On two-player games with pure strategies on intervals $ [a, \; b] $ and comparisons with the two-player, two-strategy matrix case

We consider games of two-players with utility functions which are not necessarily linear on the product of convex and compact intervals of \begin{document}$ \mathcal{R}^2 $\end{document}. An issue is how far an analogy can be drawn with two-player, two-strategy matrix games with linear utility functions, where [0, 1] registers probabilities and equilibria are at the intersection of reaction functions. Now, the idea of \begin{document}$ \delta $\end{document} functions is exploited to construct mixed strategies to look for Nash equilibria (NE). "Reaction" functions are constructed and results are obtained graphically. They are related to topological theorems on NE. The games chosen make specific points in relation to existence conditions and properties of solutions. It is a distinguishing feature that an interval [a, b] now registers both pure and mixed strategies. For NE a choice has to be justified. Also "reaction" functions are more complicated and their intersection does not guarantee an equilibrium.