Smallest tournaments not realizable by $${\frac{2}{3}}$$-majority voting

Define the predictability number α(T) of a tournament T to be the largest supermajority threshold for which T could represent the pairwise voting outcomes from some population of voter preference orders. We establish that the predictability number always exists and is rational. Only acyclic tournaments have predictability 1; the Condorcet voting paradox tournament has predictability ; Gilboa has found a tournament on 54 alternatives (i.e. vertices) that has predictability less than , and has asked whether a smaller such tournament exists. We exhibit an 8-vertex tournament that has predictability , and prove that it is the smallest tournament with predictability <  . Our methodology is to formulate the problem as a finite set of two-person zero-sum games, employ the minimax duality and linear programming basic solution theorems, and solve using rational arithmetic. D. Shepardson was supported by a NSF Graduate Research Fellowship during the course of this work.     

The NIP graph of a social welfare function

We consider the fraction of pairs of m distinct alternatives on which a social welfare function f may be nondictatorially independent and Pareto when the domain of f satisfies the free k-tuple property. When k = 4 we improve the existing upper bound to . When there are at least 26 alternatives and we obtain an original upper bound, . To obtain these results we define and analyze the graph formed from the nondictatorial independent and Pareto pairs and combine the results of this analysis with known results from extremal graph theory. The authors extend special thanks to the two reviewers and the editor for their comments.     

Arrovian impossibilities in aggregating preferences over non-resolute outcomes

Let A be a set of alternatives whose power set is . Elements of are interpreted as non-resolute outcomes. We consider the aggregation of preference profiles over into a (social) preference over . In case we allow individuals to have any complete and transitive preference over , Arrow’s impossibility theorem naturally applies. However, the Arrovian impossibility prevails, even when the set of admissible preferences over is severely restricted. In fact, we identify a mild “regularity” condition which ensures the dictatoriality of a domain. Regularity is compatible with almost all standard extension axioms of the literature. Thus, we interpret our results as the strong prevalence of Arrow’s impossibility theorem in aggregating preferences over non-resolute outcomes. This paper is part of a project entitled “Social Perception—A Social Choice Perspective”, supported by Istanbul Bilgi University Research Fund. It has been completed while Remzi Sanver was visiting Ecole Polytechnique, Paris. We are grateful to both institutions. We thank Nick Baigent, two anonymous referees and an anonymous associate editor for their valuable comments.     

Fair Groves mechanisms

We study allocation problems in which a costly task is to be assigned and money transfers are used to achieve fairness among agents. We consider a series of fairness notions (k-fairness for where n is the number of agents) of decreasing restrictiveness that are based on Rawls’ maximin equity criterion and impose welfare lower bounds. These fairness notions were introduced by Porter et al. (J Econ Theory 118:209–228, 2004) who also introduced two classes of Groves mechanisms that are 1-fair and 3-fair, respectively, and generate deficits that are bounded above. We show that these classes are the largest such classes of Groves mechanisms. We generalize these mechanisms for each and show that the corresponding mechanisms generate the smallest deficit for each economy among all k-fair Groves mechanisms. The first draft of this paper was written while we were Ph.D. students at the University of Rochester. We are grateful to William Thomson for his guidance and advice. We also thank two referees and the associate editor for their comments.     

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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The TAL-Family of Rules for Bankruptcy Problems

This paper analyzes a family of rules for bankruptcy problems that generalizes the Talmud rule (T) and encompasses both the constrained equal-awards rule (A) and the constrained equal-losses rule (L). The family is defined by means of a parameter that can be interpreted as a measure of the distributive power of the rule. We provide a systematic study of the structural properties of the rules within the family and its connections with the existing literature.     

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.     

Marriage matching and gender satisfaction

The performance of the Gale–Shapley marriage matching algorithm (Am Math Mon 16:217–222, 1962) has been studied extensively in the special case of men’s and women’s preferences random. We drop the assumption that women’s preferences are random and show that , where R _n is the men’s expected level of satisfaction, that is, the expected sum of men’s rankings of their assigned mates, when the men-propose Gale–Shapley algorithm is used to match n men with n women. This is a step towards establishing a conjecture of Knuth (Mariages Stables et leurs relations avec d’ autres problémes combinatoires, 1976, CRM Proceedings and Lecture Notes, Vol 10, 1997) of 30 years standing. Under the same assumptions, we also establish bounds on the expected rankings by women of their assigned mates.     

$${\mathcal{Q}}$$-anonymous social welfare relations on infinite utility streams

This paper studies a class of social welfare relations (SWRs) on the set of infinite utility streams. In particular, we examine the SWRs satisfying -Anonymity, an impartiality axiom stronger than Finite Anonymity, as well as Strong Pareto and a certain equity axiom. First, we characterize the extension of the generalized Lorenz SWR by combining -Anonymity with Strong Pareto and Pigou–Dalton Equity. Second, we replace Pigou–Dalton Equity with Hammond Equity for characterizing the extended leximin SWR. Third, we give an alternative characterization of the extended utilitarian SWR by substituting Incremental Equity for Pigou–Dalton Equity.     

On the Average Minimum Size of a Manipulating Coalition

We study the asymptotic average minimum manipulating coalition size as a characteristic of quality of a voting rule and show its serious drawback. We suggest using the asymptotic average threshold coalition size instead. We prove that, in large electorates, the asymptotic average threshold coalition size is maximised among all scoring rules by the Borda rule when the number m of alternatives is 3 or 4, and by -approval voting when m ≥ 5.     

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.     

How the size of a coalition affects its chances to influence an election

Since voting rules are prototypes for many aggregation procedures, they also illuminate problems faced by economics and decision sciences. In this paper we are trying to answer the question: How large should a coalition be to have a chance to influence an election? We answer this question for all scoring rules and multistage elimination rules, under the Impartial Anonymous Culture assumption. We show that, when the number of participating agents n tends to infinity, the ratio of voting situations that can be influenced by a coalition of k voters to all voting situations is no greater than $D_{m} \frac{k}{n}Since voting rules are prototypes for many aggregation procedures, they also illuminate problems faced by economics and decision sciences. In this paper we are trying to answer the question: How large should a coalition be to have a chance to influence an election? We answer this question for all scoring rules and multistage elimination rules, under the Impartial Anonymous Culture assumption. We show that, when the number of participating agents n tends to infinity, the ratio of voting situations that can be influenced by a coalition of k voters to all voting situations is no greater than , where D _m is a constant which depends only on the number m of alternatives but not on k and n. Recent results on individual manipulability in three alternative elections show that this estimate is exact for k=1 and m=3.

Arkadii SlinkoEmail:

     

A Geometric Study of Shareholders’ Voting in Incomplete Markets: Multivariate Median and Mean Shareholder Theorems

A simple parametric general equilibrium model with S states of nature and K < S firms is considered. Since markets are incomplete, at a (financial) equilibrium shareholders typically disagree on whether to keep or not the status quo production plans. Hence each firm faces a genuine problem of social choice. The setup proposed in the present paper allows to study these problems within a classical (Downsian) spatial voting model. Given the multidimensional nature of the latter, super majority rules with rate are needed to guarantee existence of politically stable production plans. A simple geometric argument is proposed showing why a 50%-majority stable production equilibrium exists when K=S−1. When the degree of incompleteness is more severe, under more restrictive assumptions on agents’ preferences and the distribution of agents’ types, equilibria are shown to exist for rates ρ smaller than Caplin and Nalebuff (Econometrica 59: 1–23, 1991) bound of 0.64: they obtain for production plans whose span contains the ‘ideal securities’ of all K mean shareholders.Hervé Crès is a member of the GREGHEC, unité CNRS, UMR 2959.     

Heights of representative systems: A proof of Fishburn's conjecture

Representative systems with n-voters are hierarchical choice functions from {-1, 0, 1} ⁿ to {-1, 0, 1} constructed as iterations of weighted majority voting. The height of a representative system is the minimal number of iterations necessary for this construction.In the paper we give an upper bound for (n), the maximal height of any n-voter representative system, and show that goes to zero as n goes to infinity, thus proving a conjecture made by Fishburn. Technically, the results are obtained by transferring the problem to the context of proper simple games, which have a similar hierarchical structure, and using known results on heights of simple games.     

Subgroup independence conditions on preferences

The concept of n-scale independence is introduced for a preference relation defined on ${\mathbb{R}^{n}=\mathbb{R}^{n_{1}}\times \cdots \times \mathbb{R}^{n_{p}}}$ . In addition to zero-independence and upper semicontinuity at zero, n-scale independence allows us to characterizate linear oligarchies as well as to offer a (semi)continuous welfarist analogue of Wilson’s theorem. We also include a characterization of the class of continuous, n-separable and n-scale independent, p ≥ 3, social orderings in terms of what we call homogeneous oligarchies.     

Gray squirrel density,habitat suitability,and behavior in urban parks

Increased density, increased intraspecific aggression, and a reduced fear of humans have been suggested as the more observable and frequently described characteristics of wildlife species undergoing synurbization, the process of becoming urbanized. The relationship among these variables and how they may be related to environmental variables that change with urbanization is poorly understood. In this paper we explore the relationship between density, intraspecific aggression, and reduced fear of humans in urban populations of gray squirrel. In the summer and fall of 2003 and 2004, we studied a park with a documented high density of gray squirrels, Lafayette Park, Washington, DC, and six urban parks in Baltimore, MD with unknown squirrel densities. We used linear regression (SAS Institute, SAS/STAT user’s guide. SAS Institute, Cary, NC, 2005) to determine if there was a relationship (P < 0.05) between squirrel density and intraspecific aggression, squirrel density and reduced fear of humans (wariness), and squirrel density and habitat suitability. We found a positive association between density and intraspecific aggression (R ² = 0.81, P < 0.00). A negative relationship between density and wariness (, P < 0.00). However, no relationship was evident between habitat suitability and squirrel density (, P = 0.437).     

Reproduction, juvenile survival and retention in an urban fox squirrel population

Fox squirrels in an urban environment had an average litter size of 2.83 (95% CI 2.5, 3.16) which was similar to rural populations. Nonetheless, the proportion of squirrels reproductively active during the summer/fall breeding season (0.73) and annually (spring + summer/fall) was higher (1.23) than previously recorded for non-manipulated rural populations. The average monthly survival rate () of urban juvenile fox squirrels during the first 14 weeks of life was higher than reported for adult populations in the area. High rates of reproduction and juvenile survival yielded high juvenile to adult ratios (juveniles/adults) averaging 0.44. Retention of juveniles after 6 months (15%) was significantly less (χ ² = 7.24, p = 0.0071) than adults/subadults (40%). Results suggest that the urban environment provides quality habitat for fox squirrels and fox squirrels in urban environments have the potential to be a source population for surrounding suburban and rural environments.     

Identifying habitat sinks: a case study of Cooper’s hawks in an urban environment

We studied a population of Cooper’s hawks (Accipiter cooperii) in Tucson, Arizona from 1994 to 2005. High rates of mortality of nestlings from an urban-related disease prompted speculation that the area represented an ecological trap and habitat sink for Cooper’s hawks. In this paper, we used estimates of survival and productivity from 11years of monitoring to develop an estimate of the rate of population change, λ, for Cooper’s hawks in the area. We used a Cormack–Jolly–Seber approach to estimate survival of breeding hawks, and a stochastic, stage-based matrix to estimate λ. Despite the urban-related disease, the estimate of λ indicated that the area does not function as a habitat sink for Cooper’s hawks (= 1.11 ± 0.047; P = 0.0073 for the null of λ ≤ 1). Because data required to reliably identify habitat sinks are extensive and difficult to acquire, we suggest that the concept of habitat sinks be applied cautiously until substantiated with reliable empirical evidence.     

Unequivocal majority and Maskin-monotonicity

The unequivocal majority of a social choice rule is a number of agents such that whenever at least this many agents agree on the top alternative, then this alternative (and only this) is chosen. The smaller the unequivocal majority is, the closer it is to the standard (and accepted) majority concept. The question is how small can the unequivocal majority be and still permit the Nash-implementability of the social choice rule; i.e., its Maskin-monotonicity. We show that the smallest unequivocal majority compatible with Maskin-monotonicity is n- ë \fracn-1m û{n-\left\lfloor \frac{n-1}{m} \right\rfloor} , where n ≥ 3 is the number of agents and m ≥ 3 is the number of alternatives. This value is equal to the minimal number required for a majority to ensure the non-existence of cycles in pairwise comparisons. Our result has a twofold implication: (1) there is no Condorcet consistent social choice rule satisfying Maskin-monotonicity and (2) a social choice rule satisfies k-Condorcet consistency and Maskin-monotonicity if and only if k 3 n- ë \fracn-1m û{k\geq n-\left\lfloor \frac{n-1}{m}\right\rfloor}.     

Notions of equal,or equivalent,opportunities

We formulate and study three concepts of equity designed to capture certain notions of equal, or equivalent, opportunities. The central concept is that of a family of choice sets. Given such a family , a feasible allocation z is alternatively required to be such that (i) there is B such that each agent i maximizes his satisfaction in B at z _i , (ii) there is B such that each agent i is indifferent between z _i and the maximizer of his satisfaction in B, (iii) for each agent i there is B _i such that z _i maximizes agent i's satisfaction in the union of the B _j and z _i is in B _i . Most of the standard concepts of equity can be obtained as particular cases of these general definitions by appropriately choosing . We identify conditions on guaranteeing that the resulting allocations be efficient. We apply the definitions to economies with only private goods, and to economies with public goods.This is a revised version of an earlier paper circulated under the title of Notions of equal opportunities. Early drafts were presented at the Conference on Economic Models and Distributive Justice, held in Bruxelles and Namur, January 1987, and at the Public Choice Society meeting in Tucson, Arizona, March 1987. The author thanks the participants, in particular T. Schwartz, for their comments, and NSF for its support, under grant No. 85 11136 and 88 09822. The suggestions of two anonymous referees, D. Diamantaras, L. Gevers, and H. Konishi are also gratefully acknowledged.