Choice under uncertainty

2020 ◽  
pp. 71-131
Author(s):  
David M. Kreps
Keyword(s):  
Utility Function ◽  
Expected Utility ◽  
Probability Distributions ◽  
Market Demand ◽  
Von Neumann ◽  
Consumption Decisions ◽  
Choice Under Uncertainty ◽  
States Of Nature ◽  
Special Case

This chapter examines how many important consumption decisions concern choices, the consequences of which are uncertain at the time the choice is made. It begins with the theory of von Neumann–Morgenstern expected utility. In this theory, uncertain prospects are modeled as probability distributions over a given set of prizes. That is, the probabilities of various prizes are given as part of the description of the object. The chapter then takes up the special case where the prizes are amounts of money; then one is able to say a bit more about the nature of the utility function that represents preferences. It discusses a few applications of this theory to the topic of market demand. Finally, the chapter turns to a richer theory, where uncertain prospects are functions from “states of nature” to prizes, and where probabilities arise subjectively, as part of the representation of a consumer's preferences.

The Expected Utility Theory and Measurement Theory of von Neumann and Morgenstern, 1944–1947

2018 ◽  
pp. 147-162 ◽  
Author(s):  
Ivan Moscati
Keyword(s):  
Decision Making ◽  
Utility Function ◽  
Expected Utility ◽  
Utility Theory ◽  
Theory Of Measurement ◽  
Measurement Theory ◽  
Expected Utility Theory ◽  
Economic Behavior ◽  
Utility Analysis ◽  
Von Neumann

Chapter 9 discusses the axiomatic version of expected utility theory (EUT), a theory of decision-making under risk, put forward by John von Neumann and Oskar Morgenstern in their book Theory of Games and Economic Behavior (1944). EUT was a changing factor in the history of utility measurement. In fact, while discussions of the measurability of utility before 1944 focused on the utility used to analyze decision-making between risk-free alternatives, after that year, discussions centered on the utility used to analyze decision-making between risky alternatives. In Theory of Games, the nature of the cardinal utility function u featured in von Neumann and Morgenstern’s EUT, and its relationship with the riskless utility function U of previous utility analysis remained ambiguous. Von Neumann and Morgenstern also put forward an axiomatic theory of measurement, which presents some similarities with Stanley Smith Stevens’s measurement theory but had no immediate impact on utility analysis.

What Is That Function?

2018 ◽  
pp. 163-176
Author(s):  
Ivan Moscati
Keyword(s):  
Utility Function ◽  
Expected Utility ◽  
Utility Theory ◽  
Expected Utility Theory ◽  
Cardinal Utility ◽  
John Von Neumann ◽  
Von Neumann ◽  
Milton Friedman

Chapter 10 reconstructs the first part of the American debate on expected utility theory (EUT), which ranges from 1947, when the second edition of John von Neumann and Oskar Morgenstern’s Theory of Games was published, to April 1950. In this period, a number of eminent American economists, including Milton Friedman, Leonard J. Savage, Jacob Marschak, Paul Samuelson, and William Baumol, wrote papers in which they took stances on the validity of EUT and the nature of the cardinal utility function u featured in the expected utility formula. Friedman, Savage, and Marschak supported EUT, although for different reasons, while Samuelson and Baumol rejected it. Regarding the nature of the cardinal utility function u, however, they all shared the view that it is interchangeable with the utility function U that the earlier utility theorists had used to analyze choices between riskless alternatives.

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

scholarly journals Coping with Loss Aversion in the Newsvendor Model

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Jianwu Sun ◽  
Xinsheng Xu
Keyword(s):  
Expected Utility ◽  
Loss Aversion ◽  
Confidence Level ◽  
Profit Maximization ◽  
Optimal Order ◽  
Market Demand ◽  
Newsvendor Model ◽  
Order Quantity ◽  
Decision Bias ◽  
Optimal Order Quantity

We introduce loss aversion into the decision framework of the newsvendor model. By introducing the loss aversion coefficientλ, we propose a novel utility function for the loss-averse newsvendor. First, we obtain the optimal order quantity to maximize the expected utility for the loss-averse newsvendor who is risk-neutral. It is found that this optimal order quantity is smaller than the expected profit maximization order quantity in the classical newsvendor model, which may help to explain the decision bias in the classical newsvendor model. Then, to reduce the risk which originates from the fluctuation in the market demand, we achieve the optimal order quantity to maximize CVaR about utility for the loss-averse newsvendor who is risk-averse. We find that this optimal order quantity is smaller than the optimal order quantity to maximize the expected utility above and is decreasing in the confidence levelα. Further, it is proved that the expected utility under this optimal order quantity is decreasing in the confidence levelα, which verifies that low risk implies low return. Finally, a numerical example is given to illustrate the obtained results and some management insights are suggested for the loss-averse newsvendor model.

If It Is Surely Better, Do It More? Implications for Preferences Under Ambiguity

2021 ◽  
Author(s):  
Soheil Ghili ◽  
Peter Klibanoff
Keyword(s):  
Expected Utility ◽  
Ambiguity Aversion ◽  
Upper Bounds ◽  
Maxmin Expected Utility ◽  
Weak Dominance ◽  
Choice Under Uncertainty ◽  
Tight Connection ◽  
Sales Agent ◽  
Utility Preferences ◽  
Sensitive Behavior

Consider a canonical problem in choice under uncertainty: choosing from a convex feasible set consisting of all (Anscombe–Aumann) mixtures of two acts f and g, [Formula: see text]. We propose a preference condition, monotonicity in optimal mixtures, which says that surely improving the act f (in the sense of weak dominance) makes the optimal weight(s) on f weakly higher. We use a stylized model of a sales agent reacting to incentives to illustrate the tight connection between monotonicity in optimal mixtures and a monotone comparative static of interest in applications. We then explore more generally the relation between this condition and preferences exhibiting ambiguity-sensitive behavior as in the classic Ellsberg paradoxes. We find that monotonicity in optimal mixtures and ambiguity aversion (even only local to an event) are incompatible for a large and popular class of ambiguity-sensitive preferences (the c-linearly biseparable class. This implies, for example, that maxmin expected utility preferences are consistent with monotonicity in optimal mixtures if and only if they are subjective expected utility preferences. This incompatibility is not between monotonicity in optimal mixtures and ambiguity aversion per se. For example, we show that smooth ambiguity preferences can satisfy both properties as long as they are not too ambiguity averse. Our most general result, applying to an extremely broad universe of preferences, shows a sense in which monotonicity in optimal mixtures places upper bounds on the intensity of ambiguity-averse behavior. This paper was accepted by Manel Baucells, decision analysis.

scholarly journals To the Problem of Constructing Analytical Indexes of Market Demand: A Variative Approach

2020 ◽  
Vol 27 (3) ◽  
pp. 65-80
Author(s):  
V. K. Gorbunov ◽  
L. A. Kozlova ◽  
A. G. Lvov
Keyword(s):  
Utility Function ◽  
World Literature ◽  
Consumer Preferences ◽  
Stable Solution ◽  
Market Demand ◽  
Classical Work ◽  
Consumer Expenditures ◽  
Price Indexes ◽  
Economic Indexes ◽  
Holistic Theory

The article develops methods for constructing economic (analytical) indexes in the framework of the holistic theory of market demand, built in recent years. By this, the economic indexes presented in the world literature within the framework of the theory of individual demand and, accordingly, related to households, acquire practical value.The introduction provides a brief overview of the main problems of modern indexology and the implementation of an economic approach dating back to the classical work of 1924 by the Soviet statistician A.A. Konüs. The properties of the most well-known «formula» indexes of Laspeyres, Paasche, and Fischer with respect to the fulfillment of the Fisher test criteria are described. These indexes play an important role in the methods proposed by the authors for constructing analytical indexes, which are determined through the function of consumer expenditures. The latter is determined by a utility function that rationalizes trade statistics. The rationalizing utility function is constructed ambiguously, and the corresponding task should be specified. Methods for its solution are proposed, developed within a non-parametric demand analysis of Afriat-Varian. The core of this analysis is the system of linear Afriat’s inequalities that determine the values of the utility function and marginal utility corresponding to statistical demand. This system can be inconsistent and unstable with respect to variations of non-exact demand statistics. In the case of compatibility, inequalities have many solutions, and the choice of different solutions of inequalities gives different values of analytical indexes. The authors suggest three types of tasks for the stable solution of Afriat’s inequalities, which define indexes with characteristics of optimism (low price indexes and high quantity indexes), pessimism (vice versa) and objectivity.Therefore, the problem of increasing the objectivity of consumer demand indexes receives a theoretically justified toolbox methods for calculating analytical market demand indexes that take into account, in contrast to formula indices, consumer preferences.

scholarly journals The extended odd family of probability distributions with practice to a submodel

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3855-3867 ◽  
Author(s):  
Hassan Bakouch ◽  
Christophe Chesneau ◽  
Muhammad Khan
Keyword(s):  
Heavy Tails ◽  
Maximum Likelihood Method ◽  
Probability Distributions ◽  
Likelihood Method ◽  
Tuning Parameter ◽  
Data Sets ◽  
New Family ◽  
Rate Functions ◽  
Special Case ◽  
Family Of Distributions

In this paper, we introduce a new family of distributions extending the odd family of distributions. A new tuning parameter is introduced, with some connections to the well-known transmuted transformation. Some mathematical results are obtained, including moments, generating function and order statistics. Then, we study a special case dealing with the standard loglogistic distribution and the modifiedWeibull distribution. Its main features are to have densities with flexible shapes where skewness, kurtosis, heavy tails and modality can be observed, and increasing-decreasing-increasing, unimodal and bathtub shaped hazard rate functions. Estimation of the related parameters is investigated by the maximum likelihood method. We illustrate the usefulness of our extended odd family of distributions with applications to two practical data sets.

An Implementation of Expected Utility Theory in Decision Based Design

1998 ◽  
Author(s):  
Matthew Marston ◽  
Farrokh Mistree
Keyword(s):  
Expected Utility ◽  
Utility Theory ◽  
Design Science ◽  
Expected Utility Theory ◽  
Simple Design ◽  
Von Neumann ◽  
Decision Based Design ◽  
Mathematics And Physics ◽  
Science Of The Artificial ◽  
The Ideal

Abstract The development of a design science rests on the ideal that design is anchored in a set of fundamental axioms similar to the more ‘traditional’ sciences of mathematics and physics. However, the axioms upon which a design science is constructed must reflect that design is a science of the artificial. It is our contention that such axioms may exist in Decision-Based Design as those formulated by von-Neumann and Morgenstern for developing utilities under conditions of risk. In this paper we have a very narrow focus: evaluating a proposed framework for applying these axioms in the context of a simple design problem through the use of Monte Carlo simulation and expected utility theory.

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