The core and superdifferential of a fuzzy TU-cooperative game

In the paper, we consider conditions providing coincidence of the cores and superdifferentials of fuzzy cooperative games with side payments. It turned out that one of the most simple sufficient conditions consists of weak homogeneity. Moreover, by applying so-called S*-representation of a fuzzy game introduced by the author, we show that for any vwith nonempty core C(v) there exists some game u such that C(v) coincides with the superdifferential of u. By applying subdifferential calculus we describe a structure of the core forboth classic fuzzy extensions of the ordinary cooperative game (e.g., Aubin and Owen extensions) and for some new continuations, like Harsanyi extensions and generalized Airport game.

Asymptotic results for non-linear processes of the McKean tagged-molecule type

McKean's tagged-molecule process is a non-linear homogeneous two-state Markov chain in continuous time, constructed with the aid of a binary branching process. For each of a large class of branching processes we construct a similar process. The construction is carefully done and the weak homogeneity is deduced. A simple probability argument permits us to show convergence to the equidistribution (½, ½) and to note that this limit is a strong equilibrium. A non-homogeneous Markov chain result is also used to establish the geometric rate of convergence. A proof of a Boltzmann H-theorem is also established.

Asymptotic results for non-linear processes of the McKean tagged-molecule type

McKean's tagged-molecule process is a non-linear homogeneous two-state Markov chain in continuous time, constructed with the aid of a binary branching process. For each of a large class of branching processes we construct a similar process. The construction is carefully done and the weak homogeneity is deduced. A simple probability argument permits us to show convergence to the equidistribution (½, ½) and to note that this limit is a strong equilibrium. A non-homogeneous Markov chain result is also used to establish the geometric rate of convergence. A proof of a Boltzmann H-theorem is also established.

The combinational structure of non-homogeneous Markov chains with countable states

LetP(s,t)denote a non-homogeneous continuous parameter Markov chain with countable state spaceEand parameter space[a,b],−∞<a<b<∞. LetR(s,t)={(i,j):Pij(s,t)>0}. It is shown in this paper thatR(s,t)is reflexive, transitive, and independent of(s,t),s<t, if a certain weak homogeneity condition holds. It is also shown that the relationR(s,t), unlike in the finite state space case, cannot be expressed even as an infinite (countable) product of reflexive transitive relations for certain non-homogeneous chains in the case whenEis infinite.