For some game theoretic solution concepts, such as dominant strategies, Nash equilibrium, and undominated strategies, only dictatorial social choice functions are implementable on a full domain of preferences with at lest three alternatives. For other solution concepts, such as the iterative removal of weakly dominated strategies, undominated Nash equilibrium, and maximin, it is possible to implement non-dictatorial social choice functions. Which characteristics of solution concepts account for these differences? We begin by proving a new impossibility theorem. This theorem shows that conditions which are significantly weaker than strategy-proofness, assure that a social choice function is dictatorial on a full domain of preferences. This helps us to identify the essential parts of the impossibility theorems and leads to two characterization of solution concepts which lead to impossibility results. Keywords: Implementation, Social Choice.