Set Functions, Games and Capacities in Decision Making

Grabisch, Michel

doi:10.1007/978-3-319-30690-2

Cited by 219 publications

(162 citation statements)

References 202 publications

(424 reference statements)

Supporting

Mentioning

159

Contrasting

Unclassified

Order By: Relevance

“…iff (a), (b) or (c) hold. Equality (44) is certainly true if A = Ω, by coherence of P , l: P (Ω) = l(Ω) = 1, and, if P (A) = 0, using also(10): P (A) = l(A) = 0.Therefore, it remains to check whenP (A) = max ωi⇒A l i , 1 − ωi⇒¬A u i for those events A such that P (A) > 0, A = Ω. Since by (33) l(A) ≤ P (A), this is equivalent to check when P (A) = ωi⇒A l i or P (A) = 1 − ωi⇒¬A u i .Taking then one such A, we investigate first whenP (A) = ωi⇒A l i .…”

mentioning

confidence: 97%

Nearly-Linear uncertainty measures

Corsato

Pelessoni

Vicig

2019

International Journal of Approximate Reasoning

View full text Add to dashboard Cite

Several easy to understand and computationally tractable imprecise probability models, like the Pari-Mutuel model, are derived from a given probability measure P0. In this paper we investigate a family of such models, called Nearly-Linear (NL). They generalise a number of well-known models, while preserving a simple mathematical structure. In fact, they are linear affine transformations of P0 as long as the transformation returns a value in [0, 1]. We study the properties of NL measures that are (at least) capacities, and show that they can be partitioned into three major subfamilies. We investigate their consistency, which ranges from 2-coherence, the minimal condition satisfied by all, to coherence, and the kind of beliefs they can represent. There is a variety of different situations that NL models can incorporate, from generalisations of the Pari-Mutuel model, the ε-contamination model and other models to conflicting attitudes of an agent towards low/high P0-probability events (both prudential and imprudent at the same time), or to symmetry judgments. The consistency properties vary with the beliefs represented, but not strictly: some conflicting and partly irrational moods may be compatible with coherence. In a final part, we compare NL models with their closest, but only partly overlapping, models, neo-additive capacities and probability intervals.

show abstract

mentioning

confidence: 97%

Nearly-Linear uncertainty measures

Corsato

Pelessoni

Vicig

2019

International Journal of Approximate Reasoning

View full text Add to dashboard Cite

show abstract

“…Let

μ : 2^{{1, 2}} goodbreakinfix\to [0, 1]

be a capacity with values

μ (\emptyset) goodbreakinfix= 0 goodbreakinfix, μ ({1}) goodbreakinfix= a goodbreakinfix, μ ({2}) goodbreakinfix= b

and

μ ({1, 2}) goodbreakinfix= 1

, where

a goodbreakinfix, b goodbreakinfix\in [0, 1]

. Then the corresponding Sugeno integral

S u_{μ} : {[0, 1]}^{2} goodbreakinfix\to [0, 1]

is given by

S u_{μ} (x_{1}, x_{2}) = (x_{1} \land a) \lor (x_{2} \land b) \lor (x_{1} \land x_{2}) .

Note that

\land

and

\lor

stand here for the minimum and maximum, respectively.…”

Section: Set‐based Extended Aggregation Functionsmentioning

confidence: 99%

Set‐based extended aggregation functions

et al. 2019

View full text Add to dashboard Cite

Inspired by the Zadeh approach to fuzzy connectives in fuzzy set theory and by some applications, we introduce and study set‐based extended functions, and in particular, set‐based extended aggregation functions. These functions reflect neither reordering nor repetition of input values, and, linking different arities, they introduce serious constraints for extended functions. A complete characterization of set‐based extended (aggregation) functions is given, and some constructions of such functions are also proposed, including several examples.

show abstract

“…4 it is enough to realize that in both formulas the value of the Sugeno integral (Su) D f dµ is obtained by the intersection point of the decumulative function µ(D ∩ {f ≥ t}) and the diagonal, no matter whether ≥ or > is taken 5 capacity is a normalized monotone measure µ, i. e., µ(X) = 1 Remark 3.2. Due to the fact that the mapping ϕ is suprema and infima preserving, the following can easily be seen Indeed, it means that − ,D f dµ is the Sugeno integral of ϕ • f with respect to ϕ • µ.…”

Section: -Associatedness-comonotonicity Preserving Operationsmentioning

confidence: 99%