interior server’s market is thus half the distance between its neighbors, wherever the server is located between those neighbors. The line segment between the boundary and the corresponding peripheral server is called a *hinterland*. The market of a peripheral server includes its hinterland in one direction, and extends half way to its neighbor in the other direction.

The game is in a *Nash equilibrium* if no server can increase its payoff by moving to a location other than its present location.

As an example, consider the basic Hotelling game which is played on the line $[0,1]$, and involves four servers, $s_i$ for $i=1,2,3,4$, who need to place themselves at different points on the line, $\loc_i$ for $i=1,2,3,4$. Consider a solution in which servers $s_1$ and $s_2$ place themselves at the point $1/4$, with $s_1$ to the left of $s_2$, and servers $s_3$ and $s_4$ place themselves at the point $3/4$, with $s_3$ to the left of $s_4$. (See Figure \[fig:Hgame\].)

![Equilibrium with 4 servers.[]{data-label="fig:Hgame"}](figs/Hgame-4p){width="\textwidth"}

It is easy to verify that this solution of the game, whose payoff is $1/4$ per server, is a Nash equilibrium (i.e., none of the servers is motivated to change its location).

#### Known Results.

Eaton 