This article has supplementary material online.

Discrete Bids and Empirical Inference in Divisible Good Auctions

I examine a model of a uniform price auction of a perfectly divisible good with private informa tion in which the bidders submit discrete bidpoints rather than continuous downward sloping demand functions. 1 characterize necessary conditions for equilibrium bidding. The characterization reveals a close relationship between bidding in multiunit auctions and oligopolistic behaviour. I demonstrate that a recently proposed indirect approach to the revenue comparisons of discriminatory and uniform price auc tions is not valid if bid functions have steps. In particular, bidders may bid above their marginal valuation in a uniform price auction. In order to demonstrate that discrete bidding can have important consequences for empirical analysis I use my model to examine a data set consisting of individual bids in uniform price treasury auctions of the Czech government. I propose an alternative method for evaluating the perfor mance of the employed mechanism. My results suggest that the uniform price auction performs well, both in terms of efficiency of the allocation and in terms of revenue maximization. I estimate that the em ployed mechanism failed to extract at most 3 basis points in terms of the annual yield of T-bills worth of expected surplus while implementing an allocation resulting in almost all the efficient surplus. Failing to account for discreteness of bids would in my application result in overestimating the unextracted revenue by more than 50%.

Optimal equilibrium bidding strategies for budget constrained bidders in sponsored search auctions

Budget optimization (selecting an optimal set of keywords to bid for, given a limited amount of budget) and bid optimization (determining optimal bids for the selected keywords) in the face of an unpredictable keyword traffic is a challenging problem facing advertisers in sponsored search auctions. Another key requirement in sponsored search auctions is stability of the bid profile, which is usually characterized through the notion of locally envy-free equilibrium. A variety of budget optimization and bid optimization ideas have been proposed in the literature; the bid profiles suggested by them have varying stability properties. In this paper, our objective is to come up with a bidding strategy for advertisers that is optimal for the budget optimization as well as the bid optimization problems and also guarantees convergence to a locally envy-free equilibrium. Our approach works with a linear program for the budget optimization problem whose optimal solution is a greedy strategy, which we call the OPT strategy. The OPT strategy yields a bid profile that maximizes the return on investment for the advertisers and ensures convergence of the bid profiles to a locally envy-free equilibrium in the asynchronous case as well as in a restricted synchronous case. We also suggest an improved strategy called the MAX strategy which produces equal or higher advertiser utilities compared to the OPT strategy while enjoying identical equilibrium convergence properties as OPT.

Sequential auctions for common value objects with budget constrained bidders

This paper analyzes sequential auctions for budget constrained bidders, for multiple heterogeneous common value objects. In most auctions, bidders not only have valuations for objects but also have budget constraints (i.e., constraints on how much they can actually spend in an auction). Given these constraints, the bidders' problem is to determine how much to bid in each auction. To this end, this paper analyzes the strategic behavior of bidders and determines the equilibrium bidding strategies for the individual auctions that constitute a series. We do this in an incomplete information setting where the bidders are uncertain about their budget constraints, and obtain equilibrium bidding strategies for the participants in individual auctions in a series.

Multi-Attribute Procurement Auctions Based on Cobb-Douglas Utility Function

In practical multi-attribute procurement auctions, the profit and winning a contract have different importance to the suppliers. This paper assumes that the suppliers weight the profit and probability of winning a contract with Cobb-Douglas utility function, obtains the suppliers’ equilibrium bidding strategies and the buyer’ expected utilities in the first- and second-score auctions, and compares them with those under risk-neutral.

Contests with private costs: Beyond two players

It was shown previously that in the symmetric contest game of two players, equilibrium bidding is lower in the case of private information than in the case of public information about the players' costs. I consider symmetric contests of an arbitrary number of players with continuously distributed private costs and discuss the existence and properties of equilibrium bidding functions. I show that with more than two players the relationship between equilibrium bids in the cases of public and private information is no longer universal. While high-cost players still bid less in the private information case, relatively low-cost players may bid above or below their corresponding public information bids.

The effects of statistically dependent values on equilibrium strategies of bilateral k-double auctions

This paper describes how introducing statistical dependency among trader values changes the equilibrium bidding strategies in bilateral k-double auctions and uses the special case of affiliation to illustrate the range of equilibrium responses to a change in the value distribution. Consistent with standard intuition, a change from independent to strictly affiliated valuations can result in high-value buyers and low-value sellers bidding closer to their actual values while low-value buyers and high-value sellers distort their bids farther from their true values. However, there also exist equilibria for which either type of trader responds to a change in the distribution of values in the opposite direction. Whether a given trader bids more aggressively or less aggressively can also be non-monotonic with respect to the traderʼs value.

Asymmetric Spite in Auctions

In many auctions, agents bid more aggressively than self-interest would prescribe. This can be explained by spite, where the agent's utility not only increases in the agent's surplus but also decreases as the other bidders' surpluses increase. Spite can stem from long-term benefits from making competitors worse off and from inherent psychological effects. There have been important recent game-theoretic analyses of spiteful bidding assuming all agents are equally spiteful. We present, to our knowledge, the first auction analysis in the more realistic setting where bidders may be spiteful to different extents. We show that the equilibrium bidding function can still be written in the same form — except that the spite factor is replaced by an expressed spite factor. This leads to bidders expressing spites that are higher or lower than their true spite depending on others' spite. Perhaps surprisingly, in the two-bidder case, the mapping from true spite to expressed spite is the same across all common auction mechanisms. Furthermore, even with two bidders, important properties of symmetric-spite settings cease to hold: the allocation can be inefficient and the revenue ranking may reverse between first- and second-price auctions. We also show that in sealed-bid auctions under asymmetric valuation distributions, there can be a "bargaining problem" in selecting bids. Finally, we study the generalization where agents can have different extents of spite toward different other bidders.

Bayesian Inference in Structural Second-Price Common Value Auctions

Structural econometric auction models with explicit game-theoretic modeling of bidding strategies have been quite a challenge from a methodological perspective, especially within the common value framework. We develop a Bayesian analysis of the hierarchical Gaussian common value model with stochastic entry introduced by Bajari and Hortacsu (2003). A key component of our approach is an accurate and easily interpretable analytical approximation of the equilibrium bid function, resulting in a fast and numerically stable evaluation of the likelihood function. The analysis is also extended to situations with positive valuations using a hierarchical Gamma model. We use a Bayesian variable selection algorithm that simultaneously samples the posterior distribution of the model parameters and does inference on the choice of covariates. The methodology is applied to simulated data and to a carefully collected dataset from eBay with bids and covariates from 1000 coin auctions. It is demonstrated that the Bayesian algorithm is very efficient and that the approximation error in the bid function has virtually no effect on the model inference. Both models fit the data well, but the Gaussian model outperforms the Gamma model in an out-of-sample forecasting evaluation of auction prices.

Signal jamming in a sequential auction

In a sequence of first-price auctions with stable private values bidders strategically conceal their private information until the last auction. We characterize equilibrium bidding and explore how such signal jamming affects the dynamics of equilibrium prices.

Asymmetric Information, Endogenous Illiquidity, and Asset Pricing with Imperfect Competition

We propose a novel and tractable equilibrium model to study how information asymmetry, competition among market makers, and investors’ risk aversion affect asset pricing, market illiquidity and welfare. The main innovation is that market makers compete through choosing simultaneously quantities to buy at the bid and to sell at the ask and accordingly market clears separately at the bid and at the ask. Equilibrium bid and ask prices, bid and ask depths, trading volume and market makers’ inventory levels are all derived in closed-form. Our model can help explain some of the puzzling empirical findings, such as bid-ask spreads can be lower with asymmetric information and can be positively correlated with trading volume. In addition, we find that information asymmetry may make informed investors worse off, may reduce the welfare loss due to market power and may increase the competition among market makers in equilibrium.

Auctioning the Right to Choose When Competition Persists

Several papers compare auctioning heterogeneous assets sequentially with sequentially selling the right to choose among assets not yet taken. Typically motivated by auctions of condos for owner occupation, these papers have assumed that each winning bidder exits, so each successive auction has less competition. In many heterogeneous-asset-sale situations, a winning bidder may still be interested in acquiring further assets. We build a simple model of persistent competition, in which the distribution of equilibrium revenue from separate sales is shown to be a mean-preserving spread of the distribution of revenue from selling rights to choose. Persistent competition reveals that a high bidder does not always select his most preferred asset, and that one asset being slightly more likely to be a favored asset discontinuously affects equilibrium bidding.

Winner-pay contests

We analyze winner-pay contests in which players select bids, the winner is determined probabilistically through a contest success function, and only the winner must pay her bid. We show that a pure-strategy Nash equilibrium exists and is unique under weak assumptions on the contest success function. We determine the Nash equilibrium for several examples. We also consider how equilibrium bids are influenced by private information about the value of winning the contest.

A weighted product method for bidding strategies in multi-attribute auctions

To eliminate computational problems involved in evaluating multi-attribute bids with different measures, this article first normalizes each individual component of a bid, and then makes use of the weighted product method to present a new scoring function that converts each bid into a score. Two kinds of multi-attribute auction models are introduced in terms of scoring rules and bidding objective functions. Equilibrium bidding strategies, procurer’s revenue comparisons and optimal auction design are characterized in these two models. Finally, this article discusses some improvement of robustness of our models, with respect to the assumptions.

Equilibrium Bidding Strategy for GSP Keyword Auctions

The generalized second-price (GSP) mechanism is the most widely-used auction format in sponsored search markets. However, figuring out how to bid on GSP auctions presents major theoretical and computational challenges due to the complex nature of the auction format and the infinite number of equilibria. Our study characterizes various equilibrium bidding behaviors in GSP auctions. We develop an algorithm to identify all pure-strategy Nash equilibria and discuss their distribution in the pure-strategy space. This equilibrium distribution can help advertisers formulate bidding strategies, and help search engines calculate their expected revenues.