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.

For $k\ge 3$, suppose two distinct occupied sites are $\ell <r$, a final firm must choose a different site, customers have value two per site and nearest-firm ties are split equally. A final firm left of both incumbents maximizes its payoff uniquely at $\ell -1$, when that site exists, obtaining $2(\ell -1)$. To the right it maximizes uniquely at $r+1$, obtaining $2(k-r)$. At every available interior site it obtains the same payoff $r-\ell$. Thus all best responses are obtained by comparing these three values, omitting regions with no available site and retaining every tied maximizing choice.

More generally, three distinct locations $a<b<c$ give their left, middle and right firms payoffs $a+b-1$, $c-a$ and $2k-b-c+1$. Counting the two midpoint boundaries proves these formulas; the middle firm's demand is the difference between the cumulative demands at the two boundaries. The formulas retain all even/odd midpoint ties automatically.