Some Characterizations of Lower Probabilities and Other Monotone Capacities through the use of Möbius Inversion

Chateauneuf, Alain; Jaffray, Jean-Yves

doi:10.1007/978-3-540-44792-4_19

Cited by 110 publications

(183 citation statements)

References 17 publications

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“…This last proposition (Chateauneuf and Jaffray 1989) gives the form of supermodularity and submodularity constraints in terms of the Möbius transform. …”

Section: Proposition 7 a Set Function ν: P(n) → R Is Supermodular (Rementioning

confidence: 96%

“…In terms of the Möbius representation of a game µ, Chateauneuf and Jaffray (1989) showed that, for any x = (x 1 , . .…”

Section: Definitionmentioning

confidence: 99%

“…The form of higher monotonicity constraints can be obtained similarly (see Chateauneuf and Jaffray 1989;Fujimoto and Murofushi 2005). In other terms, first are given the n singletons, then the n 2 pairs, then the n 3 3-element subsets, etc., and finally N itself.…”

Section: Proposition 7 a Set Function ν: P(n) → R Is Supermodular (Rementioning

confidence: 99%

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Quadratic distances for capacity and bi-capacity approximation and identification

Kojadinovic

2006

4OR

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The application of multi-attribute utility theory based on the Choquet integral requires the prior identification of a capacity if the utility scale is unipolar, or of a bi-capacity if the utility scale is bipolar. In order to implement a minimum distance principle for capacity or bi-capacity approximation or identification, quadratic distances between capacities and bi-capacities are studied. The proposed approach, consisting in solving a strictly convex quadratic program, has been implemented within the GNU R kappalab package for capacity and nonadditive integral manipulation. Its application is illustrated on two examples.Keywords Multi-attribute utility theory · Choquet integral · Capacity · Bi-capacity · Quadratic programming MSC Classification 90B50 · 91B16 · 90C20

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“…This last proposition (Chateauneuf and Jaffray 1989) gives the form of supermodularity and submodularity constraints in terms of the Möbius transform. …”

Section: Proposition 7 a Set Function ν: P(n) → R Is Supermodular (Rementioning

confidence: 96%

“…In terms of the Möbius representation of a game µ, Chateauneuf and Jaffray (1989) showed that, for any x = (x 1 , . .…”

Section: Definitionmentioning

confidence: 99%

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Quadratic distances for capacity and bi-capacity approximation and identification

Kojadinovic

2006

4OR

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“…Unlike Savage, in this model there is an algebra A of subsets of X containing all singletons, with typical element X . 3 We denote by U the set of the nonempty subsets in A, i.e.,U ≡ A \ ∅. As explained in the Introduction, the sets X ∈ U are the sets of consequences that the DM envisions as possible results of her actions.…”

Section: The Modelmentioning

confidence: 99%

“…As for the result of Proposition 6, it was first proved for belief functions by Shafer[32], and extended to any capacity by Chateauneuf and Jaffray[3].…”

mentioning

confidence: 92%

Coping with ignorance: unforeseen contingencies and non-additive uncertainty

Ghirardato¹

2001

Econ Theory

114

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In real-life decision problems, decision makers are never provided with the necessary background structure: the set of states of the world, the outcome space, the set of actions. They have to devise all these by themselves. I model the (static) choice problem of a decision maker (DM) who is aware that her perception of the decision problem is too coarse, as for instance when there might be unforeseen contingencies. I make a "bounded rationality" assumption on the way the DM deals with this difficulty, and then I show that imposing standard subjective expected utility axioms on her preferences only implies that they can be represented by a (generalized) expectation with respect to a nonadditive measure, called a belief function. However, the axioms do have strong implications for how the DM copes with the type of ignorance described above. Finally, I show that some decision rules that have been studied in the literature can be obtained as a special case of the model presented here (though they have to be interpreted differently).

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Special cases

Destercke

Dubois

2014

Wiley Series in Probability and Statistics

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Some Characterizations of Lower Probabilities and Other Monotone Capacities through the use of Möbius Inversion

Cited by 110 publications

References 17 publications

Quadratic distances for capacity and bi-capacity approximation and identification

Quadratic distances for capacity and bi-capacity approximation and identification

Coping with ignorance: unforeseen contingencies and non-additive uncertainty

Special cases

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