Prepared by:

Joseph Malkevitch

Mathematics and Computing Dept.

York College (CUNY)

Jamaica, New York 11451

Email: malkevitch@york.cuny.edu (for additions, suggestions, and corrections)

The ballots below show the rankings of 55 faculty for 5 choices of textbooks. Which book deserves to be selected as the choice for the group?

(Note: There is no indifference; higher preferences towards the top.)

There are three merchants who have

verified claims against the remaining assets of a small company. What would be a fair way to settle the claims in each case?

Example 1:

Claimant A B C (Assets Remaining: $200)

Claim $30 $100 $100

Example 2:

Claimant A B C (Assets Remaining: $200)

Claim $30 $200 $300

2,180

1,880

1,420

640

A wealthy alumna of the college has decided to donate 30 scholarships to attract new students based on the different divisions' enrollment.

What is a fair way to assign the 30 scholarships to each division?

A B

Turkish Rug $1100 $1800

Stamp Collection $1900 $1200

Book Collection $20000 $30000

Ethnic Jewelry $6000 $5000

What would be a fair way to handle the situation? (What would you do if the brothers had listed the same amount for some items?)

Towns A and B with populations of 12 and 8 (in hundreds of thousands) need to build a new water supply system.

Stand alone costs (in millions of dollars):

{A} = 160 {B} = 90

Joint cost: {A, B} = 200

How should A and B proceed?

Towns A, B, and C with populations of 12, 10, and 8 (in hundreds of thousands) need to build a new water supply system.

Example 1:

Stand alone costs (in millions of dollars):

{A} = 150 {B} = 120 {C} = 90

Joint costs:

{A, B} = 240 {A, C} = 220 {B, C} = 180

{A, B, C} = 325

How should A, B, and C proceed?

Example 2:

Stand alone costs (in millions of dollars):

{A} = 150 {B} = 120 {C} = 90

Joint costs:

{A, B} = 240 {A, C} = 220 {B, C} = 180

{A, B, C} = 315

How should A, B, and C proceed?

Is it fair for the representatives of these 5 towns to cast 7, 4, 3, 3, and 1 vote, respectively, where 10 votes are required to take action in the county legislature?

Players named Row and Column independently choose a row (choice of two) and column (choice of two), respectively. The "payoff" is shown in the cell at the intersection of the chosen row/column pair. The payoffs shown are from Row's point of view. Column's payoff is the negative of the number shown.

Is this a fair game?

How should Row and Column play the game to achieve his/her best result if the game is played:

a. Once?

b. Many times?

How famous the patient is?

How wealthy the patient is?

How old the patient is?

How sick the patient is?

How long the patient has been waiting for a transplant?

1. What principles of fairness and equity do you think are essential in dealing with the situations above?

2. If everyone is "happy" with the way the situation is resolved does this mean the solution method is a "good" one?

3. In some of the situations above the participants had some freedom to "lie" about some of the information provided. What are the consequences of this possibility?

4. What issues arise if, after a solution is effected, additional information becomes available (e.g. in the bankruptcy situation perhaps: a. Another creditor appears; b. more assets are available than originally thought)?

5. Can you think of situations where being fair is not the only concern in attempting to deal with the situations?

1. How would one measure whether the distribution of income (or wealth) is more equitable in one country than in another?

2. How can one compare the fairness (equitability) of different tax systems?

3. In English, the words "fair" and "equitable" have somewhat different connotations. What are their differences? Can a procedure be "fair" but not "equitable?" Can a procedure be "equitable" but not "fair?"

4. For different areas of knowledge (e.g. accounting, medicine, business) can you find examples of situations where being fair and equitable are matters of concern?

5. Discuss how one can design fair election districts?

Aumann, R. and S. Hart (Eds.) Handbook of Game Theory with Economic Applications, North-Holland, New York, Volume 1, (1992), Volume 2, (1994).

Balinski, M. and H.P. Young, Fair Representation, Yale University Press, New Haven, 1982.

Brams, S. and A. Taylor, Fair Division, Cambridge U. Press, New York, 1996.

Kuhn, H. (ed.), Classics in Game Theory, Princeton U. Press, Princeton, 1997.

Mc Lean, I. and A. Urken, Classics of Social Choice, U. Michigan Press, Ann Arbor, 1995.

Moulin, H., Axioms of Cooperative Decision Making, Cambridge U. Press, New York, 1988.

Robinson, J. and W. Webb, Cake-Cutting Algorithms, A K Peters, Natick, 1998.

Saari, D., Geometry of Voting, Springer-Verlag, New York, 1994.

Straffin, P., Game Theory and Strategy, Mathematical Association of America, Washington, 1993.

Taylor, A., Mathematics and Politics, Springer-Verlag, New York, 1995.

Young, H.P. (Ed.), Fair Allocation, American Mathematical Society, Providence, 1985.

Young, H.P., Equity, Princeton U. Press, Princeton, 1994.

1. National Science Foundation

2. Long Island Consortium for Interconnected Learning (administered by SUNY Stony Brook)

Director: Alan Tucker (Applied Mathematics)

3. York College (City University of New York)

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