Decomposition algorithm of searching equilibria in a dynamic game

Abstract

This paper considers a noncooperative game of several players (governments of neighboring countries) participating in emission reduction trading. A special emphasis is placed on the case of two players, one representing Eastern European countries and the other being the countries of the former Soviet Union. We perform statistical analysis of the model parameters based on real data under quadratic cost functions and logarithmic utility functions. The concepts of a noncooperative Nash equilibrium and cooperative Pareto maxima are introduced and connections between them are established. A new concept, i.e., a market equilibrium, which combines the properties of Nash and Pareto equilibria, is rigorously defined. An analytical solution of the market equilibrium problem is given. This analytical solution can serve for verification of numerical search algorithms. In addition, we propose a computational algorithm of market equilibrium search, which shifts a competitive Nash equilibrium to a cooperative Pareto maximum. The algorithm is interpreted as a repeated auction, where the auctioneer possesses no information about the cost functions and the functions of environmental effect from emission reduction of the participating countries. An auctioneer strategy leading to market equilibrium attainment is considered. From the game-theoretic viewpoint, a repeated auction describes a learning process in a noncooperative repeated game under uncertainty. We compare the results gained by the proposed computational algorithms with their analytical counterparts. And finally, numerical calculations of equilibrium and algorithm trajectories converging to the equilibrium are demonstrated.