have shown above, $s_i$ has the same payoff before moving in both games, and by moving $s_i$ can increase its payoff at least as much in the game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ as in the game $H(n,[a,b])$. Therefore, $s_i$ would also move in the game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$, contradicting the assumption. The inverse direction is similar. This concludes the proof that $(x_1,x_2,\ldots,x_n)$ is a Nash equilibrium in the game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ if and only if it is a Nash equilibrium in the game $H(n,[a,b])$.

Theorem \[thm:equiv\], in conjunction with conditions \[p1\] and \[p2\] for an equilibrium of the basic Hotelling game, yield the following conditions for an equilibrium of the Hotelling game with line disconnections.

The following conditions are sufficient and necessary for an equilibrium of the Hotelling game on the line with line disconnections.

1.  The peripheral servers are paired, and are located at an identical distance from the boundary, $x$. \[p1d\]

2.  No interior server’s whole market is smaller than $x- r \cdot x^2/2$. \[p2d\]

3.  No interior server’s half market is larger than $x- r \cdot x^2/2$. \[p3d\]

Suppose $(x_1 ,x_2 ,\ldots,x_n )$, for $x_1\leq x_2\leq\ldots\leq x_n$, is an equilibrium of $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$. Without loss of 