pair with $s_2$ on both sides leaving it with 0 payoff, and thus $s_2$ would move. If, on the other hand, $s_2$ is located at $\loc_2>1/(2r)$, then $s_1$ would locate at $1/(2r)$ and $s_3$ would pair with $s_2$. But then $s_3$ could improve its payoff and would thus move. Due to symmetry, a similar argument holds if we suppose $s_2$ is located at $\loc_2<1-1/(2r)$.

Next, consider the *four server* game $H_{{\mbox{\small \sc lf}}}(4,r,[0,1])$. Let us consider the strategy profile where the location $0<x<1$ satisfies that when $s_1$ and $s_2$ are paired at $x$ and $s_3$ and $s_4$ are paired at $1-x$ all four servers receive the same expected payoff (any server can pair with any other server and get his payoff). Since $s_1$ and $s_2$ are symmetric to $s_4$ and $s_3$ respectively it suffices to compare the expected payoffs of $s_1$ and $s_2$.

The following table compares the payoffs $p(s_1)$ and $p(s_2)$ according to possible values of $f$, as well as with the failure-free case.

      $f$       $\Prb[f]$   $ s_1 $    $ s_2 $
  ------------ ----------- --------- ------------
   no failure     $1-r$       $x$     $1/2 -x $
    $[0,x]$       $rx$      $ x/2 $   $ 1/2 -x $
   $[x,1-x]$    $r(1-2x)$     $x$     $ 1/2 -x $
   