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.     

Bourdieu, Smith and disinterested judgement

The paper presents a sympathetic critique of Bourdieu7rsquo;s work in terms of the tension between its critical intentions and its leanings towards sociological reductionism. Although Bourdieu argues against such reductionism in his methodological pronouncements, his empirical studies tend to reduce actors' putative disinterested judgements to functions of their habitus in relation to the social field and to unconscious strategies of distinction. Further, his concept of (non-monetary) forms of capital occludes the difference between use-value and exchange-value and the corresponding distinction between the pursuit of goods and the pursuit of distinction, which are vital for both explanation and critique. Moreover his suspicion of normative judgement on the part of social science and his concealment of his own normative standpoint subvert his critiques. Thus in relation to Bourdieu's analysis of the role of mis-recognition in social life I argue that this requires a delineation of the extent of justified recognition. In developing the argument I draw upon Adam Smith's analysis of moral sentiments and his critique of undeserved recognition and the pursuit of distinction. Where Bourdieu is dismissive about moral issues, Smith treats moral sentiments as irreducible to interest or instrumental action and as a significant element in the reproduction of social order. The paper concludes with some implications for the nature of critique in social theory.     

Recognition and judgement in social work

The purpose of this article is to argue for recognition as a normative ideal for social work and to confront the ideal with the reality found in the social institutions. I shall use the concept ‘judgement’ to describe institutional routines and ways of thinking which constitute barriers to recognition. In the first part, I outline the normative ideal and show its relevance for practical social work on the basis of social clients’ experiences of disrespect. In the second, I explain the concept of judgement and criticise the prevailing forms of judgement to be found in social institutions. In my interpretation these pose an obstacle to recognition.     

Strongly consistent game forms

This paper characterizes strongly consistent game forms in terms of power structure and cardinality (i.e. the number of persons, alternatives or strategies). There are three main new results in this paper. First, I give a necessary condition for a game form to be strongly consistent; it is also sufficient in several cases. Second, from the view point of informational requirements and based on the first result, I give a sharp lower bound on the cardinality of each agent's strategy space, for a game form to be strongly consistent and individually rational. Third, I show that the unanimous rule is the unique game form that attains this lower bound for strong implementation of the individually rational Pareto correspondence.     

Quasi-transitive and Suzumura consistent relations

We examine properties of binary relations that complement quasi-transitivity and Suzumura consistency in the sense that they, in conjunction with the original axiom(s), are equivalent to transitivity. In general, the conjunction of quasi-transitivity and Suzumura consistency is weaker than transitivity but in the case of collective choice rules that satisfy further properties, this conjunction implies transitivity of the social relation. We prove this observation by characterizing the Pareto rule as the only collective choice rule such that collective preference relations are quasi-transitive and Suzumura consistent, and standard social choice axioms are satisfied.     

Contraction consistent stochastic choice correspondence

We model a general choice environment via probabilistic choice correspondences, with (possibly) incomplete domain and infinite universal set of alternatives. We offer a consistency restriction regarding choice when the feasible set contracts. This condition, ‘contraction consistency’, subsumes earlier notions such as Chernoff’s Condition, Sen’s α and β, and regularity. We identify a restriction on the domain of the stochastic choice correspondence (SCC), under which contraction consistency is equivalent to the weak axiom of revealed preference in its most general form. When the universal set of alternatives is finite, this restriction is also necessary for such equivalence. Analogous domain restrictions are also identified for the special case where choice is deterministic but possibly multi-valued. Results due to Sen (Rev Econ Stud 38:307–317, 1971) and Dasgupta and Pattanaik (Econ Theory 31:35–50, 2007) fall out as corollaries. Thus, conditions are established, under which our notion of consistency, articulated only in reference to contractions of the feasible set, suffices as the axiomatic foundation for a general revealed preference theory of choice behaviour.     

Consumers' surpluses and consistent cost-benefit tests

In this paper we show that even if consumers' surpluses are known precisely, there are severe difficulties in using them to make consistent social-welfare judgements. In order for the Hicksian surpluses to be used, consumers must face the same prices and preferences must have affine parallel Engel curves. The cost-benefit rule must be the (possibly weighted) sum of the surpluses. We also analyze the Marshallian measure.We are indebted to Alan Abouchar, David Maloney, Robert Pollak, John Weymark, Robert Willig and two anonymous referees for helpful comments and criticisms. We also wish to thank the Social Sciences and Humanities Research Council of Canada for research support     

Minimal conversely consistent extension of the men-optimal solution

This study pertains to two-sided, one-to-one matching problems and considers the best-known solution concept: the men-optimal solution. The men-optimal solution fails to satisfy consistency as well as converse consistency. Furthermore, the minimal consistent extension of the men-optimal solution equals the core. In this article, we compute the minimal conversely consistent extension of the men-optimal solution as a correspondence which associates with each problem the set consisting of the men-optimal matching, and all stable and men-barterproof matchings for this problem.     

Essentially lexicographic aggregation

In the paper a new approach to lexicography is developed by which in the general framework of ordered blocks with a monotonic basis it is shown that a nontrivial ordering is translation-invariant if and only if it is essentially lexicographic of degree n. Here, the latter means that the ordering can be represented by an ordinary lexicographic ordering in n dimensions. As an application it is shown that a nontrivial social welfare ordering on Euclidean space possesses a useful invariance property (cardinality and non comparability) if and only if the ordering is essentially lexicographic of a strong kind in that it can be obtained from ordinary lexicography by permutation, cutting-off and order reversal with respect to components. This result generalizes the characterization of lexical individual dictatorship obtained by Gevers and d'Aspremont and it provides, within the social welfare approach, a strong version of Arrow's impossibility theorem by not invoking any Pareto principle at all.The author thanks K. Arrow, W. Gaertner, L. Gevers, and two anonymous referees for helpful hints and suggestions.     

Topological aggregation of inequality preorders

An inequality preorder is defined as a complete preorder on a simplex which satisfies the properties of continuity and strict Schur-convexity (the mathematical equivalent of Dalton's principle of transfers). The paper shows that it is possible to aggregate individual inequality preorders into a collective one if we are interested in continuous anonymous aggregation rules that respect unanimity. The aggregation problem is studied within a topological framework introduced by Chichilnisky.We are grateful to J. Laine for helpful discussions and to an anonymous referee of this Journal for valuable comments     

Strategy-proofness of continuous aggregation maps

 We consider continuous aggregation maps (e.g., social welfare functions). By assuming that the voters have preferences over social outcomes, we regard the social decision procedure as a noncooperative game, with the aggregation map as a game form. The map is called strategy-proof if it is a Nash equilibrium that the voters report their most preferred outcomes. We give sufficient topological conditions on the space of outcomes so that only dictatorial maps are strategy-proof. Received: 28 February 1994/Accepted: April 22, 1996     

The instability of robust aggregation

Levis robust aggregation suggests a generalization of the standard ex ante mode of aggregating individuals expected utilities into a social choice function. This generalization applies some social choice rule not only to the actual individuals but also to hypothetical individuals with all possible combinations of the actual individuals probabilities and utilities. We show that robust aggregation can lead to preference reversals in the social choice when the very same decision situation is described with different degrees of detail, thus violating a desirable stability condition. We also show that any stable robust social choice rule must collapse back into the ex ante mode without recourse to hypothetical individuals. Under sufficient conditions, such as strong Pareto optimality, robust aggregation even leads to an infinite series of reversals in group choices. By contrast, stability is ubiquitous in the standard ex ante mode and is closely related to independence of irrelevant alternatives.I wish to thank Christs College, Cambridge and the California Institute of Technology for their generous financial and academic support. I am grateful to three anonymous referees for their helpful comments. I am much indebted to Isaac Levi, Philippe Mongin and Teddy Seidenfeld for their help and encouragement.     

Information and preference aggregation

We investigate the implications of relaxing Arrow's independence of irrelevant alternatives axiom while retaining transitivity and the Pareto condition. Even a small relaxation opens a floodgate of possibilities for nondictatorial and efficient social choice. Received: 20 August 1997/Accepted: 29 September 1998     

The Representation of Geometric Cues in Infancy

There is evidence that, from an early age, humans are sensitive to spatial information such as simple landmarks and the size of objects. This study concerns the ability to represent a particular kind of spatial information, namely, the geometry of an enclosed layout—an ability present in older children, adults, and nonhuman animals (e.g., Cheng, 1986; Hermer & Spelke, 1996). Using a looking‐time procedure, 4.5‐ to 6.5‐month‐olds were tested on whether they could distinguish among the corners of an isosceles triangle. On each trial, the target corner was marked by a red dot. The stimulus (triangle with dot) appeared from different orientations across trials, ensuring that only cues related to the triangle itself could be used to differentiate the corners. When orientations were highly variable, infants discriminated the unique corner (i.e., the corner with the smaller angle and two equal‐length sides) from a nonunique corner; they could not discriminate between the two nonunique corners. With less variable orientations, however, infants did discriminate between the nonunique corners of the isosceles triangle. Implications for how infants represent geometric cues are discussed.