Feasible elimination procedures in social choice: An axiomatic characterization

Feasible elimination procedures (Peleg, 1978) play a central role in constructing social choice functions which have the following property: in the associated game form, for any preference profile there exists a strong Nash equilibrium resulting in the sincere outcome. In this paper we provide an axiomatic characterization of the social choice correspondences resulting from applying feasible elimination procedures. The axioms are anonymity, Maskin monotonicity, and independent blocking. We also show that these axioms are logically independent.     

Conditional belief types

We study type spaces where a player's type at a state is a conditional probability on the space. We axiomatize these spaces using conditional belief operators, examining three additional axioms of increasing strength. First, introspection, which requires the agent to be unconditionally certain of her beliefs. Second, echo, according to which the unconditional beliefs implied by the condition must be held given the condition. Third, determination, which says that the conditional beliefs are the unconditional beliefs that are conditionally certain. Echo implies that conditioning on an event is the same as conditioning on the event being certain, which formalizes the standard informal interpretation of conditional probability. The game-theoretic application of our model, discussed within an example, sheds light on a number of issues in the analysis of extensive form games. Type spaces are closely related to the sphere models of counterfactual conditionals and to models of hypothetical knowledge.     

Cooperative oligopoly equilibrium

Oligopoly cooperation is investigated in an industry with N firms: each firm selects its own output. We show that such oligopoly problems have the same structure as problems of allocating public goods since each firm views its own output as a private good while viewing the outputs of others as public bads. This leads to ‘cooperative oligopoly equilibria’ which are the Lindahl equilibria of the oligopoly model. Agreements are modeled by the selection of a mechanism or ‘game form’ which obeys certain axioms. Under the postulated axioms we show the equivalence of agreed allocations with the set of oligopoly equilibria.It is argued that the principle of ‘Lindahl agreements’ may have wider applications.     

Problems of fair division and the Egalitarian solution

The problem of fair division is considered in situations where the number of agents among whom the division is to take place may vary. The approach is axiomatic. Along with four familiar axioms, Weak Pareto-optimality, Symmetry, Independence of Irrelevant Alternatives, and Continuity, a new axiom, named Monotonicity with Respect to Changes in the Number of Agents, is imposed, expressing a certain form of solidarity among the agents as their number varies but the resources at their disposal remain fixed. The only solution to satisfy these axioms is the Egalitarian solution, which selects the only feasible alternative that yields equal utilities to all agents and is undominated by any other feasible alternative.     

AN ALTERNATIVE AXIOMATIZATION OF SEN'S POVERTY MEASURE

We provide an alternative axiomatization of Sen's (1976) poverty measure. We derive the measure from the general definition of a poverty measure by using a version of Sen's rank order axiom, and a substantially weaker form of his normalization axiom. These two axioms, together with a continuity axiom, lead uniquely to the measure suggested by Sen (1976).     

Multidimensional generalized Gini indices

Summary. The axioms that characterize the generalized Gini social evaluation orderings for one-dimensional distributions are extended to the multidimensional attributes case. A social evaluation ordering is shown to have a two-stage aggregation representation if these axioms and a separability assumption are satisfied. In the first stage, the distributions of each attribute are aggregated using generalized Gini social evaluation functions. The functional form of the second-stage aggregator depends on the number of attributes and on which version of a comonotonic additivity axiom is used. The implications of these results for the corresponding multidimensional indices of relative and absolute inequality are also considered.Received: 20 August 2003, Revised: 26 May 2004, JEL Classification Numbers: D63. Correspondence to: John A. WeymarkWe are grateful to our referee for his or her comments.     

A measure of aggregate power in organizations

We present axiomatically an index of aggregate power in organizations, by measuring the overall ability of the agents to induce a change through coalition formation. The axioms are closely related to those used for the Shapley Value. The resulting formula appears as the sum, taken over all possible coalitions, of the products of two probabilities for each coalition: its probability to form and its probability to become winning.     

Dominance in SSB utility theory

SSB utility theory represents preferences between probability measures by the positive part of a skew-symmetric bilinear functional φ defined on pairs of measures. Three facets of the theory are examined. First, dominance axioms are used to extend φ to an integral form. Second, the maximizing behavior of φ on subspaces of measures is investigated. Third, aspects of stochastic dominance are explored in the SSB setting.     

Group identification

I study a model of group identification in which individuals' opinions as to the membership of a group are aggregated to form a list of group members. Potential aggregation rules are studied through the axiomatic approach. I introduce two axioms, meet separability and join separability, each of which requires the list of members generated by the aggregation rule to be independent of whether the question of membership in a group is separated into questions of membership in two other groups. I use these axioms to characterize a class of one-vote rules, in which one opinion determines whether an individual is considered to be a member of a group. I then show that the only anonymous one-vote rule is self-identification, in which each individual determines for himself whether he is a member of the group.     

An axiomatic characterization of the Theil inequality ordering

We axiomatically characterize the Theil ordering of income inequality. In addition to the uncontroversial axioms of anonymity, homogeneity, replication invariance, strong directedness, and a standard continuity property, we appeal to both an independence and a decomposability axioms. These two axioms are ordinal implications of Theil decomposability, the central axiom in previous characterizations of the Theil index. To the best of our knowledge, the present is the first fully ordinal characterization of this index.     

Nest-monotonic two-stage acts and exponential probability capacities

This paper examines conditions for Choquet expected utility (CEU) to satisfy both the reduction of two-stage acts and the recursion axioms, which are taken for granted in economics. A key idea of this paper is to consider nest-monotonic two-stage acts, which share their rankings of states with those of their reduced one-stage acts. Our main theorem shows that the axioms, one of which is restricted to nest-monotonic two-stage acts, and consequentialism are satisfied if and only if the preference is exponential CEU, which is such that the probability capacity is an exponential transformation of a probability measure. This result indicates that within a specified range of decision problems, exponential CEU is the only form of CEU that derives indifference to the timing of information resolution. Furthermore, the relation between first- and second-stage exponential CEU is characterized both by the f*-Bayesian updating rule and by comonotonic dynamic consistency. Conditions to establish the law of iterated expectation for CEU are also discussed.     

The bargaining problem: axioms concerning changes in the conflict point

Two axioms concerning changes in the conflict point are introduced. A sufficient condition for a solution function to satisfy these axioms is that it satisfies the relatively weak adding requirement of Thomson-Myerson (1980).     

The measurement of Poverty: An experimental questionnaire investigation

We re-examine some of the standard axioms used in the literature on poverty measurement. Using a sample of 486 students from Australia, Israel and the USA we investigate the extent to which individuals' perceptions of poverty correspond to the axioms. We find that axioms such as anonymity, growth of the poor and monotonicity are resonably well supported. However there is very little support for the focus axiom and the principle of transfers was the least well supported of the eight specific criteria for poverty measurement that we examined.     

An axiomatic approach to choice under uncertainty with catastrophic risks

This paper analyses decision under uncertainty with catastrophic risks, and is motivated by problems emerging from global environmental risks. These are typically low-probability events with major irreversible consequences. For such risks, the Von Neumann–Morgenstern (NM) axioms for decision making under uncertainty are not appropriate, since they are shown here to be insensitive to low-probability events. The paper introduces an alternative set of axioms requiring sensitivity to both low- and large-probability events. Through a new representation theorem in functional analysis, the results characterize all the operators whose maximization leads to the fulfillment of these axioms. They involve a convex combination of expected utility and a criterion based on the desire to avoid low probability and potentially catastrophic events. It is shown that the new axioms help resolve the Allais paradox. Open questions about risk aversion, games under uncertainty and calculus of variations are discussed.     

Extending an order on a Set to the power set: Some remarks on Kannai and Peleg's approach

Kannai and Peleg have shown that given an ordering over a set, it is impossible to induce an ordering over the power set satisfying certain plausible axioms. We prove an impossibility and also a possibility result in this context with closely related sets of axioms, and argue that the dividing line between impossibility and possibility here is rather thin. Also, we distinguish three possible intuitive interpretations for the formal framework of Kannai and Peleg, and argue that the acceptability of specific formal axioms may crucialy depend on the particular interpretation that one chooses to adopt.     

Ranking opportunity sets in a simple intertemporal framework

We extend the problem of ranking subsets (opportunity sets) of a set of alternatives to an intertemporal setting, whereby the agent makes a sequence of choice decisions over time. In particular, we show in a simple two-period model that an agent who satisfies five plausible axioms ranks opportunity sets in a lexicographic fashion. That is, the agent ranks opportunity sets based solely on the relative desirability of their first-period alternatives, and only if the first-period alternatives are thought equally desirable does the agent consider the second-period alternatives. We discuss this strong time-discounting result and the role of the axioms, and also show that the popular “independence” axioms employed in the existing “static” literature are inconsistent with some reasonable axioms regarding intertemporal choice.      

Preference for flexibility in infinite horizon problems

Summary. We consider the extension of the classical problem of preference for flexibility to many periods. Preferences are defined over sets of infinite paths of choices. The main result provides a set of axioms on preferences that yield an additive representation over a subjective state space. This space is the set of preferences over choice today and feasible set tomorrow. The main new axiom, stochastic dominance, is a stronger form of the assumption of monotonicity. Received: September 11 2000; revised version: December 18, 2001     

An axiomatic definition of the leximin

For a class of multicriterion decision making problems the set of lexicographic minimax in a set of alternatives with regard to the given criterion functions is characterized by six axioms: symmetry, independence of irrelevant alternatives, domination, separability, convexity, and monotone transformation. Each of the axioms is independent of the rest.     

Cost-axiomatic regulatory pricing

The axioms of the Shapley value and semi-values of non-atomic games can be taken as the starting point for axioms on the relation between costs and prices of a regulated multiproduct monopoly. The price mechanisms that result from this system of axioms are called cost- axiomatic prices. This paper shows the applicability of such pricing to all cases of regulatory constraints that require revenue to be a given percentage of costs (including cost-plus regulation, break-even pricing or Amtrak's 50% clause). Moreover, we show that all such price mechanisms, including a fairly general system of taxation or subsidization of consumers, are demand compatible if cost-axiomatic pricing leads to deficits or profits.     

Biodiversity measures based on species-level dissimilarities: A methodology for assessment

Biodiversity is widely recognized as a valuable natural asset to conserve. Yet biodiversity is often reported to be declining worldwide. Biodiversity measures can help evaluating it and conserving it, but need to be clearly defined and assessed. In this paper, I review several biodiversity measures and develop a new one, all based on a matrix of species-level dissimilarity data. The data can be used in its raw form, regardless of its origin (e.g. studies of morphological traits, DNA hybridization experiments…) or of any graphical representation. Then, I propose a two-step assessment of the measures. First, I assess them in terms of their deviation from a strict additive law determining the contribution of each species to the diversity of the set in an ideal setting. This setting refers to a case where the data exactly determines the hierarchical ordering of the species. Second, I assess the measures based on their compliance with a list of axioms. These axioms reflect basic mathematical properties regarded as desirable for diversity measures, such as their monotonicity in species and dissimilarities. Finally, I show the importance of applying the new quantitative assessment and the axiomatic approach together when selecting a dissimilarity-based diversity measure.