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\begin{document}
\title{Math 301: Homework 1}
\author{Mary Radcliffe}
\date{due 7 May 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}
\item Complete problems 9, 10, 11 from Miscellany

\item In a voting matrix, a set of candidates $\{B_1, B_2, \dots, B_k\}$ are called {\it clones} if every voter ranks these candidates in some order, with no other candidates between them. For example, in the following matrix, candidates $C$ and $D$ are clones:

\[\begin{array}{cccc}5 & 3 & 2\\ \hline A & B & C\\ C & D & D\\ D & C & B\\ B & A & A\end{array}.\]

Notice that $C$ and $D$ appear in every column in adjacent positions. One might expect to see clone-like candidates in, for example, a jungle primary, in which two Republicans and two Democrats are both running for a top-two runoff. If each voter places the two Republicans in adjacent positions, and the two Democrats in adjacent positions, then each of these pairs of candidates are clones.

A voting system is said to be {\it clone positive} if adding a clone of candidate $A$ has the following outcome: There exists a clone $X$ of $A$ (possibly $A$ itself) such that $X$ is ranked above all candidates that $A$ was ranked above originally, and possibly more. A voting method is said to be {\it clone negative} if adding a clone of candidate $A$ has the opposite outcome: for every clone $X$ of $A$, $X$ may be ranked below a candidate that $A$ was ranked above originally, and cannot improve.

Prove that Borda Count is clone positive, and that plurality voting is clone negative.

\item In any voting matrix, a Condorcet Cycle is a sequence of candidates $A_1, A_2, \dots, A_k$ such that $A_i$ wins a head-to-head contest with $A_{i+1}$ for all $i$, and $A_k$ wins a head-to-head contest with $A_1$. Prove that a Condorcet Cycle exists if there is no Condorcet winner. Give an example to show that there can be a Condorcet Cycle even if there is a Condorcet winner.

\item Suppose we run an approval vote in which we are assured that every voter is sincere (totally unrealistic, but this is math, not sociology). \begin{enumerate}
\item Prove that this method will satisfy the Pareto Criterion.
\item Prove that if every voter makes a sincere vote with an absolute dichotomous cutoff, the method also satisfies IIA.
\item Why is this not a violation of Arrow's Theorem?
\end{enumerate}

\end{enumerate}

\end{document}