For some 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 least 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. We begin by offering simple prrofs of several of the "impossibility" results. These proofs provide intuition into the properties of a solution concept which make it impossible to implement non-dictatorial social choice functions. This allows us to provide a characterization of solution concepts which lead to impossibility results, as well as two easily checked sufficient conditions.