The aim of this paper is to introduce prospect theory in a game theoretic framework. We address t... more 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. JEL Classification: C70; D03.
In decision analysis and especially in multiple criteria decision analysis, several non additive ... more 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.
In decision analysis, and especially in multiple criteria decision analysis, several non additive... more In decision analysis, and especially in multiple criteria decision analysis, several non additive integrals have been introduced in the last years. These include the Choquet integral, the Shilkret integral and the Sugeno integral, among others. 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. The use of mentioned integrals in the aggregation process requests 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.
In decision analysis and especially in multiple criteria decision analysis, several non additive ... more 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.
The concept of universal integral, recently proposed, generalizes the Choquet, Shilkret and Sugen... more 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.
Two construction methods for aggregation functions based on a restricted a priori known decomposi... more Two construction methods for aggregation functions based on a restricted a priori known decomposition set and decomposition weighing function are introduced and studied. The outgoing aggregation functions are either superadditive or subadditive. Several examples, including illustrative gures, show the potential of the introduced construction methods. Our approach generalizes several known constructions and optimization methods, including decomposition and superdecomposition integrals.
We propose the concepts of superadditive and of subadditive transformations of aggregation functi... more We propose the concepts of superadditive and of subadditive transformations of aggregation functions acting on non-negative reals, in particular of integrals with respect to monotone measures. We discuss special properties of the proposed transforms and links between some distinguished integrals. Superadditive transformation of the Choquet integral, as well as of the Shilkret integral, is shown to coincide with the corresponding concave integral recently introduced by Lehrer. Similarly the transformation of the Sugeno integral is studied. Moreover, subadditive transformation of distinguished integrals is also discussed.
Following the idea of Even and Lehrer [3], we discuss a general approach to integration based on ... more Following the idea of Even and Lehrer [3], we discuss a general approach to integration based on decomposition of the integrated function. We distinguish 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.
Aggregation functions on [0, 1] with annihilator 0 can be seen as a generalized product on [0, 1]... more Aggregation functions on [0, 1] with annihilator 0 can be seen as a generalized product on [0, 1]. We study the generalized product on the bipolar scale [−1, 1], stressing the axiomatic point of view (compare also [9]). Based on newly introduced bipolar properties, such as the bipolar increasingness, bipolar unit element, bipolar idempotent element, several kinds of generalized bipolar product are introduced and studied. A special stress is put on bipolar semicopulas, bipolar quasi-copulas and bipolar copulas. Inspired by the truncated sum on [−1, 1] we introduce also the class of generalized bipolar sums, which differ from uninorms due to the non-associativity.
The aim of this paper is to introduce prospect theory in a game theoretic framework. We address t... more 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. JEL Classification: C70; D03.
Advances in Intelligent Systems and Computing, 2013
ABSTRACT The concept of universal integral has been recently proposed in order to generalize the ... more ABSTRACT The concept of universal integral has been recently proposed in order to generalize the Choquet, Shilkret and Sugeno integrals. We present two axiomatic foundations of the universal integral. The first axiomatization is expressed in terms of aggregation functions, while the second is expressed in terms of preference relations.
Communications in Computer and Information Science, 2014
Aggregation functions on [0, 1] with annihilator 0 can be seen as a generalized product on [0, 1]... more Aggregation functions on [0, 1] with annihilator 0 can be seen as a generalized product on [0, 1]. We study the generalized product on the bipolar scale [−1, 1], stressing the axiomatic point of view. Based on newly introduced bipolar properties, such as the bipolar increasingness, bipolar unit element, bipolar idempotent element, several kinds of generalized bipolar product are introduced and studied. A special stress is put on bipolar semicopulas, bipolar quasicopulas and bipolar copulas.
The concept of semicopula plays a fundamental role in the aggregation theory on interval [0, 1]. ... more The concept of semicopula plays a fundamental role in the aggregation theory on interval [0, 1]. Semicopulas are applied, for example, in the definition of universal integrals. We present an extension of the notion of semicopula to the case of symmetric bipolar interval [−1, 1]. We call this extension bipolar semicopula. The last definition can be used to obtain a simplified definition of the bipolar universal integral. Moreover, bipolar semicopulas allow for an extension of theory of quasi-copulas to the interval [−1, 1].
Proceedings of the 8th conference of the European Society for Fuzzy Logic and Technology, 2013
In many decision making problems evaluations of possible alternatives of choice, with respect to ... more In many decision making problems evaluations of possible alternatives of choice, with respect to several points of view (criteria) are expressed by means of h−interval (or fuzzy numbers). For example a pessimistic and an optimistic evaluation generate an interval containing the exact evaluation. These situations reflect lack of information or uncertainty on the same evaluations. In this paper we discuss h − k−aggregation functions that aggregate several h−interval evaluations into an overall evaluation, again expressed in terms of a k−interval.
Cumulative Prospect Theory of Tversky and Kahneman (1992) is the modern version of Prospect Theor... more 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 concept of universal integral, recently proposed and axiomatized, encompasses several integra... more The concept of universal integral, recently proposed and axiomatized, encompasses several integrals, including the Choquet, Shilkret and Sugeno integrals. In this paper we present two new axiomatizations of universal integrals on the scale [0, 1]. In the first characterization, we look at universal integrals on the scale [0, 1] as families of aggregation functions F satisfying some desired properties. The second characterization is given in the framing in which the original definition of universal integral was provided.
In decision analysis, and especially in multiple criteria decision analysis, several non additive... more In decision analysis, and especially in multiple criteria decision analysis, several non additive integrals have been introduced in the last years. These include the Choquet integral, the Shilkret integral and the Sugeno integral, among others. 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. The use of mentioned integrals in the aggregation process requests 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.
In decision analysis and especially in multiple criteria decision analysis, several non additive ... more 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.
The concept of universal integral, recently proposed, generalizes the Choquet, Shilkret and Sugen... more 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.
The aim of this paper is to introduce prospect theory in a game theoretic framework. We address t... more 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. JEL Classification: C70; D03.
In decision analysis and especially in multiple criteria decision analysis, several non additive ... more 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.
In decision analysis, and especially in multiple criteria decision analysis, several non additive... more In decision analysis, and especially in multiple criteria decision analysis, several non additive integrals have been introduced in the last years. These include the Choquet integral, the Shilkret integral and the Sugeno integral, among others. 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. The use of mentioned integrals in the aggregation process requests 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.
In decision analysis and especially in multiple criteria decision analysis, several non additive ... more 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.
The concept of universal integral, recently proposed, generalizes the Choquet, Shilkret and Sugen... more 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.
Two construction methods for aggregation functions based on a restricted a priori known decomposi... more Two construction methods for aggregation functions based on a restricted a priori known decomposition set and decomposition weighing function are introduced and studied. The outgoing aggregation functions are either superadditive or subadditive. Several examples, including illustrative gures, show the potential of the introduced construction methods. Our approach generalizes several known constructions and optimization methods, including decomposition and superdecomposition integrals.
We propose the concepts of superadditive and of subadditive transformations of aggregation functi... more We propose the concepts of superadditive and of subadditive transformations of aggregation functions acting on non-negative reals, in particular of integrals with respect to monotone measures. We discuss special properties of the proposed transforms and links between some distinguished integrals. Superadditive transformation of the Choquet integral, as well as of the Shilkret integral, is shown to coincide with the corresponding concave integral recently introduced by Lehrer. Similarly the transformation of the Sugeno integral is studied. Moreover, subadditive transformation of distinguished integrals is also discussed.
Following the idea of Even and Lehrer [3], we discuss a general approach to integration based on ... more Following the idea of Even and Lehrer [3], we discuss a general approach to integration based on decomposition of the integrated function. We distinguish 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.
Aggregation functions on [0, 1] with annihilator 0 can be seen as a generalized product on [0, 1]... more Aggregation functions on [0, 1] with annihilator 0 can be seen as a generalized product on [0, 1]. We study the generalized product on the bipolar scale [−1, 1], stressing the axiomatic point of view (compare also [9]). Based on newly introduced bipolar properties, such as the bipolar increasingness, bipolar unit element, bipolar idempotent element, several kinds of generalized bipolar product are introduced and studied. A special stress is put on bipolar semicopulas, bipolar quasi-copulas and bipolar copulas. Inspired by the truncated sum on [−1, 1] we introduce also the class of generalized bipolar sums, which differ from uninorms due to the non-associativity.
The aim of this paper is to introduce prospect theory in a game theoretic framework. We address t... more 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. JEL Classification: C70; D03.
Advances in Intelligent Systems and Computing, 2013
ABSTRACT The concept of universal integral has been recently proposed in order to generalize the ... more ABSTRACT The concept of universal integral has been recently proposed in order to generalize the Choquet, Shilkret and Sugeno integrals. We present two axiomatic foundations of the universal integral. The first axiomatization is expressed in terms of aggregation functions, while the second is expressed in terms of preference relations.
Communications in Computer and Information Science, 2014
Aggregation functions on [0, 1] with annihilator 0 can be seen as a generalized product on [0, 1]... more Aggregation functions on [0, 1] with annihilator 0 can be seen as a generalized product on [0, 1]. We study the generalized product on the bipolar scale [−1, 1], stressing the axiomatic point of view. Based on newly introduced bipolar properties, such as the bipolar increasingness, bipolar unit element, bipolar idempotent element, several kinds of generalized bipolar product are introduced and studied. A special stress is put on bipolar semicopulas, bipolar quasicopulas and bipolar copulas.
The concept of semicopula plays a fundamental role in the aggregation theory on interval [0, 1]. ... more The concept of semicopula plays a fundamental role in the aggregation theory on interval [0, 1]. Semicopulas are applied, for example, in the definition of universal integrals. We present an extension of the notion of semicopula to the case of symmetric bipolar interval [−1, 1]. We call this extension bipolar semicopula. The last definition can be used to obtain a simplified definition of the bipolar universal integral. Moreover, bipolar semicopulas allow for an extension of theory of quasi-copulas to the interval [−1, 1].
Proceedings of the 8th conference of the European Society for Fuzzy Logic and Technology, 2013
In many decision making problems evaluations of possible alternatives of choice, with respect to ... more In many decision making problems evaluations of possible alternatives of choice, with respect to several points of view (criteria) are expressed by means of h−interval (or fuzzy numbers). For example a pessimistic and an optimistic evaluation generate an interval containing the exact evaluation. These situations reflect lack of information or uncertainty on the same evaluations. In this paper we discuss h − k−aggregation functions that aggregate several h−interval evaluations into an overall evaluation, again expressed in terms of a k−interval.
Cumulative Prospect Theory of Tversky and Kahneman (1992) is the modern version of Prospect Theor... more 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 concept of universal integral, recently proposed and axiomatized, encompasses several integra... more The concept of universal integral, recently proposed and axiomatized, encompasses several integrals, including the Choquet, Shilkret and Sugeno integrals. In this paper we present two new axiomatizations of universal integrals on the scale [0, 1]. In the first characterization, we look at universal integrals on the scale [0, 1] as families of aggregation functions F satisfying some desired properties. The second characterization is given in the framing in which the original definition of universal integral was provided.
In decision analysis, and especially in multiple criteria decision analysis, several non additive... more In decision analysis, and especially in multiple criteria decision analysis, several non additive integrals have been introduced in the last years. These include the Choquet integral, the Shilkret integral and the Sugeno integral, among others. 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. The use of mentioned integrals in the aggregation process requests 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.
In decision analysis and especially in multiple criteria decision analysis, several non additive ... more 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.
The concept of universal integral, recently proposed, generalizes the Choquet, Shilkret and Sugen... more 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.
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