and Lipsey (1975) [@eaton1975principle] proved that the following are necessary and sufficient conditions for an equilibrium:

1.  Each peripheral server is paired. \[p1\]

2.  No server’s whole market is smaller than any other server’s half-market. \[p2\]

The proof these conditions are sufficient is a bit involved, but it is easy to see why they are necessary. If \[p1\] does not hold, a peripheral server would increase its profit by locating closer to its neighbor. \[p2\] follows from the fact that any server can obtain a market equal to any other server’s half-market by pairing with it.

By applying these equilibrium conditions, Eaton and Lipsey determined the equilibria of the game depending on the number of servers, as servers. We present their results in what follows.

*One Server.* The payoff of a single server $s_1$ is $p(s_1)=1$, wherever it is located.

*Two Servers.* As shown by Hotelling, there is a unique equilibrium, where the two servers are paired at $\loc_1=\loc_2=1/2$, with payoffs $p(s_1)=p(s_2)=1/2$. *Three Servers.* No equilibrium exists. It follows from the fact that \[p1\] and \[p2\] contradict one another in this case.

*Four Servers.* There is a unique equilibrium, with equal payoffs of $1/4$, where two servers are paired at 