The first four lessons of this chapter examined situations in which allvoters are considered equals. This lesson examines situations in whichsome voters have more votes than other voters.

Lesson 1.5

Weighted Voting and Voting Power

30

Public Hearing Set for Madison County Weighted Voting

Oneida Daily DispatchApril 11, 2013

The Madison County Board of Supervisors will hear publiccomments over upgrading weighted voting totals.

In Madison County, supervisors vote based on a weightedformula calculated by the population of their towns. Thatvoting configuration hasn’t been updated since 2002. Usingpopulation data from the 2010 Census, the number has beenrecalculated.

Under the proposed law, there are to be 1500 votes total. Thevotes are not divisible and are to be cast as one lump sum byeach town. The county uses the Banzhaf power index todetermine how many votes each supervisor has.

DM01_Final.qxp:DM01.qxp 5/9/14 1:54 PM Page 36

37Lesson 1.5 • Weighted Voting and Voting Power

Members of legislative bodies such as the United States Congress andcounty boards represent districts. The constitutional principle of oneperson, one vote requires that such districts be approximately the samesize. Thus, every 10 years, district lines are redrawn to reflect changes inpopulation. But redrawing boundaries can be difficult for many reasons.Some legislative bodies try to resolve difficulties by adopting a system inwhich some votes carry more weight than others.

Consider a simple example. A small high school has 110 students.Because of recent growth in the size of the community, the sophomoreclass is quite large. It has 50 members, and the junior and senior classeseach have 30 members.

The school’s student council is composed of a single representativefrom each class. Each of the three members is given a number of votesproportionate to the size of the class represented. Accordingly, thesophomore representative has five votes, and the junior and seniorrepresentatives each have three. The passage of any issue that is beforethe council requires a simple majority of six votes.

The student council’s voting model is an example of weightedvoting. Weighted voting occurs whenever some members of a voting bodyhave more votes than others have.

In recent years, several people have questioned whether weightedvoting is fair. Among them is John Banzhaf III, a law professor at GeorgeWashington University who has initiated several legal actions againstweighted voting procedures used in local government.

To understand Banzhaf’s objection toweighted voting, consider the number ofways that voting on an issue could occurin the student council example.

It is possible that an issue is favoredby none of the members, one of them, twoof them, or all three. In which cases doesan issue pass? The following list gives allpossible ways of voting for an issue andthe associated number of votes.

{; 0} {So; 5} {Jr; 3} {Sr; 3} {So, Jr; 8} {So, Sr; 8} {Jr, Sr; 6} {So, Jr, Sr; 11}

For example, {Jr, Sr; 6} indicates that the junior and seniorrepresentatives vote for an issue and that they have a total of six votesbetween them.

John Banzhaf III (1940– )

John Banzhaf, a law professorwho also holds anengineering degree, is a well-known consumer rightsadvocate.

Mathematician of Note

DM01_Final.qxp:DM01.qxp 5/9/14 1:54 PM Page 37

38

Each of these collections of voters is called a coalition. Those withenough votes to pass an issue are winning coalitions. The winningcoalitions in this example are those with six or more votes and are listedbelow along with their respective vote totals.

{So, Jr; 8} {So, Sr; 8} {Jr, Sr; 6} {So, Jr, Sr; 11}

The last winning coalition is different from the other three in oneimportant respect: If any one of the members decides to vote differently,the coalition still wins. No single member is essential to the coalition.Banzhaf argued that the only time a voter has power is when the voterbelongs to a coalition that needs the voter in order to pass an issue. Thecoalitions for which at least one member is essential are

{So, Jr; 8} {So, Sr; 8} {Jr, Sr; 6}.

Notice that the sophomore representative is essential to two of thecoalitions, which is also true of the junior and senior representatives. Inother words, about the same number of times, each of the representativescan be expected to cast a key vote in passing an issue.

A paradox: Although the votes have been distributed to give greaterpower to the sophomores, the outcome is that all members have the samepower!

Since distributing the votes in a way that reflects the populationdistribution does not always result in a fair distribution of power,mathematical procedures can be used to develop ways to measure actualpower in weighted voting situations.

A measure of the power of a member of a voting body is called apower index. In this lesson, a voter’s power index is the number ofwinning coalitions in which the voter is essential. For example, in thestudent council situation, the sophomore representative is essential to twowinning coalitions and thereby has a power index of 2, as do the juniorand senior representatives.

Chapter 1 • Election Theory: Modeling the Voting Process

DM01_Final.qxp:DM01.qxp 5/9/14 1:54 PM Page 38

39Lesson 1.5 • Weighted Voting and Voting Power

Exercises1. Consider a situation in which A, B, and C have 3, 2, and 1 votes

respectively, and in which 4 votes are required to pass an issue.

a. List all possible coalitions and all winning coalitions.

b. Determine a power index for each voter.

c. If the number of votes required to pass an issue is increased from4 to 5, determine a power index for each voter.

2. In a situation with three voters, A has 7 votes, B has 3, and C has 3.A simple majority is required to pass an issue.

a. Determine a power index for each voter.

b. A dictator is a member of a voting body who has all the power. Adummy is a member who has no power. Are there any dictatorsor dummies in this situation?

3. The student council example in this lesson depicts a situation withthree voters that results in equal power for all three. In Exercises 1and 2, power is distributed differently. Find a distribution of votesthat results in a power distribution among three voters that isdifferent from the ones you have already seen. How many newpower distributions in situations with three voters can you find?

A Power Index Algorithm1. List all coalitions of voters that are winning coalitions.

2. Select any voter, and record a 0 for that voter’s power index.

3. From the list in step 1, select a coalition of which the voterselected in step 2 is a member. Subtract the number of votes thevoter has from the coalition’s total. If the result is less than thenumber of votes required to pass an issue, add 1 to the voter’spower index.

4. Repeat step 3 until all coalitions of which the voter chosen instep 2 is a member are checked.

5. Repeat steps 2 through 4 until all voters are checked.

DM01_Final.qxp:DM01.qxp 5/9/14 1:54 PM Page 39

40 Chapter 1 • Election Theory: Modeling the Voting Process

4. In this lesson’s student council example, can the votes bedistributed so that the members’ power indices are proportionate tothe class sizes? Explain.

5. In this lesson’s student council example, suppose that therepresentatives of the junior and senior classes always differ onissues and never vote alike. Does this make any practical differencein the power of the three representatives? Explain.

6. (See Exercise 14 of Lesson 1.4 on page 33.) Let Cn represent thenumber of coalitions that can be formed in a group of n voters.Write a recurrence relation that describes the relationship betweenCn and Cn–1.

7. One way to determine all winning coalitions in a weighted votingsituation is to work from a list of all possible coalitions. Use A, B, C,and D to represent the individuals in a group of four voters and listall possible coalitions.

8. Weighted voting is commonly used to decide issues at meetings ofcorporate stockholders. Each member has one vote for each shareof stock held.

a. A company has four stockholders: A, B, C, and D. They own 26%,25%, 25%, and 24% of the stock, respectively, and more than50% of the vote is needed to pass an issue. Determine a powerindex for each stockholder. (Use your results from Exercise 7 asan aid.)

b. Another company has four stockholders. They own 47%, 41%,7%, and 5% of the stock. Find a power index for eachstockholder.

c. Compare the percentage of stock owned by the smallestshareholder in parts a and b. Do the same for the power indexof the smallest stockholder in each case.

DM01_Final.qxp:DM01.qxp 5/9/14 1:54 PM Page 40

41Lesson 1.5 • Weighted Voting and Voting Power

9. A landmark court decision on votingpower involved the Nassau County,New York, Board of Supervisors. In1964, the board had six members. Thenumber of votes given to each was 31,31, 21, 28, 2, and 2.

a. Determine a power index for eachmember.

b. The board was composed ofrepresentatives of five municipalitieswith these populations:

Hempstead 728,625North Hempstead 213,225Oyster Bay 285,545Glen Cove 22,752Long Beach 25,654

The members with 31 votesrepresented Hempstead. The otherseach represented the municipalitylisted in the same order as in thetable. Compare the power indices ofthe municipalities with theirpopulations.

10. A minimal winning coalition is one inwhich all voters are essential.

a. Give an example of a weightedvoting situation with a winning coalition for which at least one but not all of the voters is essential. Identify the minimal winning coalitions in this situation.

b. Is defining a voter’s power index as the number of minimalwinning coalitions to which the voter belongs equivalent to thedefinition used in this lesson? Explain.

Nassau Districting RuledAgainst Law

The New York TimesJanuary 15, 1970

Albany—TheCourt of Appealsruled today thatthe present“weighted voting”plan of theNassau CountySupervisors wasunconstitutionalbut that a newplan was notnecessary untilafter the 1970federal census.

In a unanimousopinion, thestate’s highestcourt said thecounty’s presentcharter provisionis a clear violation

of the one-man-one-vote principle,in that itspecifically deniedthe town ofHempsteadrepresentationthat reflected itspopulation.

The town, thecourt pointed out,constituted 57.12percent of thecounty’spopulation, butbecause of theweighted votingplan itsrepresentatives onthe board couldcast only 49.6percent of theboard’s vote.

DM01_Final.qxp:DM01.qxp 5/9/14 1:54 PM Page 41

42 Chapter 1 • Election Theory: Modeling the Voting Process

11. The president of the United States is chosen in the Electoral College,a system that can be considered a form of weighted voting amongthe states. The number of electors given to each state (and theDistrict of Columbia) is equal to its representation in Congress. Thatis, the number of electors equals the number of members of theHouse of Representatives plus two (the number of senators). In2000, Albert Gore won the popular vote by about half a millionvotes over George Bush, but lost in the Electoral College.

a. In the 2000 census, the population of California (the mostpopulous state) was 33,871,648. The population of Wyoming(the least populous state) was 493,782. California has 52representatives in the U.S. House. Wyoming has 1. Use thesedata to construct an argument that the electoral votedistribution is weighted in favor of small states.

b. Some reformers have proposed removing the electoral weighting.They suggest equating the number of electors with themembership in the U.S. House only. The following table showsthe Electoral College votes in the 2000 election. Would thisreform proposal change the 2000 election results? Explain.

Bush Gore

Alabama 9Alaska 3Arizona 8Arkansas 6California 54Colorado 8Connecticut 8Delaware 3D. C. 3Florida 25Georgia 13Hawaii 4Idaho 4Illinois 22Indiana 12Iowa 7Kansas 6

Bush Gore

Kentucky 8Louisiana 9Maine 4Maryland 10Massachusetts 12Michigan 18Minnesota 10Mississippi 7Missouri 11Montana 3Nebraska 5Nevada 4New Hampshire 4New Jersey 15New Mexico 5New York 33North Carolina 14

Bush Gore

North Dakota 3Ohio 21Oklahoma 8Oregon 7Pennsylvania 23Rhode Island 4South Carolina 8South Dakota 3Tennessee 11Texas 32Utah 5Vermont 3Virginia 13Washington 11West Virginia 5Wisconsin 11Wyoming 3

DM01_Final.qxp:DM01.qxp 5/9/14 1:54 PM Page 42

43Lesson 1.5 • Weighted Voting and Voting Power

c. The Electoral College is mandated in the United StatesConstitution. Why do you think the founders of the countryinstituted a system weighted in favor of small states? (Do someresearch if you are not sure.)

Computer Explorations12. Use the weighted voting software

that accompanies this book toexperiment with different weightedvoting systems when there are threevoters. Change the number of votesgiven to each voter and the numberof votes required to pass an issue.How many different powerdistributions are possible? Do thesame for weighted voting systemswith four voters.

Projects13. The Security Council of the United

Nations is composed of fivepermanent members and ten otherswho are elected to two-year terms.For a measure to pass, it mustreceive at least nine votes thatinclude all five of the permanentmembers. Determine a power indexfor a permanent member and for atemporary member. (Assume that allmembers are present and voting.)

Majority of MemberStates Back Pesticide

Ban

Copyright © by Thaves. Distributed from www.thecomics.com

European VoiceApril 29, 2013

A majority of member states todayvoted to approve a EuropeanCommission proposal to ban the useof neonicotinoid seed treatmentpesticides, thought to be harmful tobees.

The outcome was close, following afirst vote last month which wasinconclusive, with neither a majorityfor or against. Crucially, Germanychanged its position in today's vote toapprove the ban, after havingabstained last month. The 15 memberstates voting in favor gave 189weighted votes, versus 125 weightedvotes against, including those of theUK. Abstentions amounted to 33weighted votes.

DM01_Final.qxp:DM01.qxp 5/9/14 1:54 PM Page 43

44 Chapter 1 • Election Theory: Modeling the Voting Process

14. Research and report on other power indices. What, for example, isthe Shapley-Shubik power index?

15. What is the effect of the Electoral College system on the power ofindividual voters in selecting the president? Research the matterand report on the relative power of voters in different states.

16. Research and report on court decisions about weighted voting.

U.K., U.S. Wield Most CyberPower

National JournalJanuary 13, 2012

The United Kingdom and United States leadother developing countries in their ability towithstand cyberattacks and develop strongdigital economies, according to a new study byBooz Allen Hamilton and the EconomistIntelligence Unit.

The U.K. tops the rest of the Group of 20nations, including the U.S., in the "CyberPower Inex." The European Union, consideredthe 20th member of the G20, was notincluded.

The index, which put the U.S. in the No. 2spot, rates the countries on their legal andregulatory frameworks; economic and socialissues; technology infrastructure; and industry.

"Overall, the top five countries exhibitingcyber power, as measured by the index--theU.K.; the U.S.; Australia; Germany; andCanada--illustrate that developed Westerncountries are leading the way into the digitalera," the companies said in a statement.

DM01_Final.qxp:DM01.qxp 5/9/14 1:54 PM Page 44