The choquet integral is a powerful aggregation operator which lists many wellknown models as its special cases. Pdf decision modelling using the choquet integral researchgate. Pdf asymptotic methods in statistical decision theory. An interdisciplinary approach to determine how decisions are made given unknown variables and an uncertain decision environment framework.
A similar criterion of optimality, however, can be applied to a wider class of decision problems. Rung orthopair fuzzy choquet integral aggregation and. The theory by relaxing the independence axiom of anscombe and aumann 1963 in order to hold for comonotonic acts only schmeidler 1989 derives a utility theory that allows for nonadditive probabilities. Ui is an element of xi which is thought by the dm as completely unsatisfactory relatively to his. Intuitionistic trapezoidal fuzzy group decisionmaking. In this framework, is a set of players, any subset f is called a coalition, and, called the characteristic function of the game, expresses the worth i. Consumer decisionmaking models, strategies, and theories. Learnability and models of decision making under uncertainty. For example, if we consider a set of four alternatives x 1, x.
In the context of the choquet expected utility ceu model, con. The problem of portfolio allocation is central to economic decision theory and thus constitutes an acid test. Choquet expected utility theory as it applies to decisions under risk, to link this theory with some recent developments in the literature on risk assessment, and nally to describe its application to the problem of portfolio choice. A choquet integral is a subadditive or superadditive integral created by the french mathematician gustave choquet in 1953. Pdf a choquet integral representation in multicriteria. It is shown that some paradoxes of expected utility theory are. It is shown that choquet expected utility model for decision under uncertainty and rank dependent utility model for decision under risk are respectively.
Decision analysis on choquet integralbased multicriteria. We look at these special cases and provide their axiomatic analysis. In the decision theory framework, su cient statistics provide a reduction of the data without loss of information. A note on reexamining the law of iterated expectations for choquet decision makers andr e lapied pascal toquebeufy november 4, 2011 abstract this note completes the main result of zimper a. Fundamentals of decision theory university of washington. Pessimistic portfolio allocation and choquet expected utility. Considering the interrelationships among the criteria, this paper extends choquet integral to the q. My thesis presents the application of fuzzy integrals as tool for criteria aggregation in the decision problems.
In consideration of the interaction among attributes and the influence of decision makers risk attitude, this paper proposes an intuitionistic trapezoidal fuzzy aggregation operator based on choquet integral and prospect theory. Decision theory under ambiguity etner 2012 journal. A formal philosophical introduction richard bradley london school of economics and political science march 9, 2014 abstract decision theory is the study of how choices are and should be a variety of di. A choquet integral representation in multicriteria. S s symmetry article multiattribute decision making based on probabilistic neutrosophic hesitant fuzzy choquet aggregation operators songtao shao 1, xiaohong zhang 2,3, and quan zhao 1, 1 college of information engineering, shanghai maritime university, shanghai 206, china. Decisiontheory tries to throw light, in various ways, on the former type of period. Choquet expected utility theory as it applies to decisions under risk, to link. It is shown that choquet expected utility model for decision under uncertainty and rank dependent utility model.
Although, both cases are described here, the majority of this report focuses. Choquet expected utility assumes that agents have nonadditive beliefs over uncertain events. A bad decision may occasionally result in a good outcome if you are lucky. A decade of application of choquet and sugeno integrals 5 coming back to our framework of mcda, saying that ui is a bounded unipolar scale implies the existence in xi of two elements denoted by ui and pi, which have an absolute meaning.
While classical decision theory uses average to assess the value of. Pdf a selective overview of applications of choquet integrals. A choquet integral can be seen as an integral on a. It is applied specifically to membership functions and capacities. The problem of portfolio allocation is central to economic decision theory and thus in our view. Pdf in this paper, we introduce the choquet integral as a general tool for dealing with multiple criteria decision making. Decision modelling using the choquet integral springerlink. The usefulness of the choquet integral for modelling decision under risk and uncertainty is shown. Decision theory be interpreted as the longrun relative frequencies, and theexpected payo. These are notes for a basic class in decision theory.
A novel multiattribute decisionmaking method based on. Although it is now clearly an academic subject of its own right, decision theory is. The notes contain the mathematical material, including all the formal models and proofs that will be presented in class, but they do not contain the discussion of. Pdf application of the choquet integral in multicriteria decision. This survey will try to explain the fundamental concepts underlying the use of capacities and the choquet integral in mcda, as well as some recent advances. A parallel theory has recently emerged in the literature on risk assessment. It was initially used in statistical mechanics and potential theory, 2 but found its way into decision theory in the 1980s, 3 where it is used as a way of measuring the expected utility of an uncertain event. They concern main parallel to these theoretical works, the choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed. It is shown that some paradoxes of expected utility theory are solved using choquet integral.
We start with a presentation of the general approach to a decision problem under uncertainty, as well as the standard bayesian treatment and issues with this treatment. In much of recent decision theory choquet integration of a real function with. In mathematics, choquet theory, named after gustave choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set c. More specifically, decision theory deals with methods for determining the optimal course of action when a number of alternatives are available and their consequences cannot be.
Capacities and the choquet integral in decision making. The maxmin model is a staple of modern decision theory, and used extensively in economic applications where agents face uncertainty. They concern mainly a bipolar extension of both the choquet integral and the sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the choquet integral in multicriteria decision. Multicriteria group decisionmaking, multiset hesitantfuzzy sets, todim, choquet integral 1 introduction in many cases, it is dif. With the help of relevant knowledge in the field of risk management and decision theory, a common ground was found, on which the experiment is based. Roughly speaking, every vector of c should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an. For decision under uncertainty, time points are reinterpreted as states of nature and profiles as acts, for welfare theory time points are persons and profiles are. It can be seen that the choquet integral and the sugeno integral have similar construction, respectively with the tnorm bounded sum and the tconorm x and with the tnorm a and the tconorm v.
We present more general approaches choquet expected utility, maximin expected utility, smooth ambiguity and so forth that have been developed in the literature under the. With respect to a multiattribute group decisionmaking problem, the prospect value functions of intuitionistic trapezoidal fuzzy numbers are aggregated by the. Rofss as a novel effective tool can depict and handle uncertain information in a broader perspective. The application of multiattribute utility theory whose aggregation process is based on the choquet integral requires the prior identification of a capacity.
Choquet decision theory schmeidler 1986, 1989 and gilboa 1987. Many scholars have made quite a few achievements in this. A decision making approach based on 2additive choquet integral, international journal. Here we look at the topic from a formalphilosophical point of view with a focus on normative and. The aim of multiattribute utility theory maut is to model the preferences of the decision maker dm, represented by a binary relation.
Recent developments in the theory of choice under uncertainty and risk yield a pessimistic decision theory that replaces the classical expected utility criterion with a choquet expectation that accentuates the likelihood of the least favorable outcomes. In particular, any risk that can be achieved using a decision rule based on xcan also be achieved by a decision rule based on tx, as the following theorem makes precise. A note on reexamining the law of iterated expectations. A multicriteria decisionmaking approach based on todim. Introduction to decision theory decision making is an integral part of management planning, organizing, controlling and motivation processes. While choquet decision theory has mainly been developed in order to model the decision behavior of individuals who commit ellsberg paradoxes ellsberg 1961, our approach demonstrates the usefulness of choquet decision theory in describing the learning. Choquet integral versus weighted sum in multicriteria decision. Nevertheless, these integrals are very different in essence.
It is shown that choquet expected utility model for decision under uncertainty and rank dependent utility model for decision under risk are. Choquet expected utility theory as it applies to decisions under risk, to link this the ory with some recent developments in the literature on risk assessment, and nally to describe its application to the problem of portfolio choice. In cases where an axiomatization has been previously given in the literature, we connect the existing results with the framework that we have developed. The choquet integral 16 introduced by choquet is a useful tool to address the problem. Choquet expected utility theory with the added restriction that the capacity is an. Doretta vivona, in handbook of measure theory, 2002. It was initially used in statistical mechanics and potential theory, but found its way into decision theory in the 1980s, where it is used as a way of measuring the expected utility of an uncertain event. Choquet integrals to decision making under risk and uncertainty. It should also be noted that the random variable x can be assumed to be either continuous or discrete. Application of the choquet integral in multicriteria. Subjective ambiguity, expected utility and choquet expected. Multiattribute decision making based on probabilistic. One notable approach to modeling decision making under ambiguity makes use of the concept of nonadditive probabilities, called capacities. Testing choquet expected utility luiss guido carli.
Choquet integrals and capacities play a crucial role in modern decision theory. The focus is on decision under risk and under uncertainty, with relatively little on social choice. Decision theory is a set of concepts, principles, tools and techniques that help the decision maker in dealing with complex decision problems under uncertainty. A normative decision theory, adequate to such circumstances, would provide guidance on how boundedagents should represent the uncertainty they face, how they should revise their opinions as a result of. Decision theory concepts and methods 5 dependent on. This method is the one induced by the choquet integral choquet 1955. Choquet integralbased aggregation operator has been applied 8,9, and it has improved the weakness of simple weighted sum method. Comonotony is a central concept for these theories because the main property of a choquet integral is its additivity for comonotone functions.
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