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This paper studies convergence and stability properties of T. Sjéistréim's (1994, Games Econom. Behav. 6,502-511) mechanism, under the assumption that bound-edly rational players find their way to equilibrium using monotonic evolutionary dynamics and best-reply dynamics. This mechanism implements most social choice functions in economic environments using as a solution concept one round of deletion of weakly dominated strategies and one round of deletion of strictly dominated strategies. However, there are other sets of Nash equilibria, whose payoffs may be very different from those desired by the social choice function. With monotonic dynamics, all these sets of equilibria contain limit points of the evolutionary dynamics. Furthermore, even if the dynamics converge to the "right" 1 The authors are grateful to Ted Bergstrom, Ken Binmore, Sandro Brusco, Tilman Béirgers, Leo Hurwicz, and Karl Schlag for stimulating comments, as well as suggestions by an anonymous referee on earlier versions of this paper. The usual disclaimers apply. Both authors gratefully acknowledge financial support from the European Commission, under the "Human Capital Mobility " programo Financial support from the Spain's Ministry of Educa-tion, under Grant PB96-0302 (Cabrales) and from the Instituto Valenciano de Investigaciones Economicas (Ponti) is also gratefully acknowledged. This paper was written while Cabrales was visiting the Centre for Economic Learning and Social Evolution (ELSE) at the University College London. He would like to thank ELSE for hospitality and encouragement