The paper presents a new characterisation result of generalized Condorcet Winner (CW) mechanisms based upon Nash's and Arrow's versions of the Independence of Irrelevant Alternatives. (Theorem 1). In the definition of a CW mechanism a certain number of fixed ballots (one less than the number of agents) are the only endogenous parameters. Theorem 2 shows how these parameters may be chosen when the size of society varies. Next (Theorem 3), the domain of definition of individual preferences is extended to cover single plateau preferences and once more, the CW mechanisms are characterised by the two axioms of independence of irrelevant alternatives. Finally (Theorem 4) shows that no further extension of the preference domain is compatible with both the axioms.