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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!

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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."

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