Conditional expected,extensive utility

Luce and Krantz (1971) presented an axiom system for conditional expected utility. In this theory a conditional decision is a function whose domain is a non-null subevent and whose range is a subset of a set of consequences. Given a family of conditional decisions that is closed under unions of decisions whose domains are disjoint and under restrictions to non-null subevents, the second major primitive is an ordering of the family. Axioms were given that are adequate to construct a numerical utility function over decisions and a probability function over events for which the conditional expectation of the utility is order preserving. Several of the axioms are quite complex and seem a bit artificial, and the proof is very long. Here the structure is modified by adding to the set of outcomes a concatenation operation, and the representation theorem is modified by requiring that the utility function be additive over this binary operation as well as exhibiting the expected utility property. The advantages of this pair of changes are, first, it exploits the obvious fact that the union of consequences is itself a consequence; second, it reduces the mathematical burden carried by the set theoretic structure of conditional decisions and, as a result, the axioms can be made much easier to understand; and third, it permits a considerably shorter proof because one can draw more readily on known results. The major drawback of this approach is, of course, that it is inconsistent with the evidence that utility is not additive over consequences - at least, not over increasing amounts of a single good (diminishing marginal utility).This work was supported by a grant from the Alfred P. Sloan Foundation to the Institute for Advanced Study. I wish to thank P. C. Fishburn and F. S. Roberts for their comments.     

Stochastic expected utility theory

This paper proposes a new decision theory of how individuals make random errors when they compute the expected utility of risky lotteries. When distorted by errors, the expected utility of a lottery never exceeds (falls below) the utility of the highest (lowest) outcome. This assumption implies that errors are likely to overvalue (undervalue) lotteries with expected utility close to the utility of the lowest (highest) outcome. Proposed theory explains many stylized empirical facts such as the fourfold pattern of risk attitudes, common consequence effect (Allais paradox), common ratio effect and violations of betweenness. Theory fits the data from ten well-known experimental studies at least as well as cumulative prospect theory.

State-independent subjective expected lexicographic utility

By enriching the set of acts deemed available at least as objects of assessment, we obtain a significant tightening of the linear lexicographic representation described in LaValle and Fishburn (1991a). Under the state-independent assumption that every outcome is available in every state, each state must be either completely null or completely essential (rather than lexicographically essential), and the matrices characterizing subjective probabilities of the states must be square and lower triangular with positive diagonal entries. It follows that there are straightforward generalizations of real-valued-probability relationships such as Bayes' theorem. Even in the tighter case, the matrix probabilities cannot be reduced to scalar matrices or even fully diagonal matrices. Nevertheless, they are easy to work with and permit fully consequentialist decision analysis of problems in which preferences are non-Archimedean.     

Lexicographic state-dependent subjective expected utility

An additive-across-states decomposition of lexicographic linear utility is easily obtained under a mild structural assumption concerning sufficient richness of acts in the domain of preference assessment, but the vectorial nature of lexicographic utility introduces two complexities absent in the real-valued case. First, the concept of state nullity becomes lexicographic rather than binary; and second, a standard construction for obtaining subjective probabilities from real-valued, state-dependent utilities produces matrices instead of nonnegative real numbers in the lexicographic setting.     

Consequentialist foundations for expected utility

Behaviour norms are considered for decision trees which allow both objective probabilities and uncertain states of the world with unknown probabilities. Terminal nodes have consequences in a given domain. Behaviour is required to be consistent in subtrees. Consequentialist behaviour, by definition, reveals a consequence choice function independent of the structure of the decision tree. It implies that behaviour reveals a revealed preference ordering satisfying both the independence axiom and a novel form of sure-thing principle. Continuous consequentialist behaviour must be expected utility maximizing. Other plausible assumptions then imply additive utilities, subjective probabilities, and Bayes' rule.     

Boundedly rational expected utility theory

We build a satisficing model of choice under risk which embeds Expected Utility Theory (EUT) into a boundedly rational deliberation process. The decision maker accumulates evidence for and against alternative options by repeatedly sampling from her underlying set of EU preferences until the evidence favouring one option satisfies her desired level of confidence. Despite its EUT core, the model produces patterns of behaviour that violate standard EUT axioms, while at the same time capturing systematic relationships between choice probabilities, response times and confidence judgments, which are beyond the scope of theories that do not take deliberation into account.

     

Cost-benefit versus expected utility acceptance rules

A rule for the acceptance of scientific hypotheses called the principle of cost-benefit dominance is shown to be more effective and efficient than the well-known principle of the maximization of expected (epistemic) utility. Harvey's defense of his theory of the circulation of blood in animals is examined as a historical paradigm case of a successful defense of a scientific hypothesis and as an implicit application of the cost-benefit dominance rule advocated here. Finally, various concepts of dominance are considered by means of which the effectiveness of our rule may be increased.The number of friends who have kindly given me suggestions and encouragement is almost embarrassingly large, but I would like to express my gratitude to Myles Brand, Cliff Hooker, David Hull, Scott Kleiner, Hugh Lehman, Werner Leinfellner, Andrew McLaughlin and Tom W. Settle.     

Information evaluation under nonadditive expected utility

We examine the choice-of-single-stage-experiment problem (Raiffa and Schlaifer, 1961) under the assumption that the decider's (weak) preference relation satisfies Schmeidler's (1989) or Gilboa's (1987) axiomatization and is thus representable by a nonadditive expected-utility functional as a Choquet integral w.r.t. a monotone probability measure on events. The basic properties of information value, certainty equivalent of information cost, net gain of information, and optimal choice of experiment that obtain (La Valle, 1968) when satisfies the Anscombe-Aumann (1963) or Savage (1954) axiomatizations continue to obtain in the more general Schmeidler-Gilboa context-provided that there is no incentive to randomize the choice of experiment. When this proviso fails, information value can in general be assigned only to the set of available experiments.     

A test of generalized expected utility theory

In two experiments we test Machina's Hypothesis II (fanning-out). In each experiment we analyze patterns of responses to hypothetical lottery choice questions within a Marschak-Machina triangle. One set of questions involves lotteries on the border of the triangle, an the other set of questions involves lotteries in the interior of the triangle (off the border). Our results show that a large proportion of the observed patterns in the on-border treatment support Hypothesis II, with a considerable amount of fanning-out behavior observed. The patterns observed in the off-border treatment are significantly different from those in the on-border treatment. Hypothesis II performs well in the off-border treatment because expected utility theory itself, which satisfies the restrictions of Hypothesis II, performs well.This is an expanded version of a paper originally prepared for presentation at the Fifth International Conference on the Foundation and Applications of Utility, Risk, and Decision Theories, held June 9–13, 1990 at Duke University, Durham, NC. We acknowledge helpful comments made by participants at that conference, especially those of Mark Machina.     

Maximizing expected utility is a survival criterion

The conventional expected utility of a lottery establishes a tight myopic lower bound on the probability of indefinitely avoiding economic ruin, the probability of survival. The analysis presented assumes that the decision maker is able to decline lotteries that are unacceptable in the expected utility sense. Application of the result to the assessment of utility functions is discussed.     

Economic choice in generalized expected utility theory

Generalized expected utility models have enjoyed considerable success in explaining observed choices under uncertainty. However, there has been only limited progress in deriving comparative static results. This paper presents a general framework which permits the incorporation of a wide range of generalized expected utility models, but is sufficiently powerful to permit the derivation of comparative static results. The central idea is to represent preferences by the expected utility of a transformed probability distribution.     

Two-parameter decision models and rank-dependent expected utility

This paper extends the existing literature concerning the relationship between two parameter decision models and those based on expected utility in two main directions. The first relaxes Meyer's location and scale (or Sinn's linear class) condition and shows that a two-parameter representation of preferences over uncertain prospects and the expected utility representation yield consistent rankings of random variables when the decision maker's choice set is restricted to random variables differing by mean shifts and monotone meanpreserving spreads. The second shows that the rank-dependent expected utility model is also consistent with two-parameter ranking methods if the probability transform satisfies certain dominance conditions. The main implication of these results is that the simple two-parameter model can be used to analyze the comparative statics properties of a wide variety of economic models, including those with multiple sources of uncertainty when the random variables are comonotonic. To illustrate this point, we apply our results to the problem of optimal portfolio investment with random initial wealth. We find that it is relatively easy to obtain strong global comparative statics results even if preferences do not satisfy the independence axiom.     

Comparative statics for rank-dependent expected utility theory

Recently, a number of generalizations of the expected utility (EU) model have been proposed. In order to make such generalizations useful, it is necessary that they should yield sharp comparative static results, like those obtained using EU theory. In this article, rank dependent expected utility (RDEU) theory, a generalization of EU theory based on the concept of probability weighting, is examined. A number of methods of extending results from EU to RDEU are considered. It is shown that a major class of comparative static results can be extended to the RDEU model, but not to the case of general smooth preferences. This is because RDEU maintains the separation between probabilities and utilities that is abandoned in the general case.     

On the Conditional Value-at-Risk probability-dependent utility function

The Expected Shortfall or Conditional Value-at-Risk (CVaR) has been playing the role of main risk measure in the recent years and paving the way for an enormous number of applications in risk management due to its very intuitive form and important coherence properties. This work aims to explore this measure as a probability-dependent utility functional, introducing an alternative view point for its Choquet Expected Utility representation. Within this point of view, its main preference properties will be characterized and its utility representation provided through local utilities with an explicit dependence on the assessed revenue’s distribution (quantile) function. Then, an intuitive interpretation for the related probability dependence and the piecewise form of such utility will be introduced on an investment pricing context, in which a CVaR maximizer agent will behave in a relativistic way based on his previous estimates of the probability function. Finally, such functional will be extended to incorporate a larger range of risk-averse attitudes and its main properties and implications will be illustrated through examples, such as the so-called Allais Paradox.     

The arguments of utility: Preference reversals in expected utility of income models

There is a debate in the literature about the arguments of utility in expected utility theory. Some implicitly assume utility is defined on final wealth whereas others argue it may be defined on initial wealth and income separately. I argue that making income and wealth separate arguments of utility has important implications that may not be widely recognized. A framework is presented that allows the unified treatment of expected utility models and anomalies. I show that expected utility of income models can predict framing induced preference reversals, a willingness to pay-willingness to accept gap for lotteries, and choice-value preference reversals. The main contribution is a theorem. It is proved that for all utility functions where initial wealth and income enter separately, either there will be preference reversals or preferences can be represented by a utility function defined on final wealth alone.