Development of decomposition methods for auction games

The paper addresses the problem of searching maximum Pareto points which possess special decomposition characteristics, that inherits the properties of the competitive Nash equilibrium. Such point is called market equilibrium. First, the competitive Nash equilibrium is constructed for the auction game of three players and its properties are analyzed. Then the set of Pareto points is described analytically on the basis of convolution method. The main task is to construct a path from the competitive Nash equilibrium to the point of market equilibrium in the Pareto set. This path is generated by the shift algorithm which is based on analytical calculation of the players’ best replies with respect to auction prices. The auction mechanism of the search algorithm is expressed as a system of nonlinear differential equations, precisely they describe formation of auction prices and players’ best replies to these prices. Trajectories of the generated dynamic system are constructed and their properties of convergence to the market equilibrium are examined. The proposed algorithm is applied to the model of the auction game of fast growing economies and its efficiency is shown particularly for the emissions reduction problem.