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

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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.

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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

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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

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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).

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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

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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.

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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.

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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

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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.    

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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.

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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)