We investigate the implications of relaxing Arrow's independence of irrelevant alternatives axiom while retaining transitivity and the Pareto condition. Even a small relaxation opens a floodgate of possibilities for nondictatorial and efficient social choice.

For social choice rules that can select either one or two alternatives, strategy-proofness is incompatible with a weak Condorcet principle.

A social welfare function is Arrovian if it is transitive-valued and satisfies IIA. We examine the logical relation between the statements (A) If f is Arrovian with domain F then it is dictatorial if it satisfies the Pareto criterion and (W) If f is Arrovian with domain F then it is dictatorial...