On strategy-proof social choice correspondences

We introduce two new concepts of strategy-proofness for social choice correspondences based on the theory of preferences over sets of alternatives under complete uncertainty. One is based on Pattanaik and Peleg (Soc Choice Welf 1:113–122, 1984) and the other is based on Bossert et al. (Econ Theory 16:295–312, 2000). We prove that there is no social choice correspondence satisfying anonymity, neutrality, a range condition, and either of our concepts of strategy-proofness. An erratum to this article can be found at     

Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash Implementation of social choice correspondences. Define a lottery mechanism as a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. When preferences over alternatives are strict, we show that Maskin monotonicity (Maskin in Rev Econ stud 66: 23–38, 1999) is both necessary and sufficient for a social choice correspondence to be Nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice correspondences with private components. Finally, we apply our method to the issue of voluntary implementation (Jackon and Palfrey in J Econ Theory 98: 1–25, 2001).I thank Toyo Sakai for his comments on a previous draft. I also thank two anonymous referees and an editor of this journal for helpful comments that improved this paper. A previous version circulated as “A note on Maskin monotonicity”. After the results presented here were obtained, I became aware of a new unpublished paper by Benoit and Ok (2004). The result of Theorem 2 and the discussion that follows is partially similar to their Theorem 1.     

Contraction consistent stochastic choice correspondence

We model a general choice environment via probabilistic choice correspondences, with (possibly) incomplete domain and infinite universal set of alternatives. We offer a consistency restriction regarding choice when the feasible set contracts. This condition, ‘contraction consistency’, subsumes earlier notions such as Chernoff’s Condition, Sen’s α and β, and regularity. We identify a restriction on the domain of the stochastic choice correspondence (SCC), under which contraction consistency is equivalent to the weak axiom of revealed preference in its most general form. When the universal set of alternatives is finite, this restriction is also necessary for such equivalence. Analogous domain restrictions are also identified for the special case where choice is deterministic but possibly multi-valued. Results due to Sen (Rev Econ Stud 38:307–317, 1971) and Dasgupta and Pattanaik (Econ Theory 31:35–50, 2007) fall out as corollaries. Thus, conditions are established, under which our notion of consistency, articulated only in reference to contractions of the feasible set, suffices as the axiomatic foundation for a general revealed preference theory of choice behaviour.     

Two simplified proofs for Roberts’ theorem

Roberts (Aggregation and Revelation of Preferences. Papers presented at the 1st European Summer Workshop of the Econometric Society, pp. 321–349. North-Holland, 1979) showed that every social choice function that is ex-post implementable in private value settings must be weighted VCG, i.e. it maximizes the weighted social welfare. This paper provides two simplified proofs for this. The first proof uses the same underlying key-point, but significantly simplifies the technical construction around it, thus helps to shed light on it. The second proof builds on monotonicity conditions identified by Rochet (J Math Econ 16:191–200, 1987) and Bikhchandani et al. (Econometrica 74(4):1109–1132, 2006). This proof is for a weaker statement that assumes an additional condition of “player decisiveness”. Supported by grants from the Israeli Ministry of Science and the Israeli Academy of Sciences.     

Twofold optimality of the relative utilitarian bargaining solution

Given a bargaining problem, the relative utilitarian (RU) solution maximizes the sum total of the bargainer’s utilities, after having first renormalized each utility function to range from zero to one. We show that RU is “optimal” in two very different senses. First, RU is the maximal element (over the set of all bargaining solutions) under any partial ordering which satisfies certain axioms of fairness and consistency; this result is closely analogous to the result of Segal (J Polit Econ 108(3):569–589, 2000). Second, RU offers each person the maximum expected utility amongst all rescaling-invariant solutions, when it is applied to a random sequence of future bargaining problems generated using a certain class of distributions; this is recalls the results of Harsanyi (J Polit Econ 61:434–435, 1953) and Karni (Econometrica 66(6):1405–1415, 1998).     

Reconsidering two-agent Nash implementation

In this paper, we reconsider the full characterization of two-agent Nash implementation provided in the celebrated papers by Moore and Repullo (Econometrica 58:1083–1099, 1990) and Dutta and Sen (Rev Econ Stud 58:121–128, 1991), since we are able to show that the characterizing conditions are not logically independent. We prove that an amended version of the conditions proposed in these papers is still necessary and sufficient for Nash implementability. Then, by using our necessary and sufficient condition, we show that Maskin’s impossibility result can be avoided under restrictions on the outcomes and the domain of preferences much weaker than those previously imposed by Moore and Repullo (Econometrica 58:1083–1099, 1990) and Dutta and Sen (Rev Econ Stud 58:121–128, 1991).     

A note on the structure of stochastic social choice functions

This note sharpens the result of Nandeibam (J Econ Theory 68:212–233, 1996). We show that a stochastic social choice function which satisfies regularity, independence of irrelevant alternatives and weak ex-post Pareto optimality is essentially a weak random dictatorship.     

Gamson’s law and hedonic games

This note adds one celebrated coalition formation game due to Gamson (Am Soc Rev 26:373–382 1961a, Am Soc Rev 26:565–573, 1961b) in the list of applications of the theory of hedonic games explored by Banerjee et al. (Soc Choice Welf 18:135–153, 2001) and Bogomolnaia and Jackson (Games Econ Behav 38: 204–230, 2002). We apply their results to study the original Gamson game and offer extensions both to a multi-dimensional characteristics space and to an infinite number of players.     

Top-Pair and Top-Triple Monotonicity

We say that a social choice function (SCF) satisfies Top-k Monotonicity if the following holds. Suppose the outcome of the SCF at a preference profile is one of the top k-ranked alternatives for voter i. Let the set of these k alternatives be denoted by B. Suppose that i’s preference ordering changes in such a way that the set of first k-ranked alternatives remains the set B. Then the outcome at the new profile must belong to B. This definition of monotonicity arises naturally from considerations of set “improvements” and is weaker than the axioms of strong positive association and Maskin Monotonicity. Our main results are that if there are two voters then a SCF satisfies unanimity and Top-2 or Top-pair Monotonicity if and only if it is dictatorial. If there are more than two voters, then Top-pair Monotonicity must be replaced by Top-3 Monotonicity (or Top-triple Monotonicity) for the analogous result. Our results demonstrate that connection between dictatorship and “improvement” axioms is stronger than that suggested by the Muller–Satterthwaite result (Muller and Satterthwaite in J Econ Theory 14:412–418, 1977) and the Gibbard–Sattherthwaite theorem.     

Choice rules with fuzzy preferences: Some characterizations

Consider an agent with fuzzy preferences. This agent, however, has to make exact choices when faced with different feasible sets of alternatives. What rule does he follow in making such choices? This paper provides an axiomatic characterization of a class of binary choice rules called the α satisfying rule. When α=1, this rule is the Orlovsky choice rule. On the other hand, for α≤1/2, the rule coincides with the M _α rule that has been extensively analyzed in the literature on fuzzy preferences. Received: 3 August 1995/Accepted: 19 November 1997     

Brief proofs of Arrovian impossibility theorems

Since Kenneth Arrow showed the general possibility theorem, a number of social choice theorists have provided alternative proofs of it. In a recent article, Geanakoplos (Econ Theory 26:211–215, 2005) has constructed a new proof of the theorem. The present article provides alternative proofs of various Arrovian impossibility results from the 1960s to the 1970s by utilizing Geanakoplos’s method. We prove semi-order impossibility theorems, the quasi-transitive veto theorem, the quasi-transitive dictatorship theorem, the triple acyclic veto theorem, and the impossibility theorem without the Pareto principle.     

Essential alternatives and freedom rankings

We study the problem of ranking opportunity sets in terms of freedom of choice. The analysis is based on the notion of essential alternatives introduced in Puppe (J Econ Theory 68:174–199, 1996). An alternative in an opportunity set is called essential if by deleting it, the reduced opportunity set offers less freedom than the original set. We provide an axiomatic characterization of the ranking according to which an opportunity set offers more freedom than another opportunity set if its share of essential elements in their union is larger.     

Why do larger families reduce parental investments in child quality, but not child quality per se?

Becker’s Quantity–Quality model (Becker in Demographic and economic change in developed countries, Princeton University Press, Princeton, pp 209–240, 1960; Becker and Lewis in J Polit Econ 81(2): S279–S288, 1973; Becker and Tomes in J Polit Econ 84(4): S143–S162, 1976) suggests a trade-off between family size and parental investments in children. To date, only Cáceras-Delpiano (J Hum Resour 41(4): 738–754, 2006) tests this theory by considering private school enrolment. This study extends this work by using a unique data set containing a broader range of parental investments that are arguably linked to parental intentions for producing higher quality children, such as overall and non-sectarian private school enrolment, the number of computers in the home per child, and saving for the child’s education. Both studies find that fertility reduces parental investments. However, the literature generally finds that fertility has no impact on child outcomes. The study offers three potential explanations for this ‘puzzle’.     

Income Share Elasticity and Stochastic Dominance

Income share elasticity is a function π which can describe the size distribution of income (Esteban in Intern Econ Rev 27:439–444, 1986). On the other hand, the conventional density representation of the latter gives parameters of first or second order stochastic dominance (SD), widely used to describe shifts in income distribution, to which inequality measures are attached. The paper draws a link between the two, by providing conditions such that a given shift to π is equivalent to a first or second order SD shift of the distribution of income. Some applications to Lorenz rankings are also provided.     

Internalities and paternalism: applying the compensation criterion to multiple selves across time

One reason to call an activity a vice and suppress it is that it reduces a person’s future happiness more than it increases his present happiness. Gruber and Koszegi (Q J Econ 116(4):1261–1303, 2001) show how a vice tax can increase a person’s welfare in a model of multiple selves with hyperbolic preferences across time. The present paper shows that an interself analogy of the compensation criterion can justify a vice ban whether preferences are hyperbolic or exponential, but subject to the caveat that the person has a binding constraint on borrowing.     

An axiomatization of the Nash bargaining solution

I prove that ‘Disagreement Point Convexity’ and ‘Midpoint Domination’ characterize the Nash bargaining solution on the class of two-player bargaining problems and on the class of smooth bargaining problems. I propose an example to show that these two axioms do not characterize the Nash bargaining solution on the class of bargaining problems with more than two players. I prove that the other solutions that satisfy these two properties are not lower hemi-continuous. These different results refine the analysis of Chun (Econ Lett 34:311–316, 1990). I also highlight a rather unexpected link with the result of Dagan et al. (Soc Choice Welfare 19:811–823, 2002).     

On Ehrhart polynomials and probability calculations in voting theory

In voting theory, analyzing the frequency of an event (e.g. a voting paradox), under some specific but widely used assumptions, is equivalent to computing the exact number of integer solutions in a system of linear constraints. Recently, some algorithms for computing this number have been proposed in social choice literature by Huang and Chua (Soc Choice Welfare 17:143–155 2000) and by Gehrlein (Soc Choice Welfare 19:503–512 2002; Rev Econ Des 9:317–336 2006). The purpose of this paper is threefold. Firstly, we want to do justice to Eugène Ehrhart, who, more than forty years ago, discovered the theoretical foundations of the above mentioned algorithms. Secondly, we present some efficient algorithms that have been recently developed by computer scientists, independently from voting theorists. Thirdly, we illustrate the use of these algorithms by providing some original results in voting theory. Helpful comments by Philippe Clauss and his team are gratefully acknowledged.     

Is India better off today than 15 years ago? A robust multidimensional answer

This paper provides a robust multidimensional normative evaluation of the growth episode that India has experienced in the last 15 years. Specifically, the paper compares the evolution, between 1987, 1995 and 2002, of the distribution of several individual attributes on the basis of ethically robust dominance criteria. The individual attributes considered are real consumption (measured at the individual level), literacy rate, under 5 mortality and violent crime rates (all measured at the district levels). District level variables are interpreted as (local) public goods which, along with consumption, contribute to individual well-being. The robust criteria used are generalizations, to more than two attributes, of the first and second order dominance criteria of Atkinson and Bourguignon (Rev Econ Stud 49:183–201, 1982) and coincide with the unanimity of utilitarian value judgements taken over a specific class of individual utility functions. The main result of the empirical analysis is that all utilitarian rankings of distributions of the four attributes that assume that individual utility functions satisfy the assumptions of second order dominance agree that India is better off in 2002 than in 1987 or 1995 but that these rankings disagree as to how to rank 1987 and 1995. Furthermore, if one removes crime from the list of attributes, the dominance is shown to apply steadily over the whole period.     

Market arbitrage, social choice and the core

 This paper establishes a clear connection between equilibrium theory, game theory and social choice theory by showing that, for a well defined social choice problem, a condition which is necessary and sufficient to solve this problem – limited arbitrage – is the same as the condition which is necessary and sufficient to establish the existence of an equilibrium and the core. The connection is strengthened by establishing that a market allocation, which is in the core, can always be realized as a social allocation, i.e. an allocation which is optimal according to an ordering chosen by a social choice rule. Limited arbitrage characterizes those economies without Condorcet triples, and those for which Arrow’s paradox can be resolved on choices of large utility values. Received: 30 December 1994/Accepted: 12 August 1996     

Strategy-proof and anonymous rule in queueing problems: a relationship between equity and efficiency

We consider a relationship between equity and efficiency in queueing problems. We show that under strategy-proofness, anonymity in welfare implies queue-efficiency. Furthermore, by combining the result of Kayı and Ramaekers (Games Econ Behav 68:220–232, 2010) with ours, we also give a characterization of the class of rules that satisfy strategy-proofness, anonymity in welfare, and budget-balance.