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6

No, there is not. Consider a game with two players, Ann and Bob. Both choose such vectors with entries $0$ or $1$ of the form $(a_1,a_2,\ldots,a_J)$ or $(b_1,b_2,\ldots,b_J)$, respectively. If $\sum_{i=1}^J a_i+b_i$ is odd, Ann wins and Bob loses. If the number is even, Ann loses and Bob wins. Clearly, one of them loses in every profile of pure strategies, ...

2

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 ...

1

Think about an auction, where the designer is selling good and trying to sell it to the person that values it the most while collecting as much revenue as possible. A direct mechanism means that the seller asks buyers for how much they value the good and based on that decides who gets the good and how much they pay. Suppose that the designer uses a second-...

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