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The objective of this paper is to provide continuous utility representation theorems analogous to Debreu's classic utility representation theorem, albeit for preference relations that may fail to be complete and/or transitive. Specifically, we show that every (continuous and) reflexive binary relation on a (compact) metric space can be represented by means of the maxmin, or dually, minmax, of a (compact) set of (compact) sets of continuous utility functions. This notion of “maxmin multi-utility representation,” generalizes the recently proposed notions of “multi-utility representation” for preorders and “justifiable preferences” for complete and quasitransitive relations. As such, our main representation theorems lead to some new characterizations of these special cases as well.

We develop the ordinal theory of (semi)continuous multi-utility representation for incomplete preference relations. We investigate the cases in which the representing sets of utility functions are either arbitrary or finite, and those cases in which the maps contained in these sets are required to be (semi)continuous. With the exception of the case where the representing set is required to be finite, we find that the requirements of such representations are surprisingly weak, pointing to a wide range of applicability of the representation theorems reported here. Some applications to decision theory under uncertainty and consumer theory are also considered.

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 major problem of the positive theory of income taxation is to explain why statutory income tax schedules in practice are marginal-rate progressive. While it is commonly believed that this is but a simple consequence of the fact that the number of relatively poor voters exceeds that of richer voters in general, putting this contention in a voting equilibrium context is not a trivial task. We do this here in the context of nonlinear taxation and attempt to provide a formal argument in support of this heuristic claim. We first establish the existence of mixed strategy equilibria and identify certain cases in which marginal-rate progressive taxes are chosen almost surely by the political parties. Unfortunately, we also find that if the tax policy space is not artificially constrained, the support of at least one equilibrium cannot be contained within the set of marginal-rate progressive taxes.

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.

Motivated by the empirical findings concerning the importance of one's current situation on her choice behavior, the main objective of this paper is to propose a rational choice theory that allows for the presence of a status quo bias, and that incorporates the standard choice theory as a special case. We follow a revealed preference approach, and obtain two nested models of rational choice that allow phenomena like the status quo bias and the endowment effect, and that are applicable in any choice situation to which the standard (static) choice model applies.

We study axiomatically the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteries by means of a set of von Neumann-Morgenstern utility functions. It is shown that, when the prize space is a compact metric space, a preference relation admits such a multi-utility representation provided that it satisfies the standard axioms of expected utility theory. Moreover, the representing set of utilities is unique in a well-defined sense.

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.

This paper introduces a technique for mobility dominance and compares the degree of earnings mobility of men in the USA from 1970 to 1995. The highest mobility is found in the 1975–1980 or 1980–1985 periods.

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.

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.

We show that a marginal rate progressive tax always defeats a marginal rate regressive tax under pairwise majority voting (so long as the latter collects at least as much revenue as the former one) irrespective of whether the voters care at all about their relative incomes or not.

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.

We propose the principle of equal sacrifice to determine the class of “vertically inequitable” progressive taxes. A necessary condition for an income tax function to be equal sacrifice is formulated, and hence, a subclass of progressive taxes which cannot inflict the same sacrifice upon all individuals relative toanystrictly increasing and concave utility function is determined. Conversely, it is shown in a general framework that any convex (thus progressive) tax function satisfies the principle of equal sacrifice. Our findings point to the fact that equal sacrifice under progressive income taxation depends heavily upon the degree of marginal rate (as opposed to average rate) progressivity.Journal of Economic LiteratureClassification Numbers: D63, H20.

Income mobility may be seen as arising from two sources: (i) the transfer of income among individuals with total income held constant, and (ii) a change in the total amount of income available. In this paper, we propose several sensible properties defining the concept of income mobility and show that an easily applicable measure of mobility is uniquely implied by these properties. We also show that the resulting measure is additively decomposable into the two sources listed above, namely, mobility due to the transfer of income within a given structure and mobility due to economic growth or contraction. Finally, these results are compared and contrasted with other mobility concepts and measures in the literature. Journal of Economic Literature : Classification Numbers: D31, D63.

In an influential article by Young ( Journal of Economic Theory , 1988, 44, 321–335), it is shown that the traditional equal sacrifice principles in taxation can be justified from a non-utilitarian perspective by proving that equal sacrifice is, in fact, a consequence of more primitive concepts of distributive justice. This result is obtained in a distribution theoretic setting where total revenue to be raised is fixed. The present paper studies the same question in a setting where the basic problem is to determine a tax function without any reference to the total tax revenue (a setting which is familiar from the theory of progressive taxation) and shows that (i) there exists a utility function (which is continuous and strictly increasing) relative to which all agents sacrifice equally if and only if the tax function is incentive preserving; and (ii) any convex (hence progressive) and incentive preserving tax function equalizes sacrifice relative to a utility function which is continuous, strictly increasing and concave.

The generalized Hamming distance is commonly used in determining a closest crisp relation to a given fuzzy relation (which, in turn, is crucial in studying exact choices originating from fuzzy preferences). This note illustrates a difficulty attached to this practice, and proposes an alternative metric which is intuitively more appropriate in the case of fuzzy preferences. Some implications of this new metric on finding the nearest exact relation to a fuzzy preference ordering and on the understanding of the max-min transitivity property are also discussed.