Semiparametric Quantile Models for Ascending Auctions With Asymmetric Bidders

Abstract

The article proposes a parsimonious and flexible semiparametric quantile regression specification for asymmetric bidders within the independent private value framework. Asymmetry is parameterized using powers of a parent private value distribution, which is generated by a quantile regression specification. As noted in Cantillon, this covers and extends models used for efficient collusion, joint bidding and mergers among homogeneous bidders. The specification can be estimated for ascending auctions using the winning bids and the winner’s identity. The estimation is in two stage. The asymmetry parameters are estimated from the winner’s identity using a simple maximum likelihood procedure. The parent quantile regression specification can be estimated using simple modifications of Gimenes. Specification testing procedures are also considered. A timber application reveals that weaker bidders have 30% less chances to win the auction than stronger ones. It is also found that increasing participation in an asymmetric ascending auction may not be as beneficial as using an optimal reserve price as would have been expected from a result of Bulow and Klemperer valid under symmetry.

Keywords:

• Ascending auctions
• Asymmetry
• Private values
• Quantile regression
• Seller expected revenue
• Two-stage quantile regression estimation

Supplement materials

The proofs of all the results of the article as well as a simulation exercise of the estimation procedure, details of the specification tests used in the empirical application and some additional tables not displayed in the main article can be found at a supplementary material available online.

Acknowledgments

Nathalie Gimenes thanks Ying Fan and Ginger Jin for very useful comments. The authors would like to thank three anonymous referees and the Editor Chris Hansen, who made many stimulating suggestions, in particular proposing to address specification issues. Many comments from seminar and conference participants have helped to improve the article.

Notes

1 Our approach carries over with minor modifications for other quantile semiparametric specifications, such as the exponential one $F^{-1}(\tau |X)=\hspace{0.17em}\text{exp}\hspace{0.17em}(X^{\prime }\gamma (\tau ))$.

2 It is, however, possible to identify α₁, strengthening Assumption 4 to ensure $V^{(1)}(·)$ exists and is strictly positive near 0. If so and setting $V_1(\tau )=V({\tau }^{1/{\alpha }_1})$, it holds that $V_1(\tau )-V_1(0)=V^{(1)}(0){\tau }^{1/{\alpha }_1}(1+o(1))$, showing that α₁ is identified from the lower tail of the $V_1(·)$. This is left for further research.

3 Alternatively, a fixed effects specification as in Example 1 can be used provided the fixed effects α_i can only take K unknown values ${\mu }_1,\dots ,{\mu }_K$, where K is the number of types.

4 Note also that the function $t\mapsto (1-N){\widehat{\Lambda }}_N{t}^{{\widehat{\Lambda }}_N-1}+{\sum }_{i=1}^N{\widehat{\Lambda }}_{N|i}{t}^{{\widehat{\Lambda }}_{N|i}-1}$ vanishes at t = 0 as long as ${\widehat{\Lambda }}_N>1$, and at 1, since ${\widehat{\Lambda }}_N={\sum }_{i=1}^N{\widehat{\lambda }}_i$ and ${\widehat{\Lambda }}_{N|i}={\widehat{\Lambda }}_N-{\widehat{\lambda }}_i$. This also suggests that the lower and upper tails have a moderate contribution in the expected revenue integral, at least for reasonable value of N.

5 For a vector, $X\le x$ means that $X_j\le x_j$ for all j.

6 See their Footnote 26. This was also pointed to us by an anonymous Referee. These auctions represent slightly less than 8% of the sample.

7 To see this, observe that the probability of selling is the probability that the maximum private value $V_{(N)}$ is above the reserve price R. The Markov inequality gives the bound $\mathbb{E}[V_{(N)}]/R$ for the latter. A proxy for $\mathbb{E}[V_{(N)}]/R$ is the nonstrategical revenue $\Pi (0)$ when the seller value is 0, suggesting to use the bound $\Pi (0)/R$ for the probability of selling.

8 The bootstrap 95% confidence intervals for the optimal reserve price have a larger length, between 12$and 14$for an optimal reserve price between 104$and 112$. As a matter of comparison, Coey et al.’s (2017) set identified confidence bounds for the seller revenue and optimal reserve price look huge, but they also allow for affiliated values.

9 As suggested in Coey, Larsen, and Sweeney (2019), a comparison between a strategical and nonstrategical expected revenue can be fruitful to the seller due to the costs that a policy of setting an optimal reserve price may impose in practice. Recent works have highlighted the asymmetric effects on seller’s revenue due to mistakes in choosing reserve prices (see, e.g., Kim (2013), Ostrovsky and Schwarz (2016), Coey, Larsen, and Sweeney (2019) and Gimenes (2017))

10 These authors also allow for entry decision but their estimate “indicate a moderate effect of selection.”

11 See Coey, Larsen, and Sweeney (2019) for a recent econometric application to entry exogeneity.

12 Tables D.1 and D.2 in Appendix D also report the revenues obtained for an estimation of a symmetric private value model as in Gimenes (2017). Interestingly violations of Bulow and Klemperer (1996) occur for N = 2, 3 but not for larger N.

13 The downwards sloping marginal revenue condition of Bulow and Klemperer (1996) requires that $-\frac{d}{dt}\left[V_i(t)(1-t)\right]=V(t^{1/{\lambda }_i})-(1-t)\frac{t^{1/{\lambda }_i-1}}{{\lambda }_i}{V}^{(1)}(t^{1/{\lambda }_i})$ increases with t. If $V^{(1)}(0)>0$ and $1/2<{\lambda }_i<1$, the leading term when t goes to 0 of the derivative of this function is $-(1/{\lambda }_i-1)\frac{t^{1/{\lambda }_i-2}}{{\lambda }_i}{V}^{(1)}(0)$ which is negative, so that the considered condition is not compatible with our estimation of λ_L.

Additional information

Funding

Nathalie Gimenes and Emmanuel Guerre acknowledge the British Academy and Newton Fund for generously funding this project (reference code: AF150085). This project was completed during Jayeetas PhD from Queen Mary University of London (QMUL) and she thankfully acknowledges generous funding from the School of Economics and Finance, QMUL.