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\begin{document}
\title{Math 301 Homework}
\author{Mary Radcliffe}
\date{Due 13 December 2017}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Determine which of the 5 voting methods discussed in class (Plurality, Hare, Coomb's, Condorcet, Borda Count) satisfy monotonicity. Prove that your answers are correct.

\item The Minimax Condorcet method is defined as follows:

For any two candidates $A$ and $B$, define $score(A, B)= d(A,B)-d(B,A)$, where $d(A,B)$ represents the number of voters ranking $A$ above $B$. That is to say, $score(A,B)$ represents the margin between $A$ and $B$, and is positive if and only if $A$ beats $B$.

We define the winner of the vote to be $\arg\min_{X}(\max_{Y} score(Y,X))$.
\begin{enumerate}
\item What is this method doing? Explain it.
\item Show that this method satisfies the Condorcet Criterion.
\item Suppose we rephrased the Condorcet Criterion as follows: If $A$ loses every head-to-head competition, then $A$ should lose. (This is called the Condorcet Loser Criterion). Show that the Minimax Condorcet Method fails the Condorcet Loser Criterion.
\end{enumerate}

\item Prove that a Single Transferable Vote system, when used in an election with only one winner, is the same as an instant runoff using Hare Method.
\end{enumerate}
\end{document}