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Bayesian Inference in Structural Second-Price Auctions with Gamma Distributed Common Values
Stockholm University, Faculty of Social Sciences, Department of Statistics.
(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: urn:nbn:se:su:diva-57274OAI: oai:DiVA.org:su-57274DiVA: diva2:415129
Available from: 2011-05-05 Created: 2011-05-05 Last updated: 2011-05-09Bibliographically approved
In thesis
1. Bayesian Inference in Structural Second-Price Auctions
Open this publication in new window or tab >>Bayesian Inference in Structural Second-Price Auctions
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
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:nbn:se:su:diva-57278 (URN)978-91-7447-276-9 (ISBN)
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

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