Positionalist voting functions

Positionalist voting functions are those social choice functions where the positions of the alternatives in the voter's preference orders crucially influence the social ordering of the alternatives. An important subclass consists of those voting functions where numbers are assigned to the alternatives in the preference orders and the social ordering is computed from these numbers. Such voting functions are called representable. Various well-known conditions for voting functions are introduced and it is investigated which representable voting functions satisfy these conditions. It is shown that no representable voting function satisfies the Condorcet criterion. This condition and Arrow's independence condition, which are typical non-positionalist conditions, are shown to be incompatible. The Borda function, which is a well-known positionalist voting function, is studied extensively, conditions uniquely characterizing it are given and some modifications of the function are investigated.My thanks are due to professor Bengt Hansson for encouragement and several helpful suggestions.     

Nondictatorial social welfare functions with different discrimination structures

In societies where agents distinguish imperfectly among the alternatives, a necessary and sufficient condition for the existence of an Arrovian Social Welfare Function is the presence of an agent who only distinguishes two groups of alternatives or of two agents with complementary discrimination patterns. This result is based on the observation that differences in discrimination structures may lead to the absence of free triples, thus providing a way to escape Arrow's impossibility result.     

Aggregation of preferences: The fuzzy case

The basic purpose of this paper is to link both theorems of impossibility and existence by introducing fuzzy relations of preference and an exogeneous requirement, the planner's one, and then proving the fundamental part played by the extremist agents, leximin and leximax. In other words, to bring out the link between the planner's requirement and the difficulty of the transition from individual to collective, as well as the theoric relation between this requirement and the extremist agents, we define a fuzzy behavior of preference which allows us to build up two determinant fuzzy coalitions. These coalitions will be the base of the planner's requirement and the link between pessimistic results (Arrow's impossibility) and optimistic ones (May's theorem of majority choice).     

On some suggestions for having non-binary social choice functions

The various paradoxes of social choice uncovered by Arrow [1], Sen [10] and others have led some writers to question the basic assumption of a binary social choice function underlying most of these paradoxes. Schwartz [8], for example, proves an important theorem which may be considered to be a generalization of the famous paradox of Arrow, and then lays the blame for this paradox on the assumption of a binary social choice function. He then proceeds to define a type of choice functions which, like binary choice functions, define the best elements in sets of more than two alternatives on the basis of binary comparisons, but which, as he claims, have an advantage over binary choice functions, in so far as they always ensure the existence of best elements for sets of more than two alternatives irrespective of the results of binary comparisons. The purpose of this paper is to show that even a considerable weakening of the assumption of a binary social choice function does not go very far towards solving some of the paradoxes under consideration, and that if replacing the requirement of a binary social choice function by a Schwartz type social choice function solves these paradoxes, it does so only by violating the universally acceptable value judgment that in choosing from a set of alternatives, society should never choose an alternative which is Pareto inoptimal in that set (i.e., the socially best alternatives in a set should always be Pareto optimal). This argument is substantiated with the help of an extended version of Sen's [10] paradox of a Paretian liberal, and thus a by-product of our analysis is a generalization of the theorem of Sen [10]. The argument itself, however, is more general and applies also to the impossibility result proved by Schwartz [8].We are extremely grateful to Amartya Sen for his helpful comments.

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Bayesian decision theory,rule utilitarianism,and Arrow's impossibility theorem

The first part of this paper reexamines the logical foundations of Bayesian decision theory and argues that the Bayesian criterion of expected-utility maximization is the only decision criterion consistent with rationality. On the other hand, the Bayesian criterion, together with the Pareto optimality requirement, inescapably entails a utilitarian theory of morality. The next sections discuss the role both of cardinal utility and of cardinal interpersonal comparisons of utility in ethics. It is shown that the utilitarian welfare function satisfies all of Arrow's social choice postulates avoiding the celebrated impossibility theorem by making use of information which is unavailable in Arrow's original framework. Finally, rule utilitarianism is contrasted with act utilitarianism and judged to be preferable for the purposes of ethical theory.     

Even-chance lotteries in social choice theory

This paper discusses aspects of the theory of social choice when a nonempty choice set is to be determined for each situation, which consists of a feasible set of alternatives and a preference order for each voter on the set of nonempty subsets of alternatives. The individual preference assumptions include ordering properties and averaging conditions, the latter of which are motivated by the interpretation that subset A is preferred to subset B if and only if the individual prefers an even-chance lottery over the basic alternatives in A to an even-chance lottery over the basic alternatives in B. Corresponding to this interpretation, a choice set with two or more alternatives is resolved by an even-chance lottery over these alternatives. Thus, from the traditional no-lottery social choice theory viewpoint, ties are resolved by even-chance lotteries on the tied alternatives. Compared to the approach which allows all lotteries to compete along with the basic alternatives, the present approach is a contraction which allows only even-chance lotteries.After discussing individual preference axioms, the paper examines Pareto optimality for nonempty subsets of a feasible set in a social choice context with n voters. Aspects of simple-majority comparisons in the even-chance context follow, including an analysis of single-peaked preferences. The paper concludes with an Arrowian type impossibility theorem that is designed for the even-chance setting.     

Intradimensional Single-Peakedness and the Multidimensional Arrow Problem

Arrow's account (1951/1963) of the problem of social choice is based upon the assumption that the preferences of each individual in the relevant group are expressible by a single ordering. This paper lifts that assumption and develops a multidimensional generalization of Arrow's framework. I show that, like Arrow's original framework, the multidimensional generalization is affected by an impossibility theorem, highlighting not only the threat of dictatorship of a single individual, but also the threat of dominance of a single dimension. In particular, even if preferences are single-peaked across individuals within each dimension – a situation called intradimensional single-peakedness – any aggregation procedure satisfying Arrow-type conditions will make one dimension dominant. I introduce lexicographic hierarchies of dimensions as a class of possible aggregation procedures under intradimensional single-peakedness. The interpretation of the results is discussed.     

A challenge to the compound lottery axiom: A two-stage normative structure and comparison to other theories

This paper examines preferences among uncertain prospects when the decision maker is uneasy about his assignment of subjective probabilities. It proposes a two-stage lottery framework for the analysis of such prospects, where the first stage represents an assessment of the vagueness (ambiguity) in defining the problem's randomness and the second stage represents an assessment of the problem for each hypothesized randomness condition. Standard axioms of rationality are prescribed for each stage, including weak ordering, continuity, and strong independence. The Reduction of Compound Lotteries' axiom is weakened, however, so that the two lottery stages have consistent, but not collapsible, preference structures. The paper derives a representation theorem from the primitive preference axioms, and the theorem asserts that preference-consistent decisions are made as if the decision maker is maximizing a modified expected utility functional. This representation and its implications are compared to alternative decision models. Criteria for assigning the relative empirical power of the alternative models are suggested.     

On the possibility of rational policy evaluation

Arrow proved the inconsistency of a set of reasonable looking conditions on a social decision rule. These conditions are stated and their rationale explained. It is argued that the blame for the inconsistency must lie with Arrow's Collective Rationality condition. Arrow's abstract problem is generalized and interpreted in terms of individual as well as collective decision-making. His conditions are revised so that (1) cardinal - even interpersonal - utility comparisons are allowed and (2) the Collective Rationality condition - which formulates the traditional conception of rational choice as maximizing choice - is weakened to its bare bones. The revised set of conditions is still inconsistent. Once again the culprit is the Collective Rationality condition, now drastically weakened: even the bare bones of Arrow's conception of rational choice as maximizing choice is untenable. An alternative conception is proposed.     

Power of Enforcement and Dictatorship

A new Arrovian impossibility is obtained without invoking independence of irrelevant alternatives type assumptions. The new conditions leading to the impossibility are based on the concept of power of enforcement, and specify how this power can (see A3) or cannot be expanded (see A1, A2 and A4).     

How relevant are ‘Irrelevant’ Alternatives?

Arrow's Independence of Irrelevant Alternatives Condition is examined. It is shown why the standard rationale for (or against) the condition tends to be inconclusive as it fails to consider the basic game issue in social choice. Specifically it is explained how some recent results (Gibbard-Satterthwaite) on the general non-existence of strategy-proof voting procedures provide the strongest rationale for the independence condition. Also, it is shown that this rationale was exactly the one used by Condorcet in his work on decision rules for juries and elections.     

The impossibility of unbiased judgment aggregation

Standard impossibility theorems on judgment aggregation over logically connected propositions either use a controversial systematicity condition or apply only to agendas of propositions with rich logical connections. Are there any serious impossibilities without these restrictions? We prove an impossibility theorem without requiring systematicity that applies to most standard agendas: Every judgment aggregation function (with rational inputs and outputs) satisfying a condition called unbiasedness is dictatorial (or effectively dictatorial if we remove one of the agenda conditions). Our agenda conditions are tight. When applied illustratively to (strict) preference aggregation represented in our model, the result implies that every unbiased social welfare function with universal domain is effectively dictatorial.     

What are the minimal requirements of rational choice? Arguments from the sequential-decision setting

There are at least two plausible generalisations of subjective expected utility (SEU) theory: cumulative prospect theory (which relaxes the independence axiom) and Levi’s decision theory (which relaxes at least ordering). These theories call for a re-assessment of the minimal requirements of rational choice. Here, I consider how an analysis of sequential decision making contributes to this assessment. I criticise Hammond’s (Economica 44(176):337–350, 1977; Econ Philos 4:292–297, 1988a; Risk, decision and rationality, 1988b; Theory Decis 25:25–78, 1988c) ‘consequentialist’ argument for the SEU preference axioms, but go on to formulate a related diachronic-Dutch-book-style’ argument that better achieves Hammond’s aims. Some deny the importance of Dutch-book sure losses, however, in which case, Seidenfeld’s (Econ Philos 4:267–290, 1988a) argument that distinguishes between theories that relax independence and those that relax ordering is relevant. I unravel Seidenfeld’s argument in light of the various criticisms of it and show that the crux of the argument is somewhat different and much more persuasive than what others have taken it to be; the critical issue is the modelling of future choices between ‘indifferent’ decision-tree branches in the sequential setting. Finally, I consider how Seidenfeld’s conclusions might nonetheless be resisted.     

Individual rationality and the concept of social welfare

This paper, written in October 1974, deals with some game aspects of the social choice problem. The question asked is whether there exists a social decision rule satisfying the conditions imposed by Arrow over all the preference profiles that may logically arise under it (in the sense of being compatible with individual rationality). This question is answered in the affirmative. The meaning of this result is that if Arrow's condition of unrestricted domain is modified so as to exclude any profile which contradicts individual rationality, then an Arrovian social welfare function can be shown to exist (subject to the assumption that whenever the social outcome is in doubt, individuals use the maximin criterion in order to choose their voting strategy).This research was written up in October 1974 at Northwestern University and was partially supported by the National Science Foundation.     

Independent social choice correspondences

A fixed agenda social choice correspondence on outcome set X maps each profile of individual preferences into a nonempty subset of X. If satisfies an analogue of Arrow's independence of irrelevant alternatives condition, then either the range of contains exactly two alternatives, or else there is at most one individual whose preferences have any bearing on . This is the case even if is not defined for any proper subset of X.     

Conditional Desirability

Conditional attitudes are not the attitudes an agent is disposed to acquire in event of learning that a condition holds. Rather they are the components of agent's current attitudes that derive from the consideration they give to the possibility that the condition is true. Jeffrey's decision theory can be extended to include quantitative representation of the strength of these components. A conditional desirability measure for degrees of conditional desire is proposed and shown to imply that an agent's degrees of conditional belief are conditional probabilities. Rational conditional preference is axiomatised and by application of Bolker's representation theorem for rational preferences it is shown that conditional preference rankings determine the existence of probability and desirability measures that agree with them. It is then proven that every conditional desirability function agrees with an agent's conditional preferences and, under certain assumptions, every desirability function agreeing with an agent's conditional preferences is a conditional desirability function agreeing with her unconditional preferences.     

Implementation without rationality assumptions

Hurwicz (Social Choice and public decision making. Essays in honor of Kenneth J. Arrow, Cambridge University Press, Cambridge, 1986) was the first to study an approach to implementation theory based on choice functions instead of preference relations. We argue that the solution concept used by him, the generalized Nash equilibrium, is not really compatible with the idea that individual behavior is describable by a choice function. A new solution concept that better fits the choice function framework is then introduced. Using this, we investigate what behavioral assumptions are needed for the full characterizations of Nash implementable social choice correspondences to still hold. We will show that a condition known as Property α is central.     

Rationality of indecisive choice functions on triadic choice domains

Necessary and sufficient conditions are obtained for weak order representations of indecisive choice functions on domains containing all triples of (possibly non-distinct) choice alternatives, i.e., triadic choice domains. These conditions are compared with Arrow's formulation of the Weak and Strong Axioms of Revealed Preference and with Richter's Congruence Axiom.This research was supported by the National Science Foundation under Grant No. GS-35640X. The author is indebted to the referee of an earlier draft of this paper for numerous helpful comments.     

The impossibility of a paretian loyalist

Amartya Sen has argued the impossibility of the Paretian liberal. While his abstract argument is compelling, the concrete significance of the conclusion is in some doubt. This is because it is not clear how important liberalism in his sense is; in particular it is not clear that the sort of liberalism required for the impossibility result is a compelling variety. We show that even if the argument cannot be used to establish the inconsistency of Paretianism and common-or-garden liberalism, it can be adapted to prove a parallel impossibility. This is the impossibility of combining the Pareto criterion with a loyalty constraint involving certain claim-rights rather than liberty-rights. The impossibility of the Paretian loyalist is of interest in itself but it is also interesting for the light it throws on the source of Sen-style impossibilities.     

Use-value and exchange-value

Discussion of the relation between exchange-value and use-value (as defined inCapital I) is clarified by the construction of set-theoretical models of these concepts. Marx argues fallaciously for the independence of exchange-value and use-value. His fallacy is diagnosed as depending upon a mistaken assumption about the impossibility of inferring a certain linear order on a set from a certain (different) partial order on that set.