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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Notes
See Lebrun (1999)
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).
See for instance, Marshall et al. (1994)
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.
See Proposition 2.2 and Example 2.1 in Krishna (2009).
See Proposition 2.1 in Krishna (2009)
See Proposition 2.1 in Krishna (2009)
This is because \(\frac{2N^2 - N - 2}{4(N^2-1)} - \frac{(N+2)(N-1)}{2N (N+1)} = \underbrace{- \frac{(N-2)^2}{4N(N^2-1)}}_{<0}\)
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This work is based on parts of Chapter 2 of my PhD Thesis (Ceesay 2019) at the University of Naples Federico II, and my poster (available at https://siecon3-607788.c.cdn77.org/sites/siecon.org/files/media_wysiwyg/ceesay.pdf Accessed: 2021-08-03) at the 60th meeting of the Italian Economic Society in 2019. Earlier appeared under the title Suspecting Collusion.
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Muhammed, C. Collusion with Not-So-Secret Rings. J. Quant. Econ. (2024). https://doi.org/10.1007/s40953-024-00387-w
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DOI: https://doi.org/10.1007/s40953-024-00387-w