= Solution
The players are the two chains, each with the <set> of <pure strategies> $\{1,2,3,4,5\}$. A <strategy profile> $(x,y)$ gives the sites of $S$ and $C$. For later use, write the first player's <payoff> for a general street size as
$$
u(x,y)=\begin{cases}x+y-1,&x<y,\\k,&x=y,\\2k-x-y+1,&x>y.\end{cases}
$$
When $x<y$, count customers to the left of $(x+y)/2$, assigning value one rather than two to any midpoint customer. This gives $x+y-1$ whether the midpoint is an occupied building or lies between buildings. The other player receives $2k-u(x,y)$. At co-location, every customer's value is split, giving $k$ each.
For $k=5$, the <payoff matrix> for $S$ is
$$
M=\begin{pmatrix}
5&2&3&4&5\\
8&5&4&5&6\\
7&6&5&6&7\\
6&5&4&5&8\\
5&4&3&2&5
\end{pmatrix}.
$$
Rows and columns are indexed by sites $1,\ldots,5$; the full ordered <payoff> pair in cell $(x,y)$ is \b[$(M_{xy},10-M_{xy})$]. This <discrete location game> is a <constant-sum game>.
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