A fundamental structure of strategy-proof social choice correspondences with restricted preferences over alternatives

I prove that under each strategy-proof and unanimous social choice correspondence, there is at least one agent who is decisive. Because the result is established on a weak requirement on preferences over sets, the existence of a decisive agent is an underlying feature of most strategy-proof and unanimous social choice correspondences. Moreover, I consider a restriction on the space of preferences over alternatives. I prove that circular sets of preferences over alternatives are sufficient for the existence of a decisive agent.