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.     

Strategy-proof resolute social choice correspondences

We qualify a social choice correspondence as resolute when its set valued outcomes are interpreted as mutually compatible alternatives which are altogether chosen. We refer to such sets as “committees” and analyze the manipulability of resolute social choice correspondences which pick fixed size committees. When the domain of preferences over committees is unrestricted, the Gibbard–Satterthwaite theorem—naturally—applies. We show that in case we wish to “reasonably” relate preferences over committees to preferences over committee members, there is no domain restriction which allows escaping Gibbard–Satterthwaite type of impossibilities. We also consider a more general model where the range of the social choice rule is determined by imposing a lower and an upper bound on the cardinalities of the committees. The results are again of the Gibbard–Satterthwaite taste, though under more restrictive extension axioms.     

The majoritarian compromise is majoritarian-optimal and subgame-perfect implementable

It is shown that the Majoritarian Compromise of Sertel (1986) is subgame-perfect implementable on the domain of strict preference profiles, although it fails to be Maskin-monotonic and is hence not implementable in Nash equilibrium. The Majoritarian Compromise is Pareto-optimal and obeys SNIP (strong no imposition power), i.e. never chooses a strict majority's worst candidate. In fact, it is “majoritarian approving” i.e. it always picks “what's good for a majority” (alternatives which some majority regards as among the better “effective” half of the available alternatives). Thus, being Pareto-optimal and majoritarian approving, it is majoritarian-optimal. Finally, the Majoritarian Compromise is measured against various criteria, such as consistency and Condorcet-consistency. Received: 31 January 1995/Accepted: 22 July 1998     

Characterizations of two power indices for voting games with r alternatives

In the first three sections of this paper we present a set of axioms which provide a characterization of an extension of the Banzhaf index to voting games with r alternatives, such as the United Nations Security Council where a nation can vote “yes”, “no”, or “abstain”. The fourth section presents a set of axioms which characterizes a power index based on winning sets instead of pivot sets. Received: 4 April 2000/Accepted: 30 April 2001     

Aggregation of smooth preferences

Suppose p is a smooth preference profile (for a society, N) belonging to a domain P ^N . Let σ be a voting rule, and σ(p)(x) be the set of alternatives in the space, W, which is preferred to x. The equilibrium E(σ(p)) is the set {x∈W:σ(p)(x) is empty}. A sufficient condition for existence of E(σ(p)) when p is convex is that a “dual”, or generalized gradient, dσ(p)(x), is non-empty at all x. Under certain conditions the dual “field”, dσ(p), admits a “social gradient field”Γ(p). Γ is called an “aggregator” on the domain P ^N if Γ is continuous for all p in P ^N . It is shown here that the “minmax” voting rule, σ, admits an aggregator when P ^N is the set of smooth, convex preference profiles (on a compact, convex topological vector space, W) and P ^N is endowed with a C ¹-topology. An aggregator can also be constructed on a domain of smooth, non-convex preferences when W is the compact interval. The construction of an aggregator for a general political economy is also discussed. Some remarks are addressed to the relationship between these results and the Chichilnisky-Heal theorem on the non-existence of a preference aggregator when P ^N is not contractible. Received: 4 July 1995 / Accepted: 26 August 1996     

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.     

On the Arrow and Wilson impossibility theorems

A social welfare function is Arrovian if it is transitive-valued and satisfies IIA. We examine the logical relation between the statements (A) If f is Arrovian with domain 𝒫 then it is dictatorial if it satisfies the Pareto criterion and (W) If f is Arrovian with domain 𝒫 then it is dictatorial or inversely dictatorial if it is non-null and satisfies non-imposition. We exhibit a class of domains on which W implies A and another on which A implies W. We also present examples of domains on which one of the statements is true and the other is false. Received: 23 October 2001/Accepted: 19 March 2002 We thank our two referees for their suggestions for improving the exposition.     

Consistent judgement aggregation: the truth-functional case

Generalizing the celebrated “discursive dilemma”, we analyze judgement aggregation problems in which a group of agents independently votes on a set of complex propositions (the “conclusions”) and on a set of “premises” by which the conclusions are truth-functionally determined. We show that for conclusion- and premise-based aggregation rules to be mutually consistent, the aggregation must always be “oligarchic”, that is: unanimous within a subset of agents, and typically even be dictatorial. We characterize exactly when consistent non-dictatorial (or anonymous) aggregation rules exist, allowing for arbitrary conclusions and arbitrary interdependencies among premises.     

Rational choice and revealed preference without binariness

 This paper attempts to provide a unified account of the rationalization of possibly non-binary choice-functions by “Extended Preference Relations” (relations between sets and elements). The analysis focuses on transitive EPRs for which three choice-functional characterizations are given, two of them based on novel axioms. Transitive EPRs are shown to be rationalizable by sets of orderings that are “closed under compromise”; this novel requirement is argued to be the key to establish a canonical relationship between sets of orderings and choice-functions. The traditional assumption of “binariness” on preference relations or choice functions is shown to be analytically unhelpful and normatively unfounded; non-binariness may arise from “unresolvedness of preference”, a previously unrecognized aspect of preference incompleteness. Received: 28 August 1995/Accepted: 14 February 1996     

Endogenous Voting Agendas

Existence of a “simple” pure strategy subgame perfect equilibrium is established in a model of endogenous agenda formation and sophisticated voting; upper hemicontinuity of simple equilibrium outcomes is demonstrated; and connections to the set of undominated, or “core,” alternatives are examined. In one dimension with single-peaked preferences, the simple equilibrium outcome is essentially unique and lies in the core, providing a game-theoretic foundation for the median voter theorem in terms of endogenous agenda setting. Existence of equilibrium relies on a general characterization of sophisticated voting outcomes in the presence of “majority-ties,” rather than the standard tie-breaking convention in voting subgames in favor of the alternative proposed later. The model is illustrated in a three-agent distributive politics setting, and it is shown there that the standard tie-breaking convention leads to non-existence of equilibrium.     

On the multi-preference approach to evaluating opportunities

The purpose of the paper is to provide a general framework for analyzing “preference for opportunities.” Based on two simple axioms a fundamental result due to Kreps is used in order to represent rankings of opportunity sets in terms of multiple preferences. The paper provides several refinements of the basic representation theorem. In particular, a condition of “closedness under compromise” is suggested in order to distinguish the flexibility interpretation of the model from normative interpretations which play a crucial role in justifying the intrinsic value of opportunities. Moreover, the paper clarifies the link between the multiple preference approach and the “choice function” approach to evaluating opportunities. In particular, it is shown how the well-known Aizerman/Malishevski result on rationalizability of choice functions can be obtained as a corollary from the more general multiple preference representation of a ranking of opportunity sets. Received: 3 September 1996 / Accepted: 18 August 1997     

Garfinkel before ethnomethodology

A short story titled “‘Color Trouble’” by Harold Garfinkel was published inOpportunity in 1940,The Best Short Stories 1941, andPrimer for White Folks in 1945. Garfinkel wrote this short story before World War II while a research fellow at the University of North Carolina, Chapel Hill under Howard W. Odum, the founder ofSocial Forces “‘Color Trouble’” narrates poignantly the racial victimization of a young black woman traveling on a public bus through the State of Virginia. The short story provides sociologists with a different medium through which to examine the seminal interests of ethnomethodology’s founder. In a literary form, the short story depicts such ethnomethodological concepts as the breaching experiment, the “et cetera clause,” “ad hocing,” and the status degradation ceremony. Garfinkel’s “‘Color Trouble’” also suggests the way in which ethnomethodology overlaps with, as well as diverges from, Erving Goffman’s dramaturgical perspective. He received his doctoral degree from the graduate program in sociology at York University, Toronto, Ontario. His article “Autonomy and Responsibility in Social Theory” will appear inCurrent Perspectives in Social Theory, Volume 10.     

A Canadian rejoinder: Sociology north and south of the border

In response to the recent The American Sociologist special issue on Canadian sociology, this rejoinder dialogues with some of the perspectives offered there on the discipline north of the border with an eye towards lessons that American sociologists might learn from the Canadian experience. My reflections build on a larger analytic piece entitled “Canada’s Impossible Science: The Historical and Institutional Origins of the Coming Crisis of Anglo-Canadian Sociology” to be published soon in The Canadian Journal Sociology. Particular attention is paid to the different institutional arrangements of higher education in Canada and the United States, Anglo-Canadian reliance on the particularly English “weakness as strength” strategy for sociology, tensions between the cultural values of populism, egalitarianism, and excellence, and the trade-offs between professional and public intellectual work. A critique is offered of the “origin myth” of Canadian sociology as a particularly vibrant “critical sociology,” with discussion of Dorothy Smith's influence on sociology in Canada. His research interests are in sociological theory, the sociology of culture, and the study of intellectuals from the perspective of the sociology of organisations and professions. He is studying Edward Said as a “global public intellectual” as part of a Canadian government-funded interdisciplinary grant on “Globalization and Autonomy” at McMaster University. He is also working “Canadian professors as public intellectuals,” a project also funded by the Social Science and Humanities Research Council of Canada.     

Manipulability, unanimity, anonymity and merging functions

A merging (or assessment aggregation) function (see [7]) is a rule that synthesizes several individual assessments, or numerical judgements, by creating a unique “collective” assessment. Individual i can manipulate assessment aggregation if the change in the merged assessment due to a change in i's assessment from x to y depends solely on x and y. Some assumptions on the functional form capturing this dependence are put forward and their effects on aggregation investigated under unanimity and anonymity conditions using a functional equation approach. Attention is restricted to three types of results: the merging function does not exist; it is the arithmetic mean; it is dictatorial. Received: 21 November 1997/Accepted: 31 May 1999     

Money-egalitarian-equivalent and gain-maximin allocations of indivisible items with monetary compensation

Suppose that a certain quantity M of money and a finite number of indivisible items are to be distributed among n people, all of whom have equal claims on the whole. Different allocations are presented using various criteria of fairness in the special case where each player's utility function is additively separable. An allocation is “money-egalitarian-equivalent” (MEE) if each player's monetary valuation of his or her bundle is a fixed constant. We show that there is an essentially unique allocation that is MEE and Pareto-optimal; it is also envy-free. Alternatively, the “gain” of a player may be defined as the difference between how the player evaluates his bundle and an exact nth part of the whole according to his numerical evaluation of the whole. A “gain-maximin” criterion would maximize the minimum gain obtained by any player. We show that Knaster's procedure finds an allocation which is optimal under the gain-maximin criterion. That allocation is not necessarily envy-free, so we also find the envy-free allocation that is optimal under the gain-maximin criterion among all envy-free allocations. It turns out that, even though there exist allocations that are simultaneously envy-free and Pareto-optimal, this optimal allocation may fail to be Pareto-optimal, and it may also violate monotonicity criteria. Received: 30 September 1996/Accepted: 6 March 2002 The author would like to thank Professor William Thomson for a discussion on this subject; and he would like to thank the anonymous referees, who made many substantive suggestions for improving this paper – shortening it, streamlining the arguments, improving the terminology, making further ties with the literature, and improving the exposition.     

Mental Health of Non-gamblers Versus “Normal” Gamblers Among American Indian Veterans: A Community Survey

Goal This analysis was undertaken to assess the demographic and mental health characteristics of “normal” or non-problem gamblers versus non-gamblers in a representative community sample. Sample Study participants consisted of 557 North Central American Indian veterans. Data collection included a demographic and trauma questionnaire, a computer-based Diagnostic Interview Schedule for DSM-III-R, and a treatment history algorithm. Findings Univariate analyses revealed that gamblers had greater social competence (i.e., higher education, living with a spouse) and higher lifetime psychiatric morbidity. Binary regression analysis revealed that, compared to non-gamblers, gamblers were older, more highly educated, and more apt to be married. More gamblers showed evidence for lifetime risk-taking as evidenced by Antisocial Personality Disorder and Tobacco Dependence. Conclusions Social achievement and disposable income function as prerequisites for “normal” gambling in this population, although “externalizing” or “risk-taking” disorders also serve as independent contributors to at least some gambling. The increased rate of “internalizing” or emotional disorders are only indirectly related to gambling, perhaps through increasing age or through the “externalizing” disorders.     

The C1 topology on the space of smooth preference profiles

This paper defines a fine C ¹-topology for smooth preferences on a “policy space”, W, and shows that the set of convex preference profiles contains open sets in this topology.  It follows that if the dimension(W)≤v(?)−2 (where v(?) is the Nakamura number of the voting rule, ?), then the core of ? cannot be generically empty. For higher dimensions, an “extension” of the voting core, called the heart of ?, is proposed. The heart is a generalization of the “uncovered set”. It is shown to be non-empty and closed in general. On the C ¹-space of convex preference profiles, the heart is Paretian. Moreover, the heart correspondence is lower hemi-continuous and admits a continuous selection. Thus the heart converges to the core when the latter exists. Using this, an aggregator, compatible with ?, can be defined and shown to be continuous on the C ¹-space of smooth convex preference profiles. Received: 3 April 1995/Accepted: 8 April 1998     

On the equilibrium of voting games with abstention and several levels of approval

We study the core of “(j, k) simple games”, where voters choose one level of approval from among j possible levels, partitioning the society into j coalitions, and each possible partition facing k levels of approval in the output (Freixas and Zwicker in Soc Choice Welf 21:399–431, 2003). We consider the case of (j, 2) simple games, including voting games in which each voter may cast a “yes” or “no” vote, or abstain (j = 3). A necessary and sufficient condition for the non-emptiness of the core of such games is provided, with an important application to weighted symmetric (j, 2) simple games. These results generalize the literature, and provide a characterization of constitutions under which a society would allow a given number of candidates to compete for leadership without running the risk of political instability. We apply these results to well-known voting systems and social choice institutions including the relative majority rule, the two-thirds relative majority rule, the United States Senate, and the United Nations Security Council.     

‘One and a Half Dimensional’ Preferences and Majority Rule

I examine a model of majority rule in which alternatives are described by two characteristics: (1) their position in a standard, left-right dimension, and (2) their position in a good-bad dimension, over which voters have identical preferences. I show that when voters’ preferences are single-peaked and concave over the first dimension, majority rule is transitive, and the majority’s preferences are identical to the median voter’s. Thus, Black’s (The theory of committees and elections, 1958) theorem extends to such a “one and a half” dimensional framework. Meanwhile, another well-known result of majority rule, Downs’ (An economic theory of democracy, 1957) electoral competition model, does not extend to the framework. The condition that preferences can be represented in a one-and-a-half-dimensional framework is strictly weaker than the condition that preferences be single-peaked and symmetric. The condition is strictly stronger than the condition that preferences be order-restricted, as defined by Rothstein (Soc Choice Welf 7:331–342;1990).     

What is a problem?

Among others, the term “problem” plays a major role in the various attempts to characterize interdisciplinarity or transdisciplinarity, as used synonymously in this paper. Interdisciplinarity (ID) is regarded as “problem solving among science, technology and society” and as “problem orientation beyond disciplinary constraints” (cf. Frodeman et al.: The Oxford Handbook of Interdisciplinarity. Oxford University Press, Oxford, 2010). The point of departure of this paper is that the discourse and practice of ID have problems with the “problem”. The objective here is to shed some light on the vague notion of “problem” in order to advocate a specific type of interdisciplinarity: problem-oriented interdisciplinarity. The outline is as follows: Taking an ex negativo approach, I will show what problem-oriented ID does not mean. Using references to well-established distinctions in philosophy of science, I will show three other types of ID that should not be placed under the umbrella term “problem-oriented ID”: object-oriented ID (“ontology”), theory-oriented ID (epistemology), and method-oriented ID (methodology). Different philosophical thought traditions can be related to these distinguishable meanings. I will then clarify the notion of “problem” by looking at three systematic elements: an undesired (initial) state, a desired (goal) state, and the barriers in getting from the one to the other. These three elements include three related kinds of knowledge: systems, target, and transformation knowledge. This paper elaborates further methodological and epistemological elements of problem-oriented ID. It concludes by stressing that problem-oriented ID is the most needed as well as the most challenging type of ID.