Asymmetric First-price Auctions Research Articles

Numerical Solution of Asymmetric Auctions

We propose the backward indifference derivation (BID) algorithm, a new method to numerically approximate the pure strategy Nash equilibrium (PSNE) bidding functions in asymmetric first-price auctions. The BID algorithm constructs a sequence of finite-action PSNE that converges to the continuum-action PSNE by finding where bidders are indifferent between actions. Consequently, our approach differs from prevailing numerical methods that consider a system of poorly behaved differential equations. After proving convergence (conditional on knowing the maximum bid), we evaluate the numerical performance of the BID algorithm on four examples, two of which have not been previously addressed.

Assessment of Collusion Damages in First Price Auctions

We propose a structural method for estimating the revenue losses associated with bidding rings in symmetric and asymmetric first-price auctions. It is based on the structural analysis of auction data and is consistent with antitrust damage assessment methodologies: we build a but-for (competitive) scenario and estimate the differences between the two scenarios. We show in a Monte Carlo exercise that our methodology performs very well in moderate size samples. We apply it to Ohio Milk Data Set analyzed by Porter and Zona (1999) and find that damages are around 7%. Damages can be assessed without any information about unaffected markets.

Multiple equilibria in asymmetric first-price auctions

Maskin and Riley (Games Econ Behav 45:395-409, 2003) and Lebrun (Games Econ Behav 55:131-151, 2006) prove that the Bayes-Nash equilibrium of first-price auctions is unique. This uniqueness requires the assumption that a buyer never bids above his value (which amounts to the elimination of weakly dominated strategies). We demonstrate that, in asymmetric first-price auctions (with or without a minimum bid), the relaxation of this assumption results in additional equilibria that are substantial. Although in each of these additional equilibria no buyer wins with a bids above his value, the allocation of the object and the selling price may vary among the equilibria. In particular, we show that these yield higher revenue. We show that such phenomena can only occur under certain types of asymmetry in the distributions of values.

Are Structural Estimates of Asymmetric First-Price Auctions Credible? Semi- & Nonparametric Verifications: Supplemental Document

This online appendix contains figures, tables, comments, and details of payoff functions, that supplement the main article, Are Estimates of Asymmetric First-Price Auctions Credible? Semi- and Nonparametric Analyses.

An elementary proof of the common maximal bid in asymmetric first-price and all-pay auctions

We prove that the maximal bid in asymmetric first-price and all-pay auctions is the same for all bidders. Our proof is elementary, and does not require that bidders are risk neutral, or that the distribution functions of their valuations are independent or smooth.

Adaptive Learning in an Asymmetric Auction: Genetic Algorithm Approach

Agent-based simulations are performed to study adaptive learning in the context of asymmetric first-price auctions. Non-linearity of the Nash equilibrium strategies is used to investigate the effect of task complexity on adaptive learning by varying the degree of approximation the agents can handle. In addition, learning in different information environments is explored. Social learning allows agents to imitate each other's bidding strategies based on their relative success. Under individual learning agents are limited to their own experience. We observe convergence to steady states near the predicted equilibrium in all cases. The ability to learn non-linear functions helps the agents with a non-linear equilibrium strategy but hurts the agents with an almost linear one. Better information about the opponent population has a relatively modest impact. A larger number of strategies to experiment with and an ability to systematically compare strategies by holding a number of factors constant have a comparatively stronger beneficial effect.

Testing for collusion in asymmetric first-price auctions

This paper proposes a two step procedure to detect collusion in asymmetric first-price procurement (auctions). First, we use a reduced form test to short-list bidders whose bidding behavior is at-odds with competitive bidding. Second, we estimate the (latent) cost for these bidders under both competition and collusion setups. Since for the same bid the recovered cost must be smaller under collusion—as collusion increases the mark-up—than under competition, detecting collusion boils down to testing for first-order stochastic dominance, for which we use the classic Kolmogorov–Smirnov and Wilcoxon–Mann–Whitney tests. Our bootstrap based Monte Carlo experiments for asymmetric bidders confirm that the procedure has good power to detect collusion when there is collusion. We implement the tests for highway procurement data in California and conclude that there is no evidence of collusion even though the reduced form test supports collusion. This highlights potential pitfalls of inferring collusion based only on reduced form tests.

Using Economic Theory to Guide Numerical Analysis: Solving for Equilibria in Models of Asymmetric First-Price Auctions

In models of first-price auctions, when bidders are ex ante heterogeneous, deriving explicit equilibrium bid functions is typically impossible, so numerical methods are often employed to find approximate solutions. Recent theoretical research concerning asymmetric auctions has determined some properties these bid functions must satisfy when certain conditions are met. Plotting the relative expected pay-offs of bidders is a quick, informative way to decide whether the approximate solutions are consistent with theory. We approximate the (inverse-)bid functions by polynomials and employ theoretical results in two ways: to help solve for the polynomial coefficients and to evaluate qualitatively the appropriateness of a given approximation. We simulate auctions from the approximated solutions and find that, for the examples considered, low-degree polynomial approximations perform poorly and can lead to incorrect policy recommendations concerning auction design, suggesting researchers need to take care to obtain quality solutions.

Testing for Collusion in Asymmetric First-Price Auctions

This paper proposes a two step procedure to detect collusion in asymmetric first-price procurement (auctions). First, we use a reduced form test to short-list bidders whose bidding behavior is at-odds with competitive bidding; and Second, we estimate the (latent) cost for these bidders under both competition and collusion setups. Since, for the same bid the recovered cost must be smaller under collusion- as collusion increases the mark-up- than under competition, detecting collusion boils down to testing for first-order stochastic dominance, for which we use the classic Kolmogorov-Smirnov and Wilcoxon-Mann-Whitney tests. Our Bootstrap based Monte Carlo experiments for asymmetric bidders confirm that the procedure has good power to detect collusion when there is collusion. We implement the tests for Highway Procurement data in California and conclude that there is no evidence of collusion even though the reduced form test supports collusion. This highlights potential pitfalls of inferring collusion based only on reduced form tests.

Numerical simulations of asymmetric first-price auctions

The standard method for computing the equilibrium strategies of asymmetric first-price auctions is the backward-shooting method. In this study we show that the backward-shooting method is inherently unstable, and that this instability cannot be eliminated by changing the numerical methodology of the backward solver. Moreover, this instability becomes more severe as the number of players increases. We then present a novel boundary-value method for computing the equilibrium strategies of asymmetric first-price auctions. We demonstrate the robustness and stability of this method for auctions with any number of players, and for players with mixed types of distributions, including distributions with more than one crossing. Finally, we use the boundary-value method to study large auctions with hundreds of players, to compute the asymptotic rate at which large first-price and second-price auctions become revenue equivalent, and to study auctions in which the distributions cannot be ordered according to first-order stochastic dominance.

Asymmetric first-price auctions with uniform distributions: analytic solutions to the general case

In 1961, Vickrey posed the problem of finding an analytic solution to a first-price auction with two buyers having valuations uniformly distributed on (v 1 , v1) and (v 2 , v2).Todate,onlyspecialcasesoftheproblemhavebeensolved.Inthispaper, we solve this general problem and in addition allow for the possibility of a binding minimum bid. Several interesting examples are presented, including a class where the two bid functions are linear.

Asymmetric First-Price Auctions—A Dynamical-Systems Approach

We introduce a new approach for analysis and numerical simulations of asymmetric first-price auctions, which is based on dynamical systems. We apply this approach to asymmetric auctions in which players' valuations are power-law distributed. We utilize a dynamical-systems formulation to provide a proof of the existence and uniqueness of the equilibrium strategies in the cases of two coalitions and of two types of players. In the case of n different players, the singular point of the original system at b = 0 corresponds to a saddle point of the dynamical system with n − 1 admissible directions. This insight enables us to use forward solutions in the analysis and in the numerical simulations, in contrast with previous analytic and numerical studies that used backward solutions. The dynamical-systems approach provides an intuitive explanation for why the standard backward-shooting method for computing the equilibrium strategies is inherently unstable, and enables us to devise a stable forward-shooting method. In particular, in the case of two types of players, this method is extremely simple, as it does not require any shooting.

Asymmetric First-Price Auctions: A Dynamical Systems Approach

We introduce a new approach for analysis and numerical simulations of asymmetric first-price auctions, which is based on dynamical systems. We apply this approach to asymmetric auctions in which players’ valuations are power-law distributed, i.e., Fi(v) = v^{α_i} , i = 1, · · · , n. We utilize the dynamical-systems formulation to provide a simple proof of the existence and uniqueness of the equilibrium strategies in the case of two types of players. In the case of n types of players, the singularity point of the original system corresponds to a saddle point with n − 1 admissible directions of the dynamical system. This relation enables us to use forward solutions in the analysis and in the numerical simulations, in contrast with previous analytic and numerical studies which used backward solutions. The dynamical systems approach provides an intuitive explanation for why the standard backward-shooting method for computing the equilibrium strategies is inherently unstable, and enables us to devise a stable forward-shooting method. In particular, in the case of two types of players this method is extremely simple, as it does not require any shooting.

Numerical Simulations of Asymmetric First-Price Auctions - discussion paper

The standard method for computing the equilibrium strategies of asymmetric first-price auctions is the backward-shooting method. In this study we show that the backward-shooting method is inherently unstable, and that this instability cannot be eliminated by changing the numerical methodology of the backward solver. Moreover, this instability becomes more severe as the number of players increases. We then present a novel boundary-value method for computing the equilibrium strategies of asymmetric first-price auctions. We demonstrate the robustness and stability of this method for auctions with any number of players, and for players with mixed types of distributions, including both common distributions and distributions with more than one crossings. Finally, we use the boundary-value method to study large auctions with hundreds of players, to compute the rate at which large auctions first-price and second-price auctions become revenue equivalent, and to study auctions in which the distributions cannot be ordered according to first-order stochastic dominance.

Pre-Auction Offers in Asymmetric First-Price and Second-Price Auctions

We consider must-sell auctions with asymmetric buyers. First, we study auctions with two asymmetric buyers, where the distribution of valuations of the strong buyer is stretched relative to that of the weak buyer. Then, it is known that inefficient first-price auctions are more profitable for the seller than efficient second-price auctions. This is because the former favor the weak buyer. However, we show that the seller can do one better by augmenting the first-price auction by a pre-auction offer made exclusively to the strong buyer. Should the strong buyer reject the offer, the object is simply sold in an ordinary first-price auction. The result is driven by the fact that the unmodified first-price auction is too favorable to the weak buyer, and that the pre-auction offer allows some correction of this to the benefit of the seller. Secondly, we show quite generally that pre-auction offers never increase the profitability of second-price auctions, since they introduce the wrong kind of favoritism from the perspective of seller profits.

Asymmetric First-Price Auctions With Uniform Distributions: Analytic Solutions to the General Case

While auction research, including asymmetric auctions, has grown significantly in recent years, there is still little analytical solutions of first-price auctions outside the symmetric case. Even in the uniform case, Griesmer et al. (1967) and Plum (1992) find solutions only to the case where the lower bounds of the two distributions are the same. We present the general analytical solutions to asymmetric auctions in the uniform case for two bidders, both with and without a minimum bid. We show that our solution is consistent with the previously known solutions of auctions with uniform distributions. Several interesting examples are presented including a class where the two bid functions are linear. We hope this result improves our understanding of auctions and provides a useful tool for future research in auctions.

Monotonicity in asymmetric first-price auctions with affiliation

I study monotonicity of equilibrium strategies in first-price auctions with asymmetric bidders, risk aversion, affiliated types, and interdependent values. Every mixed-strategy equilibrium is shown to be outcome-equivalent to a monotone pure-strategy equilibrium under the “priority rule” for breaking ties. This provides a missing link to establish uniqueness in the “general symmetric model” of Milgrom and Weber (Econometrica 50:1089–1122, 1982). Non-monotone equilibria can exist under the “coin-flip rule” but they are distinguishable: all non-monotone equilibria have positive probability of ties whereas all monotone equilibria have zero probability of ties. This provides a justification for the standard empirical practice of restricting attention to monotone strategies.

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.

Asymmetric First-Price Auctions—A Perturbation Approach

We use perturbation analysis to obtain explicit approximations of the equilibrium bids in asymmetric first-price auctions with n bidders, in which bidders' valuations are independently drawn from different distribution functions. Several applications are presented: explicit approximations of the seller's expected revenue, the maximal bid, the optimal reserve price, inefficiency, and a consequence of stochastic dominance. We also suggest an improved numerical method for calculating the seller's expected revenue.

All Mixed Strategy Equilibria are Monotone Pure Strategy Equilibria in Asymmetric First-Price Auctions

Every mixed strategy equilibrium is outcome equivalent to a monotone pure strategy equilibrium in asymmetric first-price auctions in which n bidders have affiliated, one-dimensional, atomless types and interdependent values.