13 research outputs found
Decomposition approaches to integration without a measure
Extending the idea of Even and Lehrer [3], we discuss a general approach to
integration based on a given decomposition system equipped with a weighting
function, and a decomposition of the integrated function. We distinguish two
type of decompositions: sub-decomposition based integrals (in economics linked
with optimization problems to maximize the possible profit) and
super-decomposition based integrals (linked with costs minimization). We
provide several examples (both theoretical and realistic) to stress that our
approach generalizes that of Even and Lehrer [3] and also covers problems of
linear programming and combinatorial optimization. Finally, we introduce some
new types of integrals related to optimization tasks.Comment: 15 page
Bipolar Fuzzy Integrals
In decision analysis and especially in multiple criteria decision analysis,
several non additive integrals have been introduced in the last sixty years.
Among them, we remember the Choquet integral, the Shilkret integral and the
Sugeno integral. Recently, the bipolar Choquet integral has been proposed for
the case in which the underlying scale is bipolar. In this paper we propose the
bipolar Shilkret integral and the bipolar Sugeno integral. Moreover, we provide
an axiomatic characterization of all these three bipolar fuzzy integrals.Comment: 15 page
Robust Integrals
In decision analysis and especially in multiple criteria decision analysis,
several non additive integrals have been introduced in the last years. Among
them, we remember the Choquet integral, the Shilkret integral and the Sugeno
integral. In the context of multiple criteria decision analysis, these
integrals are used to aggregate the evaluations of possible choice
alternatives, with respect to several criteria, into a single overall
evaluation. These integrals request the starting evaluations to be expressed in
terms of exact-evaluations. In this paper we present the robust Choquet,
Shilkret and Sugeno integrals, computed with respect to an interval capacity.
These are quite natural generalizations of the Choquet, Shilkret and Sugeno
integrals, useful to aggregate interval-evaluations of choice alternatives into
a single overall evaluation. We show that, when the interval-evaluations
collapse into exact-evaluations, our definitions of robust integrals collapse
into the previous definitions. We also provide an axiomatic characterization of
the robust Choquet integral.Comment: 24 page
The bipolar Choquet integral representation
Cumulative Prospect Theory of Tversky and Kahneman (1992) is the modern version of Prospect Theory (Kahneman and Tversky (1979)) and is nowadays considered a valid alternative to the classical Expected Utility
Theory.
Cumulative Prospect theory implies Gain-Loss Separability,
i.e. the separate evaluation of losses and gains within a mixed gamble.
Recently, some authors have questioned this assumption of the theory, proposing new paradoxes where the Gain-Loss Separability is violated.
We present a generalization of Cumulative Prospect Theory which does not imply Gain-Loss Separability and is able to explain the cited paradoxes.
On the other hand, the new model, which we call the bipolar
Cumulative Prospect Theory, genuinely generalizes the original Prospect Theory of Kahneman and Tversky (1979), preserving the main features of the theory.
We present also a characterization of the bipolar Choquet
Integral with respect to a bi-capacity in a discrete setting
The bipolar Choquet integral representation
Cumulative Prospect Theory of Tversky and Kahneman (1992) is the modern version of Prospect Theory (Kahneman and Tversky (1979)) and is nowadays considered a valid alternative to the classical Expected Utility
Theory.
Cumulative Prospect theory implies Gain-Loss Separability,
i.e. the separate evaluation of losses and gains within a mixed gamble.
Recently, some authors have questioned this assumption of the theory, proposing new paradoxes where the Gain-Loss Separability is violated.
We present a generalization of Cumulative Prospect Theory which does not imply Gain-Loss Separability and is able to explain the cited paradoxes.
On the other hand, the new model, which we call the bipolar
Cumulative Prospect Theory, genuinely generalizes the original Prospect Theory of Kahneman and Tversky (1979), preserving the main features of the theory.
We present also a characterization of the bipolar Choquet
Integral with respect to a bi-capacity in a discrete setting
Discrete bipolar universal integrals
Abstract. The concept of universal integral, recently proposed, generalizes the Choquet, Shilkret and Sugeno integrals. Those integrals admit a discrete bipolar formulation, useful in those situations where the underlying scale is bipolar. In this paper we propose the concept of discrete bipolar universal integral, in order to provide a common framework for bipolar discrete integrals, including as special cases the discrete Choquet, Shilkret and Sugeno bipolar integrals. Moreover we provide two different axiomatic characterizations of the proposed discrete bipolar universal integral
On prospects and games: an equilibrium analysis under prospect theory
The aim of this paper is to introduce prospect theory in a game theoretic framework.
We address the complexity of the weighting function
by restricting the object of our analysis to a 2-player 2-strategy game, in order to derive some core results. We find that dominant and indifferent strategies are preserved under prospect theory. However, in absence of dominant strategies, equilibrium may not exist depending on parameters.
We also discuss a different approach presented by Metzger and Rieger (2009) and give some interesting interpretations of the two approaches
On prospects and games: an equilibrium analysis under prospect theory
The aim of this paper is to introduce prospect theory in a game theoretic framework.
We address the complexity of the weighting function
by restricting the object of our analysis to a 2-player 2-strategy game, in order to derive some core results. We find that dominant and indifferent strategies are preserved under prospect theory. However, in absence of dominant strategies, equilibrium may not exist depending on parameters.
We also discuss a different approach presented by Metzger and Rieger (2009) and give some interesting interpretations of the two approaches