Abstract
In this article, we study the problem of Nash implementation in private good economies with single-peaked, single-plateaued, and single-dipped preferences in the presence of at least one minimally honest agent. We prove that all solutions of the problem of fair division satisfying unanimity can be implemented in Nash equilibria as long as there are at least three agents participating in the mechanism (game form). To justify this result, we provide a list of solutions which violate the condition of no-veto power.
Acknowledgments
We would like to thank Michele Lombardi for his suggestions and helpful comments. We would also like to thank the seminar participants at Maastricht University in Netherland. We are particularly grateful to two anonymous referees of the journal for comments that greatly improved this article. Of course any error is our own responsibility.
References
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- 1
The strict weak no-veto power condition has been introduced by Doghmi and Ziad (2008a, 2008b). An SCC F satisfies strict weak no-veto power, if for i,
, and 
, for 
, 
and 
for all 
, then 
. - 2
The proof of the no-logical relationship between I-weak no-veto power and strict weak no-veto power is in Doghmi and Ziad (2013a).
- 3
For all
means that, for the agent i, to consume a share 
is as good as to consume the quantity 
. The asymmetrical part is written 
and the symmetrical part 
. - 4
The monotonic correspondences in our examples in the case of private good economies with single-peaked preferences are Pareto correspondence, no-envy correspondence, individually rational correspondence from equal division, and all intersections of them. For more details, see Thomson (1990, 2010).
- 5
In standard Nash implementation, Doghmi (2013b) and Doghmi and Ziad (2013b) proved that many important monotonic correspondences studied in private good economies with single-dipped preferences by Doghmi (2013a) become no-monotonic when we allow multiple best/worst indifferent allocations. Thus, these correspondences can be examined in partially honest environment.
©2013 by Walter de Gruyter Berlin / Boston
