52 research outputs found
Bisemivalues for bicooperative games
We introduce bisemivalues for bicooperative games and we also provide an interesting characterization of this kind of values by means of weighting coefficients in a similar way as it was given for semivalues in the context of cooperative games. Moreover, the notion of induced bisemivalues on lower cardinalities also makes sense and an adaptation of Dragan’s recurrence formula is obtained. For the particular case of (p, q)-bisemivalues, a computational procedure in terms of the multilinear extension of the game is given.Peer ReviewedPostprint (author's final draft
Weighted Banzhaf power and interaction indexes through weighted approximations of games
The Banzhaf power index was introduced in cooperative game theory to measure
the real power of players in a game. The Banzhaf interaction index was then
proposed to measure the interaction degree inside coalitions of players. It was
shown that the power and interaction indexes can be obtained as solutions of a
standard least squares approximation problem for pseudo-Boolean functions.
Considering certain weighted versions of this approximation problem, we define
a class of weighted interaction indexes that generalize the Banzhaf interaction
index. We show that these indexes define a subclass of the family of
probabilistic interaction indexes and study their most important properties.
Finally, we give an interpretation of the Banzhaf and Shapley interaction
indexes as centers of mass of this subclass of interaction indexes
Measuring the interactions among variables of functions over the unit hypercube
By considering a least squares approximation of a given square integrable
function by a multilinear polynomial of a specified
degree, we define an index which measures the overall interaction among
variables of . This definition extends the concept of Banzhaf interaction
index introduced in cooperative game theory. Our approach is partly inspired
from multilinear regression analysis, where interactions among the independent
variables are taken into consideration. We show that this interaction index has
appealing properties which naturally generalize the properties of the Banzhaf
interaction index. In particular, we interpret this index as an expected value
of the difference quotients of or, under certain natural conditions on ,
as an expected value of the derivatives of . These interpretations show a
strong analogy between the introduced interaction index and the overall
importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a
few applications of the interaction index
Convex Fuzzy Games and Participation Monotonic Allocation Schemes
AMS classifications: 90D12; 03E72Convex games;Core;Decisionmaking;Fuzzy coalitions;Fuzzy games;Monotonic allocation schemes;Weber set
Convex fuzzy games and participation monotonic allocation schemes
90D12;03E72cooperative games
Approximations of Lovasz extensions and their induced interaction index
The Lovasz extension of a pseudo-Boolean function is
defined on each simplex of the standard triangulation of as the
unique affine function that interpolates at the
vertices of the simplex. Its degree is that of the unique multilinear
polynomial that expresses . In this paper we investigate the least squares
approximation problem of an arbitrary Lovasz extension by Lovasz
extensions of (at most) a specified degree. We derive explicit expressions of
these approximations. The corresponding approximation problem for
pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then
solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an
alternative definition of Banzhaf interaction index. Similarly we introduce a
new interaction index from approximations of and we present some of
its properties. It turns out that its corresponding power index identifies with
the power index introduced by Grabisch and Labreuche (2001).Comment: 19 page
Characterizing core stability with fuzzy games
This paper investigates core stability of cooperative, TU games via a fuzzy extension of the totally balanced cover of a TU game. The stability of the core of the fuzzy extension of a game, the concave extension, is shown to reflect the core stability of the original game and vice versa. Stability of the core is then shown to be equivalent to the existence of an equilibrium of a certain correspondence.cooperative game, core, stable set, fuzzy coalition, fuzzy game, core stability
A multilinear extension of a class of fuzzy bi-cooperative games 1
Abstract. In this paper, a new class of bi-cooperative games with fuzzy bi-coalitions is proposed in multilinear extension form. The extension is shown to be unique. The solution concept discussed in [3] is investigated and characterized for this class of games
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