Numerical Solution of Nash and Stackelberg Equilibria: an Evolutionary Approach

In this paper we describe evolutionary heuristics for numerically solving systems of several, interdependent optimisation problems. They can be used for the solution of some games with simultaneous moves of the players (Nash equilibria), asynchronous moves (Stackelberg equilibria), or a mix of these situations. The application is possible in cases where the presence of non-convexities, integral variables, or other factors restrain the use of traditional methods, based on derivatives. The solution of instances of well known economic equilibrium problems with these algorithms is supplied. The results obtained for these simple cases show potential applications of the strategies, and provide limited convergence evidence. 1 Overview Applications of game theory to the simulation of natural processes has been treated in 6], through an approach there called evolutionary game theory. In that text, the assumptions usually made in game theory that players will behave rationally, and according to some criterion of self-interest, are replaced; rational-ity is substituted by the criterion of population dynamics and stability, and self interest is replaced by Darwinian tness. The \solution" to a game is given by what is called an evolutionarily stable strategy | which is essentially the same as an unbeatable strategy, in game theory. The general approach of that book is to use game theory to model and explain evolutionary processes. The approach that we follow in this text is the inverse one: we aim at using the computer simulation of evolutionary processes to model and numerically solve some game theoretic problems. Evolutionary approaches have successfully been applied to solve a large variety of problems; in particular, successful applications to the numerical solution of global optimisation problems have been repeatedly reported. In this text we propose an evolutionary approach to numerically compute equilibria: systems of interdependent optimisation problems. The algorithms for equilibria computation presented here are deeply based on a method for the solution of global optimisation problems. In our implementation we have relied on an evolutionary algorithm described in 2], although equivalent methods could be used instead.