A note on the redundancy of an axiom in the Pattanaik-Peleg characterization of the lexicographic maximin extension

P. K. Pattanaik and B. Peleg have provided an axiomatic characterization of the lexicographic maximin extension of an ordering over a finite set to an ordering over the set of all non-empty subsets of that set. This note shows that one of their axioms, namely the Union axiom, is redundant, thus obtaining a more economical characterization composed of the remaining four axioms (which are independent by virtue of the examples given by Pattanaik and Peleg).     

Reference functions and possibility theorems for cardinal social choice problems

 In this paper, we provide axiomatic foundations for social choice rules on a domain of convex and comprehensive social choice problems when agents have cardinal utility functions. We translate the axioms of three well known approaches in bargaining theory (Nash 1950; Kalai and Smorodinsky 1975; Kalai 1977) to the domain of social choice problems and provide an impossibility result for each. We then introduce the concept of a reference function which, for each social choice set, selects a point from which relative gains are measured. By restricting the invariance and comparison axioms so that they only apply to sets with the same reference point, we obtain characterizations of social choice rules that are natural analogues of the bargaining theory solutions. Received: 8 August 1994/Accepted: 12 February 1996     

Axiomatic derivation of scoring rules without the ordering assumption

Earlier derivations of scoring rules, by Smith (1973) and Young (1975), assumed that a voter can express only a rank ordering of the alternatives on his or her ballot. This paper shows that scoring rules can be derived without this ordering assumption. It is shown that a voting rule must be a scoring rule if it satisfies three basic axioms: reinforcement, overwhelming majorities, and neutrality. Other range and nonreversal axioms are also discussed.     

Congruence indicators for fuzzy choice functions

We introduce the congruence indicators WFCA(·) and SFCA(·) corresponding to fuzzy congruence axioms WFCA and SFCA. These indicators measure the degree to which a fuzzy choice function verifies the axioms WFCA and SFCA, respectively. The main result of the paper establishes for a given choice function the relationship between its congruence indicators and some rationality conditions. One obtains a fuzzy counterpart of the well-known Arrow–Sen theorem in crisp choice functions theory.     

Equality of Opportunity versus Equality of Opportunity Sets

We characterize two different approaches to the idea of equality of opportunity. Roemer’s social ordering is motivated by a concern to compensate for the effects of certain (non-responsibility) factors on outcomes. Van de gaer’s social ordering is concerned with the equalization of the opportunity sets to which people have access. We show how different invariance axioms open the possibility to go beyond the simple additive specification implied by both rules. This offers scope for a broader interpretation of responsibility-sensitive egalitarianism.     

Market arbitrage, social choice and the core

 This paper establishes a clear connection between equilibrium theory, game theory and social choice theory by showing that, for a well defined social choice problem, a condition which is necessary and sufficient to solve this problem – limited arbitrage – is the same as the condition which is necessary and sufficient to establish the existence of an equilibrium and the core. The connection is strengthened by establishing that a market allocation, which is in the core, can always be realized as a social allocation, i.e. an allocation which is optimal according to an ordering chosen by a social choice rule. Limited arbitrage characterizes those economies without Condorcet triples, and those for which Arrow’s paradox can be resolved on choices of large utility values. Received: 30 December 1994/Accepted: 12 August 1996     

The informational basis of the theory of fair allocation

The theory of fair allocation is often favourably contrasted with the social choice theory in the search for escape routes from Arrow’s impossibility theorem. Its success is commonly attributed to the fact that it is modest in its goal vis-à-vis social choice theory, since it does not aspire for a full-fledged ordering of options, and settles with a subset of ‘fair’ options. We show that its success may rather be attributable to a broadened informational basis thereof. To substantiate this claim, we compare the informational basis of the theory of fair allocation with the informational requirements of social choice theory.This paper is derived from a part of an earlier draft of our paper entitled ‘Informational requirements for social choice in economic environments’. The authors thank A. Trannoy, an associate editor and three referees for comments, and participants at seminars in University of Cergy-Pontoise, University of Rochester, Hitotsubashi University, and Waseda University, and the 5th International Meeting of the Society for Social Choice and Welfare in Alicante. Financial support from the Ministry of Education of Japan through Grant-in-Aid No. 10045010 (‘Economic Institutions and Social Norms: Evolution and Transformation’) and the 21st Century Center of Excellence Project on the Normative Evaluation and Social Choice of Contemporary Economic Systems is gratefully acknowledged.     

Pairwise Strategy-Proofness and Self-Enforcing Manipulation

“Strategy-proofness” is one of the axioms that are most frequently used in the recent literature on social choice theory. It requires that by misrepresenting his preferences, no agent can manipulate the outcome of the social choice rule in his favor. The stronger requirement of “group strategy-proofness” is also often employed to obtain clear characterization results of social choice rules. Group strategy-proofness requires that no group of agents can manipulate the outcome in their favors. In this paper, we advocate “effective pairwise strategy-proofness.” It is the requirement that the social choice rule should be immune to unilateral manipulation and “self-enforcing” pairwise manipulation in the sense that no agent of a pair has the incentive to betray his partner. We apply the axiom of effective pairwise strategy-proofness to three types of economies: public good economy, pure exchange economy, and allotment economy. Although effective pairwise strategy-proofness is seemingly a much weaker axiom than group strategy-proofness, effective pairwise strategy-proofness characterizes social choice rules that are analyzed by using different axioms in the literature.     

Choosing subsets: a size-independent probabilistic model and the quest for a social welfare ordering

“Subset voting” denotes a choice situation where one fixed set of choice alternatives (candidates, products) is offered to a group of decision makers, each of whom is requested to pick a subset containing any number of alternatives. In the context of subset voting we merge three choice paradigms, “approval voting“ from political science, the “weak utility model” from mathematical psychology, and “social welfare orderings” from social choice theory. We use a probabilistic choice model proposed by Falmagne and Regenwetter (1996) built upon the notion that each voter has a personal ranking of the alternatives and chooses a subset at the top of the ranking. Using an extension of Sen's (1966) theorem about value restriction, we provide necessary and sufficient conditions for this empirically testable choice model to yield a social welfare ordering. Furthermore, we develop a method to compute Borda scores and Condorcet winners from subset choice probabilities. The technique is illustrated on an election of the Mathematical Association of America (Brams, 1988). Received: 18 August 1995 / Accepted: 13 February 1997     

Lattices of choice functions and consensus problems

In this paper we consider the three classes of choice functions satisfying the three significant axioms called heredity (H), concordance (C) and outcast (O). We show that the set of choice functions satisfying any one of these axioms is a lattice, and we study the properties of these lattices. The lattice of choice functions satisfying (H) is distributive, whereas the lattice of choice functions verifying (C) is atomistic and lower bounded, and so has many properties. On the contrary, the lattice of choice functions satisfying (O) is not even ranked. Then using results of the axiomatic and metric latticial theories of consensus as well as the properties of our three lattices of choice functions, we get results to aggregate profiles of such choice functions into one (or several) collective choice function(s).The authors thank two anonymous referee for several useful remarks or corrections on the first version of the paper.     

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.     

Multi-profile intergenerational social choice

In an infinite-horizon setting, Ferejohn and Page showed that any social welfare function satisfying Arrow’s axioms and stationarity must be a dictatorship of the first generation. Packel strengthened this result by proving that no collective choice rule generating complete social preferences can satisfy unlimited domain, weak Pareto and stationarity. We prove that this impossibility survives under a domain restriction and without completeness. We propose an alternative stationarity axiom and show that a social welfare function on a specific domain satisfies this modified version and some standard social choice axioms if and only if it is a chronological dictatorship.     

Social compromise and social metrics

In the context of constitutional design, a committee of citizens' representatives proposes (social choice) axioms. The outcome should not necessarily be the aggregation rule, or rules, satisfying each of the axioms. Alternative procedures are recommended for aggregating a set of axioms into a single summary axiom. The summary axiom generates a single social decision function when a social metric is applied.     

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.     

Utility and the individual: An analysis of internal conflicts

Herbert Simon has argued that economic agents do not maximise a transitive ordering but rather adopt what he calls satisficing behaviour. He argues that such behaviour is rational. This paper aims to provide a theory to support Simon's viewpoint. Because of social choice problems an individual with multiple objectives may find it impossible to construct a transitive ordering and hence may have to adopt satisficing type behaviour. We show that whether or not an individual adopts satisficing behaviour depends on the information available.I would like to thank Amartya Sen, the members of a seminar in Oxford, and an anonymous referee for their comments on previous drafts of this paper.     

Social choice without the Pareto principle: a comprehensive analysis

This article provides a systematic analysis of social choice theory without the Pareto principle, by revisiting the method of Murakami Yasusuke. This article consists of two parts. The first part investigates the relationship between rationality of social preference and the axioms that make a collective choice rule either Paretian or anti-Paretian. In the second part, the results in the first part are applied to obtain impossibility results under various rationality requirements of social preference, such as S-consistency, quasi-transitivity, semi-transitivity, the interval-order property, and acyclicity.     

Freedom of choice in a social context: comparing game forms

In this paper, the set of outcomes of game forms is introduced as the relevant attribute for evaluating freedom of choice. These sets are defined as the cartesian product of every individual’s set of available options. It is argued that doing so is one way of taking into account social interactions when evaluating individual freedom. A set of axioms is introduced, each of which conveys some intuitions about how interactions affect freedom of choice, axioms by the mean of which two criteria, the Max and the MaxMin, are characterised for comparing game forms in terms of the freedom of choice they offer. These criteria are based, respectively, on the comparison of the best and the worst outcome the individual can reach in the game form.     

Psychology Implies Paternalism? Bounded Rationality may Reduce the Rationale to Regulate Risk-Taking

Behavioral economists increasingly argue that violations of rationality axioms provide a new rationale for paternalism – to “de-bias” individuals who exhibit errors, biases and other allegedly pathological psychological regularities associated with Tversky and Kahneman’s (in Science 185:1124–1131, 1974) heuristics-and-biases program. The argument is flawed, however, in neglecting to distinguish aggregate from individual rationality. The aggregate consequences of departures from normative decision-making axioms may be Pareto-inferior or superior. Without a well-specified theory of aggregation, individual-level biases do not necessarily imply losses in efficiency. This paper considers the problem of using a social-welfare function to decide whether to regulate risk-taking behavior in a population whose individual-level behavior may or may not be consistent with expected utility maximization. According to the social-welfare objective, unregulated aggregate risk distributions resulting from non-maximizing behavior are often more acceptable (i.e., lead to a weaker rationale for paternalism) than population distributions generated by behavior that conforms to the standard axioms. Thus, psychological theories that depart from axiomatic decision-making norms do not necessarily strengthen the case for paternalism, and conformity with such norms is generally not an appropriate policy-making objective in itself.     

Twofold optimality of the relative utilitarian bargaining solution

Given a bargaining problem, the relative utilitarian (RU) solution maximizes the sum total of the bargainer’s utilities, after having first renormalized each utility function to range from zero to one. We show that RU is “optimal” in two very different senses. First, RU is the maximal element (over the set of all bargaining solutions) under any partial ordering which satisfies certain axioms of fairness and consistency; this result is closely analogous to the result of Segal (J Polit Econ 108(3):569–589, 2000). Second, RU offers each person the maximum expected utility amongst all rescaling-invariant solutions, when it is applied to a random sequence of future bargaining problems generated using a certain class of distributions; this is recalls the results of Harsanyi (J Polit Econ 61:434–435, 1953) and Karni (Econometrica 66(6):1405–1415, 1998).     

Ranking opportunity sets: An approach based on the preference for flexibility

We describe a criterion to evaluate subsets of a finite set of alternatives which are considered as opportunity sets. The axioms for set comparison are motivated within the preference for flexibility framework. We assume the preference over the universal set of alternatives to be made of two disjoint binary relations. The result is the axiomatic characterization of a procedure which is formally similar to the leximax ordering, but in our case it incorporates the presence of some uncertainty about the decision-maker final tastes. Received: 20 January 1999/Accepted: 20 October 1999