Two shopowners must decide the location of their respective shops along an interval $[0,1]$. Player 1 sells cheap sports equipment, whereas Player 2 deals in elegant sports gear. As side-by-side comparison is, on average, favorable to the merchant of cheap equipment, the players face an inelastic demand; Player 1 wants to locate as close as possible to Player 2, whereas Player 2 tries to move as far as possible from Player 1. We assume that the profit functions take the following form.
$$
\begin{array}{rlrl}
\mathrm{x}_1=\mathrm{x}_2=[0,1] & & \\
\mathrm{u}_1\left(\mathrm{x}_1, \mathrm{x}_2\right) & =1-\left|\mathrm{x}_1-\mathrm{x}_2\right| & & \\
\mathrm{u}_2\left(\mathrm{x}_1, \mathrm{x}_2\right) & =\left|\mathrm{x}_1-\mathrm{x}_2\right| & & \text { if }\left|\mathrm{x}_1-\mathrm{x}_2\right| \leq \frac{2}{3} \\
& =\frac{2}{3} & & \text { if }\left|x_1-\mathrm{x}_2\right| \geq \frac{2}{3}
\end{array}
$$
Notice the negative externalities imposed by Player 1 on Player 2 vanish when their distance is at least $2 / 3$.
The pure-strategy game has no Nash equilibrium. Show that Glicksberg's theorem implies the existence of a mixed Nash equilibrium. Check that the following pair is a Nash equilibrium.
$$
\begin{aligned}
& \mu_1^*=\frac{1}{3} \delta_0+\frac{1}{6} \delta_{\frac{1}{3}}+\frac{1}{6} \delta_{\frac{2}{3}}+\frac{1}{3} \delta_1 \\
& \mu_2^*=\frac{1}{2} \delta_0+\frac{1}{2} \delta_1
\end{aligned}
$$
Neither of these two mixed strategies are completely mixed.
However $\left(\mu_1^*, \mu_2^*\right)$ share the typical property of completely mixed equilibria, namely,
$$
\begin{array}{ll}
\overline{\mathrm{u}}_1\left(\mu_1, \mu *_2\right)=\overline{\mathrm{u}}_1\left(\mu_1^*, \mu_2^*\right) & \text { al1 } \mu_1 \in \mathrm{M}_1 \\
\overline{\mathrm{u}}_2\left(\mu{ }_1^*, \mu_2\right)=\overline{\mathrm{u}}_2\left(\mu_1^*, \mu_2^*\right) & \text { al1 } \mu_2 \in \mathrm{M}_2
\end{array}
$$