The present paper continues similar results of the authors and introduces the notions of \(Q_\theta\)-majorant of \(\varnothing\) and \(Q_\theta\)-majorized correspondence in order to generalize the lower semi-continuous correspondences which are irreflexive and have open convex values. The aim of this paper is to prove an existence theorem of maximal elements for the new type of preference correspondence which are \(Q_\theta\)-majorized. A new existence theorem of equilibrium for an abstract economy and quantitative game is proved under the conditions that the intersection of constraint and preference correspondence are \(Q_\theta\)-majorized for any (countable and uncountable) set of agents in Hausdorff locally convex topological vector spaces.