W hether the parties involved are individuals,organizations, communities, states, or nations; ajoint endeavor raises questions of fairness.

Massive undertakings like monitoring the health of theearth raise many fairness issues. For example, how can thecost be shared equitably among all nations? What is a fairway to share the data that is generated?

Fair division questions arise in the simplest ofsituations. For example, you probably experiencedunfairness as a child when you felt that someone receiveda piece of cake or portion of ice cream that was better thanyours.

• How can a portion of food be divided fairly amongtwo or more children?

• Is the meaning of fairness when food is dividedamong children different from the meaning offairness when an estate is divided among heirs orwhen seats in Congress are divided among states?

• Are the methods that are commonly used to dividefood, estates, and legislatures necessarily the fairestmethods?

Discrete mathematics plays an important role inanswering these questions.

In this chapter, you will consider several fairnessissues and examine mathematical models for resolvingthem. Most of these fair division models are algorithmsthat can be expressed in a few numbered steps.

CHAPTER

Fair Division 2

Lesson 2.1A Fair Division Activity

Lesson 2.2Estate Division

Lesson 2.3Apportionment Models

Lesson 2.4MoreApportionment Models andParadoxes

Lesson 2.5Fair DivisionModels: The ContinuousCase

Lesson 2.6MathematicalInduction

ChapterExtensionEnvy-FreeDivision

Chapter 2Review

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Lesson 2.1

A Fair DivisionActivity

There are many situations inwhich the division of anobject in a fair way is impor-tant to those involved. Threeof the most common are thedivision of food among chil-dren, a house in an estateamong heirs, and the seatsin a governmental bodyamong districts. Each hascharacteristics that make itdifferent from the others.

In this lesson, you willconsider an example of eachof these three situations and propose solution models ofyour own. As is the case inelection theory, fair divisionis an area of discrete math-ematics in which importantproblems can be understoodand solved without havingconsiderable backgroundknowledge.

Nations Seek Flexible Climate Approach

ReutersMay 3, 2013

New, more flexible ways tofight climate change weresketched out on Friday atthe end of a week of talksbetween 160 nations.

The meeting of seniorofficials in Bonn,Germany, aired formulasto resolve disputesbetween rich and poor onsharing out the cost ofcurbing greenhouse gasemissions as part of a newU.N. deal, a successor tothe 1997 Kyoto Protocol.

Attempts to reachagreement have

foundered above all on afailure to agree on thecontribution developingcountries should make tocurbing the industrialemissions responsible forglobal warming.

The United Nations saidthere was a broadagreement amongdelegates that any newaccord should haveflexibility to ratchet upcurbs on emissions,without a need for furthernegotiations, if scientificfindings about floods,droughts and rising sealevels worsen in comingyears.

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57Lesson 2.1 • A Fair Division Activity

Explore ThisBelow are three fair division problems. You will find it helpful to discussone or more of them with a few other people, as time permits. Here is away to divide the problems among small groups in your class. At thedirection of your instructor, divide your class into groups of three people.Write the numbers 1, 2, or 3 on each of several slips of paper. Have aperson from each group draw one of the slips from a bag or box. Eachgroup should discuss the problem that corresponds to the number drawnby the group’s representative. Allow about 15 minutes for the discussions.

After all groups have finished their discussions, a spokesperson foreach group should present the group’s result to the class. Each group thatdiscussed Problem 1 should report first, and so forth.

In your notebook, record the method used by each group. You willneed that information for this lesson’s exercises.

1. Martha and Ray are siblings who want to divide the last piece ofthe cake that their mother baked yesterday. Propose a way to dividethe last piece of cake that will seem fair to both Martha and Ray.

2. Juan and Mary are the only heirs to their mother’s estate. The onlyobject of significant value is the house in which they were raised.Propose a way to resolve the issue of dividing the house that willseem fair to both Juan and Mary.

3. The sophomore, junior, and senior classes ofCentral High School have 333, 288, and 279members, respectively. The school’s studentcouncil is composed of 20 members dividedamong the three classes. Determine a fairnumber of seats on the council for each class.

The members of your class may not havereached consensus on the best way to solve each ofthe three problems. In this lesson’s exercises youwill consider some of the important fairness issuesin each of the three problems.

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58 Chapter 2 • Fair Division

Exercises1. a. Did any groups resolve Problem 1 by relying on the mother’s

authority? In what way or ways was this done?

b. Does the resolution of such a problem by the mother or otherauthority figure always produce a solution that seems fair toboth children? Explain.

c. Cite at least two examples of situations in which fair divisionproblems are resolved by an authority.

2. a. Did any of the groups use a random event such as a coin flip toresolve Problem 1? In what way or ways was this done?

b. Does the use of randomness in such a problem always produce asolution that seems fair to both children? Explain.

c. Cite at least two examples of situations in which randomness isused to resolve an issue.

3. a. Did any of the groups use a means of measuring the piece ofcake to resolve Problem 1? In what way or ways was this done?

b. Does the use of measurement in such problems always producea solution that seems fair to bothchildren? Explain.

c. Give an example of a situation inwhich measurement is likely to resultin an agreeable solution to a fairdivision problem.

4. A common way of resolving Problem 1 isto have one of the children cut the cakeinto two pieces and to have the otherchoose first.

a. If Martha cuts the cake, how does she feel about the two pieces?

b. If Ray chooses one of the pieces that Martha cut, how does hefeel about the two pieces?

c. If you were one of the participants in this scheme, would yourather be the cutter or the chooser? Why?

5. Write a description of what you consider to be desirable results of amodel that fairly divides a cake among any number of people.

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59Lesson 2.1 • A Fair Division Activity

6. Did any of the groups resolve Problem 2 by selling the house anddividing the cash? Why might the results of such a model beunsatisfactory to one or more of the heirs?

7. Did any of the groups use a model that considers the possibilitythat the heirs might not agree on the value of the house? In whatway or ways was this done?

8. Suppose that Juan thinks the house is worth $100,000 and Maryfeels it is worth $120,000.

a. Who do you think should receive thehouse? Explain.

b. How might the person who doesn’t get thehouse be compensated?

9. Write a description of what you consider tobe desirable results of a model that fairlydivides a house among several heirs.

10. How might the possibility of lying about thevalue of the house affect the result of adivision process?

11. Did all the groups that discussed Problem 3divide the seats among the classes in thesame way? If not, describe the differences.

12. If some of the groups that discussed Problem3 obtained different results, which of themodels do you think is the fairest? If allgroups produced the same result, do youagree that the result is fair? Why or why not?

13. Write a description of what you consider tobe desirable results of a model that fairlydivides the seats in a student council amonga school’s classes.

14. Summarize the similarities and differences inthe meaning of fairness in this lesson’s threeproblems. For each of the three problems,explain why you think it is or is not possibleto achieve fairness.

15. Read the news article on the right.

a. How is the problem of fairly sharing the cost of somethingdifferent from the three problems you considered in this lesson?

County, Feds ShareCost of RoadWidening

Galesburg Register-MailApril 14, 2012

The Knox County Board metWednesday to approve or fund aseries of projects, ranging fromroad expansion to courtroomrenovations.

Board members approved anagreement between the federalgovernment and the county forwork to be done on CountyHighway 4. Widening thehighway is slated to cost $5million, with $3 million infunding coming from the federalgovernment. The county will pay$1.7 million for the project, whilethe state will chip in $306,000.

Final construction plans for theproject have been submitted tothe Illinois Department ofTransportation, with work slatedto begin in June

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60 Chapter 2 • Fair Division

b. Which of the three problems of this lesson do you think is mostlike the problem of sharing the cost of something? Describe thecharacteristics that are similar and different.

Projects16. Pick a situation in which individuals or groups have developed a

model for settling fair division problems other than the division offood, an estate, or legislative seats. Report on the model used.Compare it with models developed in this lesson and later in thischapter.

New Airwave-Sharing Scheme Will Launch a Wireless Revolution

MIT Technology ReviewDecember 11, 2012

Aiming to boost wireless bandwidth and innovation, the U.S. FederalCommunications Commission is poised to recommend the biggestregulatory change in decades: one that allows a newly availablechunk of wireless spectrum to be leased by different companies atdifferent times and places, rather than being auctioned off to onehigh bidder.

The move in effect allocates spectrum for another Wi-Fi—a technologythat has had tremendous impact. But it is the sharing approach thatrepresents a dramatic change in unleashing bandwidth.

The move spells the beginning of the end of a system in whichspectrum is either exclusively owned by a private company, walled offfor government use, or unlicensed and crowded.

Under the new approach, a startup that wants cheap spectrum to testa breakthrough idea would no longer have to rely on clutteredunlicensed bands—such as those that handle baby monitors andgarage-door openers—to experiment with its idea in the real world.Rather, it could use a slice of choice spectrum at zero or low cost—andfor a short time period if desired.

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Lesson 2.2

Estate Division

A fair division problem can be either discrete or continuous. The problem ofdividing a house among heirs and that of dividing a student councilamong classes are examples of the discrete case. Discrete division occurswhenever the objects of the division cannot be meaningfully separatedinto pieces. Dividing a cake is an example of the continuous case becausethe cake can be divided into any number of pieces.

This lesson considers fair division of an estate among heirs. In yourdiscussions in Lesson 2.1, you may have decided that fairness is difficult todefine in some situations because different people place different valueson the same object. However, it is sometimes possible to use suchdifferences of opinion to the advantage of all those involved. The estatedivision model on the next page is an algorith that produces an appealingparadox: Each of the heirs receives a share that is larger than he or shethinks is fair.

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