Past exam of the computer science course of the University of Oxford / 2019 / Computational game theory / 1 / f

There is no pure Nash equilibrium. If the firms choose different sites, $S$ serves at least its own building, so $C$ receives less than $2k$. By moving onto $S$'s site, $C$ makes every customer equidistant from the two shops and wins every tie, obtaining $2k$. Thus any separated profile has a profitable unilateral deviation by $C$.

At any co-located profile, $S$ receives zero. It can move to a different site and capture at least that site's customers, obtaining at least two. Thus every co-located profile also has a profitable unilateral deviation. These cases exhaust all pure strategy profiles.