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Best response dynamics is one of the most explored iterative scheme to find Nash equilibria. In this presentation we investigate the global convergence of the affine relaxations of the best response algorithm in the class of two-player ratio-bounded games. Then, we present a numerical scheme (based on a direct-search method) in order to approach the Nash equilibrium of a ratio-bounded game and another one to approach the solution of a two-leader two-follower game involving a ratio-bounded game in the second stage. (with Maria Carmela Ceparano and Jacqueline Morgan).