 Discrete location game

A discrete location game can model firms choosing sites from $\{1,\dots ,k\}$ while customers at those sites choose a nearest firm. The customer masses, distance rule, permission to share a location and rule for ties are part of the model. For two firms, customer value two per site and equal splitting of ties, the payoff to a firm at $x$ against one at $y$ isFor $x<y$, counting sites strictly left of the midpoint and assigning half of an equidistant site's value gives $x+y-1$, whether $x+y$ is even or odd. The other firm receives the remainder, so this is a constant-sum game with total $2k$. Changing tie priority can fundamentally change its Nash equilibria, especially when co-location is permitted.