COMPUTATION OF THE CLOSED-LOOP STACKELBERG SOLUTION USING THE GENETIC ALGORITHM Tamer Ba¸ sar *,1 Thomas Vall´ ee ** * Coordinated Science Laboratory, University of Illinois, 1308 West Main Street, Urbana, IL 61801, USA. Email: tbasar@decision.csl.uiuc.edu ** LEN-C3E, Economics Department, Universit´ e de Nantes, Chemin de la Censive du Tertre, 44322 Nantes Cedex 3, France. Email: vallee@sc-eco.univ-nantes.fr Abstract: This paper deals with the computation of the closed-loop dynamic Stackel- berg equilibrium solution in two-player nonzero-sum dynamic games using the genetic algorithm. When the leader has access to closed-loop state information, which provides him (indirectly) with on-line information on the past actions of the follower, derivation of the Stackelberg solution is known to be a challenging one. We address here the question of whether in this context genetic algorithm techniques or their appropriately modified versions can be used as computational tools. Following demonstrations using analytic derivations in general linear quadratic games, the paper applies the tool to a nonlinear dynamic taxation problem. Keywords: Computational methods, Stackelberg games, genetic algorithms. 1. INTRODUCTION This paper deals with the computation of the closed-loop dynamic Stackelberg equilibrium so- lution in two-player nonzero-sum dynamic games using heuristic search algorithms, and in par- ticular the genetic algorithm (Goldberg, 1989). As is well known (Ba¸ sar and Olsder, 1995), this equilibrium solution models hierarchical decision scenarios where one of the players, leader, imposes his policy on the other player, follower, who is taken as a rational optimizer. When the leader has access to closed-loop state information, which provides him (indirectly) with on-line information on the past actions of the follower, derivation of the Stackelberg solution is known to be a challeng- ing one. It has in fact remained an open problem for a long time, even in a deterministic framework, and was resolved using an indirect approach that involves a particular representation of the team- 1 Research supported in part by the National Science Foundation under Grant ECS 93-12807. optimal solution of a single-criterion problem that uses the leader’s cost function, which has also strong connections with incentive design problems (Ba¸ sar and Selbuz, 1979; Ba¸ sar, 1984; Tolwin- ski, 1981; Zheng and Ba¸ sar, 1982; Ho et al., 1982). Despite the development of this indirect approach, the computation of the closed-loop Stackelberg solution still creates formidable difficulties, espe- cially for problems with nonlinear dynamics and for those where the leader does not have perfect knowledge of the cost function of the follower. Recently, heuristic search methods, such as the genetic algorithm (GA), have been proposed and used successfully to solve optimal control problems (Krishnakumar and Goldberg, 1992; Michalewicz et al., 1992), or to find the open- loop Nash equilibrium solution in dynamic games ( ¨ Ozyildirim, 1997). The question remains as to whether these tools or their appropriately mod- ified versions can be used to compute the closed- loop dynamic Stackelberg equilibrium solution. We address this question in this paper, and