### Basic Principles of Fairness (8/29/99)

Prepared by:

Joseph Malkevitch
Mathematics and Computing Department
York College (CUNY)
Jamaica, New York 11451-0001

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

Imagine that a circular pie is to be fairly divided by a third party (Nina) on behalf of Alex and Ben by drawing a diameter through the pie. The pie is 1/4 red, 1/4 white, and 1/2 green. Unknown to Ben, Alex prefers the white part of the pie and is indifferent between the red and green parts. Alex's preference is indicated by the fact that in his mind he assigns 80 points as the value of the white part of the pie while he only assigns the red and green parts a value of 10 points each. Unknown to Alex, Ben prefers the red part of the pie and is indifferent between the white and green parts. Ben's preference is indicated by the fact that in his mind he assigns 90 points to the red region and 5 points to each of the green and white regions. Nina knows the boys' preferences.

Figure 1 shows four ways Nina could divide the pie and assign pieces to Alex and Ben. Note that divisions 1 and 2 differ only in who gets which piece. In all four of these situations each of Alex and Ben get 1/2 of the pie from a physical point of view but not always 1/2 from the "psychological" view that takes into account the point assignments to the regions. If there were n people instead of 2, in order to be considered fair each person would want to receive at least 1/n of the pie with respect to his/her "psychological" point of view. This basic idea in fairness is known as the principle of proportionality .

The division in Figure 1 a will make Ben jealous (envious) of the piece that Alex got and Alex jealous (envious) of the piece that Ben got. Thus, one or more person may be envious of the piece of pie that another person got, since the person thinks that the piece that the other person got is better (bigger) than his/her own. Ideally the division should be envy-free . In the case of two people proportional solutions are envy-free and conversely. (Can you prove this?) However, for situations which involve three or more people the concepts of being envy free and proportional are not equivalent. (Can you think of any examples that show this?)

In light of the fact that Alex prefers the white part of the pie and Ben prefers the red part, the division in Figure 1 b or c would be superior to the the division in Figure 1 a. This is because in these solutions each person not only gets what he/she perceives as half the pie, but, in fact, each person gets what he/she perceives as more than one half of the pie. This shows that there are division problems where in each person's mind he/she receives more than an exactly proportional share.

Is the solution found in Figure 1 b or c the best that Nina can find? Clearly not! As shown in Figure 1 d we have another solution in which each player is happier than he was with the solution in Figure 1 b or c. In this sense the divisions of the pie in Figure 1 b or c are inefficient . In economics the solution in Figure 1 b or c would be said not to be Pareto Optimal (in honor of the Italian economist Vilfredo Pareto, who was a pioneer in studying this idea), since there is another solution which makes at least one player happier and with which the other players are no less happy. This constitutes another fairness principle, that of efficiency or Pareto Optimality .

Given the preferences of the boys in this situation, a proportional, envy-free, efficient solution is possible (Figure 1 d). (In fact, there are other solutions with these properties. Can you find them?) Unfortunately, there are problems in which these principles (i.e. proportionality, envy-free, efficient) can not be achieved simultaneously! Can you find examples for yourself?

References:

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

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

3. Sen, A., Inequality Reexamined, Harvard U. Press, Cambridge, 1992.

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

Acknowledgement

This work was prepared with partial support from the National Science Foundation (Grant Number: DUE 9555401) to the Long Island Consortium for Interconnected Learning (administered by SUNY at Stony Brook, Alan Tucker, Director).

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