EXPECTED UTILITY HYPOTHESIS

We provide a condition for an individual preference ordering to be represented by a function measured in terms of a commodity, i.e., for the commodity to be transferable utility. We also consider the relationships between conditions of the preference ordering and the utility function.

In the paper there is studied an optimal saving model in which the interest-rate risk for saving is a fuzzy number. The total utility of consumption is defined by using a concept of possibilistic expected utility. A notion of possibilistic precautionary saving is introduced as a measure of the variation of optimal saving level when moving from a sure saving model to a possibilistic risk model. A first result establishes a necessary and sufficient condition that the presence of a possibilistic interest-rate risk generates an extra-saving. This result can be seen as a possibilistic version of a Rothschilld and Stiglitz theorem on a probabilistic model of saving. A second result of the paper studies the variation of the optimal saving level when moving from a probabilistic model (the interest-rate risk is a random variable) to a possibilistic model (the interest-rate risk is a fuzzy number).

Attachment values Allais paradox Insurance paradox Ellsberg paradox Coalescing paradox Violations of stochastic dominance Cash segregation paradox Risk preferences for losses paradox a b s t r a c t The expected utility (EU) model is widely used for predicting and describing choices under uncertainty. Its usefulness, however, is limited because of its widely acknowledged inconsistencies and paradoxes. This paper describes how important EU model paradoxes can be resolved by accounting for the influences of socio-emotional goods (SEGs) embedded in word and other symbolic frames.

The first axiom of the Analytic Hierarchy Process (AHP) states that the pairwise comparison judgments are reciprocal. Here we explore the relationships between the reciprocal property and preference relations with and without the axiom of transitivity and point out some important distinctions between utility theory and the AHP.

This paper addresses the impacts of degree of risk aversion on optimal farm plans in Kazakh agriculture. The results obtained during the field research in the region of Northern Kazakhstan are presented. Calculations were carried out using data from 145 peasant farms in the region for 2017-2022 based on a risk model. It has been found that the choice of a portfolio is influenced not only by considerations of profitability and riskiness of crops, but also by grain production traditions deeply rooted among the farmers of the region, and the skills and knowledge associated with them, as well as the existing infrastructure. These circumstances constrain the wider spread of oilseed crops in the region. It seems that the size of the farm does not significantly affect the choice of portfolio, while the degree of risk aversion by the farmer affects the optimal farming plan. The potential benefits of the farming diversification are an empirical issue, and it should be addressed on a case-by-...

Risk aversion plays a crucial role in understanding how individuals make financial decisions and allocate their resources. This study analyzes the influence of risk aversion on behavioral intentions and explores the mediating role of attitudes, subjective norms, and perceived behavioral control. Additionally, it investigates the moderating effect of gender and financial literacy on behavioral intentions of investors. A sample of 400 people was collected from Indian retail investors by administering a structured questionnaire through stock brokering firms, and data were analyzed using Partial least squares -Structural equation modelling in the Smart PLS 3.3.9 software. The research found that risk aversion, attitude, subjective norms, and perceived behavioral control significantly impact an investor's intention. Among all the antecedents of behavioral intentions, perceived behavioral control (β 0.481*) was found as a significant predictor of the intention compared to attitude (β 0.154*), subjective norms (β 0.224*) and risk aversion (β 0.082*) factors. Further, mediation analysis found that attitude, subjective norms, and perceived behavioral control partially mediated the relationship between risk aversion and intention. Lastly, the multi-group analysis revealed that gender and financial literacy did not moderate the association between risk aversion and intention.

We study a Black-Scholes market with a finite time horizon and two investors: an honest and an insider trader. We analyze it with anticipating stochastic calculus in two steps. First, we recover the classical result on portfolio optimization that shows that the expected logarithmic utility of the insider is strictly greater than that of the honest trader. Then, we prove that, whenever the market is viable, the honest trader can get a higher logarithmic utility, and therefore more wealth, than the insider with a strictly positive probability. Our proof relies on the analysis of a sort of forward integral variant of the Doléans-Dade exponential process. The main financial conclusion is that the logarithmic utility is perhaps too conservative for some insiders.

The decision to transfer or share an insurable risk is critical for the decision maker's economy. This paper deals with this decision, starting with the definition of a function that represents the difference between the expected utility of insuring, with or without deductibles, and the expected utility of not insuring. Considering a constant relative risk aversion (CRRA) utility function, we provide a decision pattern for the potential policyholders as a function of their wealth level. The obtained rule applies to any premium principle, any per-claim deductible and any risk distribution. Furthermore, numerical results are presented based on the mean principle, a per-claim absolute deductible and a Poisson-exponential model, and a sensitivity analysis regarding the deductible parameter and the insurer security loading was performed. One of the main conclusions of the paper is that the initial level of wealth is the main variable that determines the decision to insure or not to insure; thus, for high levels of wealth, the decision is always not to insure regardless of the risk aversion of the decision maker. Moreover, the parameters defining the deductible and the premium only have an influence at low levels of wealth.

We model the notion of a "small world" as a context dependent state space embedded into the "grand world". For each situation the decision maker creates a "small world" reflecting the events perceived to be relevant for the act under consideration. The "grand world" is represented by an event space which is a more general construction than a state space. We retain preference axioms similar in spirit to the Savage axioms and obtain, without abandoning linearity of expectations, a subjective expected utility theory which allows for an intuitive distinction between risk and uncertainty. We also obtain separation of subjective probability and utility as in the state space models. JEL classification: D8 and G12.

We formulate a new theory of expected utility in which risk and uncertainty is modelled by the usage of a so called event space which is a natural generalisation of a state space. The basic idea is that the decision maker for each group of related decisions creates a "small world" (a local state space) from the events in the "grand world" (the event space). We introduce a set of preference axioms similar in spirit to the Savage axioms, and show that they lead to a a more general expected utility theory, where in each "small world" risk is described by an additive probability measure. All local risk measures appear as restrictions of a common integrated additive expectation functional defined on the "grand world". A benefit of the theory is that it allows for an intuitive distinction between risk and uncertainty. In particular, there is a numerical measure of the degree of uncertainty aversion associated with a given preference relation, which can be calculated. We illustrate the use of the theory for the Ellsberg paradox and for a no-trade portfolio decision problem, none of which can be captured using standard expected utility theory. JEL classification: D8 and G12.

We model the notion of a "small world" as a context dependent state space embedded into the "grand world". For each situation the decision maker creates a "small world" reflecting the events perceived to be relevant for the act under consideration. The "grand world" is represented by an event space which is a more general construction than a state space. We retain preference axioms similar in spirit to the Savage axioms and obtain, without abandoning linearity of expectations, a subjective expected utility theory which allows for an intuitive distinction between risk and uncertainty. We also obtain separation of subjective probability and utility as in the state space models. JEL classification: D8 and G12.

We formulate a new theory of expected utility in which risk and uncertainty is modelled by the usage of a so called event space which is a natural generalisation of a state space. The basic idea is that the decision maker for each group of related decisions creates a "small world" (a local state space) from the events in the "grand world" (the event space). We introduce a set of preference axioms similar in spirit to the Savage axioms, and show that they lead to a a more general expected utility theory, where in each "small world" risk is described by an additive probability measure. All local risk measures appear as restrictions of a common integrated additive expectation functional defined on the "grand world". A benefit of the theory is that it allows for an intuitive distinction between risk and uncertainty. In particular, there is a numerical measure of the degree of uncertainty aversion associated with a given preference relation, which ...

Risk management lies at the core of information security. Professionals need to assess risk and make decisions on how to treat risk. Risk perception and judgement of individuals are inherently involved in this process. This paper examines information security professionals’ attitude to risk. We conduct an online experiment and survey which solicits preferences using risky lotteries. We also test whether framing of decisions as gains, losses, or individually separated losses has an effect on their risk attitude. Framing is found to diversify professionals’ risk behaviour significantly. Our findings suggest that professionals reveal a preference for paying to reduce risk instead of paying to eliminate it. They also prefer to reduce the expected loss of threat scenarios rather than reducing the vulnerability associated with this loss. Overall, professionals are risk averse when they face lotteries with small probabilities of loss and risk seeking for lotteries with large probabilities.

Several approaches have been proposed for evaluating information in expected utility theory. Among the most popular approaches are the expected utility increase, the selling price and the buying price. While the expected utility increase and the selling price always agree in ranking information alternatives, Hazen and Sounderpandian have demonstrated that the buying price may not always agree with the other two. That is, in some cases, where the expected utility increase would value information A more highly than information B, the buying price may reverse these preferences. In this paper, we discuss the conditions under which all these approaches agree in a generic decision environment where the decision maker may choose to acquire arbitrary information bundles.

People violate expected utility theory and this has been traditionally modeled by augmenting its weight-and-add framework by nonlinear transformations of values and probabilities. Yet individuals often use one-reason decision-making when making court decisions or choosing cellular phones, and institutions do the same when creating rules for traffic safety or fair play in sports. We analyze a model of one-reason decision-making, the priority heuristic, and show that it simultaneously implies common consequence effects, common ratio effects, reflection effects, and the fourfold pattern of risk attitude. The preferences represented by the priority heuristic satisfy some standard axioms. This work may provide the basis for a new look at bounded rationality. Most descriptive theories of decision-making-and certainly those that are variants of expected value theory (EVT) or expected utility theory (EUT)-make the following psychological assumptions: 1. Independent evaluations: Every option has a value that is measured by a single number (options are not evaluated relative to other options). 2. Exhaustive search: The value of an option is calculated by using all available information (for gambles, the probabilities and values for all possible outcomes). 3. Trade-offs: To calculate an option's value, low values on one attribute (e.g., a value) can be compensated by high values on another attribute (e.g., a probability). EVT makes two additional assumptions for calculating an option's value: 4. Objective probabilities: The probabilities used to calculate an option's value are equal to the objective probabilities (the objective probabilities are not transformed in a nonlinear way).

In this work, we examine the combined problem of optimal portfolio selection rules for an insurer in a continuous-time model where the surplus of an insurance company is modelled as a compound Poisson process. The company can invest its surplus in a risk free asset and in a risky asset, governed by the Black-Scholes equation. According to utility theory, in a financial market where investors are facing uncertainty, an investor is not concerned with wealth maximization per se but with utility maximization. It is therefore possible to introduce an increasing and concave utility function φ(x, t) representing the expected utility of a risk averse investor (insurance company). Therefore, the goal of this work is not anymore to maximize the expected portfolio value or minimize the ruin probability or maximizing the expectation of the present value of all dividends paid to the shareholders up to the ruin, but to maximize the expected utility stemming from the wealth during the life contract [0, T ]. In this direction, using the Dynamic Programming Principle of the problem, we obtain the Hamilton-Jacobi-Bellman equation by our optimization problem (HJB). Finally, we present numerical solutions in some cases, obtaining as optimal strategy the well known Merton's strategy.

First, I show that the expected consumer's surplus is equivalent to ex ante compensating variation if and only if the consumer is risk neutral, and the consumer's income elasticity of demand for the commodity is zero. Moreover, the conditions are equivalent to the von Neuman - Morgenstern utility function being quasi-linear. Second, I show that the expected consumer's surplus is an approximation for the consumer's welfare, measured by expected utility, also if the expenditure share is small. Third, I propose a formula to evaluate approximately the consumer's welfare, measured both by expected utility and by ex ante compensating variation, when the above conditions are not met.

There is a long standing discussion of whether expected utility theory (EU) or prospect theory (PT) best explains the behavior with respect to risky choices. Often these two approaches are compared by putting questions to students in laboratory situations. Here we try to investigate stated preferences of farmers who are functioning under high levels of risk in real life. As part of a larger survey, four binary choices were offered during two successive years. The experimental test was performed 199 farmers in two districts in Tigray, Northern Ethiopia. Three items were central in comparing the risk attitude according to EU and PT: the asymmetry of risk perceptions, the independence axiom and the shape of the utility function. The farmers in the two different districts (Enderta and Hintalo-Wajerat) differed significantly in their risk attitude. Enderta farmers were significantly risk-averse for gains and risk-seeking for losses; their preferences conformed to the hypothesis of prospe...

Stochastic multi-armed bandits solve the Exploration-Exploitation dilemma and ultimately maximize the expected reward. Nonetheless, in many practical problems, maximizing the expected reward is not the most desirable objective. In this paper, we introduce a novel setting based on the principle of risk-aversion where the objective is to compete against the arm with the best risk-return trade-off. This setting proves to be intrinsically more difficult than the standard multi-arm bandit setting due in part to an exploration risk which introduces a regret associated to the variability of an algorithm. Using variance as a measure of risk, we introduce two new algorithms, investigate their theoretical guarantees, and report preliminary empirical results.

Strategic decision-making over valuable resources should consider risk-averse objectives. Many practical areas of application consider risk as central to decision- making. However, machine learning does not. As a result, research should provide insights and algorithms that endow machine learning with the ability to consider decision-theoretic risk. In particular, in estimating decision-theoretic risk on short dependent sequences generated from the most general possible class of processes for statistical inference and through decision-theoretic risk objectives in sequential decision-making. This thesis studies these two problems to provide principled algorithmic methods for considering decision-theoretic risk in machine learning. An algorithm with state-of-the-art performance is introduced for accurate estimation of risk statistics on the most general class of stationary–ergodic processes and risk-averse objectives are introduced in sequential decision-making (online learning) in bo...

La prise de decision strategique concernant des ressources de valeur devrait tenir compte du degre d'aversion au risque. D'ailleurs, de nombreux domaines d'application mettent le risque au cœur de la prise de decision. Toutefois, ce n'est pas le cas de l'apprentissage automatique. Ainsi, il semble essentiel de devoir fournir des indicateurs et des algorithmes dotant l'apprentissage automatique de la possibilite de prendre en consideration le risque dans la prise de decision. En particulier, nous souhaiterions pouvoir estimer ce dernier sur de courtes sequences dependantes generees a partir de la classe la plus generale possible de processus stochastiques en utilisant des outils theoriques d'inference statistique et d'aversion au risque dans la prise de decision sequentielle. Cette these etudie ces deux problemes en fournissant des methodes algorithmiques prenant en consideration le risque dans le cadre de la prise de decision en apprentissage automati...

We use data from 298 showings of the television program "Affari Tuoi," which involves contestants making decisions between risky prospects with possible prizes of up to half a million euros, to estimate three models of decision-making under risk: Expected Utility, Rank-Dependent Expected Utility and Regret-Rejoice. We find that Regret-Rejoice does not significantly improve upon Expected Utility, while Rank-Dependent outperforms it. Interestingly, we find that the CARA specification fits significantly better than the conventionally-adopted CRRA specification. Crucially, we find a significant role for unobserved heterogeneity, implying that our estimates provide more superior estimates of risk attitude and of probability weighting than other studies.

For helpful commments on an earlier version, we thank participants at a conference at Princeton, particularly Cli¤ Carrubba and John Patty. We further thank Sepher Shahshahani, Ben Johnson, and students in the Spring 2016 class "Courts" in the NYU Politics Department.

The riskiness of random processes is compared by (a) employing a decision-theoretic equivalence between processes and lotteries on pathspaces to identify the riskiness of the former with that of the latter, and (b) using the theory of comparative riskiness of lotteries over vector spaces to compare the riskiness of lotteries on a given path-space. We derive the equivalence used in step (a) and contribute a new criterion to the theory applied in step (b). The new criterion, involving a generalized form of second order stochastic dominance, is shown to be valid by establishing its equivalence to the standard decision-theoretic criterion. We demonstrate its tractability via diverse economic applications featuring risk embodied in random processes.

We provide a general method for extending social preferences defined for riskless economic environments to the context of risk and uncertainty. We apply the method to the problems of managing unemployment allowances (in the context of macroeconomic fluctuations) and catastrophic risks (in the context of climate change). The method guarantees ex post fairness and pays attention to individuals' risk attitudes, while ensuring rationality properties for social preferences, revisiting basic ideas from Harsanyi's celebrated aggregation theorem . The social preferences that we obtain do not always take the form of an expected utility criterion, but they always satisfy statewise dominance. When we require social preferences to be expected utilities, we obtain a variant of Harsanyi's result under a weak version of the Pareto principle, and a maximin criterion under a stronger Pareto requirement, whenever the ex post social ordering does not depend on people risk attitudes. We also show how non-expected utility individual preferences can be accommodated in the approach.

In the context of extreme climate change, we ask how to conduct expected utility analysis in the presence of catastrophic risks. Economists typically model decision making under risk and uncertainty by expected utility with constant relative risk aversion (power utility); statisticians typically model economic catastrophes by probability distributions with heavy tails. Unfortunately, the expected utility framework is fragile with respect to heavy-tailed distributional assumptions. We specify a stochastic economyclimate model with power utility and explicitly demonstrate this fragility. We derive necessary and sufficient compatibility conditions on the utility function to avoid fragility and solve our stochastic economy-climate model for two examples of such compatible utility functions. We further develop and implement a procedure to learn the input parameters of our model and show that the model thus specified produces quite robust optimal policies. The numerical results indicate that higher levels of uncertainty (heavier tails) lead to less abatement and consumption, and to more investment, but this effect is not unlimited.

In searching for an appropriate utility function in the expected utility framework, we formulate four properties that we want the utility function to satisfy. We conduct a search for such a function, and we identify Pareto utility as a function satisfying all four desired properties. Pareto utility is a flexible yet simple and parsimonious two-parameter family. It exhibits decreasing absolute risk aversion and increasing but bounded relative risk aversion. It is applicable irrespective of the probability distribution relevant to the prospect to be evaluated. Pareto utility is therefore particularly suited for catastrophic risk analysis. A new and related class of generalized exponential (gexpo) utility functions is also studied. This class is particularly relevant in situations where absolute risk tolerance is thought to be concave rather than linear.

We investigate the application of classification tech niques to utility elicitation. In a decision problem, two sets of parameters must generally be elicited: the prob abilities and the utilities. While the prior and condi tional probabilities in the model do not change from user to user, the utility models do. Thus it is necessary to elicit a utility model separately for each new user. Elicitation is long and tedious, particularly if the out come space is large and not decomposable. There are two common approaches to utility function elicitation. The first is to base the determination of the user's util ity function solely on elicitation of qualitative prefer ences. The second makes assumptions about the form and decomposability of the utility function. Here we take a different approach: we attempt to identify the new user's utility function based on classification rel ative to a database of previously collected utility func tions. We do this by identifying clusters of utility func tions that minimize an appropriate distance measure. Having identified the clusters, we develop a classifi cation scheme that requires many fewer and simpler assessments than full utility elicitation and is more ro bust than utility elicitation based solely on preferences. We have tested our algorithm on a small database of utility functions in a prenatal diagnosis domain and the results are quite promising.

Obtaining utility maximizing optimal portfolio in closed form is a challenging issue when the return vector follows a more general distribution than the normal one. In this note, we give closed form expression, in markets based on finitely many assets, for optimal portfolios that maximize exponential utility function when the return vector follows normal mean-variance mixture models. We then consider large financial markets based on normal mean-variance mixture models also and show that the optimal exponential utilities based on small markets converge to the optimal exponential utility in the large financial market. This shows, in particular, that to reach the best utility level investors need to diversify their investment to include infinitely many assets into their portfolio and with portfolios based on only finitely many assets they never be able to reach the optimum level of utility.

An investment opportunity whose return is perfectly predictable, hardly exists at all. Instead, investor makes his decisions under conditions of uncertainty. Theory of expected utility is the main analytical tool for description of choice under uncertainty. Critics of the theory contend that individuals have bounded rationality and that the theory of expected utility is not correct. When agents are faced with risky decisions they behave differently, conditional on their attitude towards risk. They can be risk loving, risk averse or risk neutral. In order to make an investment decision it is necessary to compare probability distribution functions of returns. Investment decision making is much simpler if one uses expected values and variances instead of probability distribution functions.

Empirically, co-skewness of asset returns seems to explain a substantial part of the cross-sectional variation of mean return not explained by beta. This finding is typically interpreted in terms of a risk averse representative investor with a cubic utility function. This comment questions this interpretation. We show that the empirical tests fail to impose risk aversion and the implied utility function takes an inverse S-shape. Unfortunately, the first-order conditions are not sufficient to guarantee that the market portfolio is the global maximum for an inverse S-shaped utility function, and our results suggest that the market portfolio is more likely to represent the global minimum than the global maximum. In addition, if we impose risk aversion, then co-skewness has minimal explanatory power.

We consider an extended two-stage risk-averse stochastic optimization problems in several formulations. Risk-aversion is reflected by risk constraints in form of stochastic-order relations, which are imposed in a time-consistent manner. The problem is analyzed under convexity assumptions. The main goal of this study is to establishing relations between the extended two-stage problem with stochastic-order constraints on the recourse function on the one hand and the two-stage problems with alternative models of risk such as utility functions, distortions, or coherent measures of risk on the other hand.

This paper extends our earlier work on reverse Bayesianism by relaxing the assumption that decision makers abide by expected utility theory, assuming instead weaker axioms that merely imply that they are probabilistically sophisticated. We show that our main results, namely, (modified) representation theorems and corresponding rules for updating beliefs over expanding state spaces and null events that constitute "reverse Bayesianism," remain valid.

We consider a rm that is faced with an uncontrollable stochastic cash ow, or random risk process. There is one investment opportunity, a risky stock, and we study the optimal investment decision for such rms. There is a fundamental incompleteness in the market, in that the risk to the investor of going bankrupt can not be eliminated under any investment strategy, since the random risk process ensures that there is always a positive probability of ruin (bankruptcy). We therefore focus on obtaining investment strategies which are optimal in the sense of minimizing the risk of ruin. In particular, we solve for the strategy that maximizes the probability o f a c hieving a given upper wealth level before hitting a given lower level. This policy also minimizes the probability of ruin. We p r o ve that when there is no risk-free interest rate, this policy is equivalent to the policy that maximizes utility f r o m t e r m i n a l w ealth, for a xed terminal time, when the rm has an exponential utility function. This validates a longstanding conjecture about the relation between minimizing probability of ruin and exponential utility. When there is a positive risk-free interest rate, the conjecture is shown to be false.

We consider a firm that is faced with an uncontrollable stochastic cash flow, or random risk process. There is one investment opportunity, a risky stock, and we study the optimal investment decision for such firms. There is a fundamental incompleteness in the market, in that the risk to the investor of going bankrupt cannot be eliminated under any investment strategy, since the random risk process ensures that there is always a positive probability of ruin (bankruptcy). We therefore focus on obtaining investment strategies which are optimal in the sense of minimizing the risk of ruin. In particular, we solve for the strategy that maximizes the probability of achieving a given upper wealth level before hitting a given lower level. This policy also minimizes the probability of ruin. We prove that when there is no risk-free interest rate, this policy is equivalent to the policy that maximizes utility from terminal wealth, for a fixed terminal time, when the firm has an exponential util...

A rigorous reconstruction of scenario-based real choice making reveals the incompleteness of decision-modeling and the practical prevalence of uncertainty. Theoretically complete models conceal it. As a remedy a scenario-based procedure of coping with uncertainty can prescribe how the boundedly rational decision-maker should proceed from her or his internal point of view. Though models of substantive rationality cannot serve as guidance for decision-making under uncertainty, we can improve prescriptions for coping with uncertainties in view of evidence of substantive success or failure. Decision theory should aim for a reflective equilibrium incorporating internalist-prescriptive and externalist-descriptive aspects to turn good into better practice.

Possibilistic de cision theory has be e n propose d twe nty years ago and has had se ve ral extensions since the n. Even though ap pe aling for its ability to handle qualitative decision proble ms, possibilistic decision the ory suffe rs from an important drawback. Qualitative possibilistic utility criteria compare acts through min and max operators, which Jeads to a drowning effect. To over come this Jack of decision power of the theory, several refinements have been proposed. Lexicographie refinements are particularly appealing since they allow to benefit from the Expected Utility background, while remaining qualitative. This article aims at extend ing lexicographie refinements to sequential decision problems i.e., to possibilistic decision trees and possibilistic Markov decision proce sse s, whe n the horizon is finite . We present two crite ria that re fine qualitative possibilistic utilitie s and provide dynarnic prograrnming algorithrns for calculating Jexicographically optimal policies.

We develop a new version of prospect theory that employs cumulative rather than separable decision weights and extends the theory in several respects. This version, called cumulative prospect theory, applies to uncertain as well as to risky prospects with any number of outcomes, and it allows different weighting functions for gains and for losses. Two principles, diminishing sensitivity and loss aversion, are invoked to explain the characteristic curvature of the value function and the weighting functions. A review of the experimental evidence and the results of a new experiment confirm a distinctive fourfold pattern of risk attitudes: risk aversion for gains and risk seeking for losses of high probability; risk seeking for gains and risk aversion for losses of low probability. Expected utility theory reigned for several decades as the dominant normative and descriptive model of decision making under uncertainty, but it has come under serious question in recent years. There is now general agreement that the theory does not provide an adequate description of individual choice: a substantial body of evidence shows that decision makers systematically violate its basic tenets. Many alternative models have been proposed in response to this empirical challenge (for reviews, see Camerer, 1989; Fishburn, 1988; Machina, 1987). Some time ago we presented a model of choice, called prospect theory, which explained the major violations of expected utility theory in choices between risky prospects with a small number of outcomes (Kahneman and Tversky, 1979; Tversky and Kahneman, 1986). The key elements of this theory are 1) a value function that is concave for gains, convex for losses, and steeper for losses than for gains, *An earlier version of this article was entitled "Cumulative Prospect Theory: An Analysis of Decision under Uncertainty." This article has benefited from discussions with Colin Camerer, Chew Soo-Hong, David Freedman, and David H. Krantz. We are especially grateful to Peter P. Wakker for his invaluable input and contribution to the axiomatic analysis. We are indebted to Richard Gonzalez and Amy Hayes for running the experiment and analyzing the data.

Acts are functions from states of nature into finite-support distributions over a set of "deterministic outcomes". We characterize preference relations over acts which have a numerical representation by the functional J(f) = min(. 1 uofdP I P e C) where f is an act, u is a von-Neumann-Morgtnstern utility over outcomes, and C is a closed and convex set of finitely additive probability measures on the states of nature.

Acts are functions from states of nature into finite-support distributions over a set of "deterministic outcomes". We characterize preference relations over acts which have a numerical representation by the functional J(f) = min(. 1 uofdP I P e C) where f is an act, u is a von-Neumann-Morgtnstern utility over outcomes, and C is a closed and convex set of finitely additive probability measures on the states of nature.

This paper explores a particularly simple model of choice under risk, based on geometric means and entropy. Despite its simplicity, it satisfies various prudence and risk aversion conditions, is consistent with the Allais paradox, and generates various insurance-related results. Within a portfolio framework with compounded reinvestments, our index fits the risks/rewards data from post-war US stock market returns and recent international markets, at least as well as does the standard deviation measures more typically used. It also generates returns that are consistent with the equity premium puzzle.

The standard power utility function is widely used to explain asset prices. It assumes that the coefficient of relative risk aversion is the inverse of the elasticity of substitution. Here I use the Kihlstrom and Mirman (1974) expected utility approach to relax this assumption. I use time consistent preferences that lead to time consistent plans. In our examples, the past does not matter much for current portfolio decisions. The risk aversion parameter can be inferred from experiments and introspections about bets in terms of permanent consumption (wealth). Evidence about the change in the attitude towards bets over the life cycle may also restrict the value of the risk aversion parameter. Monotonic transformations of the standard power utility function do not change the predictions about asset prices by much. Both the elasticity of substitution and risk aversion play a role in determining the equity premium.

We discuss a central question in the study of courts: What do judges want? We suggest three di¤erent domains that might serve as the basic preferences of a judge: case dispositions and rules, caseloads and case mixes, and social consequences. We emphasize preferences over dispositions on the grounds of plausibility and tractability. We then identify desireable properties of dispositional utility functions and the relationship between dispositional utility and expected utility for rules. We examine the impact on expected rule utility from case distributions that are sensitive to the enforced rule. We illustrate how to combine dipositional utility with e¤orts costs and time constraints. We provide examples of case spaces, dispositional utility functions, and expected utility functions for enforced rules. This essay is a draft chapter of a book-in-progress on the positive political theory of courts.

Purpose. To determine current tendencies of the effect of science on economic growth of certain countries in the world and to formulate theoretical and methodological approaches to their explanation in the context of social and economic development of the society. methodology. The research applies the totality of general scientific and specific methods and approaches of empiric and theoretical knowledge, i.e.: methods of analysis and synthesis-to study sources and tendencies of the modern economic growth; grouping and classification-while forming sampling of the countries to analyze the effect of science on their development; method of mathematical statistics-for quantitative evaluation of the effect of science and innovations on economic growth of certain countries in the world; abstracting-to identify the reasons of inconformity of theoretical conclusions of the economic growth models to the modern tendencies of social and economic growth. Findings. Effect of the factors connected with the scientific and technical progress on economic growth has been tested empirically. Basing on the Integral Science Index developed by the authors, parameters of the science effect on the economic growth of certain countries in the world have been evaluated quantitatively. Qualitative assessment of the connection between the mentioned Index and GDP per capita in different groups of countries is represented with the help of a sampling coefficient of correlation characterizing the degree of closeness of linear correlation dependence. It has been determined that there is no direct connection between the science and economic growth; the paradox is explained in detail. originality. Inconsistencies of the key conclusions of the models of economic growth with current tendencies of social and economic development have been identified. A statement concerning the uniqueness of the effect of science on the development of countries has been disproved. A necessity of going beyond the economic and mathematical set of tools, while studying economic development, in favour of complexity and multidisciplinarity has been proved. Practical value. The determined variability of the effect of science on the economic growth approves the understanding of ambiguousness and nonlinearity of the process; thus, it requires constant empiric tests of the available dependences and searching for new influences depending on the stage of civilization process which different countries experience. All that helps substantiate the fundamentals of economic policy of national governments in their favouring social and economic development by selecting more effective leverages.