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Journal of Quantitative Economics

01.03.2024 | Notes and Short Article

Collusion with Not-So-Secret Rings

verfasst von: Ceesay Muhammed

Erschienen in: Journal of Quantitative Economics

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Abstract

When collusion is analyzed for Independent private value auctions, it is implicitly assumed that ring presence is commonly known to colluding and non-colluding bidders. We drop this assumption and analyze a simple model of a first price Independent Private Value auction with uniformly distributed values where a single bidder knows privately of the existence of collusion by others. We show that this knowledge leads him to bid shading (weakly) in the first price auction compared to what he would have bid otherwise. This in turn yields the result that the second price auction dominates the first price auction in terms of seller revenue. This contrasts results from the literature showing that under our framework, when bidding is done while the presence of colluding bidders is common knowledge, the first price auction dominates the second price auction.

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Fußnoten
1
Several papers have documented this phenomenon, including (Kawai and Nakabayashi 2022; OFT 2009; Athey 2001; Baldwin et al. 1997; Asker 2010; Porter 2005; Pesendorfer 2000).
 
2
See Lebrun (1999)
 
3
Kaplan and Zamir (2012) derive closed form solutions for two bidder asymmetric auctions where bidders values are drawn from uniform distributions with different supports. Cheng (2006) derives for values drawn from distributions \(F(x)=x^{\alpha } \text { and } H(x)=(\frac{x}{\delta })^{\lambda }\), where \(\delta =\frac{\lambda (\alpha +1)}{\alpha (\lambda +1)}\), improving on Plum (1992) who analyzed the case for which \(\lambda =\alpha \) and \(\delta \) arbitrary. Other papers analyze bidding strategies via numerical methods or through a qualitative analysis of the resulting dynamical system. See for instance (Marshall et al. 1994; Gayle and Richard 2008; Fibich and Gavish 2011) and (Hubbard et al. 2015).
 
4
See for instance, Marshall et al. (1994)
 
5
Proposition 1. Kirkegaard (2012) Let \(F_{S} \text { and } F_{W}\) denote the distributions of the Strong and Weak bidder respectively such that \(\forall \) \(x<y\) in their common support, \( \frac{F_{S}(x)}{F_{S}(y)} < \frac{F_{W}(x)}{F_{W}(y)} \). If
$$\begin{aligned} \int _{x}^{F_{S}^{-1}(F_{W}(x))} (f_{W}(x)-f_{S}(\psi ))d\psi \ge 0 \end{aligned}$$
\(\forall x\), the seller’s revenue from the first price auction will be higher than the revenue from the second price auction. In our case, \(F_{S}(x)=x^{N-1}\), and \(F_{W}(x)=x\). So that
$$\begin{aligned} \int _{x}^{F_{S}^{-1}(F_{W}(x))} (f_{W}(x)-f_{S}(\psi ))d\psi = \int _{x}^{x^{\frac{1}{N-1}}} (1-(N-1)\psi ^{N-2})d\psi =0 \end{aligned}$$
implying that seller’s revenue is higher under the first price auction.
 
6
See Proposition 2.2 and Example 2.1 in Krishna (2009).
 
7
See Proposition 2.1 in Krishna (2009)
 
8
See Proposition 2.1 in Krishna (2009)
 
9
This is because \(\frac{2N^2 - N - 2}{4(N^2-1)} - \frac{(N+2)(N-1)}{2N (N+1)} = \underbrace{- \frac{(N-2)^2}{4N(N^2-1)}}_{<0}\)
 
Literatur
Asker, J. 2010. A study of the internal organization of a bidding cartel. American Economic Review 100 (3): 724–62.CrossRef
Athey, S. 2001. Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information. Econometrica 69 (4): 861–899.MathSciNetCrossRef
Baldwin, L.H., R.C. Marshall, and J. Richard. 1997. Bidder collusion at forest service timber sales. Journal of Political Economy 105 (4): 657–699.CrossRef
Ceesay, M. (2019) Collusion in Auctions: Secret vs Public Rings. PhD thesis, University of Naples Federico II.
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Fibich, N., and N. Gavish. 2011. Numerical simulations of asymmetric first-price auctions. Games and Economic Behavior 73: 479–495.MathSciNetCrossRef
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Kawai, K., and J. Nakabayashi. 2022. Detecting large-scale collusion in procurement auctions. Journal of Political Economy 130 (5): 1364–1411.CrossRef
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Marshall, R., and L. Marx. 2012. The Economics of Collusion. The MIT Press.CrossRef
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Porter, R.H. 2005. Detecting collusion. Review of Industrial Organization 26 (2): 147–167.CrossRef
Metadaten
Titel
Collusion with Not-So-Secret Rings
verfasst von
Ceesay Muhammed
Publikationsdatum
01.03.2024
Verlag
Springer India
Erschienen in
Journal of Quantitative Economics
Print ISSN: 0971-1554
Elektronische ISSN: 2364-1045
DOI
https://doi.org/10.1007/s40953-024-00387-w

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