Chad Watson
Mathematics 482
May 8, 2013
Is Our Voting System the Best Mathematics Can Offer?
In the year 2000, my friend and I were both proponents of Al Gore who supported Ralph Nader.
We were committed to encouraging everyone to vote for Nader to help the Green Party gain the
5% of the popular vote that would provide public funding and a place in the debates for 2004,
but we were aware he had less than a 1% chance of victory. I lived in California, my friend
leaved in Florida. I was able to convince a handful of folks, but it was easy; Al Gore was going
to win the California Electoral College by a landslide. The race in Florida was a much different.
Bush and Gore were a toss up. I suggested to my friend he vote for Nadar, but cease recruiting
voters. He shrugged off my suggestion and he continued to campaign, even convincing his blue
dog Democrat mother to vote green. Gore won California in a landslide but the recount in
Florida went on for over a month. While am not how many people my friend persuaded, I can
safely say that my friend did not convince 537 people to vote for Nader. Twelve states were
decided by a margin of less than 5%, which is less than the acceptable margin of error in surveys
seen on the news. Two states threw out more ballots than the margin of victory. Nine states
threw out over 2% of their vote while other states threw out no votes (Leip,
http://uselectionatlas.org/RESULTS/index.html). In a contest to pick out the leader of our
this is all a little too close for comfort.
While the 2000 election may be the most contested election in recent memory, it is not the first
in the United States’ history. William Poundstone argues since the advent of the current voting
system, upwards of 7 presidential elections were won by the candidate who did not win a
majority of the popular vote (Poundstone, 91). How can someone win without winning a
majority, by winning with a plurality. If Millard Fillmore and Franklin Pierce are both running
for comptroller of the senior class and Millard Fillmore receives 51% of votes cast while Pierce
receives 49% of votes cast then Millard Fillmore wins with a majority. Let’s says George
McGovern and Spiro Agnew get into the race and the vote cast is now 28% for Millard Fillmore,
27% for Pierce, 26% for McGovern and 19% for Agnew. Millard Fillmore wins with a plurality;
he received the most votes, but not a more than 50%.
How can this happen in a country that is a standard barer for democracy? How can someone be
elected with more than half the registered voters going against him or her? Vote splitting is one
of the problems with our current system. Vote splitting is when candidates have similar
viewpoints on an issue or issues and voters divide their choices between the two. This often
results in neither candidate winning. Most voters do not fall perfectly on one side of the binary
Republican/Democrat system. When a third party candidate enters the race they tend to take
votes away from the Republican or Democrat. For instance, in the original iteration of our
comptroller election I supported Milard Filmore and his Whig values because I detest a
Jacksonion Democrat like Pierce. The electorate is either in agreement with or stands ins
complete opposition to me. Votes change when 20th century Republican and Democrat Agnew
and McGovern, respectively, get into the race. Now there exists a broad spectrum of beliefs
within the election and the vote is now distributed among the new candidates. My vote goes to
Agnew in hopes he would better manage the finances of the class. While the class is a small part
of my life what about broader issues? Suppose we had hypothetical gubernatorial candidates
with their platform in their name, Stay the Course Sara with 51% of the vote and Free Tuition by
Eliminating Football Fred with 49% of the vote, then Four Day Weekend William enters race.
2 5% of the votes are transferred from Sara votes to William and 2% of the voters who were with
Fred are now voting for William, this Gives Free Tuition Fred a narrow victory. This is not a
silly and purely hypothetical case. Take a look at the 1998 Minnesota gubernatorial race which
pitted a Democratic Attorney General and Republican St. Paul mayor with over two decades of
combined experience against one another. Well into the race they were forced to face off against
noted World Wrestling Federation heel and conspiracy theorist Jesse “the Body” Ventura as the
Reform Party candidate. Even though Ventura admitted he was not sure how he could get things
done in the governor’s mansion he managed to appeal to 37% of the voters and won the plurality
(Poundstone, 213). In 1992 George H. W. Bush, Bill Clinton, and Ross Perot faced off for the
White House. Many analysts believe that a portion of Perot’s 19% could have turned the
election in favor of Bush. Bill Clinton won this election with over 56% of the electorate voting
against him.
It is difficult to say if Perot or Ventura took votes from the frontrunners. Perhaps they brought
people to the polls who would not have otherwise voted. Regardless, when more than two people
are running, the plurality system falls apart because it is difficult to receive a majority of the
vote. Mathematician and economist Kenneth Arrow won a Nobel Prize in economics for
bringing to the forefront a problem called Arrow’s paradox. The main problem of ranked voting
is the intransitivity of voting choices (Poundstone 38-41). We learned in elementary school that
if A > B and B>C then A>C. This not the case with ranked voting. Take four candidates, the
front runners President Obama and former Governor Romney, along with the two major third
party candidates, former Governor Gary Johnson from the Libertarian party and Dr. Jill Stein
from the Green Party. A voter may favor Johnson over Obama, Romney over Johnson, Obama
over Romney, and Stein and Obama. How does this person vote? They may ideologically
3 believe in Johnson, but they do not want Romney to be in the White House, so they vote for
Obama as the lesser of two evils. Another voter may favor Stein over others, but vote Romney
in order to change the status quo. Vote splitting is evident to some extent in all elections when a
third party enters the race. People vote strategically. Sometimes they will vote not for their
favorite candidate but of the front runner they favor most. Voters may often support a weak
candidate in the primaries to give their candidate an easier opponent. Often they vote their
conscience and the vote is split. Are there better systems? Most are susceptible to “gaming” or
strategic voting that clouds the results. But many have strong arguments that they are better
methods than the current ranked plurality system.
Kenneth Arrow went into detail in his 1950 paper “A Difficulty in the concept of Social
Welfare.” Arrow was a practitioner in the field of social choice theory. Social choice theory uses
elements of formal logic to construct axioms and social welfare functions with elements of game
theory mixed in. Suppose A is a set of outcomes and N is the number of voters. Let F: L(A)N
L(A) where L(A)N is the list of all profiles of voter and L(A) is the final outcome is the list of
society’s profile and a ordering is listed a voter’s ranking of candidates: (R1,R2,…,Rm).
“Arrow’s Paradox considers the following properties:”
(http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem)
1.) Pareto Efficiency is unanimity or the fact that no one can be made better off without
making someone else worse. If a>b for all orderings (R1,R2,…,Rm) then a>b for
F[(R1,R2,…,Rm)]. In short, common rankings should be the outcome.
2.) Non-Dictatorship states there is no individual whose preference always prevails
4 3.) The Independence of Irrelevant Alternatives: Two candidates ranking should only be
motivated by preference of one to the other. If I have B>C>D>Z B>C should hold if I
change my preference to Z>D.
Arrow’s Paradox stated that if society fields over two candidates than one of the above
properties is violated resulting in the best candidate not always having the best chance of
winning. In 1972 Arrow was awarded the Nobel Prize in Economics which brought his idea to
the masses and resulted in a boarder discussion on the validity of voting systems.
The Marquis de Condorcet published one of the first alternatives to plurality voting first in a
1785 paper (Poundstone, 141). He claimed the best system was one in which each candidate
goes up against the other in a two-way contest. One who wins in all head to head match ups is
the Condorcet winner (Poundstone, 142). Here again we run into the problem of intransitivity.
What if there is no Condorcet winner, then we have what analysts call a Condorcet cycle. If in
the current contest Romney>Obama, Johnson>Romney, Stein>Romney, then we have a
Condorcet cycle and there is no clear winner.
Another system to consider is the Borda count system. It tries to counteract vote splitting by
giving voters more than one vote via a weighted point system. This is the system used in the
Harris Interactive and USA Today Coaches Poll that are used in part to determine the Bowl
Championship Series rankings. In the USA Today Coaches poll 59 BCS school football
coaches rank the teams 1 to 25 and the teams are assigned points in inverse order, the top team
gets 25 the 25th team gets 1. Then the 59 votes are tabulated and divided by 1475, the maximum
point total (collegefootballpoll.com/bcs_explained.html) to get their place in the weekly poll. It
can be ranked in reverse with fewer points being better than more (first place gets one point).
5 The rationale for this system is the ability of the electorate to vote for more than one candidate
and the ability to rank the candidate by preference. Under this system my friend could have
voted 4 for Gore and 3 for Ralph Nader. The biggest problem with Borda count is you can bury
votes. If we had contest with George Washington, Abraham Lincoln, and Dracula using the
Borda count with a decreasing scale 3 to 1, than let’s say 14 people voted: seven gave
Washington a three and seven gave Lincoln a three, so they are tied with 21 votes each. Let’s
say each of those 14 gave their main opposition a 1 in order to damage his chances, and they
each gave Dracula a 2 thinking he is the weaker candidate and has no chance of winning. They
are taking votes away from their competition and giving to a third party, thinking they are
burying their competition. This scenario leaves Washington, Lincoln, and Dracula tied. If a
fourth or fifth candidate was introduced than it is possible Washington and Lincoln could be
buried under Dracula enough to allow Dracula to win even though no listed him as their top
candidate.
A different, perhaps better system is single transferable voting. STV is used in Parliamentary
elections in Ireland and India as well many local elections (county and municipal) in England
and Australia (Gueorguieva 21,143,163). In STV, each voter submits a preferred order of
candidates, and if a single candidate does not have a majority then the candidate with the least
number is eliminated and his/her votes are transferred to the next ranked candidate on the ballot.
If this had been the mode of voting in Florida in 2000 when it was clear that Nader was not going
get the majority then my friend’s vote would have been moved to Gore. STV is most often used
in city council elections where multi faceted issues lead to a broad spectrum of candidates and
beliefs on issues. The process of redistribution continues until the number of seats is the number
of candidates. When only one seat is up for grabs STV is called instant runoff voting. This is a
6 great method for elections concerning two major candidates and a few minor ones. Strategic
voters have nothing to gain by listing their opponents last because as weaker candidates are
eliminated, the buried ones will rise. When you have a field of moderately strong candidates
William Poundstone claims “Instant runoff voting is subject to something called the center
squeeze theorem. A popular moderate can receive relatively few first place votes but because of
vote splitting from candidates the moderate will be eliminated early on”(Poundstone 168) Much
like the squeeze theorem in calculus, when you have functions on either side of another function,
the limit of the function is “squeezed” out. What if the frontrunner did not receive a majority
and the spilt was Washington 44% Jefferson 35% Burr 21% Burr is eliminated and perhaps 16%
of his voters ranked Jefferson next while the other 5% ranked Washington next, than Jefferson
wins with 51% of the vote, some may say this is a fair and just result but since Jefferson is not
the Condorcet winner, his mandate will be questioned.
A simple system that attempts to counteract vote splitting is approval voting. Approval voting is
when the voter simply gives a 0, 1 or in some cases leaves a blank. This method allows
everyone that is approved to have one vote. My friend in Florida could have given votes to both
Nader and Gore; I could have given votes to Nader and Gore. Perhaps under modified rules
Nader could have gotten his 5% and Gore still could have gotten the electoral votes necessary.
Multiple candidates can cloud the system and you could give real competitors a 0 and less
qualified a 1, so you can bury, a process known as burying votes There is also ambiguity about
who is approved; uninformed voters may give everyone a one or a zero. If there are two strong
candidates running a majority of the electorate could approve both as happened in the 1800
election. In the 1800, the Electoral College used approval voting with every elector allowed to
vote for two candidates. Aaron Burr and Thomas Jefferson were the front-runners and they tied.
7 The House of Representatives had to vote on the winner and it took 36 ballots before a clear
majority cast votes for Jefferson (http://www.loc.gov/rr/program/bib/elections/election1800.html
) Burr’s Dilemma is when two top candidates struggle for a seat one is approved but the other is
not. This leaves a path for a lesser qualified candidate to challenge for
office(http://rangevoting.org/BurrSummary.html), a major downside of approval voting. This
has not prevented neither the American Mathematical Society from adopting this method nor the
Secretary General of the United Nations.
Cumulative voting is another proportional representation model. According to Fairvote.org “To win under choice voting, candidates need an exact number of votes called a "threshold". After transferring surplus ballots until no remaining candidate has obtained the winning threshold, the candidate with the fewest votes is eliminated.” (http://www.fairvote.org/how-­‐choice-­‐voting-­‐works#.UI6qu4XrFJg) If a candidate has
surplus votes over the threshold then those surplus ballots are transferred to the next preference.
This is done until all seats have been filled. For example if we used this to elect the House or
Representatives and there were no restriction on districts, just 435 seats to fill and 90,000,000
people were voting (90,732,693 was the turn out for the 2010 midterm
http://elections.gmu.edu/Turnout_2010G.html). The threshold would be
[90,000,000/(435+1)]+1 = 206,424 if we rounded up. Rep. Nancy Pelosi or Rep. Barney Frank
would need 206, 424 votes to get elected, any votes over the threshold would go to the second
choice on the ballot. But if we had said only 3 seats on city council and 12 people running then
we run into Poundstone’s center squeeze dilemma. Some people claim that many choices are
overwhelming and are equivalent to no good choice.
8 The final method we shall discuss is range voting, candidates are commonly given a score of 0-9.
The final tally can be expressed as averages or totals. Blanks are not incorporated into the
averages, so voters can give a score based how they much they approve the candidate rather than
a simple thumbs up or thumbs down (http://rangevoting.org/SmithWM.html). Range voting
seems more immune to strategic voting. It is used in Amazon.com, imdb, and iTunes rating,
with multiple votes allowed. In theory the voter is giving the candidate a grade. Critics say
strategic voting is possible with range voting as well. Muddling of the result is possiblewhen we
bring pre-election polls in the process. The voter we may change their candidate’s opponents
score to manipulate the average. Voters could also give everyone a maximum score, which
essentially is voting three times. So is this system really worth looking at? Let’s look at Lincoln
versus Washington versus Dracula, and let’s say 450 people vote: 150 die-hard Lincoln
supporters, 150 die-hard Washington supporters and 150 uninformed people. The range is 99-0.
The die-hard supporters give their guy 99 and their main opposition 0 and give no vote to
Dracula thinking he is of little or no concern. The unformed look at a third party and think that
would be a change if nothing else. The unformed give the “old” guys a 20 and give this rogue
reformer a 40. Under a simplified range system, Washington and Lincoln have a score of 39.7
and Dracula wins with 40, a bad result for everyone. What if three honest voters get involved:
One votes Washington 50 Lincoln 40 Dracula 0, the next votes Washington 40 Lincoln 50
Dracula 0 and the final honest voter votes Washington 40 Lincoln 60 Dracula 10. This Changes
the distribution to Washington 39.71 Lincoln 39.73 and Dracula now has 1.16. It only took 3 out
of 453 voters (less than 1%) to be honest and well informed. Warren Smith (founder of the center
for Range voting and number proponent of Range voting) ran 144 computer simulations
(Poundstone 238). He found that Range Voting has the best Bayesian regret. In a Bayesian regret
9 model voters are assigned utility values (via computer simulation) for each candidate as to who
will make the voter the happiest. Bayesian regret is a measurement of how much an outcome can disappoint a person. If you have two candidates, you can measure how satisfied a person would be with the candidates. Let us measure candidates on a basis of would be happy with their stance on an issue (1) or would not be happy with their stance on an issue (0). We will use a set of 6 values: social issues, the economy, domestic policy, foreign policy, health care, and education. If we give President Obama a score of social issues 1, the economy 1, domestic policy 1, foreign policy 0, health care 1, and education 0. He has a score of 4; this is President Obama’s utility score. We give Governor Romney a score of social issues 1, the economy 0, domestic policy 1, foreign policy 1, health care 1, and education 1. We give him a score of 5. When President Obama wins so the absolute value of the difference of the two candidates is 1, this is the Bayesian Regret for us. If Governor Romney had won our Bayesian Regret would have been zero since we gave the former Governor a higher score than the President. (http://rangevoting.org/BayRegDum.html) The
systems allowing more weighted votes offer a more nuanced ballot where someone who is
fiscally conservative and socially liberal can express their views and feel satisfied or someone
who is fiscally liberal and socially conservative can voice their displeasure with the current
office holders if need be. In STV or cumulative voting people can elect voices that echo the
populace. Maybe these alternatives are not perfect, but they demand a more thorough study by
not only academics, but also the real world.
Our system is flawed. We need to a multiple step approach. Many say let’s just get rid of the electoral college, but that will never happen since it gives smaller populated states more clout, thus they would never vote the system out. I suggest a two-­‐tiered system. 10 What if we voted for the President via range voting with a graded 100-­‐point system? The American populace has a good feel for grades and what value a grade holds. The second tier is the House and Senate. Why not a cumulative voting system for both? This would allow for a broader range of candidates. More third party candidates or candidates who may be more inclined to challenge the party line would have a better chance. More diversity could lead Democrats and Republicans to open their tents. The current system is conducive only to a two party system. We are seeing movements like the tea party and the Occupy parties but the last election showed the tea part had trouble maintaining a grasp on key offices and the Occupy movement has become an excellent relief group, but its political future is questionable. If there is more diversity in the congress and a more accurate system for picking our President than perhaps we can make progress.
11 Citations
College Football Poll. “BCS Explained” collegefootballpoll.com/bcs_explained.html CNN.com “ How we got here: A Timeline of the Florida Recount.” Dec 13, 2000.
http://archives.cnn.com/2000/ALLPOLITICS/stories/12/13/got.here/index.html)
Dueorguieva, Vassia and Simon, Rita. Voting and Elections The World Over. Maryland:
Lexington Books, 2009. Print
George Mason University. United States Election Project. http://elections.gmu.edu/Turnout_2010G.html Hodge, Jonathan and Kilma, Richard. The Mathematics Of Voting and Elections: A Hands –
On Approach. Providence, R.I. : American Mathematical Society, 2005. Print Jeter, Jon. “’The Body’ Slams into Politics.” Washington Post Oct 22, 1998. http://www.washingtonpost.com/wp-­‐
srv/politics/campaigns/keyraces98/stories/mn102298.html Reny, Philip. (2001)“Arrow’s Theorem and the Gibbard-­‐Satterthwaite Theorem: A Unified Approach,” Economics Letters, 70, v.1,99-­‐105. Leip, David Dave Leip's Atlas of U.S. Presidential Elections. http://uselectionatlas.org/RESULTS/index.html Library of Congress. Election of 1800 Resource Guide. http://www.loc.gov/rr/program/bib/elections/election1800.html Poundstone, William. Gaming the Vote. New York: Hill and Wang, 2008. Print
Saari, Donald. Chaotic Elections: A Mathematician Looks at Voting. Providence, R.I. : American Mathematical Society, 2001. Print. Smith, Warren D. “Range Voting: The Best Way to Select a Leader.” The Center for Range Voting. http://rangevoting.org/SmithWM.html The Center For Range Voting. “Bayesian Regret for Dummies.” http://rangevoting.org/BayRegDum.html The Center for Voting and Democracy. http://www.fairvote.org/how-­‐choice-­‐voting-­‐
works#.UI6qu4XrFJg Wikipedia. Arrow’s Impossibility Theorem. (http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem) 12