# 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!

## 1 Answer

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 $$m$$.

This is easy to show, think of "unraveling toward the center."