condition \[p2d\], the whole market of each interior server $s_i$, for $2\leq i\leq n$ is at least $x- r \cdot x^2/2$, and by condition \[p3d\] the half market of each interior server $s_j$ is at most $x- r \cdot x^2/2$, for $2\leq j\leq n$. By transitivity, it follows that the whole market of $s_i$ is at least the size of the half-market of $s_j$. This proves that condition \[p2\] holds.

It follows that $(x_1 ,x_2 ,\ldots,x_n )$ is an equilibrium of the game $H(n,[a,b])$, and thus, by Theorem \[thm:equiv\], $(x_1 ,x_2 ,\ldots,x_n )$ is an equilibrium of the game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ as well. This concludes the proof of the Corollary.

To summarize, this section established the following theorem.

\[thm:disconnect\] In the $n$-server Hotelling game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$

1.  For $n=1$, a Nash equilibrium exists with $\loc=1/2$.

2.  For $n=2$, there exists an equilibrium with $\loc_1=\loc_2=1/2$.

3.  For $n=3$, no equilibrium exists.

4.  For $n=4$, there exists an equilibrium with $\loc_1=\loc_2=(2-\sqrt{4-r})/r$, $\loc_3=\loc_4=1-(2-\sqrt{4-r})/r$.

5.  For $n=5$, there exists an equilibrium with $\loc_1=\loc_2=(3-\sqrt{9-4r})/(2r)$, $\loc_4=\loc_5=1-(3-\sqrt{9-4r})/(2r)$ and $\loc_3=1/2$.

6.  For $n\ge 6$, there exist infinitely many equilibria, characterized as follows: peripheral servers are paired with their neighbor and have identical hinterlands of length $x$. 