Small worlds: Modeling attitudes toward sources of uncertainty

We introduce the concept of a conditional small world event domain—an extension of Savage's [The Foundations of Statistics, Wiley, New York, 1954] notion of a ‘small world’—as a self-contained collection of comparable events. Under weak behavioral conditions we demonstrate probabilistic sophistication in any small world event domain without relying on monotonicity or continuity. Probabilistic sophistication within, though not necessarily across, small worlds provides a foundation for modeling a decision maker that has source-dependent risk attitudes. This also helps formalize the idea of source preference and suggests an interpretation of ambiguity aversion, often associated with Ellsberg-type behavior, in terms of comparative risk aversion across small worlds.     

An additively separable representation in the Savage framework

This paper proposes necessary and sufficient conditions for an additively separable representation of preferences in the Savage framework (where the objects of choice are acts: measurable functions from an infinite set of states to a potentially finite set of consequences). A preference relation over acts is represented by the integral over the subset of the product of the state space and the consequence space which corresponds to the act, where this integral is calculated with respect to an evaluation measure on this space. The result requires neither Savage's P3 (monotonicity) nor his P4 (weak comparative probability). Nevertheless, the representation it provides is as useful as Savage's for many economic applications.     

Probabilistic sophistication and variational preferences

This paper shows that in the class of variational preferences the notion of probabilistic sophistication is equivalent to expected utility as long as there exists at least one event such that the independence axiom holds for bets on that event. This extends the result of Marinacci (2002) [13] and provides a novel interpretation of his result.     

Rationalizability and the savage axioms

Summary. Suppose there is a finite set of acts defined on a finite state space and a decision maker chooses an act from the set. In this setting, the subjective expected utility model is observationally indistinguishable from all models of preference that satisfy Savage's axiom P3. The result has implications also for rationalizability in strategic games. Received: September 18, 1998; revised version: January 29, 1999     

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     

Preference aggregation under uncertainty: Savage vs. Pareto

Following Mongin [J. Econ. Theory 66 (1995) 313; J. Math. Econ. 29 (1998) 331], we study social aggregation of subjective expected utility preferences in a Savage framework. We argue that each of Savage's P3 and P4 are incompatible with the strong Pareto property. A representation theorem for social preferences satisfying Pareto indifference and conforming to the state-dependent expected utility model is provided.     

Envy and Altruism: Contrasting Bivariate and Univariate Prospect Preferences

We compare prospect ordering with and without envy and altruism. We find that envy can induce a violation of the univariate first‐degree stochastic dominance (FSD), and thus a violation of the classic expected utility monotonicity axiom. Surprisingly, altruism can also violate FSD preferences. The intuitive explanation of the result in the case of altruism hinges on the sign of the mixed derivative of the bivariate preference: the individual might prefer a certain correlation between her wealth and her peer group's wealth, and is therefore willing to violate FSD as long as the outcomes of the two parties are ordered according to her preferences. When investments are considered, envy and altruism can distort not only preferences but also actual choices.     

Axiomatization of the core of assignment games

The core of assignment games is characterized by two different lists of axioms. The first list consists of Pareto optimality, consistency, pairwise monotonicity, and individual monotonicity. The second list is obtained from the first one replacing the last axiom by population monotonicity. As a corollary, individual monotonicity and population monotonicity are equivalent under the other axioms. The core is also characterized by the second list on the restricted domain in which the worth of each pair is non-negative and every reservation value is zero.     

The projective independence axiom

Summary A new axiom for preference orderings over lotteries, called the projective independence axiom, is formulated. Given suitable continuity and monotonicity assumptions, the axiom implies that utility is either in the weighted utility class or is quadratic in probabilities. The betweenness axiom is used to distinguish between these two classes of functions.We gratefully acknowledge the financial support of the National Science Foundation and the Social Sciences and Humanities Research Council of Canada.     

Cost sharing with multiple technologies

We study cost sharing problems where gains from cooperation can come from the presence of other agents, such as when agents share their technologies. A simple model is built, where economies of scale are eliminated in order to study this effect. We use as the key axiom the property that, if an agent does not improve the technology of any coalition he joins, he should not get any part of the gain from cooperation. With properties of linearity and symmetry, this axiom characterizes a well-defined set of rules. From this set, we propose a rule derived from the familiar Shapley value. We show that it is the only rule in that set satisfying an upper-limit property on individual cost allocations or a monotonicity property when technology improves. We also derive a distinct rule using a property that ensures that no coalition has an incentive to manipulate the individual demands of its members.     

Cost sharing with multiple technologies

We study cost sharing problems where gains from cooperation can come from the presence of other agents, such as when agents share their technologies. A simple model is built, where economies of scale are eliminated in order to study this effect. We use as the key axiom the property that, if an agent does not improve the technology of any coalition he joins, he should not get any part of the gain from cooperation. With properties of linearity and symmetry, this axiom characterizes a well-defined set of rules. From this set, we propose a rule derived from the familiar Shapley value. We show that it is the only rule in that set satisfying an upper-limit property on individual cost allocations or a monotonicity property when technology improves. We also derive a distinct rule using a property that ensures that no coalition has an incentive to manipulate the individual demands of its members.     

Probabilistic choice and stochastic dominance

This paper presents an axiomatic model of probabilistic choice under risk. In this model, when it comes to choosing one lottery over another, each alternative has a chance of being selected, unless one lottery stochastically dominates the other. An individual behaves as if he or she compares lotteries to a reference lottery—the least upper bound or the greatest lower bound in terms of stochastic dominance. The proposed model is compatible with several well-known violations of expected utility theory such as the common ratio effect and the violations of betweenness. Necessary and sufficient conditions for the proposed model are completeness, weak stochastic transitivity, continuity, common consequence independence, outcome monotonicity, and odds ratio independence.     

‘Regular’ choice and the weak axiom of stochastic revealed preference

We explore the relation between two ‘rationality conditions’ for stochastic choice behavior: regularity and the weak axiom of stochastic revealed preference (WASRP). We show that WASRP implies regularity, but the converse is not true. We identify a restriction on the domain of the stochastic choice function, which suffices for regularity to imply WASRP. When the universal set of alternatives is finite, this restriction is also necessary for regularity to imply WASRP. Furthermore, we identify necessary and sufficient domain restrictions for regularity to imply WASRP, when the universal set of alternatives is finite and stochastic choice functions are all degenerate. Results in the traditional, deterministic, framework regarding the relation between Chernoff’s condition and the weak axiom of revealed preference follow as special cases. Thus, general conditions are established, under which regularity can substitute for WASRP as the axiomatic foundation for a theory of choice behavior.     

On the uniqueness of subjective probabilities

Summary The purpose of this paper is twofold: First, within the framework of Savage (1954), we suggest axiomatic foundations for the representation of event-dependent preference relations over acts. This representation has the form of expectation of event-debendent utility with respect to non-unique subjective probabilities on the set of states. Second, we give an economic-theoretic motivation for selecting a unique probability distribution as an appropriate concept of subjective probabilities. However, unlike in Savage's theory, this notion of subjective probabilities does not necessarily represent the decisions-maker's belief regarding the likelihood of events.Our approach involves a departure from Savage's postulate P4, which guarantees the completeness of Savage's likelihood relation on the set of all events. Instead, we assume the existence of a finite partition of the set of states, {S ₁,...S _n}, such that, for events within each element of this partition P4 is satisfied. This weakening of Savage's axioms suffices for the existence of an expected event-dependent utility representation, but not for the uniqueness of the subjective probabilities.In many economic problems involving decision-making under uncertainty the existence of a unique probability is presumed and, in fact, is essential for the statement of the result. An example is Arrow's (1965) finding that all risk averse decision-makers will invest in a risky asset provided its expected rate of return exceeds that of an alternative risk-free asset. We show that a unique probability distribution can be chosen so as to render such results meaningful. Namely, any risk averse decision-maker will hold a positive position in the risky asset if and only if its expected rate of return with respect to the chosen probability exceeds that of the riskless asset.The research described in this paper began while the authors visited the Mathematisches Forschungsinstitut Oberwolfach, Germany. It was carried out in part while the second author visited the Santa Fe Institute in Sante Fe, New Mexico, USA and the Autonomous University of Barcelona, Spain. The second author would also like to acknowledge the financial support by NSF grant 911873.     

Imprecise probabilistic beliefs as a context for decision-making under ambiguity

Coherent imprecise probabilistic beliefs are modeled as incomplete comparative likelihood relations admitting a multiple-prior representation. Under a structural assumption of Equidivisibility, we provide an axiomatization of such relations and show uniqueness of the representation. In the second part of the paper, we formulate a behaviorally general “Likelihood Compatibility” axiom relating preferences and probabilistic beliefs and characterize its implications for the class of “invariant biseparable” preferences that includes the MEU and CEU models among others.     

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.     

Comment on the Kannai-Peleg impossibility theorem for extending orders

Recent work by Y. Kannai and B. Peleg shows that two appealing axioms for extending a linear order on a set of six or more elements to a weak order on the family of non-empty subsets of that set are mutually incompatible. This comment shows that a modest restriction of their monotonicity axiom is compatible for extension with a generalization of their other axiom provided that the linear order on the basic set is a well ordering.     

Demand Aggregation and the Weak Axiom of Stochastic Revealed Preference

We address the problem of aggregating demand across a group of consumers, who are identical in terms of wealth and face identical price vectors, but vary in their chosen consumption bundles. We show that, when a stochastic demand function is constructed to aggregate a number of deterministic demand functions, satisfaction of the weak axiom of stochastic revealed preference by this stochastic demand function is weaker than the restriction that every underlying deterministic demand function satisfy Samuelson's weak axiom of revealed preference. Journal of Economic Literature Classification Number: D11.     

External and internal consistency of choices made in convex time budgets

We evaluate data on choices made from convex time budgets (CTB) in Andreoni and Sprenger (Am Econ Rev 102(7):3333–3356, 2012a) and Augenblick et al. (Q J Econ 130(3):1067–1115, 2015), two influential studies that proposed and applied this experimental technique. We use the weak axiom of revealed preference (WARP) to test for external consistency relative to pairwise choice, and demand, wealth and impatience monotonicity to test for internal consistency. We find that choices made by subjects in the original Andreoni and Sprenger (Am Econ Rev 102(7):3333–3356, 2012a) paper violate WARP frequently; violations of all three internal measures of monotonicity are concentrated in subjects who take advantage of the novel feature of CTB by making interior choices. Wealth monotonicity violations are more prevalent and pronounced than either demand or impatience monotonicity violations. We substantiate the importance of our desiderata of choice consistency in examining effort allocation choices made in Augenblick et al. (Q J Econ 130(3):1067–1115, 2015), where we find considerably more demand monotonicity violations, as well as many classical monotonicity violations which are associated with time inconsistent behavior. We believe that the frequency and magnitude of WARP and monotonicity violations found in the two studies pose important confounds for interpreting and structurally estimating choice patterns elicited through CTB. We encourage researchers employing CTB in present and future experiments to include consistency tests in their design and pre-estimation analysis.     

Population Monotonicity and Egalitarianism for Binary Social Choice Problems

We consider the class of binary social choice problems. A society must choose one of two public projects, money being available to perform side payments and each agent having quasi-linear preferences. Moulin (1987, Quarterly Journal of Economics 102 , 769–783) formulates the problem and characterizes the egalitarian solution on the basis of agreement . This axiom requires that changes in the preferences of some members of the society should affect the agents whose preferences have not changed in the same direction; all gain or all lose. In this paper, we present an alternative characterization of the egalitarian solution on the basis of population monotonicity. This axiom requires that upon the arrival of new agents, all of the original agents should be affected in the same direction; all gain or all lose.