Implementation, Elimination of Weakly Dominated Strategies and Evolotionary Dynamics

This paper is concerned with the realism of mechanisms that implement social choice functions in the traditional sense. Will agents actually play the equilibrium assumed by the analysis? As an example, we study convergence and stability prop-erties of Sjostrom's (1994) mechanism, on the assumption that boundedly rational players find their way to equilibrium using monotonic learning dynamics. This mechanism implements most social choice functions in economic environments using as a solution concept the iterated elimination of weakly dominated strategies (only one round of deletion of weakly dominated strategies is needed). There are, how-ever, many sets of Nash equilibria whose payoffs may be very different from those desired by the social choice function. We show that many equilibria in all the sets of equilibria we have described are the limit points of trajectories that have completely mixed initial conditions. The initial conditions that lead to these equilibria need not be very close to the limiting point. Furthermore, even if the dynamics converge to the "right" set of equilibria, it still can converge to quite a poor outcome in welfare terms.