Roberts Theorem with Neutrality: A Social Welfare Ordering Approach

arXiv:1003.1550v1 [cs.GT] 8 Mar 2010 Roberts’ Theorem with Neutrality: A Social Welfare Ordering Approach ∗ Debasis Mishra† and Arunava Sen ‡ May 25, 2018 Abstract We consider dominant strategy implementation in private values settings, when agents have multi-dimensional types, the set of alternatives is finite, monetary transfers are allowed, and agents have quasi-linear utilities. We show that any implementable and neutral social choice function must be a weighted welfare maximizer if the type space of every agent is an m-dimensional open interval, where m is the number of alternatives. When the type space of every agent is unrestricted, Roberts’ theorem with neutrality [23] becomes a corollary to our result. Our proof technique uses a social welfare ordering approach, commonly used in aggregation literature in social choice theory. We also prove the general (affine maximizer) version of Roberts’ theorem for unrestricted type spaces of agents using this approach. Keywords: Dominant strategy mechanism design; Roberts’ theorem; affine maximizers; social welfare ordering JEL Classification: D44. ∗We are extremely grateful to Sushil Bikhchandani, Juan Carlos Carbajal, Claude d’Aspremont, Mridu Prabal Goswami, Amit Goyal, Ron Lavi, Thierry Marchant, Herv´e Moulin, Arup Pal, David Parkes, Kevin Roberts, Maneesh Thakur, Rakesh Vohra, and seminar participants at the “Multidimensional Mechanism Design” workshop in Bonn, Universitat Aut`onoma de Barcelona, and Delhi School of Economics for their comments. †Corresponding Author. Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi - 110016. ‡Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi - 110016. 1 1 Introduction The well-known Gibbard-Satterthwaite [12, 26] Impossibility Theorem in mechanism design asserts that in unrestricted domains, every implementable social choice function which has at least three alternatives in its range, must be dictatorial. A crucial aspect of the unre- stricted domain assumption is that monetary transfers are not permitted. However, models where monetary transfers are admissible are very important. Both the auction setting and the standard public good model assume that agents can receive monetary transfers (either positive or negative) and that the underlying utility function of every agent is quasi-linear in money. This paper is a contribution to the literature which investigates the structure of social choice functions which can be implemented in dominant strategies in these settings. [28, 8, 13] showed that efficient social choice functions can be implemented by a unique family of transfer rules, now popularly known as Vickrey-Clarke-Groves (VCG) transfer schemes. Remarkably, when the domain is unrestricted (as in the Gibbard-Satterthwaite setup) and the range of the mechanism contains at least three alternatives, the only (dom- inant strategy) implementable social choice functions are affine maximizers. These social choice functions are generalizations of weighted efficiency rules. This result was proved by [23] in a seminal paper. It can be seen as the counterpart to the Gibbard-Satterthwaite theorem for quasi-linear utility environments. As in the literature without money, the literature with quasi-linear utility has since tried to relax various assumptions in Roberts’ theorem. [24] shows that a certain cycle monotonic- ity property characterizes dominant strategy implementable social choice functions. Though this characterization is very general - works for any domains and any set of alternatives (finite or infinite) - it is not as useful as the Roberts’ theorem since it does not give a functional form of the class of implementable social choice functions. Along the lines of [24], [2] and [25] have shown that a weak monotonicity property characterizes implementable social choice functions in auction settings, a severely restricted domain, when the set of alternatives is finite and the type space is convex 1. Again, the precise functional form of the implementable social choice functions are missing in these characterizations. A fundamental open question is the following: What subdomains allow for a functional form of implementable social choice functions? Several attempts have been made recently to simplify, refine, and extend Roberts’ theo- rem. Using almost the same structure and approach, [16] reduced the complexity of Roberts’ original proof. [11] also provide an alternate (modular) proof of Roberts’ theorem for unre- stricted domain. Building on the technique of [16], [5] extend Roberts’ theorem to continuous domains. Other proofs of Roberts’ theorem can be found in (for unrestricted domains) [15] 1See also [21] and [1]. [1] prove the converse of this result also. 2 and [29]. 1.1 Our Contribution Our paper contributes to the literature in two ways. First, we characterize restricted domains where the affine maximizer theorem holds in the presence of an additional assumption on social choice functions, that of

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