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This paper begins by observing that any reflexive binary (preference) relation (over risky prospects) that satisfies the independence axiom admits a form of expected utility representation. We refer to this representation notion as the coalitional minmax expected utility representation. By adding the remaining properties of the expected utility theorem, namely, continuity, completeness, and transitivity, one by one, we find how this representation gets sharper and sharper, thereby deducing the versions of this classical theorem in which any combination of these properties is dropped from its statement. This approach also allows us to weaken transitivity in this theorem, rather than eliminate it entirely, say, to quasitransitivity or acyclicity. Apart from providing a unified dissection of the expected utility theorem, these results are relevant for the growing literature on boundedly rational choice in which revealed preference relations often lack the properties of completeness and/or transitivity (but often satisfy the independence axiom). They are also especially suitable for the (yet overlooked) case in which the decision‐maker is made up of distinct individuals and, consequently, transitivity is routinely violated. Finally, and perhaps more importantly, we show that our representation theorems allow us to answer many economic questions that are posed in terms of nontransitive/incomplete preferences, say, about the maximization of preferences, the existence of Nash equilibrium, the preference for portfolio diversification, and the possibility of the preference reversal phenomenon.

This paper develops axiomatically a revealed preference theory of reference-dependent choice behavior. Instead of taking the reference for an agent as exogenously given in the description of a choice problem, we suitably relax the Weak Axiom of Revealed Preference to obtain, endogenously, the existence of reference alternatives as well as the structure of choice behavior conditional on those alternatives. We show how this model captures some well-known choice patterns such as the attraction effect. (JEL D11, D81)

For a society that consists of at least three individuals, we show that a social choice rule is Maskin monotonic if and only if it is Nash implementable by means of a mechanism that is stochastic or a mechanism that contains (arbitrary) awards. In equilibrium, the mechanisms do not have any stochastic elements and do not involve any awards. Thus, loosely speaking, one can drop the no-veto power postulate from Maskin's classic theorem on Nash implementability, provided that the notion of a mechanism is suitably generalized, thereby narrowing the gap between the properties of Maskin monotonicity and Nash implementability. Moreover, using the standard notion of a mechanism, we prove that Nash implementability is equivalent to Maskin monotonicity with renegotiation: this is a pure improvement over a well-known result of Maskin and Moore [Maskin, E., Moore, J., 1999. Implementation and renegotiation, Rev. Econ. Studies 66, 39–56].

We derive a representation theorem for time preferences (on the prize-time space) which identifies a novel notion of relative discounting as the key ingredient. This representation covers a variety of time preference models, including the standard exponential and hyperbolic discounting models and certain non-transitive time preferences, such as the similarity-based and subadditive discounting models. Our axiomatic work thus unifies a number of seemingly disparate time preference structures, thereby providing a tractable mathematical format that allows for investigating certain economic environments without subscribing to a particular time preference model. This point is illustrated by means of an application to sequential bargaining theory.

We study if (and when) the intersections and unions of social choice rules that are implementable with respect to a certain equilibrium concept are themselves implementable with respect to that equilibrium concept. Our results for dominant strategy equilibrium are mostly of negative nature; similarly, the intersection of Nash implementable SCCs need not be Nash implementable. On the other hand, we find that the union of any set of Nash implementable social choice rules is Nash implementable (for societies of at least three constituents). This last observation allows us to formulate the notion of the largest Nash implementable subcorrespondence of a social choice rule.

A commonly held belief in economics is that an individual's preferences that are revealed by her choices must be complete. This paper takes issue with this position by showing that one may be able to distinguish between indifference and indecisiveness of an agent upon observing her choice behavior. We relax the standard Weak Axiom of Revealed Preferences (WARP) and show that a potent theory of individual choice (with and without risk) can be founded on this weaker axiom when it is coupled with some other standard postulates. The most notable features here are that an agent is allowed to be indifferent between certain alternatives and indecisive about others. Moreover, an outside observer can identify which of these actually occur upon examining the (observable) choice behavior of the decision maker. As an application, we also show how our theory may be able to cope with the classical preference reversal phenomenon.

We weaken the no-veto power condition of Maskin [Maskin, E.,1999. Nash equilibrium and welfare optimality. Rev. Econ. Stud. 66, 23–38] to limited veto power, and prove that any monotonic social choice rule is Nash implementable if it satisfies this weaker condition. The result is obtained by using the canonical Maskin mechanism without modification . An immediate corollary is that the weak core is Nash implementable in any coalitional game environment. An example is given to show that the strong core need not be implementable, even when it is monotonic.

We consider the problem of representing a (possibly) incomplete preference relation by means of a vector-valued utility function. Continuous and semicontinuous representation results are reported in the case of preference relations that are, in a sense, not “too incomplete.” These results generalize some of the classical utility representation theorems of the theory of individual choice and paves the way towards developing a consumer theory that realistically allows individuals to exhibit some “indecisiveness” on occasion. Journal of Economic Literature Classification Number: D11.

We introduce a procedural model of risky choice in which an individual is endowed with a core preference relation that may be highly incomplete. She can, however, derive further rankings of alternatives from her core preferences by means of a procedure based on the independence axiom. We find that the preferences that are generated from an initial set of rankings according to this procedure can be represented by means of a "set" of von Neumann-Morgenstern utility functions, thereby allowing for incompleteness of preference relations. The proposed theory also yields new characterizations of the stochastic dominance orderings. Copyright 2002 by the Economics Department of the University of Pennsylvania and Osaka University Institute of Social and Economic Research Association

We study the evolution of preferences via payoff monotonic dynamics in strategic environments with and without complete information. It is known that, with complete information and subgroup matching, empirically plausible interdependent preference relations may entail the local instability of individualistic preferences (which target directly the maximization of material payoffs/fitness). In contrast, under incomplete information (unobservability of preference types), we show that independent preferences are globally stable in a large set of environments and locally stable in essentially any standard environment, provided that the number of subgroups that form in the society is large. Since these results are obtained within the context of a very general model, they may be thought of as providing an evolutionary rationale for the prevalence of individualistic preferences. Journal of Economic Literature Classification Numbers: C72, D42.

We axiomatically investigate the problem of rationalizing bargaining solutions by social welfare functions that are linear in every rank-ordered subset of R n + . Such functions, the so-called Choquet integrals, have been widely used in the theories of collective and individual choice. We refer to bargaining solutions that can be rationalized by Choquet integrals as Choquet bargaining solutions. Our main result is a complete characterization of Choquet bargaining solutions. As a corollary of our main result, we also obtain a characterization of the generalized Gini bargaining solutions introduced by Blackorby et al. (1994, Econometrica 62 , 1161–1178). Journal of Economic Literature Classification Numbers: D71, C78.

We study certain classes of supermodular and submodular games which are symmetric with respect to material payoffs but in which not all players seek to maximize their material payoffs. Specifically, a subset of players have negatively interdependent preferences and care not only about their own material payoffs but also about their payoffs relative to others. We identify sufficient conditions under which members of the latter group have a strategic advantage in the following sense: at all intragroup symmetric equilibria of the game, they earn strictly higher material payoffs than do players who seek to maximize their material payoffs. The conditions are satisfied by a number of games of economic importance. We discuss the implications of these findings for the evolutionary theory of preference formation and the theory of strategic delegation. Journal of Economic Literature Classification Numbers: C72, D62.

This paper studies the conditions under which the basic results of the revealed preference theory can be established on the domain of choice problems which include non-convex feasible sets; the exercise is closely related to the works of Peters and Wakker (1991) and Bossert (1994). We show that while no continuous choice function can satisfy strong Pareto optimality over the class of all compact and comprehensive choice problems, strong Pareto optimality, Arrow's choice axiom, upper hemicontinuity and a weak compromisation postulate turn out to be necessary and sufficient to represent choice correspondences by continuous, strictly increasing and quasiconcave real-valued functions. Some applications of our main findings to axiomatic bargaining theory are also studied.

The sequential generalized Lorenz (SGL) ordering is introduced by Atkinson and Bourguignon (1987) to rank heterogeneous income distributions. It is well-known that this ordering has a strong utilitarian support. In this note, we show that one does not have to be a utilitarian to accept this welfare ordering: the SGL ordering is supported by all increasing SWFs which record an increase in overall welfare when a (cardinal) welfare transfer is made from less needy to needier.

The Lorenz–Pareto optimal frontier of a collective choice problem identifies a subset of all Pareto optimal outcomes which are not inegalitarian according to the Lorenz criterion. An inequality averse choice function must thus be Lorenz–Pareto optimal in the sense that it should never `choose' an outcome outside the Lorenz–Pareto frontier of any choice problem. We study the basic properties of the Lorenz–Pareto optimal choice functions and, in particular, obtain necessary and sufficient conditions for rationalizability and representability of such choice functions. Several applications to the theory of Nash bargaining are also outlined.

I study the problem of ranking distributions of opportunity sets on the basis of equality. The analysis begins by developing alternative ways of formulating the notion of equalizing transfers in the context of opportunity distributions. It is readily observed that any such formulation must rely on agents' preferences over individual opportunity sets, and therefore, a theory of opportunity inequality measurement must be conditional upon the “social” opportunity set ranking one postulates. Unfortunately, the results make it clear that such a theory can be constructed only when one uses the cardinality ordering of Pattanaik and Xu (1990). More precisely put, I show that it is essentially impossible to extend Dalton's transfer principle to the context of opportunity distributions unless one is content with ranking the opportunity sets only on the basis of their cardinalities. Journal of Economic Literature Classification Numbers: D31, D63, D71.

This paper starts from the premise that the concept of income inequality is ill-defined, and hence, it studies the measurement of income inequality from a fuzzy set theoretical point of view. It is argued that the standard (fuzzy) transitivity concepts are not compatible with fuzzy inequality orderings which respect Lorenz ordering. For instance, we show that there does not exist a max-min transitive fuzzy relation on a given income distribution space which ranks distributions unambiguously according to the Lorenz criterion whenever they can actually be ranked by it. Weakening the imposed transitivity concept, it is possible to escape from the noted impossibility theorems. We introduce some alternative transitivity concepts for fuzzy relations, and subsequently, construct a class of fuzzy orderings which preserve Lorenz ordering and satisfy these alternative transitivities. It is also shown that fuzzy measurement can be used to construct confidence intervals for the crisp conclusions of inequality indices.

We consider the problem of ranking distributions of opportunity sets on the basis of equality. First, conditional on a given ranking of individual opportunity sets, we define the notion of an equalizing transformation. Then, assuming that the opportunity sets are ranked according to the cardinality ordering, we formulate the analogues of the notions of the Lorenz partial ordering, equalizing (Dalton) transfers, and inequality averse social welfare functions – concepts which play a central role in the literature on income inequality. Our main result is a cardinality-based analogue of the fundamental theorem of inequality measurement: one distribution Lorenz dominates another if and only if the former can be obtained from the latter by a finite sequence of rank preserving equalizations, and if and only if the former is ranked higher than the latter by all inequality averse social welfare functions. In addition, we characterize the smallest monotonic and transitive extension of our cardinality-based Lorenz inequality ordering.

This paper studies the possibility of progressive income taxation of heterogeneous populations. While a result due to Moyes and Shorrocks (1994) indicates that there does not exist a universally inequality-reducing tax structure which distinguishes between at least two subpopulations (in the sense of applying a different tax function to each subclass), it is shown here that a minimal refinement of the universality of inequality reduction leads one to a possibility conclusion. Informally stated, we prove the existence of uncountably many differentiated tax structures which are strictly progressive almost everywhere.