Diversification preferences in the theory of choice

Diversification represents the idea of choosing variety over uniformity. Within the theory of choice, desirability of diversification is axiomatized as preference for a convex combination of choices that are equivalently ranked. This corresponds to the notion of risk aversion when one assumes the von Neumann–Morgenstern expected utility model, but the equivalence fails to hold in other models. This paper analyzes axiomatizations of the concept of diversification and their relationship to the related notions of risk aversion and convex preferences within different choice theoretic models. Implications of these notions on portfolio choice are discussed. We cover model-independent diversification preferences, preferences within models of choice under risk, including expected utility theory and the more general rank-dependent expected utility theory, as well as models of choice under uncertainty axiomatized via Choquet expected utility theory. Remarks on interpretations of diversification preferences within models of behavioral choice are given in the conclusion.     

Context dependent beliefs

This paper examines a model where the set of available outcomes from which a decision maker must choose alters his perception of uncertainty. Specifically, this paper proposes a set of axioms such that each menu induces a subjective belief over an objective state space. The decision maker’s preferences are dependent on the realization of the state. The resulting representation is analogous to state-dependent expected utility within each menu; the beliefs are menu dependent and the utility index is not. Under the interpretation that a menu acts as an informative signal regarding the true state, the paper examines the behavioral restrictions that coincide with different signal structures: elemental (where each element of a menu is a conditionally independent signal) and partitional (where the induced beliefs form a partition of the state space).     

Decision analysis using targets instead of utility functions

A common precept of decision analysis under uncertainty is the choice of an action which maximizes the expected value of a utility function. Savage's (1954) axioms for subjective expected utility provide a normative foundation for this principle of choice. This paper shows that the same set of axioms implies that one should select an action which maximizes the probability of meeting an uncertain target. This suggests a new perspective and an alternate target-based language for decision analysis. We explore the implications and the advantages of this target-based approach for both individual and group decision-making.     

Axiomatic approach to approximate solutions in multiobjective optimization

An axiomatic approach is introduced in order to study the approximate solution map of a vector optimization problem in the image space. We investigate the possibility to formulate an appropriate notion of approximate solutions that is compatible with von Neumann–Morgenstern utility theory. An impossibility result is proved in the sense that, whenever all of the axioms are satisfied, either the set of the approximate solutions is a subset of the exact solutions of the problem, or it coincides with the whole admissible set. Moreover, the geometry of the approximate solution map is studied in some special cases.      

A simplified approach to subjective expected utility

I provide a novel simplified approach to Savage’s theory of subjective expected utility. Such an approach is based on abstract integral representation theorems in the space of measurable functions. The advantage of such an approach is that these results can be used to easily obtain variations on Savage’s theorem, such as representations with state-dependent utility or probability measures that can have atoms. Finally, I discuss how such an approach can be used in other settings such as decision making under ambiguity.     

A von Neumann–Morgenstern representation result without weak continuity assumption

In the paradigm of von Neumann and Morgenstern (1947), a representation of affine preferences in terms of an expected utility can be obtained under the assumption of weak continuity. Since the weak topology is coarse, this requirement is a priori far from being negligible. In this work, we replace the assumption of weak continuity by monotonicity. More precisely, on the space of lotteries on an interval of the real line, it is shown that any affine preference order which is monotone with respect to the first stochastic order admits a representation in terms of an expected utility for some nondecreasing utility function. As a consequence, any affine preference order on the subset of lotteries with compact support, which is monotone with respect to the second stochastic order, can be represented in terms of an expected utility for some nondecreasing concave utility function. We also provide such representations for affine preference orders on the subset of those lotteries which fulfill some integrability conditions. The subtleties of the weak topology are illustrated by some examples.     

Differentiability of von Neumann–Morgenstern utility functions

This paper studies necessary and sufficient preference-based conditions for differentiability of risk averse (prudent, or temperate) von Neumann–Morgenstern utility functions. The very idea to devise those conditions is based on the reverse claim of an old observation by Arrow that a risk-averse expected-utility maximizer will always accept a sufficiently small stake in any positive expected-value bet if her von Neumann–Morgenstern utility function is differentiable.     

Decision making with imprecise probabilistic information

We develop an axiomatic approach to decision under uncertainty that explicitly takes into account the information available to the decision maker. The information is described by a set of priors and a reference prior. We define a notion of imprecision for this informational setting and show that a decision maker who is averse to information imprecision maximizes the minimum expected utility computed with respect to a subset of the set of initially given priors. The extent to which this set is reduced can be seen as a measure of imprecision aversion. This approach thus allows a lot of flexibility in modelling the decision maker attitude towards imprecision. In contrast, applying Gilboa and Schmeidler [J. Math. Econ. 18 (1989) 141] maxmin criterion to the initial set of priors amounts to assuming extreme pessimism.     

Effective utility functions induced by organizational target‐based incentives

Many companies set performance targets for their divisions to decentralize the decision‐making process and communicate with outside investors. This paper analyzes the effects of performance targets on the decision‐making behavior of the divisions. We introduce the notion of an ‘effective utility function’—a function that a division should use in its selection of projects if it wishes to maximize the probability of achieving its targets. We show that many target‐based incentives induce S‐shaped utility functions and discuss the organizational problems they may pose. We then show how an organization can set targets that induce expected utility maximization. Copyright © 2008 John Wiley & Sons, Ltd.     

Dynamic theory of preferences: Habit formation and taste for variety

We analyze individual preferences over infinite horizon consumption choices. Our axioms provide the foundation for a recursive representation of the utility function that contains as particular cases the classical Koopmans representation (Koopmans (1960)) as well as the habit formation specification.We examine some of the consequences of our axiomatization by considering a standard consumer choice problem, and show that typically in the space of concave utility functions satisfying our axioms the consumer displays a taste for variety. The latter means that such a consumer selects optimally time variant consumption programs for any given time invariant sequence of commodities’ relative prices and for all possible sequences of market discount factors. In contrast, if a concave utility function satisfies Koopmans’ axioms the consumer does not display a taste for variety.     

Observable probabilistic beliefs

Under the assumption that asset markets are possibly incomplete, conditions are examined under which an agent's asset demand correspondence can be used to recover without ambiguity his subjective joint distribution of returns of the various assets and/or his von Neumann– Morgenstern cardinal utility index.     

Simple axioms for countably additive subjective probability

This paper refines Savage’s theory of subjective probability for the case of countably additive beliefs. First, I replace his continuity axioms P6 and P7 with a simple modification of Arrow’s (1970) Monotone Continuity. Second, I relax Savage’s primitives: in my framework, the class of events need not be a σ-algebra, and acts need not have finite or bounded range. By varying the domains of acts and events, I obtain a unique extension of preference that parallels Caratheodory’s unique extension of probability measures. Aside from subjective expected utility, I characterize exponential time discounting in a setting with continuous time and an arbitrary consumption set.     

Coalitional stability in the location problem with single-dipped preferences: An application of the minimax theorem

This study considers a situation in which agents choose the location of a public facility from a street according to a given mechanism. Agents have single-dipped preferences over a set of feasible locations. We analyze coalitional behavior for any given mechanism for this situation. We identify a necessary and sufficient condition for a mechanism to possess a strong Nash equilibrium by applying the minimax theorem of von Neumann and Morgenstern (1944). We introduce a class of core solutions and show that these solutions are characterized by strong Nash implementability. As a byproduct of these results, we propose a simple mechanism that implements any core solution in strong Nash equilibria.     

Eliciting preferences and the construction of indifference maps: A comparative empirical evaluation of two measurement methodologies

The paper reports results of experiments conducted to evaluate the following methods of preference measurement: (a) the method of eliciting certainty equivalents to gambles, for obtaining a von Neumann Morgenstern utility function; and (b) the dominance method for obtaining indifference maps. The following attributes were used for method comparison: (1) test-retest consistency; (2) linearity of trade-offs; (3) confidence in the method; (4) ease of judgment required by method; and (5) goodness of method predictions. The study also investigated the associations between method reliability and several behavioural and experimental factors such as subjects' acceptance of rationality axioms, propensities of subjects for judicial modes of evaluation, perceived realism of scenarios and subjects' discrimination bands for probabilities.     

Resource allocation, equity and public risk: Dying one at a time vs dying all together

This paper focuses on the evaluation, from an individual and societal perspective, of risk in terms of possible loss of life due to an exposure to two different types of events over a period of time. The two types are: risk of death from a catastrophic event (a sudden death of many people in a disaster at a yet unknown point in time) expected to occur during a planning period, or risk of death from another event (e.g. disease, road accident, etc) which claims fewer lives each year. but for which the expected total number of deaths over the planning period is equal to the expected number of deaths from the catastrophic event. Our analysis considers the extreme case in which these two types of events have the same probabilities of death every year and the same expected number of fatalities over the planning period. The individual's decision problem is described using a von-Neumann Morgenstern (vNM) utility function. The model suggests that the choice between these types of events depends on the value of the following variables: the probability of death over the planning period, the length of the planning period, the individual's time preference pattern, and the utility of being in different anxiety states. Stochastic extensions that may direct the public decision making process (involving aggregated preferences) are discussed. We also discuss issues of implementation.     

Risk neutrality regions

An Expected Utility maximizer can be risk neutral over a set of nondegenerate multivariate distributions even though her NM (von Neumann Morgenstern) index is not linear. We provide necessary and sufficient conditions for an individual with a concave NM utility to exhibit risk neutral behavior and characterize the regions of the choice space over which risk neutrality is exhibited. The least concave decomposition of the NM index introduced by Debreu (1976) plays an important role in our analysis as do the notions of minimum concavity points and minimum concavity directions. For the special case where one choice variable is certain, the analysis of risk neutrality requires modification of the Debreu decomposition. The existence of risk neutrality regions is shown to have important implications for the classic consumption–savings and representative agent equilibrium asset pricing models.     

Are CEOs expected utility maximizers?

Are individuals expected utility maximizers? This question represents much more than academic curiosity. In a normative sense, at stake are the fundamental underpinnings of the bulk of the last half-century’s models of choice under uncertainty. From a positive perspective, the ubiquitous use of benefit-cost analysis across government agencies renders the expected utility maximization paradigm literally the only game in town. In this study, we advance the literature by exploring CEO’s preferences over small probability, high loss lotteries. Using undergraduate students as our experimental control group, we find that both our CEO and student subject pools exhibit frequent and large departures from expected utility theory. In addition, as the extreme payoffs become more likely CEOs exhibit greater aversion to risk. Our results suggest that use of the expected utility paradigm in decision making substantially underestimates society’s willingness to pay to reduce risk in small probability, high loss events.     

Intergenerational justice when future worlds are uncertain

Let there be a positive (exogenous) probability that, at each date, the human species will disappear. We postulate an Ethical Observer (EO) who maximizes intertemporal welfare under this uncertainty, with expected-utility preferences. Various social welfare criteria entail alternative von Neumann Morgenstern utility functions for the EO: utilitarian, Rawlsian, and an extension of the latter that corrects for the size of population. Our analysis covers, first, a cake-eating economy (without production), where the utilitarian and Rawlsian recommend the same allocation. Second, a productive economy with education and capital, where it turns out that the recommendations of the two EOs are in general different. But when the utilitarian program diverges, then we prove it is optimal for the extended Rawlsian to ignore the uncertainty concerning the possible disappearance of the human species in the future. We conclude by discussing the implications for intergenerational welfare maximization in the presence of global warming.     

A short proof of Deb’s Theorem on Schwartz’s rule

Schwartz in (Nous,7, 1972, Definition, 3) introduces a generalization of the Condorcet criterion, which is the classical approach to rational choice in the context of cycles, and he defines the Schwartz set. Deb (J Econ Theory 16:103–110, 1977) shows that the Schwartz set consists of the maximal elements according to the transitive closure of the asymmetric part of a binary relation corresponding to a choice process or representing the decision maker’s preferences. This note provides a short and simple proof of Deb’s theorem on the characterization of the Schwartz set.