An impossibility result for virtual implementation with status quo
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 -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 be the finite set of outcomes, where is an outcome designated as the status quo. Let . Let be the finite set of agents, and .
Let the lottery space be , where 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 , with ; 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 is -close to if, for any , and . Definition 4 An SCF is virtually implementable with -admissible support if for all there exists a
Avoiding the constancy result
In defining the domain of preferences , 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 are such that (i) ; (ii) there is an agent , such that ; 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 . Suppose that is such that: for some
Main result
Lemma 4 allows us to focus on SCFs with the range that does not involve ; that is, on . Lemma 4 Suppose an SCF is unanimous and satisfies monotonicity. Suppose that there is , such that . Then and .
Proof Suppose that there exists , and , such that . First, note that . Otherwise, pick an arbitrary outcome and construct a profile by making every other outcome worse than . By Lemma 2, . However,
Further result
In this section, we show that if we restrict the domain of an SCF to , and the lottery space to , we can strengthen our impossibility result. Note that such a restriction on 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 and . If an SCF is unanimous and monotonic, then 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).
References (17)
- et al.
Strategy-proof probabilistic rules for expected utility maximizers
Math. Social Sci.
(1998) Strategic manipulation in voting games when lotteries and ties are permitted
J. Econom. Theory
(2002)- et al.
Virtual Nash implementation with admissible supports
J. Math. Econom.
(2010) A new approach to the implementation problem
J. Econom. Theory
(1988)- et al.
The equivalence of strong positive association and strategy-proofness
J. Econom. Theory
(1977) Arrow’s theorem and the Gibbard–Satterthwaite theorem: a unified approach
Econom. Lett.
(2001)On constant Maskin monotonic social choice functions
J. Econom. Theory
(1987)- et al.
Virtual implementation in Nash equilibrium
Econometrica
(1991)