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Social Choice and Welfare
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07-07-2021 | Original Paper

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

Authors: Xun Chen, Shanmin Li, Dazhong Wang

Published in: Social Choice and Welfare | Issue 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.
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Appendix
Available only for authorised users
Footnotes
1
We say a bidding strategy is (quasi) strictly increasing if it is strictly increasing in all participating types.
 
2
The multiplicative form captures the complementarity between the operator’s productivity and the seller’s input. Kogan and Morgan (2010) adopt a similar specification.
 
3
See Riordan and Sappington (1988) and DeMarzo et al. (2005).
 
4
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.
 
5
Without confusion, “effort” here has a general meaning, including input, investment, capacity, (see, Dixit 1987).
 
6
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.
 
7
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.
 
8
See Bagnoli and Bergstrom (2005).
 
9
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.
 
10
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 \(s_r \in (1/2, 1)\).
 
11
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 \(\beta (\theta _i) = 1-\frac{\uplambda k}{\theta _i^2}-\frac{\uplambda k}{G(\theta _i)\theta _i}\int _{\theta _r}^{\theta _i}\frac{G(\tau )}{\tau ^2}d\tau = \beta ^{I}(\theta _i)\).
 
12
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, \(\frac{ 2 \epsilon }{\theta _r} - \ln \frac{\theta }{\theta _r}< \epsilon ^{\frac{3}{4}} - 2 \ln \frac{\theta }{2\epsilon ^{\frac{1}{4}}} < 1 - 2 \ln \frac{\theta }{2\epsilon ^{\frac{1}{4}}}\).
 
Literature
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Metadata
Title
Optimal revenue-sharing mechanisms with seller commitment to ex-post effort
Authors
Xun Chen
Shanmin Li
Dazhong Wang
Publication date
07-07-2021
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 1/2022
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-021-01351-w

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