The influence of variables on pseudo-Boolean functions with applications to game theory and multicriteria decision making

in Discrete Applied Mathematics (2000), 107(1-3), 139-164

The power of players in a collective decision process is a central issue in game theory. For this reason, the concept of influence of players on a simple game has been introduced. More generally, the ... [more ▼]

The power of players in a collective decision process is a central issue in game theory. For this reason, the concept of influence of players on a simple game has been introduced. More generally, the influence of variables on Boolean functions has been defined and studied. We extend this concept to pseudo-Boolean functions, thus making it possible to appraise the degree of influence of any coalition of players in cooperative games. In the case of monotone games, we also point out the links with the concept of interaction among players. Although they do not have the same meaning at all, both influence and interaction functions coincide on singletons with the so-called Banzhaf power index. We also define the influence of variables on continuous extensions of pseudo-Boolean functions. In particular, the Lovász extension, also called discrete Choquet integral, is used in multicriteria decision making problems as an aggregation operator. In such problems, the degree of influence of decision criteria on the aggregation process can then be quite relevant information. We give the explicit form of this influence for the Choquet integral and its classical particular cases. [less ▲]

An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria

in IEEE Transactions on Fuzzy Systems (2000), 8(6), 800-807

The most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean. In many problems however, the criteria considered interact, and a substitute to ... [more ▼]

The most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean. In many problems however, the criteria considered interact, and a substitute to the weighted arithmetic mean has to be adopted. We show that, under rather natural conditions, the discrete Choquet integral is an adequate aggregation operator that extends the weighted arithmetic mean by the taking into consideration of the interaction among criteria. The axiomatic that supports the Choquet integral is presented and an intuitive approach is proposed as well. [less ▲]

Le chaos de Wiener et l'intégrale de Skohorod

Presentation (2000, November 15)

On perfect, amicable, and sociable chains

E-print/Working paper (2000)

Let $\bfx = (x_0,\ldots,x_{n-1})$ be an $n$-chain, i.e., an $n$-tuple of non-negative integers $< n$. Consider the operator $s: \bfx \mapsto\bfx' = (x'_0,\ldots,x'_{n-1})$, where $x'_j$ represents the ... [more ▼]

Let $\bfx = (x_0,\ldots,x_{n-1})$ be an $n$-chain, i.e., an $n$-tuple of non-negative integers $< n$. Consider the operator $s: \bfx \mapsto\bfx' = (x'_0,\ldots,x'_{n-1})$, where $x'_j$ represents the number of $j$'s appearing among the components of $\bfx$. An $n$-chain $\bfx$ is said to be perfect if $s(\bfx) = \bfx$. For example, (2,1,2,0,0) is a perfect 5-chain. Analogously to the theory of perfect, amicable, and sociable numbers, one can define from the operator $s$ the concepts of amicable pair and sociable group of chains. In this paper we give an exhaustive list of all the perfect, amicable, and sociable chains. [less ▲]

On the associativity functional equation

in Fuzzy Sets and Systems (2000), 114(3), 381-389

Let [a,b] be any bounded closed real interval. The class of all continuous, nondecreasing, associative functions M : [a,b]^2 --> [a,b] fulfilling the boundary conditions M(a,a)=a and M(b,b)=b is described.

Mentale Modelle beim Verstehen von Personbeschreibungen: Konsistenz der Information und Kohärenz des Textes.

Scientific Conference (2000, September)

What can Evaluative Learning tell us about Implicit Learning?

Poster (2000, September)

Steigert Selbstvertrauen die Leistung beim Bewegungslernen

in dvs-Informationen (2000), 15

The Sectio Aurea in a Fugue by Darius Milhaud

Presentation (2000, July 17)

On the entropy of non-additive weights

Scientific Conference (2000, July)

We consider a Choquet capacity, that is a set function which describes the importance of every subset of criteria in a MCDA problem. The following question is approached: what is the generalized ... [more ▼]

We consider a Choquet capacity, that is a set function which describes the importance of every subset of criteria in a MCDA problem. The following question is approached: what is the generalized counterpart of the Shannon entropy (defined for a probabilistic measure) for such a capacity? The extension that is proposed depends on the scale type. In the cardinal case, the entropy is defined in terms of the first derivatives of the non-additive measures. In the ordinal case, it refers to the cardinality of the scale values that appear in the set of all capacities. Both generalized entropies are symmetric functions of the capacities and their extreme values (max entropy and min entropy) are characterized. An application to the determination of weights is given when interacting criteria are considered. [less ▲]