 Solution

Neither player has a weakly dominant strategy among its pure strategies. Against an opponent at site one, the unique best response is site two, yielding $2k-2$. Co-location yields only $k$, and moving further right reduces one's payoff. Against an opponent at site $k$, the unique best response is $k-1$, by reflection. For $k\ge 4$, these two sites differ. No single pure strategy can therefore be a best response to both opposing actions, as a weakly dominant strategy would require.