Von Neumann-Morgenstern Utility Function Research Articles

Risk Aversion in the Small and in the Large. When Outcomes are Multidimensional

The paper discusses criteria for comparing risk of decision makers when outcomes are multidimensional. A weak concept, commodity specific greater risk aversion, is based on the comparison of risk premia paid in a specified commodity. A stronger concept, uniformly greater risk aversion is based on the comparison of risk premia regardless of what commodities are used for payment. Neither concept presumes that von Neumann-Morgenstern utility functions are ordinally equivalent. Nonincreasing consumption specific risk is shown to be sufficient to make randomization undesirable in an agency problem with hidden characteristics.

A multiunit generalization of a first price auction

This paper studies equilibrium bidding in a first price multiunit auction, in which all agents have constant elasticity Von Neumann-Morgenstern utility functions.

Aggregation under complete markets

This paper proves a complete market aggregation result for a multiple good economy in which the consumers have time-additive, von Neumann–Morgenstern utility functions. This result applies to all concave intraperiod functions, and provides a contrast to many of the results in the microeconomics literature which depend on such stringent assumptions as linear Engel curves.

Expected utility theory without the completeness axiom

We study 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.

Power indices and the veil of ignorance

We provide an axiomatic foundation of the expected utility preferences over lotteries on roles in simple superadditive games represented by the two main power indices, the Shapley-Shubik index and the Banzhaf index, when they are interpreted as von Neumann-Morgenstern utility functions. Our axioms admit meaningful interpretations in the setting proposed by Roth in terms of different attitudes toward risk involving roles in collective decision procedures under the veil of ignorance. In particular, an illuminating interpretation of `efficiency', up to now missing in this set up, as well as of the corresponding axiom for the Banzhaf index, is provided.

A Smooth Model of Decision Making under Ambiguity

We propose and axiomatize a model of preferences over acts such that the decision maker evaluates acts according to the expectation (over a set of probability measures) of an increasing transformation of an act's expected utility. This expectation is calculated using a subjective probability over the set of probability measures that the decision maker thinks are relevant given her subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective information, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes towards risk are characterized by the shape of the von Neumann-Morgenstern utility function, as usual, while attitudes towards ambiguity are characterized by the shape of the increasing transformation applied to expected utilities. We show that the negative exponential form of this transformation is the special case of constant ambiguity aversion. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures. This characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by Epstein-Zhang (2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's Paradox, etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion. Two illustrative applications to portfolio choice are offered.

A Model of Procedural Decision Making in the Presence of Risk*

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.

Manipulation of Preferences and Relative Utilitarianism

Given n agents with von Neumann-Morgenstern utility functions who wish to divide m commodities, consider the n-person noncooperative game with strategies consisting of concave, increasing von Neumann-Morgenstern utility functions, and whose outcomes are the relative utilitarian solution. It is shown that any constrained equal-income competitive equilibrium allocation for the true utilities is a Nash equilibrium outcome for the noncooperative game. Conditions are presented under which these are the only pure strategy equilibrium outcomes.

A General Equilibrium Interpretation of Damage-Contingent Securities

ABSTRACT Cass, Chichilnisky, and Wu (1996) show in an endowment economy that mutual insurance and securities contingent on aggregate states support optimal risk-sharing. We extend their result to a model with production in which risk is endogenous and beliefs about the aggregate state vary across individuals. We use the model to interpret the role of new securities that are contingent on measures of total damage from natural catastrophes. Plausible special cases of the model predict the trade pattern in such securities across diverse regions and predict that such securities will not represent actuarially fair gambles. INTRODUCTION Cass, Chichilnisky, and Wu (1996) show in an endowment economy that if the structure of uncertainty resembles an individual insurance situation, then a combination of mutual insurance and securities contingent on aggregate states supports optimal risk-sharing. In other words, in such settings, a complete set of Arrow-Debreu markets is not needed. We extend their result to a model with production in which the amount of risk depends on individual decisions--as when houses are built in areas prone to hurricanes. Our model is of a static economy in which the aggregate amount of resources subject to risk is the result of individual investment decisions and in which beliefs about the aggregate state vary across individuals. For that model, we show that a Pareto efficient allocation can be supported by a competitive equilibrium with two types of securities: mutual insurance that provides for risk sharing among individuals who are ex ante similar, and a set of state-contingent securities that are indexed by the aggregate exogenous state. This extends the result in Cass, Chichilnisky, and Wu (1996) to the case of a productive economy where the level of aggregate risk is endogenous. We go on to show that the state-contingent securities can be replaced by securities contingent on aggregate losses rather than the exogenous state, despite the fact that aggregate losses are endogenous. For special cases of the model, we describe the implied pattern of trade in the contingent contracts and how they are priced. We show that there is no theoretical basis for believing that the damage-contingent securities are priced in an actuarially fair manner. If individuals have identical beliefs, then the price of a damage contingent security will in general include a risk premium that rewards individuals for taking on this risk. After presenting those results, we conclude with a brief discussion of recent policy initiatives and related discussion concerning losses from natural catastrophes. A MODEL WITH ENDOGENOUS DAMAGE We present a static model that is designed to represent, in a simple way, regions which end up being subject to different amounts of risk. The economy consists of a finite number of islands that are indexed by h from 1 to H. Island h is inhabited by [N.sub.h] people of type h. Each island is perfectly round and has a plateau in the middle of it. The land on the coast is subject to the risk of damage (from storms), while land located on the plateau is safe. Each person on island h is endowed with an ex ante identical slice of land, which includes some coastal land and some plateau land, and with a resource, [y.sub.h], which can be interpreted as labor. The only use of the resource is as an input into production of a single good, rice, on the coastal land or on the plateau land owned by that person. All type h people are identical in terms of von Neumann-Morgenstern utility function, endowments, and technologies. We let [u.sub.h]: [R.sub.+] [right arrow] R denote a type h person's von Neumann-Morgenstern utility function (for rice consumption), and we assume that [u.sub.h] is strictly increasing, strictly concave, and continuously differentiable. The technologies determine the distribution of output resulting from the decision about dividing [y. …

A Note on Asset Proportions, Stochastic Dominance, and the 50% Rule

In this note we analyze the composition of an optimal portfolio by considering the cumulative conditional expected outcome of two dependent assets. We develop a conditional stochastic dominance relation and show that for any concave von Neumann-Morgenstern utility function, the proportion of wealth invested in the dominant asset will be greater than 50%.

Differentiable von Neumann-Morgenstern utility

Differentiability is a convenient property of von Neumann-Morgenstern utility functions which is almost always imposed but has not been translated into behavioral terms. In applications, expected utility is usually maximized subject to a constraint, and the maximization is carried out by differentiating the utility function. This paper presents two sets of necessary and sufficient conditions for a risk averse von Neumann-Morgenstern utility function to be differentiable. The first of them is formulated in terms of the equivalent risk premia of small gambles. It says, in brief, that the equivalent risk premium is of a smaller order of magnitude than the risk itself, as measured by the expectation of the absolute value of the risk. The second set of necessary and sufficient conditions is formulated in terms of the probability premium of small lotteries. It says, essentially, that the probability premium for small binary lotteries goes to zero as the size of the lottery goes to zero.

Non-Archimedean subjective probabilities in decision theory and games

To allow conditioning on counterfactual events, zero probabilities can be replaced by infinitesimal probabilities that range over a non-Archimedean ordered field. This paper considers a suitable minimal field that is a complete metric space. Axioms similar to those in Anscombe and Aumann [Anscombe, F.J., Aumann, R.J., 1963. A definition of subjective probability, Annals of Mathematical Statistics 34, 199–205.] and in Blume et al. [Blume, L., Brandenburger, A., Dekel, E., 1991. Lexicographic probabilities and choice under uncertainty, Econometrica 59, 61–79.] are used to characterize preferences which: (i) reveal unique non-Archimedean subjective probabilities within the field; and (ii) can be represented by the non-Archimedean subjective expected value of any real-valued von Neumann–Morgenstern utility function in a unique cardinal equivalence class, using the natural ordering of the field.

ANXIETY AND DECISION MAKING WITH DELAYED RESOLUTION OF UNCERTAINTY

In many real-world gambles, a non-trivial amount of time passes before the uncertainty is resolved but after a choice is made. An individual may have a preference between gambles with identical probability distributions over final outcomes if they differ in the timing of resolution of uncertainty. In this domain, utility consists not only of the consumption of outcomes, but also the psychological utility induced by an unresolved gamble. We term this utility anxiety. Since a reflective decision maker may want to include anxiety explicitly in analysis of unresolved lotteries, a multiple-outcome model for evaluating lotteries with delayed resolution of uncertainty is developed. The result is a rank-dependent utility representation (e.g., Quiggin, 1982), in which period weighting functions are related iteratively. Substitution rules are proposed for evaluating compound temporal lotteries. The representation is appealing for a number of reasons. First, probability weights can be interpreted as the cognitive attention allocated to certain outcomes. Second, the model disaggregates strength of preference from temporal risk aversion and thus provides some insight into the old debate about the relationship between von Neumann–Morgenstern utility functions and strength of preference value functions.

Linear aggregation of SSB utility functionals

A necessary and sufficient condition for linear aggregation of SSB utility functionals is presented. Harsanyi's social aggregation theorem for von Neumann–Morgenstern utility functions is shown to be a corollary to this result. Two generalizations of Fishburn and Gehrlein's conditional linear aggregation theorem for SSB utility functionals are also established.

Non-Archimedean Subjective Probabilities in Decision Theory and Games

To allow conditioning on counterfactual events, zero probabilities can be replaced by infinitesimal probabilities that range over a non-Archimedean ordered field. This paper considers a suitable minimal field that is a complete metric space. Axioms similar to those in Anscombe and Aumann (1963) and in Blume, Brandenburger and Dekel (1991) are used to characterize preferences which: (i) reveal unique non-Archimedean subjective probabilities within the field; and (ii) can be represented by the non-Archimedean subjective expected value of any real-valued von Neumann-Morgenstern utility function in a unique cardinal equivalence class, using the natural ordering of the field.

10 Development of a clinical prediction model for an ordinal diagnostic outcome

For choice with deterministic consequences, the standard rationality hypothesis is ordinality, i.e., maximization of a weak preference ordering. For choice under risk (resp. uncertainty), preferences are assumed to be represented by the objectively (resp. subjectively) expected value of a von Neumann-Morgenstern utility function. For choice under risk, this implies a key independence axiom; under uncertainty, it implies some version of Savage’s sure-thing principle. This chapter investigates the extent to which ordinality, independence, and the sure thing principle can be derived from more fundamental axioms concerning behavior in decision trees. Following Cubitt (1996), these principles include dynamic consistency, separability, and reduction of sequential choice, which can be derived in turn from one consequentialist hypothesis applied to continuation subtrees as well as entire decision trees. Examples of behavior violating these principles are also reviewed, as are possible explanations of why such violations are often observed in experiments.

Paretian aggregation of asymmetric utility functionals

An analogue of Harsanyi's social aggregation theorem for von Neumann-Morgenstern utility functions is presented for asymmetric utility functionals.

Structuring and assessing linear lexicographic utility

We outline an approach for assessing preferences in the form of a finite-dimensional, linear-lexicographic utility function when they lack the continuity properties necessary for representing them in the form of a real-valued, von Neumann-Morgenstern utility function. This approach takes the decision consequences as general descriptions, but it can exploit an available and conformable multiattribute structuring.

Decomposition of Utility Functions on Subsets of Product Sets

The standard decomposition theorem for additive and multiplicative utility functions (Pollak [Pollak, R. O. 1967. Additive von Neumann-Morgenstern utility functions. Econometrica 35 485–494.], Keeney [Keeney, R. L. 1974. Multiplicative utility functions. Opns. Res. 22 22–34.]) assumes that the outcome set is a whole product set. In this paper this assumption is relaxed, and the question of whether or not a natural revision of this theorem necessarily holds is investigated. This paper proves that two additional conditions are needed for the decomposition theorem to hold in the context where the outcome set is a subset of a Cartesian product. It is argued that these two new conditions are satisfied by a large family of subsets corresponding to significant real-world problems. Further research avenues are suggested including a generalization of this new decomposition result to nonexpected utility theories.

Changes in Background Risk and Risk Taking Behavior

ECONOMIC DECISION MAKING UNDER UNCERTAINTY often takes place in the presence of multiple and in markets that are less than complete. As a consequence, choices about endogenous sometimes must be made while simultaneously facing one or more immutable exogenous risks that are not under the control of the agent, and that are independent of endogenous risks. It is somehow natural to assume that an exogenous deterioration in background wealth will cause an individual to take more care elsewhere. If we define a deterioration, for example, as making the individual poorer by removing a fixed amount of initial wealth, we know from Pratt (1964) that decreasing absolute risk aversion (DARA) of an individual's von Neumann-Morgenstern utility function yields this natural result. If, on the other hand, background wealth becomes riskier due to the addition of a zero-mean risk, that is also statistically independent of all other risks, behavior will be more risk averse if and only if preferences are risk vulnerable as defined by Gollier and Pratt (1996). Risk vulnerability (described below in Section 4) is a stronger notion than DARA and includes proper risk aversion (Pratt and Zeckhauser (1987)) and standard risk aversion (Eeckhoudt and Kimball (1992), Kimball (1993)) as particular cases. But a deterioration in background wealth may encompass more complicated distribution changes than the introduction of another statistically independent risk. In this paper, we examine background wealth deteriorations that take the form of both general firstand second-degree stochastic dominance changes in risk (FSD and SSD respectively). In particular, we determine conditions that are both necessary and sufficient for each of these two types of background risk changes to imply more risk-averse behavior on the part of the individual. For the case of FSD changes, this condition turns out to be Ross' stronger characterization of decreasing absolute risk aversion. In the case of general SSD changes in the distribution of background wealth, the condition derived is a stronger version (in Ross' sense) of the conditions characterizing preferences that are locally risk vulnerable in the sense of Gollier and Pratt. The necessary and sufficient conditions derived are fairly restrictive upon preferences. However, if we take as positive behavior that individuals act in a more risk-averse manner whenever the distribution of background wealth deteriorates, these conditions place canonical limits upon appropriate utility representations. At the very least, they