Understanding Subjective Probability Theory for Decision Making

Muth’s Rational Expectations Hypothesis, That “…The Subjective Probability Distributions Are Distributed Around an Objective Probability Distribution, for a given Information Set…” is an Oxymoron. It Can Be Corrected by Amending It to Read “…The Subjective Probability Distributions Are Distributed Around a Subjective Consensus Probability Distribution, for a given Information Set…”

Neoclassical economists use expected utility theory to explain, predict, and prescribe choices under risk, that is, choices where the decision-maker knows – or at least deems it suitable to act as if she knows – the relevant probabilities. Expected utility theory has been subject to both empirical and conceptual criticism. This chapter reviews expected utility theory and the main criticism it has faced. It ends with a brief discussion of subjective expected utility theory, which is the theory neoclassical economists use to explain, predict, and prescribe choices under uncertainty, that is, choices where the decision-maker cannot act on the basis of objective probabilities but must instead consult her own subjective probabilities.

In Service-Oriented Computing (SOC) environments, the trustworthiness of each service provider is critical for a service client when selecting one from a large pool of service providers. The trust value of a service provider is usually in the range of [0, 1] and is evaluated from the ratings given by service clients, which represent the subjective belief of service clients on the satisfaction of delivered services. So a trust value can be taken as a subjective probability, by which one party believes that another party can perform an action in a certain situation. Hence, subjective probability theory should be adopted in trust evaluation. In addition, in SOC environments, a service provider usually can invoke the services from other service providers forming a composite service. Thus, the global trust of a composite service should be evaluated based on both the subjective probability property of trust and complex invocation structures. In this paper, we first interpret the trust dependency caused by direct service invocations as conditional probability. Then, on the basis of trust dependency, we propose a Subjective probability based deductive (SELECTIVE) approach to evaluate the subjective global trustworthiness of a composite service. All these processes follow subjective probability theory and keep the subjective probability property of trust in evaluations. Our experimental results demonstrate that when compared with existing approaches our proposed SELECTIVE approach can yield more reasonable results.

Recent work on risky choice modelling has sought to address the shortcomings of expected utility theory (EUT) by using non-expected utility theoretic (non-EUT) approaches. However, to date these approaches have been merely tested on stated choice data which is flexible and cheap. In this study, we empirically investigate the feasibility and validity of non-EUT approaches in a revealed preference (RP) context in which travel time distribution is extracted from observed historical travel time data, and subsequently present systematic comparisons between EUT, weighted utility theory, rank-dependent expected utility theory, and prospect theory (PT). The empirical evidence indicates that each non-EUT model has important behavioural insights to offer, moreover, EUT as well as non-EUT models can be applied to the RP context. However, the EUT and non-EUT model fits are generally similar with only PT providing a marginally improved model fit over EUT. The key findings presented in this study reinforce the importance of exploring non-EUT models within a revealed preference context before they can be applied reliably to modelling risky choices in the real world.

Chapter 11 studies the second phase of the debate on expected utility theory (EUT), which commenced in May 1950, when Paul Samuelson, Leonard J. Savage, Jacob Marschak, Milton Friedman, and William Baumol initiated an intense exchange of letters. These economists argued about the exact assumptions underlying EUT, quarreled over whether these assumptions are compelling requisites for rational behavior under risk, and debated the nature of the cardinal utility function u featured in EUT. This correspondence modified the views of all five economists and transformed Samuelson into a supporter of EUT. In a prominent conference in Paris in May 1952, Friedman, Savage, Marschak, and Samuelson advocated EUT in the face of attacks from Maurice Allais and other opponents of the theory. The Paris conference and the publication of an Econometrica symposium on EUT in October 1952 marked the emergence of EUT as the mainstream economic model of decision-making under risk.

Expected utility theory, and the Bayesian probability theory on which it is based, form the normative basis of most work in medical decision analysis. Recent work in the psychology of judgments and decisions indicates that people do not conform to the axioms of this theory and that these deviations occur in clinical reasoning as well as in the psychology laboratory. At issue is what to do now. The authors argue that the important next steps lie at the interface between descriptive, prescriptive, and normative accounts, all of which affect each other. They point to examples in which the simplest application of supposedly normative theory seems inappropriate, and suggest ways in which the tension between normative and descriptive models may be resolved.

The von Neumann-Morgenstern ‘expected utility hypothesis’ as presented in Chapter I of The Theory of Games and Economic Behavior (1944) was the first attempt at a rigorous axiomatization of a fundamental concept of economics, namely, a general theory of utility. There is no a priori notion of ‘rationality’ in the theory; rather, that is a concept to be derived and given meaning to from the theory. The theory is ‘absolutely convincing’ in the sense that if behavior deviates from that predicated by the theory, after explanation of the theory, behavior would be adjusted. The von Neumann-Morgenstern ‘expected utility theory’ was the first approximation to an undoubtedly much richer and far more complicated reality. Critics of the theory, such as Allais and others, have centered their criticism of the theory not on the internal consistency of the axioms of which the theory is composed but rather on the validity of the axioms themselves. In that sense, the attacks on the theory are misplaced. It is common knowledge throughout scientific circles that axiomatic systems are often modified as axioms are more precisely specified, qualified, or, at times, replaced, as our understanding of physical laws is expanded. Not unexpectedly, the evolution of the von Neumann-Morgenstern ‘expected utility theory’ over the past thirty years is in complete concordance with scientific progress.

Housing is a complex commodity and housing market choices carry with them substantial economic and social consequences for the households making them. Housing market decisions are complex, uncertain and involve expectations-formation. This paper argues that the standard economic theory of decision-making under uncertainty – expected utility theory – is particularly ill-suited as the basis for understanding such complexity. The paper then explores alternative avenues for potential development, reviewing the key characteristics of owner-occupied housing markets and housing search, and examining how the resources of institutional and behavioural economics could be used to inform our understanding of the residential mobility process. The paper concludes by outlining an agenda for empirical research.

Theoretical models as Expected Utility Theory (EUT) start from the premise that decision making individuals follow subjective rules which they found e. g. on their personal probability judgements. Thus, EUT requires that the decision maker has a complete knowledge of all relevant states of the world and consequences of possible actions. Often, there is a lack of information basis to attribute probabilities to states of the world and to evaluate the utility of potentially occurring consequences. In such cases, alternative decision theories to EUT are to be applied for a reality adequate explanation of decision-making processes. Such a theory, being presented in the following, is Case-Based Decision Theory (CBDT), which will be analysed experimentally with respect to its empirical validity. The results shall substantiate the validity of the premises and the methods of the CBDT in the context of a repeated-choice problem.

Direct experimental tests of expected utility theory (EUT), in which subjects are asked to choose among alternative gambles, or to make judgements as to their willingness to pay (WTP), and/or willingness to accept (WTA) payment for a gamble, have not been kind to EUT. As noted in the survey by Paul Slovic and Sarah Lichtenstein (1983), the results of these interrogations are remarkably consistent in a wide variety of contexts and are robust under examinations designed to determine the effect of monetary incentives, experience, and other factors that might have accounted for the discrepency between subject responses and the predictions of EUT. On the other hand, experimental studies of individual and market behavior based upon EUT models of market decision making have yielded results showing high consistency with the predictions of these models (see the references in Smith, 1985). Are individual revealed preferences in some market contexts more likely to be rational (consistent with EUT) than individual responses to choices among alternative prospects? Several studies designed to solicit WTP and WTA responses for a variety of goods have found a wide disparity between the buying price and selling price measures of individual value (see the study and the citations therein by Jack Knetsch and J. A. Sinden, 1984; hereafter K-S). Values of WTA obtained in this way are frequently an order of magnitude greater than values of WTP although theoretically WTP and WTA differ by no more than a presumed small income effect. These results should not be dismissed by economists on the grounds of poor subject motivation because the experiments include some (see those in K-S) that have carefully introduced actual monetary payments and cash compensations and have not relied on hypothetical choices. These results are explained by K-S (and by most of the authors of such studies) in terms of the Daniel Kahneman and Amos Tversky (1982) framing paradigm in which people reveal that they are less willing to spend wealth which they consider to be part of their endowment than wealth not so considered. We would emphasize that such behavior is irrational only in the narrow sense of EUT as a behavioral hypothesis which may not only be a poor predictor of individual choice, but may not be a satisfactory guide to action. For example, the differential treatment of wealth that has become part of one's endowment may have important survival value which is imprinted in decision behavior. However, K-S (p. 508, fn. 3) report one, perhaps significant, experiment in which they do not get the usual WTA-WTP disparity. In this experiment, cash payments and offers for a lunch made to respondents entering an office cafeteria showed a much smaller, statistically insignificant, disparity. In this case, the respondents were about to participate in a familiar market for the commodity being evaluated. On this same theme an important new study by Don Coursey et al. (1984) uses a repetitive series of Vickrey (second price) auctions to determine market WTA and WTP prices for entitlements to an unfamiliar item, which they compare with hypothetical measures of WTA and WTP. They find that although individual bids in the first of a sequence of Vickrey auctions show a large WTA-WTP disparity, ending bids, after a *Graduate Student in Finance, University of Pennsylvania, Philadelphia, PA 19104; Professor of Economics, University of Arizona, Tucson, AZ 85721; Assistant Professor of Economics, Indiana.University, Bloomington, IN 47401. Research support from the National Science Foundation is gratefully acknowledged.

Decision making investigates choices that have uncertain consequences and that cannot be completely predicted. Rational behavior may be described by the so-called expected utility theory (EUT), whose aim is to help choosing among several solutions to maximize the expectation of the consequences. However, Kahneman and Tversky developed an alternative model, called prospect theory (PT), showing that the basic axioms of EUT are violated in several instances. In respect of EUT, PT takes into account irrational behaviors and heuristic biases. It suggests an alternative approach, in which probabilities are replaced by decision weights, which are strictly related to the decision maker’s preferences and may change for different individuals. In particular, people underestimate the utility of uncertain scenarios compared to outcomes obtained with certainty, and show inconsistent preferences when the same choice is presented in different forms. The goal of this paper is precisely to analyze a real case study involving a decision problem regarding the Streicker Bridge, a pedestrian bridge on Princeton University campus. By modelling the manager of the bridge with the EUT first, and with PT later, we want to verify the differences between the two approaches and to investigate how the two models are sensitive to unpacking probabilities, which represent a common cognitive bias in irrational behaviors.

Bayesian statistical techniques are applicablewhen the information anduncertainty with respect to the parameters or hypotheses in question can be expressed by a probability distribution. This prior probability is also the focus of most of the criticism against the Bayesian school. My starting point is to join the critics in attacking a certain aspect of the prior probability: The probability attached to an uncertain event does not reflect the heuristic amount of information that led to the assignment of that probability. For example, when the information on the occurrence of two events is symmetric they are assigned equal prior probabilities. If the events are complementary the probabilities will be 1/2, independently of whether the symmetric information is meager or abundant.

9 - Prospect theory, asset pricing, and market dynamics

Savage's expected utility theory has been criticized for various reasons. Two of these are: (1) Subjective probabilities may be more or less ambiguous; this may affect the preference order. Savage's theory rules out any such effect. (2) Savage's theory rules out any influence of certain parameters of a probability distribution on the preference order. Such parameters are fractiles or pairs of fractiles, e.g., which have gained widespread attention as risk measures.

Structural measures of higher order risk attitudes have well‐developed foundations in Expected Utility Theory (EUT), but little is known about their empirical magnitudes. We introduce a novel experimental design and a companion econometric model that allows us to structurally estimate indices of risk aversion, prudence, and temperance under EUT without imposing restrictions on their interdependence. We find that indices of absolute risk aversion, prudence, and temperance exhibit distinct patterns of variation over income, and that predicted risk premia under EUT and Rank‐Dependent Utility Theory gradually converge as the order of risk increases. These findings are obscured by regular parametric utility functions, which inherently bias results toward prudence and temperance when subjects are risk averse. The results remain robust in subsamples of moderate size, which suggests that our approach can be adopted in broader studies that link higher order risk attitudes to other domains of latent individual preferences and economic behavior.