= Discrete location game
A <discrete location game> can model firms choosing sites from $\{1,\ldots,k\}$ 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 $y$ is
$$
u(x,y)=\begin{cases}x+y-1,&x<y,\\k,&x=y,\\2k-x-y+1,&x>y.\end{cases}
$$
For $x<y$, 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.
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