Use the following information to answer Questions
Consider the following distribution of 34 voters at eight different positions over the interval [0, 1]. $$\begin{array} { | l | l | l | l | l | l | l | l | l | }
\hline \text { Location } & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \
\hline \begin{array} { l }
\text { Number of } \
\text { voters }
\end{array} & 2 & 3 & 5 & 7 & 7 & 5 & 3 & 2 \
\hline
\end{array}$$
-Candidate A announces a policy position of 0.3 and Candidate B announces a policy position of 0.8. Is there a 2/3-separation opportunity for a third Candidate C to enter the race?

Question

Use the following information to answer Questions
Consider the following distribution of 34 voters at eight different positions over the interval [0, 1]. $$\begin{array} { | l | l | l | l | l | l | l | l | l | } 
\hline \text { Location } & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \
\hline \begin{array} { l } 
\text { Number of } \
\text { voters }
\end{array} & 2 & 3 & 5 & 7 & 7 & 5 & 3 & 2 \
\hline
\end{array}$$
-Candidate A announces a policy position of 0.3 and Candidate B announces a policy position of 0.8. Is there a 2/3-separation opportunity for a third Candidate C to enter the race?

Quizplus · Accepted Answer

A 2/3-separation opportunity exists if there is a location x such that at least 2/3 of voters prefer candidates whose policy positions are strictly less than x, and at least 2/3 of voters prefer candidates whose policy positions are strictly greater than x.

For Candidate A's position of 0.3, the proportion of voters with policy positions strictly less than 0.3 is 2/34, which is less than 2/3. Similarly, for Candidate B's position of 0.8, the proportion of voters with policy positions strictly greater than 0.8 is 2/34, which is also less than 2/3. Therefore, there is no 2/3-separation opportunity for a third candidate to enter the race.

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