In the paper under review, the author generalizes, in some aspects, the quasi fixed-point theorem due to \textit{I. Lefebvre} [Set-Valued Anal. 9, No. 3, 273--288 (2001; Zbl 0986.54051)] and proves the following Theorem. Let \(I\) and \(J\) be any index sets. For each \(i\in I\) and \(j\in J\), let \(X_i\) and \(Y_j\) be non-empty compact convex subsets of locally convex Hausdorff topological vector spaces \(E_i\) and \(F_j\), respectively. Let \(X:=\prod_{i\in I}X_i\), \(Y:=\prod_{j\in J}Y_j\), and \(Z:=X\times Y\). Let \(\phi_i:Z\to 2^{X_i}\) be a convex valued multimap with open fibers and a perfectly normal proper domain. Let \(\psi_j:Z\to 2^{Y_j}\) be a u.s.c.\ multimap with non-empty closed convex values. Then there exists a point \((\bar x,\bar y)\in Z\) such that for each \(i\in I\), either \(\phi_i(\bar x,\bar y)=\emptyset\) or \(\bar x_i\in \phi_i(\bar x,\bar y)\), and for each \(j\in J\), \(\bar y_j\in \psi_j(\bar x,\bar y)\). The main tool in the proof is the continuous selection theorem due to \textit{N.~C.\ Yannelis} and \textit{N.~D.\ Prabhakar} [J.\ Math.\ Econ.\ 12, 233--245 (1983; Zbl 0536.90019)] and the Fan--Glicksberg fixed point theorem. As a consequence of the above result, a new equlibrium existence theorem for a generalized quasi-game is obtained. The author gives one simple example of a generalized quasi-game with a finite number of agents where the previous known equilibrium existence theorems cannot be applicable.