Elsevier

Economics Letters

Volume 122, Issue 3, March 2014, Pages 380-385
Economics Letters

An impossibility result for virtual implementation with status quo

https://doi.org/10.1016/j.econlet.2013.12.015Get rights and content

Highlights

  • We study the implementation of social choice functions (SCFs) on unrestricted domain.

  • We use virtual implementation mechanisms restricted to a fixed outcome (a status quo).

  • On the unrestricted domain, unanimity and monotonicity lead to nearly dictatorial SCFs.

Abstract

In this paper, virtual implementation is restricted to deliver, on the equilibrium path, either a socially optimal outcome or a status quo: an outcome fixed for all preference profiles. Under such a restriction, for any unanimous and implementable social choice function there is a dictator, who obtains her most preferable outcome as long as all agents prefer this outcome to the status quo. Further restrictions on the lottery space and the range of social choice functions allow the dictator to impose her most preferred outcome even when other agents prefer the status quo to this outcome.

Introduction

Suppose that a society is interested in devising a mechanism to implement a rule that prescribes a single socially optimal outcome in each state of the world. We call such a rule a social choice function (SCF). If an SCF needs to cover all possible states of the world, the following impossibility result arises: with at least three alternatives, an SCF that is unanimous and exactly implementable has to be dictatorial. However, if the designer is allowed to implement an SCF approximately, any SCF is implementable.

This paper studies a virtual implementation problem when the designer is restricted on how she can approximate an SCF: instead of using an arbitrary outcome, she is restricted to using a single fixed outcome, which we call status quo. Under this restriction, unanimity and implementability again imply a form of dictatorship.

The key condition for exact implementation is Maskin-monotonicity (Maskin, 1999). Suppose that we are interested in implementing an SCF that prescribes an alternative for every possible profile of preferences (unrestricted domain of preferences); in such a case, the only implementable SCFs are those that prescribe the same alternative for every preference profile (Saijo, 1987). This constancy result relies on the profiles in which an agent is indifferent to the alternatives. If only strict preferences are considered, some non-constant SCFs can also be implemented. Suppose that, in addition to Maskin-monotonicity, we require unanimity: whenever all agents agree on the most desirable outcome, an SCF chooses that outcome. SCFs that are unanimous and Maskin-monotonic are once again restricted to a narrow class of dictatorial SCFs (Muller and Satterthwaite, 1977).1 These are the SCFs in which the preferences of one individual (a dictator) fully determine social outcomes.

One way to avoid this negative result is to introduce a mild relaxation of the implementability requirement. Suppose that society is willing to tolerate a mechanism that delivers, on the equilibrium path, a non-socially optimal outcome with small probability. This relaxation, called virtual implementation, is extremely powerful in evading the restriction to dictatorial rules: it allows the implementation of nearly every SCF (Abreu and Sen, 1991, Matsushima, 1988). However, it comes with a caveat: it places no restrictions on what non-socially optimal outcomes can be delivered on the equilibrium path. It is possible that some of these outcomes are socially intolerable.

A natural response to this problem is to restrict the type of outcomes that can be delivered on the equilibrium path. Bochet and Maniquet (2010) study this problem. They call the outcomes that are admissible as equilibrium outcomes an “admissible support” (an admissible outcome could be different for two different preference profiles). This formulation includes both exact implementation and virtual implementation. To obtain the former, one restricts an admissible support to the socially optimal outcome only. To obtain the latter, an admissible support includes every alternative.

Bochet and Maniquet (2010) derive a necessary and sufficient condition for virtual implementation with admissible support and provide several examples of correspondences that are not exactly implementable, but virtually implementable with a natural (and small) admissible support. They further show how to construct, for any given SCF, a minimal admissible support that allows the implementation of this SCF.

In this paper, we examine virtual implementation with admissible support where the support is restricted to two outcomes only: a socially optimal outcome and a fixed outcome, which we interpret as a status quo. That is, on the equilibrium path, the mechanism can either “succeed”, with large probability, and result in a socially optimal outcome, or “fail”, with small probability, and result in the status quo. We call this virtual implementation with status quo.

We show that virtual implementation with status quo escapes the constancy result of Saijo (1987), even though we allow for indifferences. Yet, the negative result of Muller and Satterthwaite (1977) partially extends to our notion of implementation. We show that the dictator is able to impose her preferences on society as long as everyone is better off with her choice than with the status quo.

Although we are using the framework of Bochet and Maniquet (2010), we are not able to exploit the characterization that they obtain for implementation with admissible support. The reason for this is that Bochet and Maniquet (2010) do not assume expected utility (assuming instead the much weaker notion of monotonicity in probabilities) and, without loss of generality, restrict their attention to ordinal mechanisms. Since we assume expected utility in this paper, the restriction to ordinal mechanism is untenable (e.g., see a discussion after Lemma 3). Therefore, we re-derive the necessary and sufficient condition. By doing so, we are effectively applying insights from Benoit and Ok (2008), Bochet (2007) and Sanver (2006) to the present paper.

Some of the proof techniques of this paper are adopted from Serrano (2004), who, in turn, adopts the proofs of Reny (2001) and Mas-Colell et al. (1995). However, as the domain of preferences considered in this paper is restricted by the assumption that preferences over lotteries can be represented by expected utility, the proofs of this paper are not implied by this earlier work. The impossibility results have been extended to lottery space in Barberà et al. (1998) and Benoit (2002), but both papers consider problems different from ours.

In the next section, we present the notation and state properties of an SCF. In Section  3, we derive a monotonicity condition relevant for the present paper. In Section  4, we discuss the constancy result. In Section  5, we collect supplementary lemmas. In Section  6, we show our main result that monotonicity and unanimity imply a q-constrained dictatorship. In Section  7, we outline a stronger result when the lottery space and the range of SCFs are further restricted. We conclude in Section  8.

Section snippets

Preliminaries

Let A{q} be the finite set of outcomes, where q is an outcome designated as the status quo. Let |A|=k. Let N be the finite set of agents, and |N|=n.

Let the lottery space be L=Δ(A{q}), where Δ is the probability simplex. We assume that preferences over lotteries satisfy standard assumptions (completeness, transitivity, continuity, and independence), so that they allow the standard von Neumann–Morgenstern expected utility representation.

Denote the set of possible profiles by Θ, with Θk×n; a

A monotonicity condition

Bochet and Maniquet (2010) introduce a notion of virtually Nash implementable SCF with admissible support, which we use in the present paper. We provide the relevant definitions from their paper, adopted to our settings, and refer the reader to Bochet and Maniquet (2010) for further details and discussion.

An SCF fϵ is ϵ-close to f if, for any θΘ,(a,p)=f(θ)(a,p)=fϵ(θ)A×[0,1], and |pp|<ϵ.

Definition 4

An SCF f is virtually implementable with q-admissible support {f(θ),q} if for all ϵ>0 there exists a

Avoiding the constancy result

In defining the domain of preferences Π+, we allow agents to be indifferent between two or more alternatives. Saijo (1987) shows that if the unrestricted domain of preferences allows indifferences, then Maskin-monotonicity implies that an SCF is constant. This observation can be illustrated by a simple example. Suppose that preference profiles θ,ϕΘ are such that (i) f(θ)=a,f(ϕ)=b; (ii) there is an agent iN, such that a=a(k,i,θ),b=a(k,i,ϕ); and (iii) all other agents’ preferences are identical

Supporting lemmas

This section contains the lemmas that catalog the preference changes performed while proving the main result. Lemma 2, Lemma 3 are trivial and their proofs are omitted; however, they are stated to make the main proof more concise.

Lemma 1 states that by improving a social choice outcome in someone’s preferences while keeping everyone else’s preferences constant, the social choice outcome cannot change.

Lemma 1

Suppose that, for some θΠ++,f(θ)=(a,p) . Suppose that ϕΠ++ is such that:

  • for some i,ui(a;ϕ)>ui

Main result

Lemma 4 allows us to focus on SCFs with the range that does not involve q; that is, on f:Π+A.

Lemma 4

Suppose an SCF f:Π+(A{q})×(0,1] is unanimous and satisfies monotonicity. Suppose that there is θΠ++, such that (a,p)=f(θ) . Then aq and p=1.

Proof

Suppose that there exists θΠ++,aA{q}, and p<1, such that f(θ)=(a,p). First, note that aq. Otherwise, pick an arbitrary outcome bA and construct a profile γ by making every other outcome cA{b} worse than q. By Lemma 2, f(γ)=f(θ)=q. However, b=a(k,i,γ)

Further result

In this section, we show that if we restrict the domain of an SCF to A, and the lottery space to L=A{q}×(0,1], we can strengthen our impossibility result. Note that such a restriction on L implies that the designer is not allowed to use arbitrary lotteries off the equilibrium path of the mechanism.

Theorem 2

Suppose that the number of pure alternatives |A|3 and L=A{q}×(0,1] . If an SCF f:Π+A is unanimous and monotonic, then f is dictatorial.

Outline of the proof

Steps  1–6 are the same as in the proof of Theorem 1. In

Conclusion

In this paper, we have shown that when the designer is restricted to a single outcome in constructing mechanisms for virtual implementation, an impossibility result arises. It would be interesting to investigate further how the impossibility result changes when we expand the number of dimensions along which the designer can approximate an SCF by increasing the number of status quo outcomes.

Acknowledgments

I thank an anonymous referee for their comments and questions that greatly benefited this paper. I thank Claudio Mezzetti and Roberto Serrano for their suggestions and acknowledge support through NSF grant SES-0133113. Some early results have been previously circulated in Artemov (2011).

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