which the location of a single server is inconsequential, in this setting the optimal location of a single server is at the center of the line, $\loc_1=1/2$, as we show next. Let $\loc_1\in [0,1]$ be the location of the server $s_1$. If no failure occurs, the payoff is 1. If the line is disconnected at $0<f<\loc_1$, then the payoff is $1-f$. If the line is disconnected at $\loc_1<f<1$, then the payoff is $f$. It follows that the payoff of $s_1$ is $$p_{{\mbox{\small \sc lf}}}(s_1)=\Expct[p(s_1)]=(1-r)\cdot 1+ r\cdot\left[\int_{0}^{\loc_1}(1-f)df+\int_{\loc_1}^{1}f df\right]
= 1-\frac r2+r\loc_1-r\loc_1^2~.$$

That is, $ p_{{\mbox{\small \sc lf}}}(s_1) $ is a function of $\loc_1$ that attains its maximum at $\loc_1=1/2$. Hence the only equilibrium is when the server is at the center of the market.

We next consider the *two server* variant. Without loss of generality let $\loc_1<\loc_2$, i.e., $s_1$ is located to the left of $s_2$. Observe that the payoff of $s_1$ is $p_{{\mbox{\small \sc lf}}}(s_1)=L_{{\mbox{\small \sc lf}}}(s_1)+R_{{\mbox{\small \sc lf}}}(s_1)$, where $L_{{\mbox{\small \sc lf}}}(s_1)$ is the length of its hinterland, and $R_{{\mbox{\small \sc lf}}}(s_1)$ is the length of its half-market on the right. (See Fig. \[fig:Hgame-2p\].)

-   **Calculating $\mathbf{L_{{\mbox{\small \sc lf}}}(s_1)}$:** if the line is disconnected at location $0<f<\loc_1$, then the 