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Social Choice and Welfare

Erschienen in:

07.07.2021 | Original Paper

Optimal revenue-sharing mechanisms with seller commitment to ex-post effort

verfasst von: Xun Chen, Shanmin Li, Dazhong Wang

Erschienen in: Social Choice and Welfare | Ausgabe 1/2022

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Abstract

This study considers the characterization and implementation of the optimal revenue-sharing mechanism when the seller’s ex-post effort affects the final outcome. In the optimal revenue-sharing mechanism with the seller’s full commitment to effort exertion, the seller commits to the first-best effort. Under the regularity condition guaranteed by the assumption of log-concave density, the optimal mechanism selects the bidder with the highest type, provided that the associated virtual surplus with the seller’s commitment to the first-best effort is positive. A first-price share auction with an appropriate reserve share and an effort commitment scheme can implement the optimal revenue-sharing mechanism. However, a second-price share auction might fail for implementation, such as in a case with two bidders and a uniform type distribution. Lastly, we introduce a sealed-bid share auction called the first-second price share auction, which can implement the optimal mechanism in dominant strategy.

Vorheriger Artikel Does the approval mechanism induce the efficient extraction in common pool resource games?

Nächster Artikel John Stuart Mill, soft paternalist

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We say a bidding strategy is (quasi) strictly increasing if it is strictly increasing in all participating types.

The multiplicative form captures the complementarity between the operator’s productivity and the seller’s input. Kogan and Morgan (2010) adopt a similar specification.

See Riordan and Sappington (1988) and DeMarzo et al. (2005).

The assumption regarding the quadratic form of disutility does not cause a loss of economic insight and is only for expositional ease. Our results could be extended to general convex forms of disutility.

Without confusion, “effort” here has a general meaning, including input, investment, capacity, (see, Dixit 1987).

See DeMarzo et al. (2005) for the argument regarding limited liability. The full-rent-extracting mechanism proposed by Riordan and Sappington (1988) generally violates the limited liability constraint.

The term “full commitment” means that the seller commits to an allocation rule, an effort commitment scheme, and a revenue-sharing rule. Without full commitment, the seller’s sequential rationality leads to moral hazard issues, and the revelation principle fails (see, Bester and Strausz 2001; Doval and Vasiliki 2020). We will consider this issue extensively in a future study.

See Bagnoli and Bergstrom (2005).

A bidding strategy for a bidder is (quasi) strictly increasing if a participating type’s bid specified by this strategy increases as the type rises.

Since \(J^*(\theta _r) \ge 0\), it holds that \(\frac{\theta _r^2}{\uplambda k} \ge 2+ \frac{2}{\theta _r}\cdot \frac{1-F(\theta _r)}{f(\theta _r)}\cdot \frac{\int ^{{\overline{\theta }}}_{{\theta }_r}\theta f(\theta ) d\theta }{\theta _r [1-F(\theta _r)]}\), thus we have \(s_r \in (1/2, 1)\).

In any equilibrium with the symmetric (quasi) strictly increasing bidding strategy of the proposed first-price auction, each bidder i with type \(\theta _i \ge \theta _r\) earns an equilibrium expected payoff \(\pi _i(\beta (\theta _i), \theta _i) = k\theta _i \int _{\theta _r}^{\theta _i} \frac{g(\tau )}{\tau }d\tau\) obtained by Condition IC2 and the zero-payoff argument for type \(\theta _r\). Accordingly, the equilibrium bid for type \(\theta _i\) must be \(\beta (\theta _i) = 1-\frac{\uplambda k}{\theta _i^2}-\frac{\uplambda k}{G(\theta _i)\theta _i}\int _{\theta _r}^{\theta _i}\frac{G(\tau )}{\tau ^2}d\tau = \beta ^{I}(\theta _i)\).

As \(\epsilon <1\), the type threshold \(\epsilon< \theta _r = \sqrt{\frac{\epsilon + \sqrt{\epsilon ^2 + 4\epsilon (1+\epsilon )^2}}{2}} < 2\epsilon ^{\frac{1}{4}}\). Thus, \(\frac{ 2 \epsilon }{\theta _r} - \ln \frac{\theta }{\theta _r}< \epsilon ^{\frac{3}{4}} - 2 \ln \frac{\theta }{2\epsilon ^{\frac{1}{4}}} < 1 - 2 \ln \frac{\theta }{2\epsilon ^{\frac{1}{4}}}\).

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Liu T, Bernhardt D (2021) Rent extraction with securities plus cash. J Financ (Forthcoming)

Liu T (2016) Optimal equity auctions with heterogeneous bidders. J Econ Theory 166:94–123CrossRef

Liu T, Bernhardt D (2019) Optimal equity auctions with two-dimensional types. J Econ Theory 184:104913CrossRef

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Titel: Optimal revenue-sharing mechanisms with seller commitment to ex-post effort
verfasst von: Xun Chen
Shanmin Li
Dazhong Wang
Publikationsdatum: 07.07.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 1/2022
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-021-01351-w

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