the following theorem. (Hereafter, proofs are deferred to the full paper.) Let $a=r x_1^2/2$ and $b=1-r x_n^2/2$, hence $$[a,b] ~=~ \left[\frac{r \cdot x_1^2}2,1-\frac{r \cdot (1-x_n)^2}2\right]~.$$

\[thm:equiv\] Let $0\leq x_1\leq x_2\leq\ldots\leq x_n\leq 1$. The configuration $(x_1,x_2,\ldots,x_n)$ is a Nash equilibrium of the game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ if and only if it is a Nash equilibrium of the game $H(n,[a,b])$.

![Illustration of Theorem \[thm:equiv\]. The top line refers to the configuration in the game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ and the bottom line refers to the configuration in the game $H(n,\left[\frac{r \cdot x_1^2}2,1-\frac{r \cdot (1-x_n)^2}2\right])$ . []{data-label="fig:Hgame-np-Disconnections"}](figs/Hgame-np-Disconnections){width="\textwidth"}

Suppose that the servers $s_1,s_2,\ldots, s_n$ are located at $x_1,x_2,\ldots,x_n$ respectively. Note first that both peripheral servers $s_1$ and $s_n$ must be paired with their respective neighbors. In the forward direction of the claim this holds by Observation \[obs:disconnect\]; in the inverse direction this follows from condition \[p1\].

We start by showing that the payoff of each server given the configuration $(x_1,x_2,\ldots,x_n)$ is the same in both games $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ and $H(n,[a,b])$, i.e., for each $s_i$, $p_{{\mbox{\small \sc lf}}}(s_i)=p(s_i)$. Consider an interior server $s_i$, for $2\leq i\leq n-1$. In the game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$, its payoff is $$p_{{\mbox{\small \sc lf}}}(s_i)=\frac{x_{i+1}-x_{i-1}}2~,$$ by Observation \[obs:disconnect\]. In the game 