Each peripheral pair is separated from the closest server by a distance twice as long as $x- r \cdot x^2/2$. The interior servers are paired or isolated such that no server’s whole market is smaller than $x- r \cdot x^2/2$, and no server’s half-market is larger than $x- r \cdot x^2/2$.

\[thm:disconnect\] In the $n$-server Hotelling game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ with a single line disconnection occurring with probability $0<r<1$ at a location chosen uniformly at random:

1.  When there is a single server, a Nash equilibrium exists for which the server is located at the center of the line, i.e., $\loc=1/2$.

2.  When two servers compete, there exists an equilibrium for which they are paired at the center of the line, i.e., $\loc_1=\loc_2=1/2$.

3.  If there are three servers, then no equilibrium exists.

4.  With four servers, there exists an equilibrium for which the servers are organized in two pairs, at $$\loc_1=\loc_2=\frac{2-\sqrt{4-r}}{r}~,
            \mbox{\hskip 1cm}
            \loc_3=\loc_4=1-\frac{2-\sqrt{4-r}}{r}~.$$

5.  If $n=5$, then there exists an equilibrium for which four of the servers are paired at $$\loc_1=\loc_2=\frac{3-\sqrt{9-4r}}{2r}~,
            \mbox{\hskip 1cm}
            \loc_4=\loc_5=1-\frac{3-\sqrt{9-4r}}{2r}~,$$ and an isolated server is located at $\loc_3=1/2$.

6.  When more than five servers occupy the line, there exist infinitely many equilibria, characterized as 