two servers are separated by a distance of $\varepsilon$ they are said to be *paired*. We say two servers $s_i$ and $s_j$ are *paired at location* $\loc$ if $\loc_i=\loc-\varepsilon/2$ and $\loc_j=\loc+\varepsilon/2$; with a slight abuse of notation, we hereafter denote this as $\loc_i=\loc_j=\loc$. Conversely, a server is *isolated* if it is not paired to another server.

Each server $s_i$ divides its market into two sides, referred to as *half-markets*. We denote by $L(s_i)$ and $R(s_i)$ the lengths of $s_i$’s left and right half-markets respectively. Therefore, the payoff of $s_i$ is $p(s_i)=L(s_i)+R(s_i)$. $$p(s_i)=L(s_i)+R(s_i).$$ Two servers are said to be *neighbors* if no server is located between them. A server that has neighbors on both sides is called an *interior server*. A server that has one neighbor is called a *peripheral server*. That is, the two peripheral servers are the server closest to 0 and the server closest to 1. (See Fig. \[fig:Hgame-defs\].)

![A possible configuration of the game. Servers $s_1$ and $s_n$ are the peripheral servers, $s_1$ and $s_2$ are paired, and the half-markets of $s_3$ are marked in the figure.[]{data-label="fig:Hgame-defs"}](figs/Hgame-defs){width="\textwidth"}

By definition, the market of an interior server extends half the distance to its two neighbors. The length of an 