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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 ...


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