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

Entropy of capacities on lattices and set systems

We propose a definition for the entropy of capacities defined on lattices. Classical capacities are monotone set functions and can be seen as a generalization of probability measures. Capacities on lattices address the general case where the family of subsets is not necessarily the Boolean lattice of all subsets. Our definition encompasses the classical definition of Shannon for probability measures, as well as the entropy of Marichal defined for classical capacities. Some properties and examples are given

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

On Cores and Stable Sets for Fuzzy Games

AMS classifications: 90D12; 03E72;cooperative games;decision making;fuzzy games

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

Cooperative investment games or population games

The model of a cooperative fuzzy game is interpreted as both a population game and a cooperative investment game. Three types of core- like solutions induced by these interpretations are introduced and investigated. The interpretation of a game as a population game allows us to define sub-games. We show that, unlike the well-known Shapley- Shubik theorem on market games (Shapley-Shubik) there might be a population game such that each of its sub-games has a non-empty core and, nevertheless, it is not a market game. It turns out that, in order to be a market game, a population game needs to be also homogeneous. We also discuss some special classes of population games such as convex games, exact games, homogeneousgames and additive games.investment game, population game, fuzzy game, core-like solution, market game

Fuzzy Cores and Fuzzy Balancedness

We study the relation between the fuzzy core and balancedness for fuzzy games. For regular games, this relation has been studied by Bondareva (1963) and Shapley (1967). First, we gain insight in this relation when we analyse situations where the fuzzy game is continuous. Our main result shows that any fuzzy game has a non-empty core if and only if it satisfies all (fuzzy) balanced inequalities. We also consider deposit games to illustrate the use of the main result.Cooperative fuzzy games;fuzzy balancedness;fuzzy core

Measuring the interactions among variables of functions over the unit hypercube

By considering a least squares approximation of a given square integrable function $f\colon[0,1]^n\to\R$ by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of $f$. 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 $f$ or, under certain natural conditions on $f$, as an expected value of the derivatives of $f$. 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

Fuzzy Clan Games and Bi-monotonic Allocation Rules

Clan game;Big boss game;Core;Decision making;Fuzzy coalition;Fuzzy game;Monotonic allocation rule