# Hotelling’s Continuous Model - game theory

Hotelling’s Continuous Model: Consider Hotelling’s model, in which citizens are a continuum of voters on the interval $$A = [−a, a]$$, with uniform distribution $$U (a)$$.

Show that for a general distribution $$F (\cdot )$$ over $$[−a, a]$$ the unique Nash equilibrium is where each candidate chooses the policy associated with the median voter.

I know that the median is $$a^m=0$$ but I don't understand what does a general distribution mean and what does "associated with median voter" means exactly. If somebody could clarify what this exercise asks me to do that would be great!

Pick an arbitrary cdf $$F$$ that is supported on $$[-a,a]$$. The median, $$m$$, must satisfy $$\int_{-a}^{m}dF \geq \frac{1}{2}, \quad \text{and} \quad \int_{m}^{a}dF \geq \frac{1}{2}$$
Note that $$m$$ is not generally $$0$$.
The phrase you mention, "the unique NE is where both candidates chooses the policy associated with the median voter," simply means that the unique NE is where both candidates choose a policy located on $$m$$.