Optimization of Coalitions - The Mutational Approach

Abstract

In cooperative game theory as well as in some domains of economic regulation by shortages (queues or unemployment), one is confronted to the problem of optimizing coalitions of players or economic agents. Since coalitions are subsets and cannot be represented by vectors -- except if we embed subsets in the family of fuzzly sets, which are functions -- the need to adapt the theory of optimization under constraint for coalitions or subsets instead of vectors did emerge. The "power spaces" in which coalitions, images, shapes, etc. have to be chosen are metric spaces without a linear structure. However, one can extend the differential calculus to a mutational calculus for maps from one metric space to another, as we shall explain in this paper. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by "transitions" with which we can also define differential quotients of a map. Their various limits are called "mutations" of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. This is the purpose of this paper