Convex fuzzy games and participation monotonic allocation schemes.
Convex cooperative fuzzy games are studied and compared with classical (crisp) convex games. Sections 2 and 3 provide the necessary background on crisp and fuzzy games and, especially, the notions of marginal vector, Shapley value, Weber set and core in both cases, as well as four characterizations of crisp convex games. Also the multilinear and crisp operators, which respectively transform a crisp game into a fuzzy game (its multilinear extension) and a fuzzy game into a crisp game (its restriction to the vertices of the unit cube), are introduced. While the core of a fuzzy game is, in general, only contained in the core of its corresponding crisp game, the Shapley value and the Weber set of both games coincide. Moreover, the core of a fuzzy game is always a subset of its Weber set as it happens for crisp games. Section 4 starts by defining convex fuzzy games as those whose characteristic function is supermodular (as in the classical case) but also coordinate-wise convex (convex with regard to each coordinate). These games form a convex cone, and the crisp game of any convex fuzzy game is also convex. After a series of propositions that play the role of lemmas, the first main result states that a fuzzy game is convex if, and only if, it satisfies the increasing average marginal return property, which reminds the classical increasing marginal contribution property. The second main result establishes that, for convex games, (i) all marginal vectors are in the core and hence the Shapley value holds a central position in the core, (ii) the core coincides with the Weber set, and (iii) the core coincides with the core of the associated crisp game. However, the equivalence between being convex and the coincidence of the core and the Weber set, valid for crisp games, doesn't hold here. Finally, it is shown that the core, superadditive in general, is additive in the convex case. Section 5 is devoted to the participation monotonic allocation schemes (pamas, for short). Based on \textit{Y. Sprumont}'s [``Population monotonic allocation schemes for cooperative games with transferable utility'', Games Econ. Behav. 2, 378--394 (1990; Zbl 0753.90083)] previous idea of population monotonic allocation scheme, a pamas is defined as a scheme of allocations, one for each fuzzy coalition and each player, which satisfies two conditions: stability, in the sense that the restricted scheme belongs to the core of any restricted game, and a participation monotonicity condition. A necessary but in general not sufficient condition for the existence of a pamas for a fuzzy game is the no emptiness of the core of all restricted subgames; however, it is also sufficient in the convex case, as is stated and shown in the main result of this section by using the idea of pamas extendable core element, since, for convex games, all core members are pamas extendable. Finally, some interesting open questions are mentioned in the concluding section.
scientific article; zbMATH DE number 2005399
scientific article