$H(n,[a,b])$, its payoff is also $$p(s_i)=\frac{x_{i+1}-x_{i-1}}2~.$$

Consider the payoff of the peripheral server $s_1$. The servers $s_1$ and $s_2$ are paired, and thus, by Observation \[obs:disconnect\], the payoff of $s_1$ in the game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ is $$p_{{\mbox{\small \sc lf}}}(s_1)=x_1-\frac{r \cdot x_1^2}2~.$$ In the game $H(n,[a,b])$, the left border of the line is at $a=\frac{r \cdot x_1^2}2$ and thus the payoff of $s_1$ is also $$p(s_1)=x_1-\frac{r \cdot x_1^2}2~.$$ Similarly, the server $s_n$ gets the same payoff in both games. This proves that, in the given configuration $(x_1,x_2,\ldots,x_n)$, each server gets the same payoff in both games $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ and $H(n,[a,b])$.

We next show that for each move available to the server $s_i$, for $1\leq i \leq n$, in one of the two games $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ and $H(n,[a,b])$, there exists a corresponding move in the other game for which $s_i$ obtains a greater or equal payoff.

Consider first a move within the interval defined by $s_i$’s present neighbors. If $s_i$ is an interior server then in both games $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ and $H(n,[a,b])$ such a move does not affect $s_i$’s payoff, as we have previously shown. If $s_i$ is a peripheral server, since it must be paired with its 