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                     \vspace*{12pt}

\begin{center}
{\large \textbf{Should Candidates Flip a Coin if the Difference in
Their
Votes is Small?    } \\[0pt]
} \bigskip January 25, 2003 \\[0pt]
\bigskip Eric Rasmusen \\[0pt]
 
\bigskip
 
 \textit{Abstract}
\end{center}

A coin flip can be a good way to settle an election if the margin of
victory is small and it is known that there is a good chance of fraud
by one candidate. In that case, however, an even better rule is to
award victory to the apparent loser. Even this rule will not entirely
eliminate
the incentive to acquire illegal votes.
\bigskip  

\noindent
{\small \hspace*{20pt}
  Indiana University Foundation Professor,
Department of Business Economics and Public Policy, Kelley School of
Business, Indiana University, BU 456, 1309 E. 10th Street,
Bloomington, Indiana, 47405- 1701. Office: (812) 855-9219. Fax: 812-
855-3354. Erasmuse@indiana.edu. Php.indiana.edu/$\sim$erasmuse.
Keywords: Social Choice,  Voting, Elections, Fraud, Illegal Votes,
Supermajorities,
Bias.  JEL
Classifications:
C11, C44, D70, D72, D81. Copies of this paper can be found at
Php.indiana.edu/$\sim$erasmuse/papers/coinflip.pdf. }

\noindent
   {\small I thank John Matsusaka and participants in workshops at the
business school of Indiana University and the University of Southern
California for helpful comments, and Harvard Law School's Olin Center
and the University of Tokyo's Center for International Research on the
Japanese Economy for their hospitality. }

%%-----------------------------%----------------

\newpage

\noindent
  \textit{1. Introduction}

	The 2000 U.S. Presidential Election was the subject of
numerous cries of unfairness. A week before the election, George Bush
seemed likely to beat Al Gore handily, but Gore surprised everyone by
catching up in the last few days. The election turned on who won
Florida. Bush was ahead by 1,831 votes, less than one-tenth of one
percent of the total Florida vote. After the required recount, his
margin had dropped to 784 votes. There ensued a battle of the lawyers
that ended a month later with a state-certified Bush margin of 537
votes and his election as president (see David Rusin [2001] for
details.)

	Many Democrats were outraged. How could Bush become president
when the margin was so close? Surely there should be a revote. A
number of journalists, including Stephen Jay Gould (2000), suggested,
perhaps humorously, flipping a coin.

	It was just not just the closeness that gave rise to
indignation. Both sides claimed that the official count was improper.
The Democrats objected that   many of their voters  voted  by
mistake  for more than one candidate or for none at all, which cost
thousands of votes for Gore. Republicans objected--- more quietly,
since they were leading--- that many of the Democratic votes were cast
illegally by felons or people not registered to vote  and that Gore
was  trying to get getting dishonest judges to invalidate overseas
absentee ballots.

	What about flipping a coin? Would it be good to have a
pre-set policy of doing this whenever the margin of victory was small?
Whether a policy is good depends on the objective, of course.   I will
take as given the conventional objective: to maximize the probability
that the candidate desired by a majority of those legally voting wins.

The reason usually given for a coin toss is that the voting procedures
have random error, so that if the margin were close and the election
were repeated, a different candidate might well win. This is a bad
reason, as we will see below. If the error is unbiased, then the
conventional  vote count is an unbiased estimator of the  legitimate
vote, and adding noise to an estimator cannot help (although the
higher the variance of the estimator, the less the noise will hurt).
And, of course, suggesting a coin toss only  after  the official count
is known is hardly playing fair.

	If, however, we are setting up a voting rule before we know
who will have the winning margin, there are indeed situations where a
coin toss helps. This will be the case if we can confidently  predict
that the official count will  be subject to fraud of some kind. The
model below will show why this is so.

     The argument will not be based on the existing literature in
economics or political science, because to my knowledge no one has
studied optimal  voting rules in the presence of fraud.  May (1952)
long ago showed the optimality of the simple majority victory rule in
the basic setting which, except for the presence of illegitimate
votes, I will use here. Subsequent articles in the social choice
literature, while they may have considered changing the required
margin for victory from 0,  have  not concerned themselves with
illegitimate votes as a motivation (e.g., Ferejohn \& Grether  [1974], 
García-Laprest \&  Llamazares [2001]).   Political science journals do
publish articles on
vote fraud, but these seem to be more historical, unrelated to the
normative literature on voting rules (e.g., Cox \& Kousser [1981],
Baum \& Hailey [1994]).  The present article
will try to link the two ideas that vote fraud can sway elections and
that simple majority voting may not be the best victory rule.


 
  
 
\bigskip

\noindent 
 \textit{2. The Model}

	Let us imagine that we are constructing rules for elections
between two candidates who we will label as ``dishonest''   and
``honest''.   In advance, we do not know who will be honest and who
will be dishonest, so we cannot use a rule such as ``The dishonest
candidate wins only if his margin is at least 500 votes. Otherwise the
honest candidate wins.'' We can, however, use a rule such as ``A
candidate wins if his margin is at least 500 votes. If the margin is
less, the election is decided by a coin toss.''

	It of course  often realistic that neither or both candidates
are dishonest. In the model below, what will matter is the difference
between their dishonest vote gains, so the reader should understand
``the dishonest candidate'' to mean ``the more dishonest,''  and his
illegal votes to be his superiority in number of illegal votes. Also,
illegal votes can be interpreted not just as illegitimate additional
votes for the dishonest candidate, but as legitimate votes for the
honest candidate that have been illegitimately suppressed.    

	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  his margin  of illegal votes by $N$.  Both $m$ and
$x$
can be negative, indicating a positive margin for the honest
candidate. By definition, $m= x+N$. 

Let $x$ be distributed by density $f(x)$ with cumulative density $F(x)
$. We will make $x$  a continuous variable for neatness, so the
probability of exact ties will be zero and we will not need to clutter
the analysis with special rules for tie-breaking.

\noindent Assume:

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

Figure 1 shows a number of densities which satisfy assumption (A1).
Figure 1a is a well-behaved density of the kind I think most
applicable. The density is greatest at $x=0$, meaning a tie is the
mode, and declines symmetrically on each side, but not to infinity,
since there are only a finite number of voters. Figure 1b shows a
bimodal asymmetric density where the mode has the dishonest candidate
winning by large margin. Figure 1c shows a density in which the honest
candidate has a solid base that enables it to win by a particular
large margin 30 percent of the time, a probability atom, but otherwise
the candidates are symmetric. Figure 1d shows a density which is
unimodal, but with the mode at a win for the dishonest candidate. (All
four examples have bounded supports because winning margins cannot
exceed the size of the voting population. but bounded support will not
be necessary for the conclusions below.)

 
 \bigskip

 

 
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 \bigskip
 
 

Figure 2 shows three distributions that do not satisfy assumption A1.
In Figure 2a, the density slopes down rather than up over the interval
$[-2N,0]$ . In Figure 2b, the distribution is uniform, so $f(x)$ is
constant rather than decreasing. In Figure 2c, the distribution's
support is less than $2N$, so the density is constant at 0 for part of
the interval $[-2N,0]$.

 
 \bigskip


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  \bigskip
 
Let us denote a victory for the dishonest candidate by $V=1$ and a
victory for the honest candidate by $V=0$. Our problem is to choose a
"victory rule": a rule which awards victory to one candidate or the
other. This rule depends on the observed margin of votes, and takes
the form $V(m)=p$, where $  m$ is the dishonest candidate's margin and
$p$ is his probability of victory given that margin.

Assume society's objective is to maximize the probability of a
legitimate victory, defined as the candidate with the most legal votes
being declared the victor. We will denote a legitimate victory by $L$,
where
\begin{equation}  \label{e2a}
\begin{array}{ll}
L & =1\; if \; x \geq 0 \; and \; V=1 \\ 
& =1\; if \; x < 0 \; and \; V=0 \\ 
& =0 \;otherwise.
\end{array}
\end{equation}
If society knew which candidate was dishonest, which we have ruled
out, the optimal victory rule would simply replicate the objective by
subtracting $N$ votes from the dishonest candidate's margin and
declaring as winner whoever had the most legal votes, i.e.,

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

We will require, however, that any victory rule be symmetric, since we
do
not know the identity of the dishonest candidate in advance:

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

\noindent
The conventional victory rule is:

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

Under the conventional victory rule, the probability that the
dishonest
candidate wins is 
\begin{equation}  \label{e3}
Prob(m>0) = Prob(x+N>0)= Prob (x > -N) = 1-F(-N).
\end{equation}
The probability that the dishonest candidate is the legitimate winner
is
\begin{equation}  \label{e4}
Prob(x >0)= 1-F(0).
\end{equation}
The probability that the honest candidate wins is 
\begin{equation}  \label{e5}
Prob(m<0) =Prob(x+N<0) = Prob (x < -N) = F(-N).
\end{equation}

	Expression (\ref{e5}) is also the probability that the honest
candidate wins legitimately, since he never wins except by having a
majority. The probability of a legitimate victory is thus
\begin{equation}  \label{e6} 1-F(0)+ F(-N). \end{equation} The
probability of a legitimate victory decreases in $N$, since bigger $N$
means smaller $F(-N)$.

\bigskip

The conventional victory rule is a special case of the following
``Coin Flip Rule'', with $T=0$. Figure 3  is a graphic illustration of
how the rule works.

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

 \bigskip


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 \bigskip

	Under the coin flip rule, the probability the dishonest
candidate is
the legitimate winner is not just the probability he is legitimate,
because sometimes, due to the coin toss, he fails to win even if he is
legitimate. The probability that he is the legitimate winner and also
wins under the victory rule is
\begin{equation}  \label{e8}
\begin{array}{l}
Prob (x>0,m>T) + .5 prob (x>0, -T<m <T) \\ 
=Prob (x>0, x+N>T) + .5 prob (x>0, -T<x+N <T) \\ 
= Prob (x>0, x >T-N) + .5 prob (x>0, -T-N<x <T-N).
\end{array}
\end{equation}

The probability that the honest candidate is the legitimate winner and
also
wins under the coin flip  rule is 
\begin{equation}  \label{e9}
\begin{array}{l}
Prob (x<0, m<-T) + .5 prob (x<0, -T<m <T) \\ 
Prob (x<0, x+N<-T) + .5 prob (x<0, -T<x+N <T) \\ 
= Prob (x<0, x <-T-N) + .5 prob (x<0, -T-N<x <T-N).
\end{array}
\end{equation}

We need to consider two cases: $T \geq N$, and $T<N$.

\noindent (1) $T \geq N$ (threshold greater than the number of
dishonest
votes). The probability that the dishonest candidate is the legitimate
winner and also wins under the victory rule is 
\begin{equation}  \label{e10}
\begin{array}{ll}
Prob(Dis.\; leg.\; win ) & = Prob (x>0, x+N > T) + .5 prob (x>0, -
T<x+N <T)
\\ 
& = Prob (x>0, x \geq T-N) + .5 prob (x>0, -T-N<x <T-N) \\ 
& = Prob (x >T-N) + .5 prob (0 <x <T-N) \\ 
& = [1- Prob(x <T-N)] + .5 [Prob(x<T-N) - Prob(x<0)] \\ 
& = 1- F(T-N) + .5 [F(T-N) - F(0)] \\ 
& = 1- .5F(T-N) - .5F(0).
\end{array}
\end{equation}

The probability that the honest candidate is the legitimate winner and
also
wins under the victory rule is 
\begin{equation}  \label{e11}
\begin{array}{ll}
Prob (hon\; leg. \; win) = & Prob (x<0, x+N<-T) + .5 prob (x<0, -T<x+N
<T)
\\ 
& = Prob (x<0, x <-T-N) + .5 prob (x<0, -T-N<x <T-N) \\ 
& = Prob ( x <-T-N) + .5 prob (-T-N<x <0) \\ 
& = F(-T-N) + .5 [F(0) -F(-T-N)] \\ 
& = .5 F(-T-N) + .5 F(0).
\end{array}
\end{equation}

The probability of a legitimate victory is thus 
\begin{equation}  \label{e12}
\pi= [1- .5F(T-N) - .5F(0)]+ [.5 F(-T-N) + .5F(0)]= 1-.5F(T-N) + .5
F(- T- N)
\end{equation}

The optimal $T$ maximizes this. The first order condition is 
\begin{equation}  \label{e13}
d\pi/dT= - .5 f(T-N) -.5 f(-T-N) =0.
\end{equation}

Expression (\ref{e13}) cannot be solved. The derivative is negative
for all $ T$ in the interval $[N, \infty]$ that we are considering so
the smaller $T$ is, the better. Thus, the optimum is $T^*=N$ if it is
in this interval.

\bigskip

\noindent 
 (2) $T<N$ (threshold less than the number of dishonest
votes). The probability that the criminal candidate is the legitimate
winner and also wins under the victory rule is
\begin{equation}  \label{e14}
\begin{array}{ll}
Prob (dis \;legit \;winner ) & = Prob (x>0, x+N>T) + .5 Prob (x>0, -
T<x+N
<T) \\ 
& = Prob (x>0, x >T-N) + .5 prob (x>0, -T-N<x <T-N) \\ 
& = Prob (x >0) + 0 \\ 
& = [1- Prob(x <0)] \\ 
& = 1- F(0)
\end{array}
\end{equation}

The probability that the honest candidate is the legitimate winner and
also
wins under the victory rule is 
\begin{equation}  \label{e15}
\begin{array}{ll}
Prob (Honest \;legit \;winner ) & = Prob (x<0, x+N<-T) + .5 prob (x<0,
-T<x+N <T) \\ 
& = Prob (x<0, x <-T-N) + .5 prob (x<0, -T-N<x <T-N) \\ 
& = Prob ( x <-T-N) + .5 prob (-T-N<x <T-N) \\ 
& = F(-T-N) + .5 [F(T-N) - F(-T-N)] \\ 
& = .5 F(-T-N) + .5 F(T-N).
\end{array}
\end{equation}

The probability of a legitimate victory is thus 
\begin{equation}  \label{e16}
\pi = 1- F(0)+ .5 F(-T-N) + .5 F(T-N).
\end{equation}

The optimal $T$ maximizes this. The first order condition with respect
to $N$
is 
\begin{equation}  \label{e17}
d\pi/dT= -.5 f(-T-N) + .5 f(T-N) =0.
\end{equation}

The derivative in (\ref{e17}) is always positive, because $f(-T-N)$ is
always less than $f(T-N)$, as shown in Figure 4. Both winning margins
$x$ are negative numbers in this case, but $T-N$ is closer to 0, where
the density is greater under our assumptions.

Thus, $T^*=N$ is the optimum.

  \bigskip

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 \bigskip

We can conclude that when we think that one candidate will have $N$
illegal votes, the optimal coin flip rule flips a coin if the margin
of victory is less than $N$.

One implication of this is that if neither candidate has a
margin of illegal votes, so $N=0$, then a coin should not be
flipped--- the optimal coinflip rule never flips the coin. This
reflects the idea--- perhaps trivial here, but still worth pointing
out---that adding noise to an unbiased estimator cannot improve it,
and, indeed, results in worse decisions. Notice too that this
conclusion in no way flows from risk aversion of the decisionmaker,
something we have not assumed here. Rather, it flows from the increase
in the probability of a wrong decision.

\bigskip

\textit{A Bayesian Approach.} This result can also be interpreted in
Bayesian terms. If society observes margin $m$, what should its
posterior belief be of the probability that the legal margin $x$ is
also positive? On observing $m=m^{\prime }$, society knows that either
(a) $x=m^{\prime }-N$ or (b) $x=-m^{\prime }-N$, depending on which
candidate is the dishonest one. If $m^{\prime }>N$, then in case (a),
$x>0$, and in case (b), $x<0$, so the posterior should be that with
probability 1 the apparent winner is the legitimate winner. This is
why $T^{\ast }$ should not exceed $N$.

  
If $m^{\prime}\in [-N,N]$, then society cannot deduce with certainty
who was the legitimate winner. In that case, if the apparent winner is
the dishonest candidate, the apparent winner is not legitimate, but if
it is the honest winner, the apparent winner is indeed legitimate. The
posterior probability that the apparent winner is the legitimate
winner is, by Bayes's Rule,
\begin{equation}  \label{e18}
P(m^{\prime}) = \frac{f(-m^{\prime}-N)} {f(m^{\prime}-N)+f(-
m^{\prime}- N)}.
\end{equation}

If $P(m^{\prime})$ is greater than .5---i.e., if $f(-m^{\prime}-N)>
f(m^{\prime}- N) $---then victory ought to be awarded to the apparent
winner. Assumption A1 tells us that that is false, however, because
both $ - m^{\prime}-N$ and $ m^{\prime}- N$ are in the interval $[-2N,
0]$ over which the density is increasing. Thus, for margins between 0
and $N$, our posterior is that the apparent winner is probably
\textit{not} the legitimate winner! Specific numbers may make this
clearer. Suppose $N= 500$, and the winning margin is 100. If the
dishonest candidate is the apparent winner, with $m= 100 $, then $x= -
400$, and we would like a rule that reverses his victory. If the
honest candidate is the apparent winner, with $ m=- 100$, then $x= -
600 $, and we want a rule that confirms the apparent winner. Which is
more probable, $m=100$ or $m=-100$? It is $m=100$ that is more
probable, because it arises when $x= -400$, which is more probably
than $x=- 600$ given assumption (A1). In short: if a candidate wins by
too few votes, the most likely explanation is that he actually lost
the legal vote and only flipped the result by virtue of illegal votes.
This suggests that the following ``reversal rule,'' illustrated in
Figure 5, is superior to the coin flip rule in maximizing the
objective function.

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

   \bigskip


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  \bigskip

\bigskip

 We have seen that the optimal rule has $T^*=N$. Let us compare the
optimal Coin Flip Rule with the Reversal Rule using the following
general rule (called ``general'' only for convenience; note that it
takes the threshold $T=N$ as given).

\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 $m < -N$ . 


If $z=.5$, the General Rule is identical to the optimal Coinflip Rule;
if $z=0$, it is identical to the Reversal Rule. Let us determine the
optimal level of $z$. The probability that the dishonest candidate is
the legitimate winner and wins under this victory rule is
 \begin{equation} \label{e19} 
 \begin{array}{l} z Prob( x>0,
-N<m <0) + (1-z) Prob( x>0, 0<m <N)+ Prob (x>0,m>N) \\
 = Prob (x+N>N) \\ =
Prob (x >0) 
\end{array} 
\end{equation}
      Equation (\ref{e19}) is telling us that if the dishonest
candidate wins legitimately, the General Rule always awards him
victory, so $z$ is irrelevant to his probability of being the
legitimate winner and also winning under this victory rule. The
probability that the honest candidate is the legitimate winner and
also wins under the victory rule
 is 
   \begin{equation} \label{e20} 
 \begin{array}{l} 
 Prob
(x<0,m<-N)+ (1-z) Prob( x<0, -N<m <0) + z Prob( x>0, 0<m <N) \\ 
  = Prob ( x+N<-N)+ (1-z) Prob( -N<x+N <0) \\
   = Prob ( x <-2N)+ (1-z) Prob( -2N<x <0) \\
\end{array}
  \end{equation} 

Thus, the probability of the legitimate winner
winning under the General Rule is 
 \begin{equation} \label{e21} 
Prob (x >0)+ Prob ( x <-2N)+ (1-z) Prob( -2N<x <0), 
  \end{equation} 
  which is clearly maximized by setting $z=0$ and using the Reversal
Rule.

 \bigskip 

The Reversal Rule    has the peculiar implication that if $N=500$ and
the dishonest
candidate knew he was going to have a ``winning'' margin of 100 votes,
he would do well to throw away 150 votes. But in our model, the
dishonest candidate cannot do that. He obtains the $N$ illegal votes
before he discovers the winning margin on election day, and he cannot
give them back. The optimality of the Reversal Rule is also
counterintuitive because the objective function in this problem is out
of the ordinary. Voting is a winner-take-all tournament, not an
attempt to measure the winning legal margin with minimal mean squared
error. This is best seen by comparison with a similar problem. Suppose
we have a scale that we know is either 40 or -40 milligrams off, with
equal probability, and we are measuring an object from a population
whose weights are unimodally and symmetrically distributed with mean
5000 milligrams. Our measurement is 5010 milligrams. We deduce that
the true weight is therefore either 5050 or 4070 milligrams.
Typically, our objective is to come up with an estimate for the weight
which is unbiased with minimum variance, or perhaps which might be
biased but has minimum mean squared error. In both cases, the estimate
would be somewhere between 4070 and 5000 milligrams, since 4070 is
more probable than 5050 as the true weight, but 5050 also has positive
probability. If, however, our objective was to maximize the
probability of estimating the weight absolutely correctly, or to
maximize the probability of choosing an estimate in the correct
interval $[0,5000]$ or $[5000, \infty]$ our best estimate would be
4070. It is this second kind of objective that was assumed for the
election problem.

\bigskip
 \noindent 
\textit{3. Endogenous Vote Stealing}


  Suppose the dishonest candidate decides in light of the victory rule
whether to incur some cost necessary to acquire the $N$ illegal votes?
What will be the effect of the various rules on
his incentives to steal?

   Figure 6 shows how the  dishonest candidate's  margin of observed
votes changes as a result of   adding the $N$ illegal votes.
  Adding $N$   votes to his margin shifts the support of the
margin   distribution to the right.   Under  the conventional victory
rule, this increases his probability of victory, the area under the
density to the right of 0,  by area B.
  
Under the coin flip rule,  the extra $N$ votes give the dishonest
candidate a gain of   .5 (A+B) in the probability of winning, because
if the margin falls in the interval $[-N,N]$, which  has probability
A+B, a coin will be flipped, but without vote stealing all of that
probability would have gone to the honest candidate. The gain of
.5(A+B), however, is less than     B because assumption A1 implies
that A is less than B.  Thus,  the coin flip rule has reduced the
dishonest candidate's incentive to steal  votes.

Under the  reversal  rule,  the extra $N$ votes give the dishonest
candidate a gain of   A   in the probability of winning, because if
the margin falls in the interval $[-N,0]$, which  has probability A,
he will win,     but without vote stealing all of that probability
would have gone to the honest candidate. The area B was part
of the honest candidate's probability of winning when there was no
stealing, and remains part of his probability of winning when there is
stealing but the reversal rule is in effect.
  The gain of A   is less than  .5(A+B),     given assumption A1, so
the reversal rule has reduced the dishonest candidate's incentive to
steal  votes compared to the coinflip rule or the conventional rule.
Even the reversal rule, however, still leaves the dishonest candidate
better off stealing votes than not stealing  them, unless the cost of
stealing votes  is too high.

 \bigskip

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 \bigskip

	Another possibility would be to steal  (or suppress)
additional votes after the margin arising from the distribution
$f(\cdot)$ is known--- as with the much-storied late returns from safe
wards in Chicago.  This adds to the attractiveness of the coin flip
rule compared to either the convention or reversal rules. The coin
flip rule has the advantage of weakening incentives to acquire more
votes, whether by fair means or foul. The expected payoff still
changes discontinuously, at $-N$ and $N$, but the changes are not as
large as with the other rules. Consider the incentive  for the
dishonest candidate to falsify  enough votes to possibly change the
outcome under the reversal rule. He can move from zero probability of
victory to certainty of victory by adding to his vote total to move
from slightly below $-N$ to slightly above or from slightly below $N$
to above;  or by reducing his vote total to move from slightly above 0
to slightly below.  Under the coin flip rule, on the other hand, there
are only two situations in which he can change his probability of
victory, and both are smaller changes. He can move from zero
probability of victory to  a fifty percent probability    by adding to
his vote total to move from slightly below $-N$ to slightly above, and
he can move from fifty percent probability of victory to   certainty
by adding to his vote total to move from slightly below $N$ to
slightly above. Thus, there are vote-stealing cost levels such that he
would steal additional votes after the election under the reversal
rule but not under the coin flip rule.\footnote{Note that under the
conventional rule, the dishonest candidate has incentive to steal
additional votes in one situation: if the margin is slightly below 0.
This is a move from zero to one hundred percent probability, and the
probability of that close a margin may be larger than the sum of the
probabilities of margins near $-N$ and $+N$, however, so it is hard to
say much about the temptation to steal votes then compared to under
the coin flip rule.}



 \bigskip
 \noindent 
\textit{4. Concluding Remarks}

     This analysis has arrived at  a paradox: when it is known in
advance that one candidate is stealing votes, but not which one, it is
optimal  under simple conditions to reverse the election and award
victory to the apparent loser.    There is a simple intuition that
helps one to understand the paradox.    If we knew that  the dishonest
candidate would  win with certainty under the conventional victory
rule of
having a margin of at least one counted vote, then we would do better
by awarding victory to the apparent loser. More generally, if the
counted margin is less than the number of votes we think have been
stolen, then that margin was more likely than not acquired by fraud,
and we should also award victory to the apparent loser.


 This intuition survives relaxing the model to allow for $N$, the
number of illegal votes, to be stochastic or not known to the social
planner with certainty. A more interesting
   extension of the model would be to  allow the number of stolen
votes to
vary along the continuum rather than just equalling 0 or $N$. One
might then investigate  what number of votes would be stolen  if they
had to be stolen (a) before the victory rule was chosen, or (b) after
the victory rule was chosen.  I have chosen to keep the present model
simple, however, since  its setting is plausible, though special; a
U.S.
state might, for example, have one large city which if controlled by
one candidate will at low cost generate a certain number of illegal
votes. A state in which the number of votes that might be stolen in
equilibrium varies more smoothly is an equally plausible case.  I
have chosen the simpler case because
the reversal rule's optimality is  a striking enough result that it
ought not to be obscured by other complications. 


    Is there any chance of the reversal rule being adopted in real
elections? One's immediate response is ``Of course not!'',  but it
interesting to ask why not. I will not give a satisfactory answer, but
I will speculate in the hope that others may give a better answer.  A
first objection to the reversal rule    is that $N$ differs across
elections and we do not know at what level to set it in advance. It is 
not hard, however,  to see that the result in this article could
survive
uncertainty over the size of $N$. The threshold could be set at our
best guess for the typical election.  Even if we knew $N$, however,  
we might still find the reversal rule objectionable---  even  if 
by ``we'' is meant people who understand the  present argument.  The
problem may be that we do not simply want to maximize the probability
of a legitimate victory.  Instead, we also wish to avoid sometimes
accidentally rewarding a candidate who has stolen votes, even if our
victory rule ends up hurting him on average.  But why this is so, and
whether it is for good reason, I cannot say.

  \newpage

 \bigskip
 \noindent
REFERENCES 


  \hangindent=5em \hangafter=1 
  Baum, Dale  \& James L. Hailey (1994) ``Lyndon Johnson's Victory in
the 1948 Texas Senate Race: A Reappraisal,'' {\it Political Science
Quarterly}, 109:595-613 (Autumn  1994).

  \hangindent=5em \hangafter=1 
  Cox, Gary W.  \&  J. Morgan Kousser (1981) ``Turnout and Rural
Corruption: New York as a Test Case,'' {\it American Journal of
Political Science},  25:646-663 (November  1981).

  \hangindent=5em \hangafter=1 
Ferejohn, J.A. \& D.M. Grether  (1974) ``On a Class of Rational Social
Decisions Procedures,'' {\it Journal of Economic Theory}, 8: 471-482.

  \hangindent=5em \hangafter=1 
García-Lapresta, José Luis   \& Bonifacio Llamazares (2001) ``Majority
Decisions Based on Difference of Votes,'' {\it Journal of Mathematical
Economics}, 35: 463-481 (June 2001).

  \hangindent=5em \hangafter=1 
Gould, Stephen J. (2000) ``Heads or Tails?'' \textit{Boston Globe},
November 30, 2000.

  \hangindent=5em \hangafter=1 
May, K.O. (1952) ``A Set of Independent Necessary and Sufficient
Conditions for Simple Majority Decision,'' {\it  Econometrica},  20:
680-684.

  \hangindent=5em \hangafter=1 
Rusin, David J. (2001) ``Likelihood of Altering the Outcome of the
Florida 2000 Presidential Election by Recounting, ''Northern Illinois
University Dept. of Mathematics, January 5, 2001, http:
//www.math.niu.edu/~rusin/uses-math/recount/index.html (February 23,
2001).


 
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