Based on the paper "On the Existence of Pure Nash Equilibria of Finite Games in Strategic Form; the Hotelling Bi-matrix game" (March 6, 2014) by Pierre van Mouche (Wageningen University) and Willem Pijnappel (Tegelen).
We study the Nash equilibrium problem in pure strategies for a specific finite game in strategic form, referred to as the Hotelling bi-matrix game. General results that guarantee the existence of an equilibrium of this game (for all parametric values) do not seem to exist. However, we are able to prove equilibrium existence by determining (by hand) the equilibrium set. In addition we conjecture that, with R the best reply correspondence, R2 has an increasing singleton-valued selection, which if true automatically guarantees equilibrium existence.