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Bayesian Inference in Structural Second-Price Auctions
Stockholm University, Faculty of Social Sciences, Department of Statistics.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The aim of this thesis is to develop efficient and practically useful Bayesian methods for statistical inference in structural second-price auctions. The models are applied to a carefully collected coin auction dataset with bids and auction-specific characteristics from one thousand Internet auctions on eBay. Bidders are assumed to be risk-neutral and symmetric, and compete for a single object using the same game-theoretic strategy. A key contribution in the thesis is the derivation of very accurate approximations of the otherwise intractable equilibrium bid functions under different model assumptions. These easily computed and numerically stable approximations are shown to be crucial for statistical inference, where the inverse bid functions typically needs to be evaluated several million times.

In the first paper, the approximate bid is a linear function of a bidder's signal and a Gaussian common value model is estimated. We find that the publicly available book value and the condition of the auctioned object are important determinants of bidders' valuations, while eBay's detailed seller information is essentially ignored by the bidders. In the second paper, the Gaussian model in the first paper is contrasted to a Gamma model that allows intrinsically non-negative common values. The Gaussian model performs slightly better than the Gamma model on the eBay data, which we attribute to an almost normal or at least symmetrical distribution of valuations. The third paper compares the model in the first paper to a directly comparable model for private values. We find many interesting empirical regularities between the models, but no strong and consistent evidence in favor of one model over the other. In the last paper, we consider auctions with both private-value and common-value bidders. The equilibrium bid function is given as the solution to an ordinary differential equation, from which we derive an approximate inverse bid as an explicit function of a given bid. The paper proposes an elaborate model where the probability of being a common value bidder is a function of covariates at the auction level. The model is estimated by a Metropolis-within-Gibbs algorithm and the results point strongly to an active influx of both private-value and common-value bidders.

Place, publisher, year, edition, pages
Stockholm: Department of Statistics, Stockholm University , 2011. , 11 p.
Keyword [en]
Asymmetry, Bid function approximation, Common values, Gamma model, Gaussian model, Markov Chain Monte Carlo, Private values, Variable selection, Internet auctions
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
URN: urn:nbn:se:su:diva-57278ISBN: 978-91-7447-276-9OAI: oai:DiVA.org:su-57278DiVA: diva2:415150
Public defence
2011-06-10, hörsal 3, hus B, Universitetsvägen 10 B, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Epub ahead of print. Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Manuscript.

Available from: 2011-05-12 Created: 2011-05-05 Last updated: 2013-07-12Bibliographically approved
List of papers
1. Bayesian Inference in Structural Second-Price Common Value Auctions
Open this publication in new window or tab >>Bayesian Inference in Structural Second-Price Common Value Auctions
2011 (English)In: Journal of business & economic statistics, ISSN 0735-0015, E-ISSN 1537-2707, Vol. 29, no 3, 382-396 p.Article in journal (Refereed) Published
Abstract [en]

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. 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.

Keyword
Bid function approximation, eBay, Internet auctions, Likelihood inference, Markov chain Monte Carlo, Normal valuation, Variable selection
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-66979 (URN)10.1198/jbes.2011.08289 (DOI)000292316500005 ()
Note
authorCount :2Available from: 2011-12-28 Created: 2011-12-22 Last updated: 2013-07-12Bibliographically approved
2. Bayesian Inference in Structural Second-Price Auctions with Gamma Distributed Common Values
Open this publication in new window or tab >>Bayesian Inference in Structural Second-Price Auctions with Gamma Distributed Common Values
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Our paper explores possible limitations of the Gaussian model in Wegmann and Villani (2011) due to intrinsically non-negative values. The relative performance of the Gaussian model is compared to an extension of the Gamma model in Gordy (1998) within the symmetric second price common value model. A key feature in our approach is the derivation of an accurate approximation of the bid function for the Gamma model, which can be inverted and differentiated analytically. This is extremely valuable for fast and numerically stable evaluations of the likelihood function. The general MCMC algorithm in WV is utilized to estimate WV's eBay dataset from $1000$ auctions of U.S. proof coin sets, as well as simulated datasets from the Gamma model with different degrees of skewness in the value distribution. The Gaussian model fits the data slightly better than the Gamma model for the particular eBay dataset, which can be explained by the fairly symmetrical value distribution. The superiority of the Gamma to the Gaussian model is shown to increase for higher degrees of skewness in the simulated datasets.

National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-57274 (URN)
Available from: 2011-05-05 Created: 2011-05-05 Last updated: 2011-05-09Bibliographically approved
3. Bayesian Comparison of Private and Common Values in Structural Second-Price Auctions
Open this publication in new window or tab >>Bayesian Comparison of Private and Common Values in Structural Second-Price Auctions
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We compare the performance of the Gaussian second-price common value (CV) model in Wegmann and Villani (2011) to a comparable independent private value (IPV) version of that model. The two models are contrasted on a dataset from $1050$ Internet coin auctions at eBay. The models are evaluated along several dimensions, such as parameter inference, in-sample fit, and accuracy of out-of-sample predictive density forecasts. Both models fit the eBay data well with a slight edge for the more robust CV model. We do not find any evidence of a winner's curse effect in the eBay data, which speaks in favor of the IPV model. However, the optimal minimum bids in the CV model are clearly closer to the actual minimum bids in the eBay data than the optimal choice of no minimum bid in the IPV model. The IPV model predicts auction prices slightly better in most auctions, while the CV model is much better at predicting auction prices in more unusual auctions. The robustness of the CV model is also supported by a small simulation study, where the CV model performs relatively better on simulated data from the IPV model than the IPV model fitted to CV data.

National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-57275 (URN)
Available from: 2011-05-05 Created: 2011-05-05 Last updated: 2011-05-09Bibliographically approved
4. Bayesian Inference in Structural Second-Price Auctions with both Private-Value and Common-Value Bidders
Open this publication in new window or tab >>Bayesian Inference in Structural Second-Price Auctions with both Private-Value and Common-Value Bidders
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Auctions with asymmetric bidders have been actively studied in recent years. Tan and Xing (2011) show the existence of monotone pure-strategy equilibrium in auctions with both private-value and common-value bidders. The equilibrium bid function is given as the solution to an ordinary differential equation (ODE). We approximate the ODE and obtain a very accurate, approximate inverse bid as an explicit function of a given bid. This results in fast and numerically stable likelihood evaluations, which is an extremely valuable property for inference. We propose a model where the valuations of both common-value and private-value bidders are functions of covariates. The probability of being a common-value bidder is modeled by a logistic regression model with Bayesian variable selection. The model is estimated on a dataset of eBay coin auctions. We analyze the model using Bayesian methods implemented via a Metropolis-within-Gibbs algorithm. The posterior inference of the common-value part of the model is similar to the ones obtained from a model with only common-value bidders, whereas the private-value part of the model is more affected by the introduction of common-value bidders. There is on average a slightly larger probability for a bidder to be a common-value bidder, but this probability depends very little on the auction-specific covariates.

National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-57276 (URN)
Available from: 2011-05-05 Created: 2011-05-05 Last updated: 2011-05-09Bibliographically approved

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