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2008, Physica A-statistical Mechanics and Its Applications
The maximum entropy principle can be used to assign utility values when only partial information is available about the decision maker's preferences. In order to obtain such utility values it is necessary to establish an analogy between probability and utility through the notion of a utility density function. According to some ] the maximum entropy utility solution embeds a large family of utility functions. In this paper we explore the maximum entropy principle to estimate the utility function of a risk averse decision maker.
International Journal of Applied Decision Sciences, 2013
Methodologies related to information theory have been increasingly used in studies in economics and management. In this paper, we use generalised maximum entropy as an alternative to ordinary least squares in the estimation of utility functions. Generalised maximum entropy has some advantages: it does not need such restrictive assumptions and could be used with both well and ill-posed problems, for example, when we have small samples, which is the case when estimating utility functions. Using linear, logarithmic and power utility functions, we estimate those functions and confidence intervals and perform hypothesis tests. Results point to the greater accuracy of generalised maximum entropy, showing its efficiency in estimation.
Journal of Applied Statistics, 2013
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Operations Research, 2006
This paper presents a method to assign utility values when only partial information is available about the decision maker’s preferences. We introduce the notion of a utility density function and a maximum entropy principle for utility assignment. The maximum entropy utility solution embeds a large family of utility functions that includes the most commonly used functional forms. We discuss the implications of maximum entropy utility on the preference behavior of the decision maker and present an application to competitive bidding situations where only previous decisions are observed by each party. We also present minimum cross entropy utility, which incorporates additional knowledge about the shape of the utility function into the maximum entropy formulation, and work through several examples to illustrate the approach.
Theory and Decision, 2012
Expected utility maximization problem is one of the most useful tools in mathematical finance, decision analysis and economics. Motivated by statistical model selection, via the principle of expected utility maximization, Friedman and Sandow (J Mach Learn Res 4:257–291, 2003a) considered the model performance question from the point of view of an investor who evaluates models based on the performance of
AIP Conference Proceedings, 2004
Recent literature in the last Maximum Entropy workshop introduced an analogy between cumulative probability distributions and normalized utility functions [1]. Based on this analogy, a utility density function can de defined as the derivative of a normalized utility function. A utility density function is non-negative and integrates to unity. These two properties of a utility density function form the basis of a correspondence between utility and probability, which allows the application of many tools from one domain to the other. For example, La Place's principle of insufficient reason translates to a principle of insufficient preference. The notion of uninformative priors translates to uninformative utility functions about a decision maker's preferences. A natural application of this analogy is a maximum entropy principle to assign maximum entropy utility values. Maximum entropy utility interprets many of the common utility functions based on the preference information needed for their assignment, and helps assign utility values based on partial preference information. This paper reviews maximum entropy utility, provides axiomatic justification for its use, and introduces further results that stem from the duality between probability and utility, such as joint utility density functions, utility inference, and the notion of mutual preference.
Procedia Economics and Finance, 2015
We present in the first part of the article types of utility functions that can describe the behavior of the investor and their applications to optimize portfolio. The second part of the paper refers to applications in calculating insurance premiums aggregated risk in zero utility principle.
Systems, 2020
The uncertainty, or entropy, of an atom of an ideal gas being in a certain energy state mirrors the way people perceive uncertainty in the making of decisions, uncertainty that is related to unmeasurable subjective probability. It is well established that subjects evaluate risk decisions involving uncertain choices using subjective probability rather than objective, which is usually calculated using empirically derived decision weights, such as those described in Prospect Theory; however, an exact objective-subjective probability relationship can be derived from statistical mechanics and information theory using Kullback-Leibler entropy divergence. The resulting Entropy Decision Risk Model (EDRM) is based upon proximity or nearness to a state and is predictive rather than descriptive. A priori EDRM, without factors or corrections, accurately aligns with the results of prior decision making under uncertainty (DMUU) studies, including Prospect Theory and others. This research is a first step towards the broader effort of quantifying financial, programmatic, and safety risk decisions in fungible terms, which applies proximity (i.e., subjective probability) with power utility to evaluate choice preference of gains, losses, and mixtures of the two in terms of a new parameter referred to as Prospect. To facilitate evaluation of the EDRM against prior studies reported in terms of the percentage of subjects selecting a choice, the Percentage Evaluation Model (PEM) is introduced to convert choice value results into subject response percentages, thereby permitting direct comparison of a utility model for the first time.
2016
Dedicated to Professor Andrzej Lasota on his 70th birthday Expected utility maximization problems in mathematical finance lead to a generalization of the classical definition of entropy. It is demonstrated that a necessary and sufficient condition for the second law of thermodynamics to operate is that any one of the generalized entropies should tend to its minimum value of zero.
Journal of Economics and Business, 2009
Using US market data, this paper sheds new empirical light on properties of the utility function. In particular, employing theoretical relations between Stochastic Discount Factors, state prices, and state probabilities, we are successful in recovering the following four functions: (i) Absolute Risk Aversion (ARA); (ii) Absolute Risk Tolerance (ART); (iii) Absolute Prudence (AP); and (iv) Absolute Temperance (AT). Our statistical analysis points out, unequivocally, that the ARA function is decreasing and convex, the ART function is convex, AT is greater than ARA, and the AP function is not decreasing. These empirical results are analyzed in light of established theory concerning, inter-alia, precautionary saving and prudence as well as the way risk attitudes are affected by the presence of "background risks" and by investors' investment horizon. discuss the recoverability of preferences from observed asset prices and an agent's consumption choice. Jackwerth is the first to use estimates of state prices and physical probabilities to recover A RA functions. We extend on that by characterizing additional properties of the vNM utility function, and relying on a more general approach for estimating state probabilities. Our approach is shown to have non-trivial consequences.
Systems
Risk perception can be quantified in measurable terms of risk aversion and sensitivity. While conducting research on the quantization of programmatic risk, a bridge between positive and normative decision theories was discovered through the application of a novel a priori relationship between objective and subjective probabilities and the application of Bernoulli’s expected utility theory. The Entropy Decision Risk Model (EDRM) derived using the Kullback–Liebler entropy divergence from certainty serves as a translation between objective and subjective probability, referred to as proximity, and has proven its applicability to various positive decision theories related to Prospect Theory. However, EDRM initially assumes the validity of the standard exponential power utility function ubiquitous to positive decision theory models as the magnitude of a choice to isolate and validate proximity. This research modifies the prior model by applying Daniel Bernoulli’s expected utility as the m...
Open Systems & Information Dynamics, 2008
The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied in [37], is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based on the idea, borrowed from mathematical finance, of maximising the expected utility of the terminal wealth of an investor. Moreover, u-entropy is also relevant in thermodynamics, as it can replace the standard Boltzmann-Shannon entropy in the Second Law. If the utility function is logarithmic or isoelastic (a power function), then the well-known notions of Boltzmann-Shannon and Rényi relative entropy are recovered. We establish the principal properties of relative and discrete u-entropy and discuss the links with several related approaches in the literature.
Natural Science, 2014
In this paper, it is discussed a framework combining traditional expected utility and weighted entropy (EU-WE)-also named mean contributive value index-which may be conceived as a decision aiding procedure, or a heuristic device generating compositional scenarios, based on information theory concepts, namely weighted entropy. New proofs concerning the maximum value of the index and the evaluation of optimal proportions are outlined, with emphasis on the optimal value of the Lagrange multiplier and its meaning. The rationale is a procedure of maximizing the combined value of a system expressed as a mosaic, denoted by characteristic values of the states and their proportions. Other perspectives of application of this EU-WE framework are suggested.
Operations Research, 2008
Information measures arise in many disciplines, including forecasting (where scoring rules are used to provide incentives for probability estimation), signal processing (where information gain is measured in physical units of relative entropy), decision analysis (where new information can lead to improved decisions), and finance (where investors optimize portfolios based on their private information and risk preferences). In this paper, we generalize the two most commonly used parametric families of scoring rules and demonstrate their relation to well-known generalized entropies and utility functions, shedding new light on the characteristics of alternative scoring rules as well as duality relationships between utility maximization and entropy minimization. In particular, we show that weighted forms of the pseudospherical and power scoring rules correspond exactly to measures of relative entropy (divergence) with convenient properties, and they also correspond exactly to the solutions of expected utility maximization problems in which a risk-averse decision maker whose utility function belongs to the linear-risk-tolerance family interacts with a risk-neutral betting opponent or a complete market for contingent claims in either a one-period or a two-period setting. When the market is incomplete, the corresponding problems of maximizing linear-risk-tolerance utility with the risk-tolerance coefficient are the duals of the problems of minimizing the pseudospherical or power divergence of order between the decision maker's subjective probability distribution and the set of risk-neutral distributions that support asset prices.
Entropy, 2007
We review a decision theoretic, i.e., utility-based, motivation for entropy and Kullback-Leibler relative entropy, the natural generalizations that follow, and various properties of these generalized quantities. We then consider these generalized quantities in an easily interpreted special case. We show that the resulting quantities, share many of the properties of entropy and relative entropy, such as the data processing inequality and the second law of thermodynamics. We formulate an important statistical learning problem-probability estimation-in terms of a generalized relative entropy. The solution of this problem reflects general risk preferences via the utility function; moreover, the solution is optimal in a sense of robust absolute performance.
European Journal of Operational Research, 2006
This paper describes a parametric family of utility functions for decision analysis. The parameterization is obtained by embedding the HARA class in a four-parameter representation for the risk aversion function. The resulting utility functions have only four shapes: concave, convex, S-shaped, and reverse S-shaped. This makes the family suited for both expected utility and prospect theory. We also describe an alternative technique to estimate the four parameters from elicited utilities, which is simpler and easier to implement than standard fitting by minimization of the mean quadratic error.
Journal of Moral Philosophy, 2008
Th eories of rationality advance principles that diff er in topic, scope, and assumptions. A typical version of the principle of utility maximization formulates a standard rather than a procedure for decisions, evaluates decisions comprehensively, and relies on idealizations. I generalize the principle by removing some idealizations and making adjustments for their absence. Th e generalizations accommodate agents who have incomplete probability and utility assignments and are imperfectly rational. Th ey also accommodate decision problems with unstable comparisons of options.
2017
At present, the choice of the best solutions out of many possible under conditions of uncertainty is the actual economic task, arising and to be solved in many economic situations. Famous classical approaches to its solution are based on various assessments of decision-making practical situations. However, they often give insufficiently accurate or incorrect results, and do not satisfy sustainability requirements, when the only invariant calculation result relative to calculation methodology is a reliable one and a corresponding to the reality result. This article describes an alternative approach to the justification of decisions under conditions of uncertainty without the construction and use of assumptions about the decision-making situation and in conformity with the approaches of the stability theory. The problem of multi-criteria decision-making in conditions of complete uncertainty, wherein structuring of alternatives is performed using the fuzzy entropy, has been formulated ...
Crimson Publishers, 2020
In the present communication Markowitz's method of mean-variance efficient frontier has been explained. Some introductory entropy models and concepts related to risk in investments have been discussed. Risk aversion index and Pareto-optimal sharing of risk have been defined. A new measure of risk based on maximum entropy principle has been studied in detail. Mathematics Subject Classification 2000: 91b24 and 94a15.
2003
We introduce an axiomatic approach to the problem of inferring a complete and transitive weak ordering representing the agent's preferences given a set of observed constraints. The axioms characterize a unique inference rule, which amounts to the constrained maximization of a certain formula we derive. The formula can be interpreted as the entropy of the agent's preference ordering, and its unique maximand identifies the simplest rationalization of the observed behavior.
Decision-making Process, 2009
Ce chapitre d'ouvrage collectif a pour but de présenter les bases de la modélisation de la prise de décision dans un univers risqué. Nous commençons par dé…nir, de manière générale, la notion de risque et d'accroissement du risque et rappelons des dé…nitions et catégorisations (valables en dehors de tout modèle de représentation) de comportements face au risque. Nous exposons ensuite le modèle classique d'espérance d'utilité de von Neumann et Morgenstern et ses principales propriétés. Les problèmes posés par ce modèle sont ensuite discutés et deux modèles généralisant l'espérance d'utilité brièvement présentés. Mots clé: risque, aversion pour le risque, espérance d'utilité, von Neumann et Morgenstern, Paradoxe d'Allais. JEL: D81
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