Policies over time and pareto optimality

This paper is concerned with sequences of policies that occur over time in voting models and planning procedures. The framework for our analysis includes assumptions that are satisfied by models in the corresponding literatures, together with other standard assumptions for microeconomic analysis that involve time. The starting point for our analyses is the prespective that results from combining the following (widely held) views: 1) certain voting models and planning procedures can be interpreted as being non-tatonnement or sequential processes (where each policy in the sequence that is generated is actually experienced by voters or consumers) and 2) an alternative being Pareto optimal in any given period (temporal Pareto optimality) is the appropriate efficiency criterion only if the alternative is the final outcome from a tatonnement process-and that, otherwise, one should examine the efficiency of the entire path (using intertemporal Pareto optimality). Our first observation about the planning literature is that is has (by and large) neglected the efficiency criterion that is appropriate for the discrete-time procedures that can be interpreted as non-tatonnement or sequential processes-and that, what's more, such trajectories will (in general) fail to meet this criterion. Our second observation identifies some results that can be used to establish that some of these trajectories will at least be ultimately intertemporally Pareto optimal. In our discussion of voting theory, we review Buchanan's opposition to requiring (social) choice consistency for voting procedures-and his argument for this position on (Pareto) efficiency grounds. We then consider voting procedures that can be interpreted as non-tatonnement or sequential processes and arrive at the conclusion that, in these cases, (i) majority rule cycles are intertemporally Pareto inefficient and (ii) achieving intertemporal Pareto optimality requires choice consistency. We then go on to show that related observations apply to Kramer's normative conclusions about his dynamical model of political equilibrium — and identify some further references where similar observations apply. In the final part of our discussion of voting models, we arrive at the further conclusion that, in the most relevant cases, a trajectory that stays in the temporal Pareto set is not necessarily more desirable (on efficiency grounds) than one that doesn't.This paper has been improved by helpful comments and suggestions that have been provided by Kenneth Arrow, an anonymous referee, and the editor who handled the paper.Peter Coughlin gratefully acknowledges financial support provided by (i) National Science Foundation Grant No. SES-8409352, and (ii) the Center for Advanced Study in the Behavioral Sciences at Stanford (with support from National Science Foundation Grant No. BNS-8700864 and the Alfred P. Sloan Foundation).     

Social aggregators

This paper proposes a general framework for analyzing a class of functions called social aggregators, which map profiles of linear orders to a set of binary relations. This class of aggregators includes aggregators that yield a preference relation (social welfare functions) and those which yield a choice of an alternative (social choice functions). Equipped with this framework, I identify a property called Preference Reversal (PR) such that any Pareto efficient aggregator having this property must be dictatorial. This allows me to state a general impossibility theorem, which includes Arrows Theorem and the Gibbard Satterthwaite Theorem as two special examples. Furthermore, I show that monotonicity and IIA are closely linked, by demonstrating that both are actually special cases of PR in specific environments.Received: 15 October 2001     

The direct social preference relation in the theory of public expenditure

It is shown that if there is a finite number of private goods, a single public good, and the individual preferences satisfy certain monotonicity and separability conditions then there is a unique and complete social preference relation defined on the set of allocations by the requirement that the relation is individualistic in terms of the individual ordinal preferences. This relation, called the direct social preference relation, is derived without imposing interpersonal welfare comparisons and all its ordinal properties are inherited directly from those of the individual preferences. However, an allocation which is maximal in terms of the direct social preference relation on the set of feasible allocations is in general not Pareto optimal and the relation may therefore not be suitable as a guide to optimizing social choice.This version of the paper has benefitted from J. Weymark's insightful and detailed comments. In addition, valuable suggestions have been received from G. C. Archibald, C. Blackorby, D. Donaldson and D. Primont. All remaining errors are the sole responsibility of the author     

Strategy-proof allocation of multiple public goods

This paper characterizes strategy-proof social choice functions (SCFs), the outcome of which are multiple public goods. Feasible alternatives belong to subsets of a product set . The SCFs are not necessarily “onto”, but the weaker requirement, that every element in each category of public goods A _k is attained at some preference profile, is imposed instead. Admissible preferences are arbitrary rankings of the goods in the various categories, while a separability restriction concerning preferences among the various categories is assumed. It is found that the range of the SCF is uniquely decomposed into a product set in general coarser than the original product set, and that the SCF must be dictatorial on each component B _l . If the range cannot be decomposed, a form of the Gibbard–Satterthwaite theorem with a restricted preference domain is obtained.     

Equilibrium agenda formation

We develop a definition of equilibrium for agenda formation in general voting settings. The definition is independent of any protocol. We show that the set of equilibrium outcomes for any Pareto efficient voting rule is uniquely determined, and in fact coincides with that of the outcomes generated by considering all full agendas. Under voting by successive elimination (or amendment), the set of equilibrium outcomes corresponds with the Banks set. We also examine the implications in several specific settings and show that studying equilibrium agendas can lead to sharp predictions, in contrast with well-known chaos theorems.Financial support under NSF grant SES-9986190 and an RTDF grant from the University of Warwick are gratefully acknowledged. We thank John Duggan, Martin Osborne, and an anonymous referee for valuable suggestions, and participants at the Sixth International Meeting of the Society for Social Choice and Welfare for helpful comments.     

Generic difference of expected vote share and probability of victory maximization in simple plurality elections with probabilistic voters

In this paper I examine single member, simple plurality elections with n ≥ 3 probabilistic voters and show that the maximization of expected vote share and maximization of probability of victory are “generically different” in a specific sense. More specifically, I first describe finite shyness (Anderson and Zame in Adv Theor Econ 1:1–62, 2000), a notion of genericity for infinite dimensional spaces. Using this notion, I show that, for any policy in the interior of the policy space and any candidate j, the set of n-dimensional profiles of twice continuously differentiable probabilistic voting functions for which simultaneously satisfies the first and second order conditions for maximization of j’s probability of victory and j’s expected vote share at is finitely shy with respect to the set of n-dimensional profiles of twice continuously differentiable probabilistic voting functions for which satisfies the first and second order conditions for maximization of j’s expected vote share.     

Nash implementation and the bargaining problem

We study Nash implementation of the bargaining solutions in a cake sharing set up. We argue that the minimal Pareto optimal, symmetric and Nash implementable SBS is the one inducing all Pareto optimal and midpoint-dominating utility vectors in each state.     

Existence of a multicameral core

When a single group uses majority rule to select a set of policies from an n-dimensional compact and convex set, a core generally exists if and only if n = 1. Finding analogous conditions for core existence when an n-dimensional action requires agreement from m groups has been an open problem. This paper provides a solution to this problem by establishing sufficient conditions for core existence and characterizing the location and dimensionality of the core for settings in which voters have Euclidean preferences. The conditions establish that a core may exist in any number of dimensions whenever n ≤ m as long as there is sufficient preference homogeneity within groups and heterogeneity between groups. With m > 1 the core is however generically empty for . These results provide a generalization of the median voter theorem and of non-existence results for contexts of concern to students of multiparty negotiation, comparative politics and international relations.     

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.     

Two-stage discrete aggregation: the Ostrogorski paradox and related phenomena

Motivated by certain paradoxa that have been discussed in the literature (Ostrogorski paradox), we prove an impossibility theorem for two-stage aggregation procedures for discrete data. We consider aggregation procedures of the following form: The whole population is partitioned into subgroups. First we aggregate over each subgroup, and in a second step we aggregate the subgroup aggregates to obtain a total aggregate. The data are either dichotomous (1 — 0; yes-no) or take values in a finite ordered set of possible attributes (e.g., exam grades A, B,...F). Examples are given by multistage voting procedures (indirect democracy, federalism), or by the forming of partial grades and overall grades in academic examinations and similar evaluation problems (sports competitions, consumer reports). It is well known from standard examples that the result of such a two-stage aggregation procedure depends, in general, not only on the distribution of attributes in the whole population, but also on how the attributes are distributed across the various subgroups (in other words: how the subgroups are defined). This dependence leads to certain paradoxa. The main result of the present paper is that these paradoxa are not due to the special aggregation rules employed in the examples, but are unavoidable in principle, provided the aggregators satisfy certain natural assumptions. More precisely: the only aggregator functions for which the result of a two-stage (a fortiori: multi-stage) aggregation does not depend on the partitioning are degenerate aggregators of the following form: there exists a partial order (dominance) on the set of possible attributes such that the aggregate over any collection of data is always equal to the supremum (w.r.t. dominance) of the attributes occurring in the data, regardless of the relative frequnencies of these occurrences. In the voting context, degeneracy corresponds to the unanimity principle. Our theorem is true for arbitrary partitionings of arbitrary (finite) sets and generalizes the results of Deb & Kelsey (for the matrix case with dichotomous variables and majority voting) to general two-stage aggregation procedures for attributes belonging to a finite ordered set. The general result is illustrated by some examples.This paper was completed during a visit to the University of Bielefeld. I am much indebted to the Faculty of Economics there for its hospitality; in particular I should like to thank Gerhard Schwödiauer and Walter Trockel for their support.     

Complete characterization of functions satisfying the conditions of Arrow’s theorem

Arrow??s theorem implies that a social welfare function satisfying Transitivity, the Weak Pareto Principle (Unanimity), and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are also allowed, a dictatorial social welfare function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions, since non-strict preferences of the dictator are not necessarily followed. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow??s theorem, in the case of non-strict preferences, does not provide a complete characterization of all social welfare functions satisfying Transitivity, the Weak Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow??s theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow??s and Wilson??s result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Weak Pareto Principle). Additionally, we derive formulae for the number of functions satisfying these conditions.     

Optimal commodity taxation for reduction of envy

This paper derives optimal commodity taxes in a two-class economy based on -equitability. An allocation is -equitable if no agent envies a proportion of the bundle of any other agent. We examine the properties of Pareto undominated allocations in a class of -equitable allocations. We first show that the consistency between envy reduction and enhancement of the low-skilled agents utility depends on 1) the income elasticity of leisure 2) the degree of preference and skill differences. On the optimal tax rule, we can show the following. When the envy constraint is binding, ceteris paribus, the goods preferred by the low skilled agent and of high Hicksian elasticities are taxed more heavily. The optimal commodity tax is uniform if, in addition to weak separability, sub-utility of the goods is homothetic.     

Some implications of an intensity measure of envy

This article proposes a cardinal or an intensity measure of interpersonal envy by directly measuring the distance between the envied person's consumption bundle and the envious person's equilibrium indifference curve. The individual's intensity of envy is aggregated to form a social envy index, which in turn is combined with the Pareto criterion to form a social choice rule. This social choice rule generates a non-empty choice set with the fair set (envy-free and Pareto efficient social states) as a proper subset of the choice set. The intensity measure of envy is then applied to the Rawlsian problem of the worst-off person. It turns out that the person with the greatest envy need not be the poorest person. Finally, it is shown that if individuals have homothetic and identical preferences the measure of envy collapses into a measure of income inequality. This measure of income inequality satisfies soem appealing axioms.I am grateful to William Vickrey and Duncan Foley for orginally encouraging the idea of intensity of envy. This version has benefited greatly from suggestions and criticisms made by F. William McElroy     

A law of large numbers for weighted plurality

Consider an election between $k$ candidates in which each voter votes randomly (but not necessarily independently) for a single candidate, and suppose that there is a single candidate that every voter prefers (in the sense that each voter is more likely to vote for this special candidate than any other candidate). Suppose we have a voting rule that takes all of the votes and produces a single outcome and suppose that each individual voter has little effect on the outcome of the voting rule. If the voting rule is a weighted plurality, then we show that with high probability, the preferred candidate will win the election. Conversely, we show that this statement fails for all other reasonable voting rules. This result is an extension of one by Häggström, Kalai and Mossel, who proved the above in the case $k=2$ .     

Coalition-proofness under weak and strong Pareto dominance

This article examines the structure of pure strategy coalition-proof Nash equilibria under weak and strong Pareto dominance in games with strategic complementarities and isotone externalities. The analysis is particularly focused on the following issues: (i) the inclusion relation between the set of coalition-proof Nash equilibria under weak Pareto dominance and the set of coalition-proof Nash equilibria under strong Pareto dominance; (ii) the conditions for the coincidence of the strong (resp. weak) Pareto dominance refinement and the coalition-proofness refinement under strong (resp. weak) Pareto dominance; (iii) the conditions for the uniqueness of a coalition-proof Nash equilibrium under both notions of Pareto dominance. Dual results are stated for games with antitone externalities.     

Ensuring a representative referendum outcome: the daunting task of setting the quorum right

The majority of the participating voters in referenda does not necessarily reflect the majority of the whole population since voters can abstain. This paper shows that although the quorum can be set such that the outcome of the referendum coincides with the population preference, the necessary assumptions make practical implementation unrealistic. Moreover, a second equilibrium can exist in which the proposal is always rejected. When insufficient information makes the optimal quorum unknown, it is in general more harmful to set the quorum too high than too low. Robustness of the results is analyzed by allowing pressure groups to encourage or discourage participation after the quorum is set.     

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.     

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.     

Characterizing Paretian preferences

A characterization of a property of binary relations is of type M if it can be stated in terms of ordered M-tuples of alternatives. A characterization of finite type provides an easy test of whether preferences over a large set of alternatives possesses the property characterized. Unfortunately, there is no characterization of finite type for Pareto representability in . A partial result along the same lines is obtained for Pareto representability in , k>2.

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