Yann Rébillé

Decision making over necessity measures through the Choquet integral criterion

We consider a decision maker who ranks necessity measures according to their Choquets expected utilities. We provide an axiomatization of such preferences. If the decision maker follows our set of axioms then there exists a fuzzy set such that any necessity measure can be reduced to a bet on being perfectly informed of the state which occurs or being totally ignorant, where the degree of information he will get is given by its Choquet expectation of the fuzzy set. This approach is similar to von Neuman and Morgensterns method under risk where a rational decision maker should evaluate a lottery according to its expected utility.

Sequentially continuous non-monotonic choquet integrals

Schmeidler (Proc. Amer. Math. Soc. 97 (2) (1986) 255-261) established an integral representation theorem through the Choquet integral for functionals satisfying monotonicity and a weaker condition than additivity, namely comonotonic additivity. Murofushi-Sugeno-Machida (Fuzzy Sets and Systems 64 (1994) 73-86) generalize this representation to the case of bounded variation functionals omitting the monotonicity condition. We give an alternative approach which is based on sequential continuity and tolerates non-monotonicity. This later condition is equivalent to @s-additivity in measure theory.

A Yosida-Hewitt decomposition for totally monotone games

We prove for totally monotone games defined on the set of Borel sets of a locally compact s-compact topological space a similar decomposition theorem to the famous Yosida-Hewitts one for finitely additive measures. This way any totally monotone decomposes into a continuous part and a pathological part which vanishes on the compact subsets. We obtain as corollaries some decompositions for finitely additive set functions and for minitive set functions.

Some characterizations of non-additive multi-period models

Abstract Building upon the works of Gilboa [Econometrica 57 (1989) 1153], Shalev [Math. Soc. Sci. 33 (1997) 203], and De Waegenaere and Wakker [J. Math. Econ. 36 (2001) 45], we show that a simple version of variation aversion, jointly with a myopia axiom allows to derive in an infinite setting a meaningful expression for evaluating income streams. Furthermore, we prove that the usual additive discounted expectation introduced by Koopmans [Koopmans, T.C., 1972. Representations of preference orderings over time. In: McGuire, C.B., Radner, R. (Eds.), Decisions and Organizations. North-Holland, Amsterdam, pp. 79–100] can be accommodated in a non-additive way.

A Yosida--Hewitt decomposition for totally monotone set functions on locally compact σ -compact topological spaces

We prove for totally monotone set functions defined on the set of Borel sets of a locally compact @s-compact topological space a similar decomposition theorem to the famous Yosida-Hewitts one for finitely additive measures. This way any totally monotone decomposes into a continuous part and a pathological part which vanishes on the compact subsets. We obtain as corollaries some decompositions for finitely additive set functions and for minitive set functions.

Autocontinuity and convergence theorems for the Choquet integral

We provide convergence theorems for the Choquet integral for various notions of convergence. Monotone almost uniform convergence from above/below is associated to monotone autocontinuity from above/below. The dominated almost uniform convergence is associated to monotone autocontinuity. Autocontinuity can be associated to the dominated convergence in measure, in mean or in strict-measure.

A Yosida--Hewitt decomposition for minitive set functions

We prove for minitive set functions defined on a @s-algebra, a similar decomposition theorem to the Yosida-Hewitts one for classical measures, this way any minitive set function can be decomposed in a fuzzy minitive measure part and a purely minitive part. A particular attention is given for the countable case where the canonical description of any @s-continuous necessity is fully elicited and provide a simple way to compute the Choquet integral of any bounded sequence.