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.

Decision Under Risk and Uncertainty | Springer Nature Link Skip to main content Advertisement Decision Under Risk and Uncertainty Chapter First Online: 14 June 2016 pp 281–323 Cite this chapter Save chapter View saved research Set Functions, Games and Capacities in Decision Making Abstract This chapter opens the part of the book on applications of set functions in decision making. The foundations of decision making are mainly due to John von Neumann and Oskar Morgenstern, although the concepts of utility function and expected utility go back to Daniel Bernoulli and Blaise Pascal. The area of decision making which is addressed in this chapter is decision under risk and uncertainty. It deals with situations where the decision maker is faced with uncertainty: the consequences of his possible decisions depend on contingencies which are out of his control. The occurrence or the non-occurrence of these contingencies determine what is called the states of nature. If probabilistic information on the states of nature is available, one speaks of decision under risk. Otherwise, it is assumed that the decision maker has a personal, subjective probability measure on the states of nature in his mind, in which case one speaks of decision under uncertainty. While classical models solely rely on probability measures (expected utility), the observation of various paradoxes, unexplained by expected utility, has lead to considering capacities (viewed as nonadditive probabilities) and the Choquet integral in decision making. This chapter tries to show the emergence of these new models. It does not pretend to a full exposition of decision under risk and uncertainty, which would require a whole book. For this reason, and because already many textbooks exist on this subject, proofs of results are not given, except for some results which either are not so well known, or for which it is less easy to find comprehensive references. Download preview PDF. Unable to display preview. Download preview PDF. Similar content being viewed by others Too much or too little information: how unknown uncertainty fuels time inconsistency Article Open access 19 January 2022 Risk and Uncertainty Chapter © 2018 Decision Making Under Uncertainty and Problem Solving Chapter © 2021 Explore related subjects Discover the latest articles, books and news in related subjects, suggested using machine learning. Decision Making Operations Research and Decision Theory Probability Theory Risk Theory Stochastic Calculus Philosophy of Probability Author information Authors and Affiliations Paris School of Economics, Université Paris I Panthéon-Sorbonne, Paris, France Michel Grabisch Authors Michel Grabisch View author publications Search author on: PubMed   Google Scholar Rights and permissions Reprints and permissions Copyright information © 2016 Springer International Publishing Switzerland About this chapter Cite this chapter Grabisch, M. (2016). Decision Under Risk and Uncertainty. In: Set Functions, Games and Capacities in Decision Making. Theory and Decision Library C, vol 46. Springer, Cham. Copy shareable link to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Keywords Utility Function Risk Aversion Prospect Theory Stochastic Dominance Standard Gamble These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The topic of this thesis is decision-making under risk. I focus my analysis on expected utility theory by von Neumann and Morgenstern. I am especially interested in modeling risk attitudes represented by Bernoulli utility functions that belong to the following classes: Constant Absolute Risk Aversion, Decreasing Absolute Risk Aversion (understood as strictly decreasing) and in particular a subset thereof - Constant Relative Risk Aversion. I build a theory of buying and selling price for a lottery, the concepts defined by Raiffa, since such theory proves useful in analyzing a number of interesting issues pertaining to risk attitudes' characteristics within expected utility model. In particular, I analyze the following: - Chapter 2 - expected utility without consequentialism, buying/selling price gap, preference reversal, Rabin paradox - Chapter 3 - characterization results for CARA, DARA, CRRA, simple strategies and an extension of Pratt result on comparative risk aversion - Chapter 4 - riskiness measure and its intuition, extended riskiness measure and its existence, uniqueness and properties

We obtain several variants of the classic von Neumann–Morgenstern expected utility theorem with and without the completeness axiom in which the derived Bernoulli utility functions are Lipschitz. The prize space in these results is an arbitrary separable metric space, and the utility functions are allowed to be unbounded. The main ingredient of our results is a novel (behavioral) axiom on the underlying preference relations, which is satisfied by virtually all stochastic orders. The proof of the main representation theorem is built on the fact that the dual of the Kantorovich–Rubinstein space is (isometrically isomorphic to) the Banach space of Lipschitz functions that vanish at a fixed point. An application to the theory of nonexpected utility is also provided.

Resumen La toma de decisiones bajo riesgo e incertidumbre ha sido estudiada desde hace cientos de años con el objetivo de establecer un consenso normativo en el verdadero comportamiento de las personas ante este tipo de situaciones. En el siguiente trabajo, se presenta una reseña bibliográfica desarrollando, de manera cronológica, la evolución de las funciones de utilidad postuladas por los autores más relevantes en esta rama de estudio. Si bien son numerosos los autores que han abordado el tema, se comenzará explicando la teoría del valor esperado, la paradoja de San Petersburgo, la teoría de la utilidad esperada, el aporte tanto de Friedman y Savage como de Markowitz, para culminar detallando la teoría prospectiva de Kahneman y Tversky. Palabras clave: toma de decisiones, funciones de utilidad, riesgo e incertidumbre. Abstract Academics have been studying decision making under risk and uncertainty for hundreds of years in order to determine the actual behavior of individuals. Throughout this paper, we present, chronologically, the most relevant state of the art in utility functions developed by the most well-known authors. Even though there are numerous authors who have studied the subject, we begin explaining expected value theory (Huygens, 1657), St. Petersburg Paradox (Bernoulli, 1738), expected utility theory (von Neumann & Morgenstern, 1944), contributions not only from Friedman and Savage (1948) but also from Markowitz (1952), and we end up detailing prospect theory (

Utility representation theorem In economics, a utility representation theorem shows that, under certain conditions, a preference ordering can be represented by a real-valued utility function, such that option A is preferred to option B if and only if the utility of A is larger than that of B. The most famous example of a utility representation theorem is the Von Neumann–Morgenstern utility theorem, which shows that any rational agent has a utility function that measures their preferences over lotteries. ## Background Suppose a person is asked questions of the form "Do you prefer A or B?" (when A and B can be options, actions to take, states of the world, consumption bundles, etc.). If the agent prefers A to B, we write A ≻ B {\displaystyle A\succ B}. The set of all such preference-pairs forms the person's preference relation. Instead of recording the person's preferences between every pair of options, it would be much more convenient to have a single utility function- a function u that assigns a real number to each option, such that u(B)}" xmlns="http://www.w3.org/1998/Math/MathML"> u ( A ) > u ( B ) {\displaystyle u(A)>u(B)} if and only if A ≻ B {\displaystyle A\succ B}. Not