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MA 267 The mathematics of Sustainability Project

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A Mathematical Look into Elections

Luwei Zeng

Instructor: Professor Emma Previato

2013/06/26

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Part I:

On February 1, 2013, Senator John Kerry announced his resignation from the United

States Senate effective. The vacancy prompted the 2013 United States Senate special Election

Massachusetts held on June 25, 2013. This election was to fill the Massachusetts Class II UnitedStates Senate seat for the remainder of the term ending January 3, 2013.

On April 30, a party primary election took place to determinate the nominees of each

party for the general election. Democratic Ed Markey and Republican Gabriel E.Gomez won

their respective primaries. On June 25, Ed Markey with popular vote1642,988 and the

percentage 54.8% won the election, while Gabriel Gomez with popular vote 525,080 and the

percentage 44.8% lost the election.

In this part, I will focus on the results of the primary election. The goal of the research is

to a more reasonable measure of power for voters within a weighted voting system by applying a

mathematical model.

2013 Republican Primary - United States Senate special election in Massachusetts

Party Candidate Votes Percentage

Republican Gabriel Gomez 96,276 51.0%

Republican Michael J. Sullivan 67,918 36.0%

Republican Daniel Winslow 24,630 13.0%

1Popular vote: actual number of ballots that are cast by the voters for a candidate.

http://en.wikipedia.org/wiki/Republican_Party_(United_States)http://en.wikipedia.org/wiki/Republican_Party_(United_States)http://en.wikipedia.org/wiki/Republican_Party_(United_States)http://en.wikipedia.org/wiki/Republican_Party_(United_States)http://en.wikipedia.org/wiki/Republican_Party_(United_States)http://en.wikipedia.org/wiki/Republican_Party_(United_States)http://en.wikipedia.org/wiki/Republican_Party_(United_States)http://en.wikipedia.org/wiki/Republican_Party_(United_States)http://en.wikipedia.org/wiki/Republican_Party_(United_States)

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Gomez

Sullivan

Winslow

2013 Democratic Primary - United States Senate special election in Massachusetts

Party Candidate Votes Percentage

Democratic Ed Markey 309,854 57.4%

Democratic Stephen Lynch 229,973 42.6%

http://en.wikipedia.org/wiki/Democratic_Party_(United_States)http://en.wikipedia.org/wiki/Democratic_Party_(United_States)http://en.wikipedia.org/wiki/Democratic_Party_(United_States)http://en.wikipedia.org/wiki/Democratic_Party_(United_States)http://en.wikipedia.org/wiki/Democratic_Party_(United_States)http://en.wikipedia.org/wiki/Democratic_Party_(United_States)

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Markey

Lynch

Among the Republican candidates, Although Gomez won the support from a much larger

part of area than Sullivan, as we can see from the map, his number of votes just 15% higher than

Sullivan. Also, Markey obtain 14,8% more support from the population than Lynch, while on the

map, it shows the area Markey occupied is much more than 14.8%. Why it is like this?

The population in the west of Massachusetts is rare, so even the candidates won the large

part of the west Massachusetts, we cannot guarantee that they could win the election. However,

we cannot say that the relationship between the voting power and the number of the population is

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positively proportional (i.e y=ax). In order to figure out the true relationship between the number

of voters and the voting power, we will begin with introducing an mathematical concept.

Banzhaf power indexDifferent voters will have differencepowerover the results of elections. We would expect

that the more votes a voter has, the more power that the voter has over the results of any vote.

Banzhaf Powe index, introduced by John F.Banzhaf III is a relatively simple way to compute the

power a voter holds.

Definition:

Suppose there are n votersA,, Anwhose voting weights are w(A),, w(An).There are

2possible line-ups of the voters into yes and no groups. In each line-up, we make a note of

those voters whose votes are crucial. Let c(Ai)be the number of yes-no line-ups in which Ais

crucial, ands(Ai)the number whereAis semi-crucial.

Banzhaf power indexof voter:

Banzhaf power of a single voter is usually meaningless. However, it is useful to compare

the power of voters.

1. Voter number n is even.

Let n = 2p

There cant be a line-up in which there is a crucial voter. These line-ups can be divided

into the following four categories:

a) A is part of a winning coalition of p+1 yes voters; the no coalition will have p-1

members

b) A is part of a winning coalition of p+1 no voters; the yes coalition will have p-1members

c) A is a member of a tied coalition of p yes voters

d)A is a member of a tied coalition of p no voters

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Table 2

4 semi-crucial

=

=

Yes voters No voters

a) A -------------------

1 p

------------------

p-1

b) ------------------

p-1

A ------------------

-

1 pc) A -------------------

1 p-1

------------------

p

d)------------------

p

A ------------------

-

1 p-1

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=

=

=

()

= ()

2. Voter number n is odd

Let n = 2p+1

There cant be a line-up in which there is a semi-crucial voter. These line-ups can bedivided into the following two categories:

a) A is part of a winning coalition of p+1 yes voters; the no coalition will have p members

b) A is part of a winning coalition of p+1 no voters; the yes coalition will have p members

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=

=

=

=

()

Therefore, we can see that the voting power and the number of voters are not in the relation y=ax.

Yes voters No voters

a) A -------------------

1 p

------------------

p

b) ------------------

P

A ------------------

1 p

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Furthermore, lets look deep into the statistics. Lets take the republican primary election

in the city of Boston as an example. We need to Compare the voting power of ward 8 and ward

12, Using the formula given, first of all, we divide the BPIsof the vote cast in ward28 over the

on in ward 12, the result of which is 1.27.Then we divide the number of votesin ward 9 over the

one in ward 12, the result of which 2.42. If we say the contribution ward 12 made to let Gomez

to win is 1, we cannot say the contribution from ward 9 is 1.27. From the data, we can see that 20

people choose Gomez in ward 8 and 5 choose him in ward 12. The contribution ward 8 make is 4

times than that in ward 12. One of the reasons is that there are many other factors which can

affect the power of the voting. The number of votes is just one of the factors.

Therefore, we should consider the number of voters as one of the determinants of the

voting power and include other objective factors.

2

A ward has one person who makes decisions for their city.

A precinct has everyone in the city vote to make decisions.

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Part II

Boston was redistricted on December 21, 2012. Boston mayoral election will hold on

November 5, 2013. Will the redistricting produce great difference of whom will be the next

Mayor.

In the Election 2013: Next Boston Mayor May Be Black or Hispanic, Kirk Clay asks

the question: will the next mayor be black or Hispanic? He states that the population of black or

Hispanic increased significantly in the last ten years. Boston grew 4.8 % to 617,593 and it

currently has over 85,000 Key Voters of Color. Also, there are historical reasons which may

lead to black or Hispanic people win more votes than before.

After redistricting in 2012:

#1343

ApprovedData

Districts White All White VAP3 Black VAP Hisp VAP Asian VAP

1 54.9 60.1 3.1 30.2 4.7

2 68.9 71.9 5 7.1 15.1

3 37.7 41.7 22.9 13 14

4 8.2 9.9 62.6 18 4.5

5 27.6 31.4 46.3 18.3 1.9

6 63.5 66.8 10.4 15.2 6.2

7 25 30.3 39.7 19.6 6

8 67.3 68.6 5.9 8.2 169 66.1 68.2 4.6 8.8 16

In 2002, the total voting age population (VAP) is 472,582, among which, Hispanic is

57,258 and Black/African is 96,897. Hispanic consists of 12.1% and Black/African is 20.5%.

They occupy totally 32.5%. In 2012, from the table, we can calculate that the Black and Hispanic

VAP consist of 26.24% of the total VAP. Although, the percentage is decreased, we cannot

conclude that Black and Hispanic people will not be possible the mayor. We cannot simply say

the larger number of the Black or Hispanic people are the more votes Black or Hispanic

candidates can obtain. We should look at the demographic changes in some districts.

Now, lets look at if the Black or Hispanic people contribute more or less to the ballots to

the black or Hispanic candidates than before. We will take education level as an important factor

3VAP: votING age population

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to see the voting power of the Black and Hispanic people. From the U.S census data, we know

that people in the district 8 are averagely educated and the education level increase at the average

level of some other districts. We assume that people with different have the equal opportunity to

obtain education. In 2002, the Hispanic VAP consists of 7.6% and Black/African VAP consists

of 6.3%. In 2012, Hispanic VAP consists of 4.94% and Black/African consists of 3.55%. As the

percentage decreases dramatically and the education level is at the average, we may think this

part of Black VAP or Hispanic VAP would contribute less than before to the

There are certainly other factors can determine the voting power of certain district. As we

discussed in the PART I, it is impossible to quantify the voting power by some simple factors.

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Work Cited

CONCENPTS OF MATHEMATICAL MODELING, WALTER J.MEYER,DOVER

PUBLICATIONS,INC

http://politic365.com/2013/04/27/election-2013-next-boston-mayor-may-be-black-or-hispanic/

http://www.cityofboston.gov/citycouncil/committees/census.asp

http://www.cityofboston.gov/citycouncil/committees/census.asp

http://www.ams.org/samplings/feature-column/fcarc-voting-introduction

http://politic365.com/2013/04/27/election-2013-next-boston-mayor-may-be-black-or-hispanic/http://politic365.com/2013/04/27/election-2013-next-boston-mayor-may-be-black-or-hispanic/http://www.cityofboston.gov/citycouncil/committees/census.asphttp://www.cityofboston.gov/citycouncil/committees/census.asphttp://www.cityofboston.gov/citycouncil/committees/census.asphttp://www.cityofboston.gov/citycouncil/committees/census.asphttp://www.ams.org/samplings/feature-column/fcarc-voting-introductionhttp://www.ams.org/samplings/feature-column/fcarc-voting-introductionhttp://www.ams.org/samplings/feature-column/fcarc-voting-introductionhttp://www.cityofboston.gov/citycouncil/committees/census.asphttp://www.cityofboston.gov/citycouncil/committees/census.asphttp://politic365.com/2013/04/27/election-2013-next-boston-mayor-may-be-black-or-hispanic/