generality, let the server $s_i$ be located at $x_i$, for $1\leq i\leq n$. By Theorem \[thm:equiv\], it follows that $(x_1 ,x_2 ,\ldots,x_n )$ is an equilibrium of $H(n,[a,b])$, where $a=r x_1^2/2$ and $b=1-r x_n^2/2$ as above. Therefore, conditions \[p1\] and \[p2\] must hold for this configuration in $H(n,[a,b])$.

By condition \[p1\], the servers $s_1$ and $s_2$ are paired and the servers $s_{n-1}$ and $s_n$ are paired. By condition \[p2\], the hinterlands are equal in length, so $$x_1-\frac{r \cdot x_1^2}2=(1-x_n)-\frac{r \cdot (1-x_n)^2}2~,$$ which yields $$x_1=(1-x_n)~.$$ This proves condition \[p1d\] holds.

By condition \[p2\], no interior server’s whole market is smaller than any other server’s half market, including the hinterland. This proves condition \[p2d\] holds.

By condition \[p2\], no interior server’s half-market is larger than any other server’s half market, including the hinterland. This proves condition \[p3d\] holds.

Conversely, suppose the configuration $(x_1 ,x_2 ,\ldots,x_n )$ satisfies conditions \[p1d\],\[p2d\] and \[p3d\], for $x_1\leq x_2\leq\ldots\leq x_n$, and without loss of generality, the server $s_i$ is located at $x_i$, for $1\leq i\leq n$. We show that $(x_1 ,x_2 ,\ldots,x_n )$ is an equilibrium in the game $H(n,[a,b])$.

Given condition \[p1d\], condition \[p1\] follows immediately in the game $H(n,[a,b])$. Furthermore, it holds that $$x_1=(1-x_n)=x~.$$ By 