44 research outputs found
Axiomatizations of signed discrete Choquet integrals
We study the so-called signed discrete Choquet integral (also called
non-monotonic discrete Choquet integral) regarded as the Lov\'asz extension of
a pseudo-Boolean function which vanishes at the origin. We present
axiomatizations of this generalized Choquet integral, given in terms of certain
functional equations, as well as by necessary and sufficient conditions which
reveal desirable properties in aggregation theory
Axiomatizations of Lov\'asz extensions of pseudo-Boolean functions
Three important properties in aggregation theory are investigated, namely
horizontal min-additivity, horizontal max-additivity, and comonotonic
additivity, which are defined by certain relaxations of the Cauchy functional
equation in several variables. We show that these properties are equivalent and
we completely describe the functions characterized by them. By adding some
regularity conditions, these functions coincide with the Lov\'asz extensions
vanishing at the origin, which subsume the discrete Choquet integrals. We also
propose a simultaneous generalization of horizontal min-additivity and
horizontal max-additivity, called horizontal median-additivity, and we describe
the corresponding function class. Additional conditions then reduce this class
to that of symmetric Lov\'asz extensions, which includes the discrete symmetric
Choquet integrals
On three properties of the discrete Choquet integral
Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, these functions coincide with the Lovász extensions vanishing at the origin, which subsume the discrete Choquet integrals
The problem of collective identity in a fuzzy environment
Producción CientíficaWe define the problem of group identication in a fuzzy environment. We concentrate on the case where the society is required to self-determine the belongingness of each member to a speci_c group, characterized by a single attribute. In general terms, this case consists of a collective identity issue that can be regarded as an aggregation problem of individual assessments within a group. Here we introduce the possibility that both the original assessments and the resulting output attach partial memberships to the collectivity, for each potential member. We propose relevant classes of rules, and some are axiomatically characterized. Our new approach provides a way to circumvent classical impossibility results like Kasher and Rubinstein's.Ministerio de Economía, Industria y Competitividad (Project ECO2012-32178
Axiomatizations of the Choquet integral on general decision spaces
PhDWe propose an axiomatization of the Choquet integral model for the
general case of a heterogeneous product set X = X1 Xn. Previous
characterizations of the Choquet integral have been given for
the particular cases X = Y n and X = Rn. However, this makes
the results inapplicable to problems in many fields of decision theory,
such as multicriteria decision analysis (MCDA), state-dependent
utility (SD-DUU), and social choice. For example, in multicriteria decision
analysis the elements of X are interpreted as alternatives, characterized
by criteria taking values from the sets Xi. Obviously, the
identicalness or even commensurateness of criteria cannot be assumed
a priori. Despite this theoretical gap, the Choquet integral model is
quite popular in the MCDA community and is widely used in applied
and theoretical works. In fact, the absence of a sufficiently general
axiomatic treatment of the Choquet integral has been recognized several
times in the decision-theoretic literature. In our work we aim to
provide missing results { we construct the axiomatization based on
a novel axiomatic system and study its uniqueness properties. Also,
we extend our construction to various particular cases of the Choquet
integral and analyse the constraints of the earlier characterizations.
Finally, we discuss in detail the implications of our results for the
applications of the Choquet integral as a model of decision making
An Ordinal Approach to Risk Measurement
In this short note, we aim at a qualitative framework for modeling multivariate risk. To this extent, we consider completely distributive lattices as underlying universes, and make use of lattice functions to formalize the notion of risk measure. Several properties of risk measures are translated into this general setting, and used to provide axiomatic characterizations. Moreover, a notion of quantile of a lattice-valued random variable is proposed, which shown to retain several desirable properties of its real-valued counterpart.lattice; risk measure; Sugeno integral; quantile.
Attitude toward information and learning under multiple priors
This paper studies learning under multiple priors by characterizing the decision maker's attitude toward information. She is incredulous if she integrates new information with respect to only those measures that minimizes the likelihood of the new information and credulous if she uses the maximum likelihood procedure to update her priors. Both updating rules expose her to dynamic inconsistency. We explore different ways to resolve this problem. One way consists to assume that the decision maker's attitude toward information is not relevant to characterize conditional preferences. In this case, we show that a necessary and sufficient condition, introduced by [Epstein L. and Schneider M., 2003. Recursive multiple priors. Journal of Economic Theory 113, 1-31], is the rectangularity of the set of priors. Another way is to extend optimism or pessimism to a dynamic set-up. A pessimistic (max-min expected utility) decision maker will be credulous when learning bad news but incredulous when learning good news.Conversely, an optimistic (max-max expected utility) decision maker will be credulous when learning good news but incredulous when learning bad news. It allows max-min (or max-max) expected utility preferences to be dynamically consistent but it violates consequentialism because conditioning works with respect to counterfactual outcomes. The implications of our findings when the set of priors is the core of a non-additive measure are explored.Multiple priors ; Learning ; Dynamic consistency ; Consequentialism ; Attitude toward information
A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid
The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. 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 multi-criteria decision aid. 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.Choquet integral, Sugeno integral, capacity, bipolarity, preferences
Unambiguous events and dynamic Choquet preferences.
This paper explores the relationship between dynamic consistency and existing notions of unambiguous events for Choquet expected utility preferences. A decision maker is faced with an information structure represented by a filtration. We show that the decision maker’s preferences respect dynamic consistency on a fixed filtration if and only if the last stage of the filtration is composed of unambiguous events in the sense of Nehring (Math Social Sci 38:197–213, 1999). Adopting two axioms, conditional certainty equivalence consistency and constrained dynamic consistency to filtration measurable acts, it is shown that the decision maker respects these two axioms on a fixed filtration if and only if the last stage of the filtration is made up of unambiguous events in the sense of Zhang (Econ Theory 20:159–181, 2002).Choquet expected utility; Unambiguous events; Filtration; Updating; Dynamic consistency; Consequentialism;
Attitude toward information and learning under multiple priors
This paper studies learning under multiple priors by characterizing the decision maker's attitude toward information. She is incredulous if she integrates new information with respect to only those measures that minimizes the likelihood of the new information and credulous if she uses the maximum likelihood procedure to update her priors. Both updating rules expose her to dynamic inconsistency. We explore different ways to resolve this problem. One way consists to assume that the decision maker's attitude toward information is not relevant to characterize conditional preferences. In this case, we show that a necessary and sufficient condition, introduced by [Epstein L. and Schneider M., 2003. Recursive multiple priors. Journal of Economic Theory 113, 1-31], is the rectangularity of the set of priors. Another way is to extend optimism or pessimism to a dynamic set-up. A pessimistic (max-min expected utility) decision maker will be credulous when learning bad news but incredulous when learning good news.Conversely, an optimistic (max-max expected utility) decision maker will be credulous when learning good news but incredulous when learning bad news. It allows max-min (or max-max) expected utility preferences to be dynamically consistent but it violates consequentialism because conditioning works with respect to counterfactual outcomes. The implications of our findings when the set of priors is the core of a non-additive measure are explored