Rationalizability of choice functions on general domains without full transitivity

The rationalizability of a choice function by means of a transitive relation has been analyzed thoroughly in the literature. However, not much seems to be known when transitivity is weakened to quasi-transitivity or acyclicity. Such weakenings are particularly relevant in the context of social choice. We describe the logical relationships between the different notions of rationalizability involving, for example, the transitivity, quasi-transitivity, or acyclicity of the rationalizing relation. Furthermore, we discuss sufficient conditions and necessary conditions for rational choice on arbitrary domains. Transitive, quasi-transitive, and acyclical rationalizability are fully characterized for domains that contain all singletons and all two-element subsets of the universal set.     

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).     

A characterization of the nucleolus without homogeneity in airport problems

We consider the problem of sharing the cost of a public facility among agents who have different needs for the facility. We show that the nucleolus is the only rule satisfying equal treatment of equals, last-agent cost additivity, and consistency.     

Measurement theory and the foundations of utilitarianism

This article reconsiders the Harsanyi–Sen debate concerning whether Harsanyi is justified in interpreting his Aggregation and Impartial Observer Theorems as providing axiomatizations of utilitarianism. Sen's criticism and its formalization by Weymark are based on the claim that von Neumann–Morgenstern utility theory is ordinal, whereas Harsanyi's utilitarian conclusions require cardinal utility. Proposals for overcoming Sen's objection that appeal to formal measurement theory are considered. It is argued that one of these proposals due to Broome and Risse rightly points to a feature of expected utility theory that was ignored by Sen and Weymark, but that this proposal does not provide a normatively compelling justification for cardinal utility. The other proposal due to Broome is shown to make use of a strength of preference relation in addition to the axioms of expected utility theory.     

Egalitarianism and utilitarianism in committees of representatives

We consider a committee of representatives that makes dichotomous choices (acceptance/rejection) by vote. Given the size of each group represented, what is the most adequate voting rule for the committee? We provide answers based on each of the two principles commonly used to make normative assessments in different contexts: egalitarianism and utilitarianism. To that end, we introduce utilities into the model and adopt a normative approach.     

Reciprocity,transitivity, and the mysterious three-cycle

Reciprocity and transitivity are the two most important structural mechanisms underlying friendship network evolution. While on their own they are understood in great detail, the relation between them is rarely studied systematically. Are friendships outside of social groups more or less likely to be reciprocated than friendships embedded in a group? Using a theoretical framework that focusses on the situations in which friends interact and the social structures that stabilise one-sided friendships, I propose that the tendency towards reciprocation of friendships within transitive groups is usually lower than outside of transitive groups. In a meta-analysis of two datasets including 29 friendship networks using stochastic actor-oriented models (SAOMs), the interaction between reciprocity and transitivity is analysed. Supporting the theoretical reasoning, the interaction is consistently negative. Second, the tendency against forming three-cycles in friendship networks, which was consistently found in previous studies, is shown to be spurious and a result of neglecting to control for the tendency against reciprocation in transitive groups. The tendency against three-cycles is commonly seen as an indicator that unreciprocated friendships indicate local hierarchy differences between individuals; this proposition has to be re-evaluated in light of the findings of this study. Future studies that analyse the evolution of friendship networks should consider modelling reciprocation in transitive triplets and potentially omit modelling three-cycles.     

Removing the anonymity axiom in assessing pro-poor growth

The recent focus on ‘pro-poor growth’ led also to an intense debate on how exactly to define and to measure pro-poor growth. All suggested measures have in common that they are based on the anonymity axiom. Such a perspective may provide a very incomplete picture given that the common objective of most studies investigating the pro-poorness of growth is to test whether specific policy reforms where beneficial to the initially poor or not. I suggest a new concept of pro-poor growth which removes the anonymity axiom, and, using an illustration based on data from Indonesia and Peru, I check whether the assessment of pro-poor growth is different when an anonymous and a non anonymous approach to pro-poor growth is used. I also suggest an original decomposition of poverty changes over time which links both concepts. The results show that the choice of the approach has a drastic impact on the interpretation of the data.     

A new approach to rights in social choice theory which incorporates utilitarianism

The paper axiomatises a generalised utilitarian aggregation rule, under which different weights are assigned to utilities depending on the different rights involved. The relationship between actions, rights and the evaluation of utilities is investigated. Application is made to a famous example, Edwin-Angelina-the Judge, which appears in the social choice literature.     

Weighting the transitivity of undirected weighted social networks with triadic edge dissimilarity scores

When an individual is socially connected to two others, the resulting triplet can be closed (if the two social partners are themselves connected) or open (if they are not connected). The proportion of closed triplets, referred to as the binary network transitivity, is a classic measure of the level of interconnectedness of a social network. However, in any given triplet, if the closing link is weak, or indeed if any of the links in the triplet is weak, then the triplet should not contribute as much to network transitivity as if all three links were equally strong. I propose two ways to weight the contribution of each triplet according to the dissimilarity between the three links in the triplet. Empirically, the resulting new metrics conveyed information not picked up by any other network-level metric. I envision that this approach could prove useful in studies of triadic mechanisms, i.e., situations where pre-existing social ties influence the interactions with third parties. These metrics could also serve as repeatable synthetic variables that summarize information about the variability of the strength of social connections.     

A characterization of the Walras rule

This paper has three purposes. First, we refine the characterization of the Walras rule proposed by Nagahisa (JET 1991) over a more natural and simple domain than the one he employed. We show that the Walras rule is the only social choice rule defined over the domain and satisfying Individual Rationality, Pareto Efficiency, and Local Independence. Second, assuming endowments to be collectively owned, we show that the Walras rule operated from equal division is the only social choice rule satisfying No Envy, Pareto Efficiency, and Local Independence. Third, we show that for every social choice rule satisfying Individual Rationality and Pareto Efficiency, Local Independence is equivalent to a condition of Nash implementation with a game form satisfying convexity.This article is a revised version of Toyama University Working Paper No. 141. We are grateful to Professors William Thomson, Shinsuke Nakamura, Tomoichi Shinotsuka and two anonymous referees for their detailed comments. Nagahisa is grateful for hospitality of the economics department of the University of Rochester.     

A characterization of the single-crossing domain

We characterize single-crossing preference profiles in terms of two forbidden substructures, one of which contains three voters and six (not necessarily distinct) alternatives, and one of which contains four voters and four (not necessarily distinct) alternatives. We also provide an efficient way to decide whether a preference profile is single-crossing.     

A characterization of the single-peaked domain

We provide in this paper two properties that are both necessary and sufficient to characterize the domain of single-peaked preference profiles. This characterization allows for a definition of single-peaked preference profiles without using an ad hoc underlying order of the alternatives and also sheds light on the structure of single-peaked profiles. Considering the larger domain of value-restricted preference profiles (Sen, Econometrica 34:491–499, 1966) we also provide necessary and sufficient conditions for a preference profile to be single-caved or group-separable. Our results show that for single-peaked, single-caved and group-separable profiles it is sufficient to restrict to profiles containing of either three individuals and three alternatives or two individuals and four alternatives.     

A characterization of the Nash bargaining solution

We characterize the Nash bargaining solution replacing the axiom of Independence of Irrelevant Alternatives with three independent axioms: Independence of Non-Individually Rational Alternatives, Twisting, and Disagreement Point Convexity. We give a non-cooperative bargaining interpretation to this last axiom. Received: 4 September 2000/Accepted: 6 September 2001     

A prudent characterization of the Ranked Pairs Rule

We show that the Ranked Pairs Rule is equivalent to selecting the maximal linear orders with respect to a DiscriMin relation, which is a natural refinement of the Min relation used to define Arrow and Raynaud’s prudent orders. We provide an axiomatic characterization of the Ranked Pairs Rule by building on an earlier characterization of the prudent order ranking rule. We conclude that a monotonicity criterion is the main distinction between the two ranking rules.     

A characterization of prudent choices

We define and axiomatize prudent choices for two criteria. Given two criteria, the alternatives chosen by the prudent choice procedure are the ones maximizing some composition of the criteria. This composition is such that (1) it contains the first criterion and a part of the second one, and (2) the new binary relation is not cyclic and cannot be enlarged with preferences of the second criterion without becoming cyclic. We also make the link between prudent choices, classical rational choices, sequentially rational choices (Manzini and Mariotti in Am Econ Rev 97(5):1824–1839, 2007a) and lexicographic binary choice rational choices (Tadenuma in J Econ Theory 104(2):462–472, 2002).