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\noindent \textbf{Should Candidates Flip a Coin if the Difference in Their
Votes is Small?}\newline
(April 14, 2002. Eric Rasmusen. Erasmuse@Indiana.edu, Php.indiana.edu/$\sim$%
erasmuse)

We are constructing rules for elections between a dishonest candidate and an
honest candidate. Denote the dishonest candidate's margin of votes (votes
for him minus votes for the honest candidate) by $m$, his margin of legal
votes by $x$, and the number of illegal votes by $N$. Both $m$ and $x$ can
be negative, indicating a positive margin for the honest candidate, and $m=
x+N$.

Let $x$ be a continuous variable distributed by density $f(x)$ with
cumulative density $F(x) $.

\noindent

\noindent (A1) The true winning margin density $f(x)$ is strictly increasing
in the range $[-2N,0]$.

The victory rule takes the form $V(m)=p$, where $m$ is the dishonest
candidate's margin and $p$ is his probability of victory given that margin.

Society's objective is to maximize the probability of a legitimate victory, $%
L=1$, defined as the candidate with the most legal votes being declared the
victor.

\noindent \textit{Symmetry Requirement.} If $V(m) =p$, then $V (-m) = 1- p. $

\noindent\textit{The Full-Information Rule.} $V = 1$ if $m-N \geq 0$; and V=
0 otherwise. (violates symmetry)

\noindent\textit{The Conventional Rule.} $V = 1$ if $m \geq 0$; and V= 0
otherwise.

This is a special case, with $T=0$, of the following:

\noindent\textit{The Coin Flip Rule.} $V = 1$ if $m \geq T$; $V = 0$ if $m
\leq -T$; and $V=.5$ otherwise.

\noindent\textit{The Reversal Rule.} $V = 1$ if $m \in [-N, 0]$ or $m> N$ ; $%
V = 0$ otherwise.

\noindent\textit{The General Rule.} $V = z$ if $m \in [-N, 0]$; $V = 1$ if $%
m>N$ ;$V = 1- z$ if $m \in [0, N]$; $V = 0$ if $m < -N$ .

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\begin{center}
\textbf{A Model with Endogenous Vote Buying }
\end{center}

Let the legitimate vote margin of the dishonest candidate, $x$, be
distributed by a symmetric and unimodal density $f(x)$ with cumulative
density $F(x) $. The dishonest candidate, however, begins with an advantage
of $x_0$ legal votes, so the modal outcome in the absence of illegal votes
is that he wins by a margin of $x_0$, or, in the notation, $F(-x_0) =.5$. We
allow $x_0$ to be negative.

Each side can try to buy votes. Candidate 1 (the dishonest candidate) starts
out with an advantage of $\underline{N}>0$ in illegitimate votes. To shift
the balance of illegitimate votes, $N$, the two candidates spend $c_1$ and $%
c_2$. The result is 
\begin{equation}
N = \underline{N} + h(c_1) - \alpha h(c_2),
\end{equation}
where $h$ is increasing and strictly concave, and $\alpha \leq 1$ is  a
shift parameter representing a possible disadvantage of Candidate 2  in
buying votes.

The first order condition for Candidate 1 is 
\begin{equation}  \label{e103c}
\begin{array}{ll}
\frac{d \pi_1}{dc_1} & = - f(-N)h^{\prime}+ f(-N-T)h^{\prime}+
f(T-N)h^{\prime}-1=0
\end{array}
\end{equation}

I would like to answer the following questions:

\begin{enumerate}
\item  Will a candidate spend more on illegal votes if he has a cost
advantage in doing so?

\item  Will a candidate spend more on illegal votes if he starts with an
advantage in legitimate votes?

\item  Does the optimal reversal rule set the threshold $T$ at less than,
equal to, or greater than the number of illegal votes chosen in equilibrium?
\end{enumerate}

I might later add that with probability $\beta$ a candidate does not buy
illegal votes. I will have to change all the notation in that draft, since
the idea of this is that with probability $\beta$ a candidate is honest.
Adding this uncertainty will make the coinflip rule less attractive.

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