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\begin{document}

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\centerline{Math 36 --- Weekly Homework}

\centerline{Assigned: 9/17} 

\centerline{Due: 9/24}
  

\begin{questions}

\question 
Consider the following collection of votes:

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
2 & 6 & 3 & 4 \\
\hline 
\hline
A & E & C & D \\
\hline
D & B & A & C \\
\hline
E & A & B & B \\
\hline
B & C & D & E \\
\hline
C & D & E & A \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}[label = (\alph*)]
\item Who is the plurality winner?

\item Is there a Condorcet winner? If so, who?

\item Draw the preference graph for these votes.

\item Find an elimination tournament in which A is the winner.

\item Find a set of Borda weights to make B the winner.
\end{enumerate}

\vspace{2 in}
\question 
Find vote counts so that the plurality, Borda, and Condorcet winners are all different. (Use standard Borda weights and ensure that a Condorcet winner exists.) Be sure to say which candidate wins in each voting method.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& & & & & \\
\hline
\hline
A & A & B & B & C & C \\
\hline
B & C & A & C & A & B \\
\hline
C & B & C & A & B & A \\
\hline

\end{tabular}
\end{center}

\vspace{2 in}
\question 
Suppose that instead of a line, our 1D spatial model of voting was on a circle. How do you define distance between two points? What are the implications of this modeling choice? Is this a reasonable model? Why or why not?

\vspace{2 in}
\question
Describe what you think the probability density distribution of the US population in the 1D liberal-conservative model looks like. How many peaks does it have? Are there lots of moderates or extremists? It may useful to draw a rough sketch. Cite at least one reputable source.

You don't need to give an exact equation for the distribution, just describe it qualitatively. 





\end{questions}


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