A counterexample to a conjecture of Schwartz

In 1990, motivated by applications in the social sciences, Thomas Schwartz made a conjecture about tournaments which would have had numerous attractive consequences. In particular, it implied that there is no tournament with a partition A, B of its vertex set, such that every transitive subset of A is in the out-neighbour set of some vertex in B, and vice versa. But in fact there is such a tournament, as we show in this article, and so Schwartz’ conjecture is false. Our proof is non-constructive and uses the probabilistic method.     

Social network structure and the trade-off between social utility and economic performance

We put forward a computational multi-agent model capturing the impact of social network structure on individuals’ social trust, willingness to cooperate, social utility and economic performance. Social network structure is modeled as four distinct social capital dimensions: degree, centrality, bridging and bonding social capital. Model setup draws from socio-economic theory and empirical findings based on our novel survey dataset. Results include aggregate-level comparative statics and individual-level correlations. We find, inter alia, that societies that either are better connected, exhibit a lower frequency of local cliques, or have a smaller share of family-based cliques, record relatively better aggregate economic performance. As long as family ties are sufficiently valuable, there is a trade-off between aggregate social utility and economic performance, and small world networks are then socially optimal. We also find that in dense networks and trustful societies, there is a trade-off between individual social utility and economic performance; otherwise both outcomes are positively correlated in the cross section.     

A simple proof of Maskin's theorem on Nash implementation

This paper gives a simple proof of the following result, which is due to Maskin [1]: If an n-person social choice rule f, with n3, satisfied monotonicity and no veto power, then f can be implemented in Nash equilibria.     

Choice functions over a finite set: A summary

A choice function picks some outcome(s) from every issue (subset of a fixed set A of outcomes). When is this function derived from one preference relation on A (the choice set being then made up of the best preferred outcomes within the issue), or from several preference relations (the choice set being then the Pareto optimal outcome within the issue, or the union of the best preferred outcomes for each preference relation)? A complete and unified treatment of these problems is given based on three functional properties of the choice function. None of the main results is original.     

Internal cohesion of ls sets in graphs

Let G be a finite connected graph. A set of vertices H ? V(G) is called a LS set if for every proper subset K ? H, there are more edges linking K to H ? K than there are linking K to V(G)-H. Since “cliques” in social networks have usually been seen informally as sets of individuals more closely tied to each other than to outsiders, LS sets provide a natural realization of the “clique” concept. In this paper, it is shown that LS sets in social networks have cohesive properties that make them even more useful for empirical analyses. In particular, subgraphs induced by LS subsets remain connected even after several edges have been removed. Results bounding the number of edges that can be so removed are used to get an upper bound for the diameter of subgraphs induced by LS subsets.     

Identification of domain restrictions over which acyclic, continuous-valued, and positive responsive social choice rules operate

We consider a social choice problem in various economic environments consisting of n individuals, 4≤n<+∞, each of which is supposed to have classical preferences. A social choice rule is a function associating with each profile of individual preferences a social preference that is assumed to be complete, continuous and acyclic over the alternatives set. The class of social choice rules we deal with is supposed to satisfy the two conditions; binary independence and positive responsiveness. A new domain restriction for the social choice rules is proposed and called the classical domain that is weaker than the free triple domain and holds for almost all economic environments such as economies with private and/or public goods. In this paper we explore what type of classical domain that admits at least one social choice rule satisfying the mentioned conditions to well operate over the domain. The results we obtained are very negative: For any classical domain admitting at least one social choice rule to well operate, the domain consists only of just one profile.     

Public information and social choice

We examine the value of public information when a society uses a social choice rule to decide among a set of outcomes. We require that a social choice function satisfies unrestricted domain, non-decisiveness and the Pareto principle. We show that there exist payoff structures for every social choice function, such that an arbitrary subset of voters is worse off by public information. We apply the proposition to collective information acquisition and to irreversible investments. Received: 2 June 1997/Accepted: 30 September 1998     

Nondictatorially independent pairs

Suppose that social welfare function f satisfies the Pareto condition and has complete and transitive values. On a domain satisfying the free quadruple property, if the outcome set X has m< outcomes, then the set of pairs of distinct alternatives on which f satisfies both independence and nondictatorship contains at most the fraction 2/m of all pairs. On a domain satisfying the free six-tuple property, if X is a subset of Euclidean space then the set of pairs on which f satisfies both independence and nondictatorship has Lebesgue measure zero.Campbell's research was financed by the National Science Foundation, grant SES 9209039.     

Distribution of coalitional power in randomized multi-valued social choice

This paper considers the distribution of coalitional influence under probabilistic social choice functions which are randomized social choice rules that allow social indifference by mapping each combination of a preference profile and a feasible set to a social choice lottery over all possible choice sets from the feasible set. When there are at least four alternatives in the universal set and ex-post Pareto optimality, independence of irrelevant alternatives and regularity are imposed, we show that: (i) there is a system of additive coalitional weights such that the weight of each coalition is its power to be decisive in every two-alternative feasble set; and (ii) for each combination of a feasible proper subset of the universal set and a preference profile, the society can be partioned in such a way that for each coalition in this partition, the probability of society's choice set being contained in the union of the best sets of its members is equal to the coalition's power or weight. It is further shown that, for feasible proper subsets of the universal set, the probability of society's choice set containing a pair of alternatives that are not jointly present in anyone's best set is zero. Our results remain valid even when the universal set itself becomes feasible provided some additional conditions hold. Received: 10 May 1999/Accepted: 18 June 2000 I would like to thank Professor Prasanta Pattanaik for suggesting to me the line of investigation carried out in this paper. I am solely responsible for any remaining errors and omissions.     

The Bordes-LeBreton exceptional case

Bordes and LeBreton have found a set D of weak orderings of X that includes all linear (strong) orderings of X but where there exists a social welfare function f on D ^N satisfying all of Arrow's (non-domain) conditions. In this paper we characterize all such D and f possibilities.     

The core of social choice problems with monotone preferences and a feasibility constraint

We consider collective choice with agents possessing strictly monotone, strictly convex and continuous preferences over a compact and convex constraint set contained in ₊^k . If it is non-empty the core will lie on the efficient boundary of the constraint set and any policy not in the core is beaten by some policy on the efficient boundary. It is possible to translate the collective choice problem on this efficient boundary to another social choice problem on a compact and convex subset of ₊^c (c<k) with strictly convex and continuous preferences. In this setting the dimensionality results in Banks (1995) and Saari (1997) apply to the dimensionality of the boundary of the constraint set (which is lower than the dimensionality of the choice space by at least one). If the constraint set is not convex then the translated lower dimensional problem does not necessarily involve strict convexity of preferences but the dimensionality of the problem is still lower. Broadly, the results show that the homogeneity afforded by strict monotonicity of preferences and a compact constraint set makes generic core emptyness slightly less common. One example of the results is that if preferences are strictly monotone and convex on ² then choice on a compact and convex constraint exhibits a version of the median voter theorem.I thank Donald Saari for helpful comments on an earlier version of this paper.     

Minimal manipulability and interjacency for two-person social choice functions

This paper investigates the number of profiles at which a social choice function is manipulable. It is found that there is a lower bound on this number when the social choice function is assumed non-dictatorial. Also, any number between this lower bound and the maximal number of profiles may be obtained as the number of manipulable profiles of some social choice function. Received: 30 November 1993 / Accepted: 14 April 1997     

Social choice theory in the case of von Neumann-Morgenstern utilities

Part I of this paper offers a novel result in social choice theory by extending Harsanyi's well-known utilitarian theorem into a multi-profile context. Harsanyi was contented with showing that if the individuals' utilities u _i are von Neumann-Morgenstern, and if the given utility u of the social planner is VNM as well, then the Pareto indifference rule implies that u is affine in terms of the u _i. We provide a related conclusion by considering u as functionally dependent on the u _i, through a suitably restricted social welfare functional (u ₁,...,u _n)u=f(u ₁,...,u _n). We claim that this result is more in accordance with contemporary social choice theory than Harsanyi's single-profile theorem is. Besides, harsanyi's initial proof of the latter was faulty. Part II of this paper offers an alternative argument which is intended to be both general and simple enough, contrary to the recent proofs published by Fishburn and others. It finally investigates the affine independence problem on the u _i discussed by Fishburn as a corollary to harsanyi's theorem.The authors are indebted to L. Haddad, A. Sen and two anonymous referees for useful written comments. They also benefited from stimulating remarks in seminars and helpful conversations with their colleagues. The usual caveat of course applies. One of the authors acknowledges partial financial support from the ARI Communication of the C.N.R.S., Paris.     

Nondictatorially independent pairs and Pareto

Suppose a social welfare function f has complete and transitive values and the set X of outcomes has size m>3. If the domain D of f has the free quadruple property then the fraction of pairs on which f satisfies nondictatorship, independence, and Pareto is less than or equal to . Received: 9 June 1999/Accepted: 30 May 2000     

A characterization of strategy-proof social choice functions for economies with pure public goods

We characterize strategy-proof social choice functions when individuals have strictly quasi-concave, continuous and satiated utility functions on convex subsets of IR ^m , representing preferences for the provision of m pure public goods. When specialized to the case m=1, these assumptions amount to requiring that preferences are single peaked, and for such a domain there exists a wide class of strategy-proof social choice functions. These were studied by Moulin (1980) under strong additional assumptions. Our first results characterize the complete class, after an appropriate extension of the single-peakedness condition. The new characterization retains the flavour of Moulin's elegant representation theorem. For the general m-dimensional case, previous results have shown that there is no efficient, strategy-proof, nondictatorial social choice function, even within the domain restrictions under consideration (Border and Jordan 1983; Zhou 1991). In fact, Zhou's powerful result indicates that nondictatorial strategy-proof s.c.f.'s will have a range of dimension one. This allows us to conclude with a complete characterization of all strategy-proof s.c.f.'s on IR ^m , because restrictions of preferences from our admissible class to one dimensional subsets satisfy the slightly generalized notion of single-peakedness that is used in our characterization for the case m=1. We feel that a complete knowledge of the class of strategy-proof mechanisms, in this as well as in other contexts, is an important step in the analysis of the trade-offs between strategy-proofness and other performance criteria, like efficiency.This paper was written while both authors were visiting GREMAQ, Université des Sciences Sociales de Toulouse. We are thankful for its hospitality and good research atmosphere. Barberà's work is supported by the Instituto de Estudios Fiscales and by research grant PB89-0294 from the Secretaría de Estado de Universidades e Investigación, Spain. Jackson acknowledges the support of NSF grant SES8921409. We thank Jacques Crémer, Beth Allen, John Weymark and two anonymous referees for helpful comments on earlier drafts.     

Note on social choice allocation in exchange economies with Cobb–Douglas preferences

We show that in a pure exchange economy with two agents and a finite number of goods, there exists no strategy-proof, Pareto-efficient, and non-dictatorial social choice allocation function on any local Cobb–Douglas preference domain. This is a slight extension of a result proved by Hashimoto (2008).     

The production of social capital in US counties

A growing belief exists that social capital contributes to economic growth of communities. In this paper, we identify inputs into the production of social capital at the level of US counties, using an array of individual and community factors that are theoretically important determinants of social capital. We use data from the Bureau of the Census, County Business Patterns, USA Counties on CD, National Center for Charitable Statistics, and the Regional Economic Information System for two time periods. Ethnic homogeneity, income inequality, attachment to place, education, age, and female labor force participation are strongly associated with levels of social capital across US counties.     

A theory of policy differentiation in single issue electoral politics

Voter preferences are characterized by a parameter s (say, income) distributed on a set S according to a probability measure F. There is a single issue (say, a tax rate) whose level, b, is to be politically decided. There are two parties, each of which is a perfect agent of some constituency of voters, voters with a given value of s. An equilibrium of the electoral game is a pair of policies, b ₁ and b ₂, proposed by the two parties, such that b _i maximizes the expected utility of the voters whom party i represents, given the policy proposed by the opposition. Under reasonable assumptions, the unique electoral equilibrium consists in both parties proposing the favorite policy of the median voter. What theory can explain why, historically, we observe electoral equilibria where the ‘right’ and ‘left’ parties propose different policies? Uncertainty concerning the distribution of voters is introduced. Let {F(t)} _{t ε T} be a class of probability measures on S; all voters and parties share a common prior that the distribution of t is described by a probability measure H on T. If H has finite support, there is in general no electoral equilibrium. However, if H is continuous, then electoral equilibrium generally exists, and in equilibrium the parties propose different policies. Convergence of equilibrium to median voter politics is proved as uncertainty about the distribution of voter traits becomes small.     

Choosing subsets: a size-independent probabilistic model and the quest for a social welfare ordering

“Subset voting” denotes a choice situation where one fixed set of choice alternatives (candidates, products) is offered to a group of decision makers, each of whom is requested to pick a subset containing any number of alternatives. In the context of subset voting we merge three choice paradigms, “approval voting“ from political science, the “weak utility model” from mathematical psychology, and “social welfare orderings” from social choice theory. We use a probabilistic choice model proposed by Falmagne and Regenwetter (1996) built upon the notion that each voter has a personal ranking of the alternatives and chooses a subset at the top of the ranking. Using an extension of Sen's (1966) theorem about value restriction, we provide necessary and sufficient conditions for this empirically testable choice model to yield a social welfare ordering. Furthermore, we develop a method to compute Borda scores and Condorcet winners from subset choice probabilities. The technique is illustrated on an election of the Mathematical Association of America (Brams, 1988). Received: 18 August 1995 / Accepted: 13 February 1997     

A measure of distance between judgment sets

In the literature on judgment aggregation, an important open question is how to measure the distance between any two judgment sets. This is relevant for issues of social choice: if two individuals hold different beliefs then we might want to find a compromise that lies somewhere between them. We propose a set of axioms that determine a measure of distance uniquely. This measure differs from the widely used Hamming metric. The difference between Hamming’s metric and ours boils down to one axiom. Given judgment sets A and B, this axiom says that if the propositions in ${A \cap B}$ jointly imply that the propositions in A?B share the same truth value, then the disagreement between A and B over those propositions in A?B should be counted as a single disagreement. We consider the application of our metric to judgment aggregation, and also use the metric to measure the distance between preference rankings.