Dynamical Connectionist Network and Cooperative Games

Socio-economic networks, neural networks and genetic networks describe collective phenomena through constraints relating actions of several players, coalitions of these players and multilinear connectionist Operators acting on the set of actions of each coalition. Static and dynamical cooperative games also involve coalitions. Allowing “coalitions to evolve” requires the embedding of the finite set of coalitions in the compact convex subset of “fuzzy coalitions”. This survey present results obtained through this strategy. We provide first a class of control systems governing the evolution of actions, coalitions and multilinear connectionist Operators under which the architecture of a network remains viable. The controls are the “viability multipliers” of the “resource space” in which the constraints are defined. They are involved as “tensor products” of the actions of the coalitions and the viability multiplier, allowing us to encapsulate in this dynamical and multilinear framework the concept of Hebbian learning rules in neural networks in the form of “multi-Hebbian” dynamics in the evolution of connectionist Operators. They are also involved in the evolution of coalitions through the “cost” of the constraints under the viability multiplier regarded as a price, describing a “nerd behavior”. We use next the viability/capturability approach for studying the problem of characterizing the dynamic core of a dynamic cooperative game defined in a characteristic function form. We define the dynamic core as a set-valued map associating with each fuzzy coalition and each time the set of imputations such that their payoffs at that time to the fuzzy coalition are larger than or equal to the one assigned by the characteristic function of the game and study it.