Kalai-smorodinsky balances for N-tuples of interfering elements

Abstract

The study proposed here builds up a game model (with associated algorithms) in a specific non-linear interfering scenario, with n possible interacting elements. Our examination provides optimal Kalai-Smorodinsky compromise solution n-tuples, for the game, whose components indicate active principle quantity percentages. We solve the problem by using the Carfì’s payoff analysis method for differentiable payoff functions. Moreover we implement Matlab algorithms for the construction and representation of the payoff spaces and for the finding of Kalai-Smorodinsky solutions. The software for the determination of graphs are adopted, but not presented here explicitly. The core section of the paper, completely studies the game in the n-dimensional case, by finding the critical zone of the game in its Cartesian form. At this aim, we need to prove a theorem and a lemma about the Jacobian determinant of the n-game. In the same section, we write down the intersection of the critical zone and the Kalai-Smorodinsky straight-line. In the Appendix 1 we solve the problem in closed form for the 2 dimensional case and numerically for n> 2. Our methods works also for games with non-convex payoff space. Finally, in a particular highly symmetrical case, we solve analytically the Kalai-Smorodinsky compromise problem in all cases. We provide some applications of the obtained results, particularly to economic problems.