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.     

$${\mathcal{Q}}$$-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.     

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.     

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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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.     

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.     

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.     

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:

     

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.     

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.     

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.     

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.     

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).     

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.     

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.     

Invariance axioms for preferences: applications to social choice theory

I investigate the role played by the combination of two invariance axioms for preferences in utility theory; namely, those of zero-independence and scale-independence, respectively. I provide a characterization of the preference relations on ${\mathbb{R}^{n}}$ that satisfy these two axioms as those which are either trivial, or what I call a two-serial total preorder on ${\mathbb{R}^{n}}$ . This result is then applied in social choice theory to characterize those social welfare functions that satisfy IIA and PI. Other characterizations involving the usual Pareto concepts are also provided.     

Rethinking reification

Conclusion By now this essay has accumulated over twenty alleged meanings, senses, or aspects of reification; there seems not much point in listing them. Some of them are mutually consistent, almost overlapping; others are incompatible. Some are in accord with the dictionary definition, others not. With some, the sense in which some entity is being converted into a res is evident; with others, try as I might, I cannot find such a sense. The dictionary entry itself is rather slipshod, and interpreters have expanded the term's meaning almost indefinitely beyond it. The main ways in which Lukács and Berger and Luckmann use the word do not fit into the dictionary definition at all. They make the meaning of reification dependent on concepts like can and could that are themselves heavily dependent on the particular context of their use. Lukács's and Berger and Luckmann's discussions are confusing and very probably confused. The whole thing is a swamp.Can this concept be saved? And should it?Within the limits of the dictionary definition, the concept does its work well enough. When Stephen Gould, say, criticizes psychologists for reifying intelligence by assuming that the I.Q. test must be testing something, he and the concept perform a clear and useful function. But if the main concerns are those of Lukács and Berger and Luckmann, in my opinion the term mystifies more than it reveals. Their concerns could be accommodated within the dictionary definition as reification of persons, in the sense of denying people's capacity for agency. But although Lukács and Berger and Luckmann do occasionally use the word this way, for the most part they do not. So articulating their concerns within this sense of the word would require extensive rewriting of their arguments. Would political theorists who share those concerns not do better to abandon the concept?I would unhesitatingly advise it, except for one, crucial consideration. There really is something going on among us that we urgently need to think and talk about, and that Lukács's and Berger and Luckmann's conceptions of reification were meant to address. People do feel trapped, in a way that makes Kafka's little fable so perfectly emblematic for our experience. Despite the prevalence in modern society of all of Berger and Luckmann's social circumstances that favor dereification, very many people do feel helpless to influence the conditions that constrain their lives. Millions of Americans turn their backs on politics, judging that engagement in it would make no significant difference. Millions of members of the underclass feel worthless - though also filled with diffuse rage - because society seems to have no use for them. Almost all of us function in large organizational systems, whether as parts of the machinery or materials being processed, and have learned to take that condition for granted. We function within an economy that depends on a system of international banking and finance that everyone knows to be in constant danger of collapse. Almost all of us submit without question to the technological imperative that daily exhausts our resources, destroys our health, and poisons the earth. And we march like sleepwalkers down the road marked deterrence and nonproliferation, toward nuclear doom. Experts and critics offer various diagnoses of our condition, but whatever measures are actually taken to treat it seem only to make it worse.This familiar litany of troubles suggests a malaise far too extensive and too grave for the powers of political prudence. When a society, or an entire civilization, or even the whole human species seems bent on self-destruction, one suspects systematic, pervasive, fundamental derangement in people's patterns of both thought and conduct. Calling on political prudence here is almost bound to mean calling for more of the same. Here what is needed is a more basic realignment of assumptions, of the sort that has traditionally been associated with great political theory.Wading through the dismal swamp of reification theory, as this essay does, can leave one feeling that such concepts, and political theory itself, are hopelessly abstracted from reality and of no practical use in relation to our urgent political problems, so that political prudence is the only hope. But political reality itself, and the prudence by which gifted actors know how to move within it, always presuppose and depend on theoretical frameworks - if not self-conscious, deliberate theorizing, then unexamined, inherited theory or, more likely, fragments of theories that may well be outdated or mutually inconsistent. So if we seem today bereft of political prudence and judgment, close to our wits' end, that may be because our wits are operating out of such incoherent fragments of inherited assumptions.The message to be derived from the familiar litany of our troubles and our sleepwalking, then, is not the familiar exhortation to, For God's sake, do something before it is too late! For while we may feel inert, we are already doing something - a lot of things - and they are the source of our troubles. Like Kafka's mouse, we run and run. Berger and Luckmann's and Lukács's concept of reification was meant to address precisely those troubles that are the large-scale outcome of our myriad activities, sustained and enlarged by nothing more than what we do. The problem is how to stop, how to do something else, what else to do.That is a problem as much for thought as for action, a problem for action informed and empowered by new thought. Part of the value of Berger and Luckmann's - and even more of Lukács's - discussion of reification is that they tried to provide a general theory of the nature and roots of our condition, orienting us to likely avenues for action, feasible ways and means, probable allies and opponents. Their efforts, this essay has argued, were confused and deeply flawed. The concept of reification is probably not a good tool for the job, and bad tools mean sloppy work. But better sloppy work with a bad tool than no work at all. Those of us who persist in reaching for the word reification as a tool should probably employ greater care. We should require ourselves to specify in each case precisely what we mean, and attend to whether and how our various meanings in various contexts are interrelated. But whether we revise the concept of reification, or abandon it, or just let it continue to slop along in its present state of dishevelment doesn't much matter. What matters is that we continue to think - hard and critically, theoretically and politically - about the conditions that Lukács and Berger and Luckmann were trying to address.Our thinking here must be simultaneously theoretical and political: theoretical in the sense of radical, cutting through conventions and cliches to the real roots of our troubles, seeing social arrangements large-scale and long-range, as if from the outside, which may be what Lukács meant by intending totality. Yet the thinking must also be political, in the sense of oriented to action, practical, speaking in a meaningful way to those capable of making the necessary changes, those Lukács called the we of genesis. For Lukács, of course, that meant the proletariat. But one need not be a Marxist to see the need for locating such a we, and the point of seeking it among those with an objective interest in the right sort of change and the potential power to bring it about. The aim is not some new doctrine to save us from ourselves, but a transformed way of seeing what we already tacitly know and do, which restores us to our real world - the concrete here and now, as Lukács puts it - and our real, living selves, including our capacities for action. That would mean not some access to mysterious, infinite powers, but the appropriation of our actual powers, recognizing the present moment as the moment of decision, the moment of the birth of the new, as Lukács says, out of which we jointly make the future. That is no return to Hegelian idealism, but a recovery of the practical, political Marx.Thinking both theoretically and politically in this way is no easy task; indeed, it is almost a contradiction in terms. Yet it may well be our best hope, and the world is in a hurry. Despite all of the political and philosophical difficulties, unless we undertake this task we may well guarantee our own entrapment, assuring that we will end up like Kafka's mouse, rather than human and free.

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Discussion of Anna Ornstein's paper

Summary I have used only a few themes in Dr. Ornstein's rich paper to suggest another range of perspectives. While the theories of self and ego psychology differ, there are in every person overlapping elements which we have called self and ego. We need to continue to advance our understanding of each in the service of advancing our capacities to treat.The Shirley Greenberg Memorial Lecture delivered at Simmons College, School of Social Work Boston, Massachusetts on June 9, 1983