neighbor, such a move reduces $s_i$’s payoff.

Second, consider a move to an interior interval between two new neighbors $s_j$ and $s_{j+1}$. In both games, the payoff is equal to half the length of the interval, and thus the move yields the same payoff in both games.

Finally, consider a move to either one of the hinterlands. In both games, the best a server $s_i$ could do is pair with the peripheral server, in which case $s_i$ gets the payoff of this peripheral server. But, as we have shown above, the payoff of each peripheral server is the same in both games $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ and $H(n,[a,b])$, and thus $s_i$ obtains the same payoff by making this move in both game. This proves that for each move $s_i$ may make in one of the games $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ and $H(n,[a,b])$, there exists a move in the other game for which $s_i$ gets greater or equal payoff.

Assume towards contradiction that $(x_1,x_2,\ldots,x_n)$ is a Nash equilibrium in the game $H_{{\mbox{\small \sc lf}}}(n,r,[0,1])$ and but not in the game $H(n,[a,b])$. It follows that there exists a server $s_i$ that can improve his payoff by moving in the game $H(n,[a,b])$. But, as we 