Asymmetric Auctions Research Articles

Revenue Equivalence of Large Asymmetric Auctions

The paper analyzes the revenue of auctions with asymmetric bidders with a large, but finite number of players. We explicitly calculate the seller’s expected revenue in large asymmetric first-price, second-price, and optimal auctions to O(1/n 3) accuracy, where n is the number of players. These calculations show that the revenue differences among these three auction mechanisms scale as ∈2/n 3, where e is the level of asymmetry (heterogeneity) among the distributions of bidders’ valuations. This novel scaling law shows that bidders’ asymmetry already has a negligible effect on revenue ranking of auctions with several (e.g., n = 6) bidders. In contrast, previous results studied only the limiting case n → ∞. We also show that bidders’ asymmetry always reduces the expected revenue in large auctions, but not necessarily in small ones. Finally, we extend the asymptotic O(∈2/n 3) revenue equivalence to a broader class of asymmetric auctions.

An Experimental Investigation of Procurement Auctions with Asymmetric Sellers

Electronic reverse auctions are a commonly used procurement mechanism. Research to date has focused on suppliers who are ex ante symmetric in that their costs are drawn from a common distribution. However, in many cases, a seller's range of potential costs depends on their own operations, location, or economies of scale and scope. Thus, understanding how different bidder types impact auction outcomes is key when designing an auction. This study reports the results of the first controlled laboratory experiment designed to compare prices between first‐price and second‐price procurement auctions for homogeneous goods when seller cost types are asymmetric and the number of bidders varies. The results indicate that first‐price auctions generate lower prices regardless of market composition. The results also reveal that first‐price auctions are at least weakly more efficient than second‐price auctions despite the theoretical prediction that the reverse should hold in asymmetric auctions. Post hoc analysis of individual bidders' behavior in first‐price auctions revealed evidence that bidders systematically underbid when their cost realizations were close to the lower bound. Furthermore, bidders adjust their behavior based on the type of the other bidders in the market in a manner inconsistent with theory. Consequently, adding a third bidder to a two‐bidder market is not advantageous to the buyer unless that third bidder is a low‐cost type.

Incentive Mechanism Design for Heterogeneous Crowdsourcing Using All-Pay Contests

Many crowdsourcing scenarios are heterogeneous in the sense that, not only the workers’ types (e.g., abilities or costs) are different, but the beliefs (probabilistic knowledge) about their respective types are also different. In this paper, we design an incentive mechanism for such scenarios using an asymmetric all-pay contest (or auction) model. Our design objective is an optimal mechanism, i.e., one that maximizes the crowdsourcing revenue minus cost. To achieve this, we furnish the contest with a prize tuple which is an array of reward functions each for a potential winner. We prove and characterize the unique equilibrium of this contest, and solve the optimal prize tuple. In addition, this study discovers a counter-intuitive property, called strategy autonomy (SA), which means that heterogeneous workers behave independently of one another as if they were in a homogeneous setting. In game-theoretical terms, it says that an asymmetric auction admits a symmetric equilibrium. Not only theoretically interesting, but SA also has important practical implications on mechanism complexity, energy efficiency, crowdsourcing revenue, and system scalability. By scrutinizing seven mechanisms, our extensive performance evaluation demonstrates the superior performance of our mechanism as well as offers insights into the SA property.

Revenue Equivalence of Large Asymmetric Auctions

Revenue Equivalence of Large Asymmetric Auctions

Optimal dynamic mechanism design with deadlines

A seller maximizes revenue from selling an object in a dynamic environment, with buyers that differ in their patience: Each buyer has a privately known deadline for buying and a privately known valuation. First, we derive the optimal mechanism, neglecting the incentive constraint for the deadline. The deadline of the winner determines the time of the allocation and therefore also the amount of information available to the seller when he decides whether to allocate to a buyer. Depending on the shape of the markup that the seller uses, this can lead to a violation of the neglected incentive constraint. We give sufficient conditions on the type distribution under which the neglected constraint is fulfilled or violated. Second, for the case that the constraint cannot be neglected, we consider a model with two periods and two buyers. Here, the optimal mechanism is implemented by a fixed price in period one and an asymmetric auction in period two. The asymmetry, which is introduced to prevent the patient type of the first buyer from buying in period one leads to pooling of deadlines at the top of the type space.

An Asymmetric Job Auction Model

In this paper we investigate a labor market with search frictions. Workers with heterogeneous productivity apply to different types of job vacancies by bidding the optimal profits that they can offer the firms. In the steady state equilibrium, the most capable workers separate themselves by only bidding at jobs requiring high skills, while applicants in the middle of the productivity distribution pool with the low-productivity workers and apply for jobs requiring low skills.

Sequential Bidding in Asymmetric First Price Auctions

We study asymmetric first price auctions in which bidders place their bids sequentially, one after the other and only once. We show that with a strong bidder and a weak bidder (in terms of first order stochastic dominance of their valuations distribution function), when the asymmetry between the bidders is large enough the expected revenue in the sequential bidding first price auction (when the strong bidder bids first) is higher than in the simultaneous bidding first price auction as well as in the second price auction. The expected payoff of the weak bidder is also higher in the sequential first price auction. Therefore a seller interested in increasing revenue facing asymmetric bidders may find it beneficial to order them and let them bid sequentially instead of simultaneously. In terms of efficiency, both the simultaneous first price auction and the sequential first price auction cannot guarantee full efficiency (as opposed to a second price auction which guarantees full efficiency). The sequential bidding auction when the stronger bidder bids first achieves lower efficiency than the simultaneous auction. However, when the order is reversed and bidders are asymmetric enough the sequential first price auction achieves higher efficiency than the simultaneous one.

Asymmetry and revenue in second-price auctions: a majorization approach

Bidder asymmetries are common in certain auctions where heterogeneous bidders coexist. In this paper, we employ the tool of majorization to study the effect of bidder asymmetry on revenue in second-price auctions. As a preorder on vectors of real components, the majorization is used to rank the asymmetry of bid distributions from different auctions. Our analysis indicates that bidders’ asymmetries may raise revenue under certain classes of value distributions (e.g., proportional reversed hazard rate models) and may not in others (e.g., proportional hazard rate models). These results are further validated through supportive evidence that is based on simulated three-bidder auctions. The use of majorization allows us to extend and strengthen the results obtained in the existing literature under more general conditions. This study adds to the literature a useful analytical technique for future research on asymmetric auctions.

On the analysis of asymmetric first price auctions

We provide new tools for studying asymmetric first price auctions, connecting their equilibria to the ρ-concavity of the underlying type distributions, and showing how one can use surplus expressions for symmetric auctions to bound equilibrium behavior in asymmetric auctions. We apply these tools to studying procurement auctions in which, as is common in practice, one seller is given an advantage, reflecting for example better reliability or quality. We show conditions under which for any given first price handicap auction, there exists a superior second price auction with bonuses.

Are Structural Estimates of Asymmetric First-Price Auction Models Credible? Semi- and Nonparametric Verifications

Structural first-price auction estimation methods, built upon Bayesian Nash Equilibrium (BNE), have provided prolific empirical findings. However, due to the unobserved nature of underlying valuations, the assumption of BNE is not feasibly testable with field data, a fact that evokes harsh criticism of the literature. To respond to skepticism regarding credibility, we provide a focused answer by analyzing estimates derived from experimental asymmetric auction data in which researchers observe valuations. We test the statistical equivalence between the estimated and true value distributions. The Kolmogorov-Smirnov test fails to reject the distributional equivalence, strongly supporting the credibility of structural asymmetric auction estimates. The Supplement for this paper is available at the following URL: http://ssrn.com/abstract=2400104

Ranking asymmetric auctions: Filling the gap between a distributional shift and stretch

I consider first-price auctions (FPA) and second-price auctions (SPA) with two asymmetric bidders. The FPA is known to be more profitable than the SPA if the strong bidder's distribution function is convex and the weak bidder's distribution is obtained by truncating or horizontally shifting the former. In this paper, I employ a new mechanism design result to show that the FPA remains optimal if the weak bidder's distribution falls between the two benchmarks in a natural way. The same conclusion holds if the strong bidder's distribution is concave, but with a vertical shift replacing the horizontal shift. A result with a similar flavor holds if the strong bidder's distribution is neither convex nor concave. The dispersive order and the star order prove useful in comparing the weak bidder's distribution to the benchmarks. A key step establishes a relationship between these orders and reverse hazard rate dominance.

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.

Experimental evidence on the performance of emission trading schemes in the presence of an active secondary market

As auction based emission trading schemes (ETS) become more common in addressing climate change, it is of interest to study the effects of bargaining power in resale markets on original as well as post-resale allowance allocations in terms of prices and efficiency. This article provides an experimental study of first price private value asymmetric auctions followed by a -double auction resale market opportunity. We compare the -double auction to other resale regimes with an uneven distribution of market power and contrast initial bids, resale prices and efficiency. Despite the conventional wisdom that full efficiency requires the absence of market power, we find that -double auction resale markets lead to lower efficiency than the monopsony resale regime. The level of efficiency achieved, however, is close to the highest across mechanisms that have differential bargaining power at the resale stage.

Ranking asymmetric auctions

We compare the expected revenue in first- and second-price auctions with asymmetric bidders. We consider “close to uniform” distributions with identical supports and show that in the case of identical supports the expected revenue in second-price auctions may exceed that in first-price auctions. We also show that asymmetry over lower valuations has a stronger negative impact on the expected revenue in first-price auctions than in second-price auctions. However, asymmetry over high valuations always increases the revenue in first-price auctions.

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.

Adaptive learning in an asymmetric auction: genetic algorithm approach

Adaptive learning in an asymmetric auction: genetic algorithm approach

Partial identification in asymmetric auctions in the absence of independence

This paper examines identification in second‐price and ascending auctions within the private‐values framework. The first part of the paper considers an arbitrary type of dependence of bidders’ values and analyses identification under several observational scenarios, in which the highest bid is never observed. In a basic scenario, only the winner’s identity and the winning price are observed. The most informative is the scenario in which all the identities and all the bids except for the highest bid are known. Using results from Athey and Haile (2002), the joint distribution of bidders’ values in these scenarios is not identified. The paper uses the information available in auctions’ outcomes to construct bounds on the joint distribution of values for any subset of bidders. The second part of the paper takes a different tack by showing how bounds can be improved under different types of positive dependence of bidders’ values.

Revenue Comparison in Asymmetric Auctions with Discrete Valuations

AbstractWe consider an asymmetric auction setting with two bidders such that the valuation of each bidder has a binary support. First, we characterize the unique equilibrium outcome in the first price auction for any values of parameters. Then we compare the first price auction with the second price auction in terms of expected revenue. Under the assumption that the probabilities of low values are the same for the two bidders, we obtain two main results: (i) the second price auction yields a higher revenue unless the distribution of a bidder’s valuation first-order stochastically dominates the distribution of the other bidder’s valuation “in a strong sense” and (ii) introducing reserve prices implies that the first price auction is never superior to the second price auction. In addition, in some cases, the revenue in the first price auction decreases when all the valuations increase.

A NOTE ON THE EFFECT OF ASYMMETRY ON REVENUE IN SECOND-PRICE AUCTIONS

We compare the seller's expected revenue in asymmetric second-price auctions with the benchmark case where all bidders have the average distribution. We show that with two bidders, asymmetry has a negative effect on revenue. However, for n > 2 bidders there is no clear observation we can make. We prove that in the case of weak asymmetry, sellers prefer asymmetry over low valuations and symmetry over high valuations. In addition, we show that a good approximation for the expected revenue in the case of weak asymmetry can be obtained by calculating the revenue of the symmetric auction with identical distributions equal to the geometric or arithmetic average.

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.