In this paper I investigate the properties of social welfare functions defined on domains where the preferences of one agent remain fixed. Such a domain is a degenerate case of those investigated, and proved Arrow consistent, by Sakai and Shimoji :435–445, 2006). Thus, they admit functions from them to a social preference that satisfy Arrow’s conditions of Weak Pareto, Independence of Irrelevant Alternatives, and Non-dictatorship. However, I prove that according to any function that satisfies these conditions on such a domain, (...) for any triple of alternatives, if the agent with the fixed preferences does not determine the social preference on any pair of them, then some other agent determines the social preference on the entire triple. (shrink)