In 1959, Nikaidô established a remarkable coincidence theorem in a compact Hausdorff topological space, which generalizes and gives a unified treatment to the results of Gale regarding the existence of economic equilibrium and theorems in game problems. The main purpose of the present paper is to deduce several generalized key results based on this very powerful result together with some KKM property. Indeed, we shall simplify and reformulate a few coincidence theorems on acyclic multifunctions as well as some Gòrniewicz-type fixed point theorems. Beyond the realm of monotonicity nor metrizability, the results derived here generalize and unify various earlier ones from classic optimization theory. In the sequel, we shall deduce two versions of Nikaidô’s coincidence theorem about Vietoris maps from different approaches.