equilibrium for every $x$ such that $\varepsilon \leq (1-3x)-3x \leq 2x$, $$\varepsilon \leq (1-3x)-3x \leq 2x~,$$ or rather $1/8 \leq x < 1/6$. (See Figure \[fig:Hgame-6p\].)

![Equilibrium with 6 servers.[]{data-label="fig:Hgame-6p"}](figs/Hgame-6p){width="\textwidth"}

Hotelling Game on the Line with Link Failures {#sec:disconnect}
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Let us now consider the game with environmental failures. For concreteness, let us assume that the only possible failure is a disconnection of the line at point $f$, chosen uniformly from $[0,1]$, which severs the line into two separate markets, and forces some of the clients (specifically, those disconnected from their originally chosen server) to change their server selection. For simplicity, assume at most one disconnection may occur, with constant probability $0<r<1$, at a location $f\in[0,1]$ chosen uniformly at random. We call this game the *Line Failure Hotelling game* and denote it as $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$. The new payoff function is denoted as $p_{{\mbox{\small \sc lf}}}$ and becomes the expected profit under these assumptions (i.e., the payoff of player $s_i$ is $1-r$ times its payoff in the fault-free case plus $r$ times its expected payoff in case a disconnection occurred at point $0<f<1$).

Let us begin by considering this game with only *one server*. Unlike the basic Hotelling game, in 