DOI 10.1515/mel-2013-0019 | Math. Econ. Lett. 2013; 1(2–4): 69–74 Ahmed Doghmi* Nash Implementation in Rationing Problems with Single-Crossing Preferences Abstract: In this paper we study the rationing problems in using the issue of Nash implementation in an envi- ronment of single-crossing preferences. We show that strict monotonicity () implies strict weak no-veto power and unanimity and () is equivalent to Maskin monotonicity, which is vacuously checked in this domain. We show that any social choice correspondence that has full range can be implemented in Nash equilibria. Keywords: Nash implementation, Rationing problem, Single-crossing preferences. JEL: C72, D71 || *Corresponding Author: Ahmed Doghmi, University of Rabat, Mohamma- dia School of engineering, the QSM Laboratory, Avenue Ibnsina B. P. 765 Agdal, 10100 Rabat, Morocco; and University of Caen, Center for Research in Economics and Management (UMR CNRS 6211), 19 Rue Claude Bloch 14032 Caen, Cedex, France; and National Institute of Statistics and Applied Economics, Madinat Al Irfane, Rabat Institutes, 10100 Rabat, Morocco, e-mail: ahmeddoghmi@hotmail.com 1 Introduction Since the work of [1], the domain of single-peaked preferences has been used in order to explore many impos- sibility results like Arrow’s impossibility theorem and the existence of voting equilibria. In the same objec- tive, other domains restrictions have been emerged, we cite in particular the single-dipped domain of [2], the domain of single-plateaued preferences and other sup-domains of single-peaked preferences.¹ Nevertheless, there exists a number of interesting models where none of these domains can be held. This is due to some lim- itations which are, for example, related either to the behavior of agents or to the dimension of spaces. These domains restrictions have only a property of preferences, and so they neglect the character of the agents’ heterogeneity to make a choice. Moreover, these domains do not work in multidimensional policies while the political economy is interesting with these policies. To resolve these problems, other restrictions on individual preferences have been proposed, the very known are single-crossing preferences. This preference domain has been exploited in many political-economic settings by several authors like [19], [20], [21], and [22] in order to state the problem of income taxation and redistribution, and to guarantee the existence of majority vot- ing equilibria in one-dimensional space and the non-manipulable choice rules in multi-dimensional choice spaces. In implementation theory, there are few works that have explored this domain restriction. In his celebrate paper, [23, 24] pointed out that Maskin monotonicity is vacuously checked by any social choice correspon- dence in a single-crossing environment. But, He has not provided a characterizing result in this area. [25] studied the implementation problem in principal-agent models of adverse selection in the single-crossing environments. They established the relationship between Maskin and Bayesian monotonicity and the single- crossing property which are necessary for Nash and Bayesian implementation. They showed that, with com- 1 For more details, see for example [3–18]. Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 7/18/15 8:48 PM