(See Fig. \[fig:Hgame-2p-max\].)

![$s_1$ is located such that its payoff is maximal, and moving closer to $s_2$ would reduce $s_1$’s payoff. However, $s_2$ would maximize its payoff by moving to the left of $s_1$.[]{data-label="fig:Hgame-2p-max"}](figs/Hgame-2p-Disconnections-Equil){width="\textwidth"}

Consequently, As a result of the above calculations, for the general game, with $n\geq 3$ servers, we have the following.

\[obs:disconnect\] In the Hotelling game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$, on the line with edge disconnections, the following holds:

1.  A peripheral server located at distance $\loc$ from the boundary has a hinterland with an expected length of $\loc - r\loc^2/2$. $$\loc - \frac{r\cdot \loc^2}2~.$$

2.  Two neighboring servers at distance $x$ gain a half-market of expected length $x/2$ each in the direction of the other (as in the basic Hotelling game).

3.  Each peripheral server would increase his hinterland up to $1/(2r)$. That is, condition \[p1\] holds unless the neighbor of a peripheral server is at a distance of more than $1/(2r)$ from the boundary.

In light of the observation above, if the game is played with *three servers*, then no equilibrium exists. To see why, observe that if the interior server $s_2$ is located between $1-1/(2r)$ and $1/(2r)$, then the peripheral servers $s_1$ and $s_3$ would 