Elections: Who Deserves to Win?
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
All 55 students in a small school take the same 5 classes together. The students produced the rankings for the 5 classes they took as shown below:
The ballot symbolism above is interpreted as suggested below. Consider
Figure 2 shows the ballot produced by one voter, who had exactly 4 choices to pick from and chose to rank all 4 of these choices. Furthermore, the voter was not indifferent between any pair of the alternatives that he/she could choose between. The interpretation of this ballot, called an ordinal or preferential ballot, is that D is preferred strictly to C, B, and A, while C is strictly preferred to B and A, and B is preferred to A. Figure 1 shows a voting situation where 5 different choices were being ranked or evaluated. In that figure, the numbers under the figure show the numbers of different people who selected this ballot.
Based on the votes shown in Figure 1, which of the classes that the students were taking should the teacher view as the "favorite" of the group?
From the point of view of a teacher there are many ramifications to calling students' attention to problems of this type. One of the appeals of mathematical modeling is that it offers ways to link mathematical techniques to situations where mathematics is used. Hence, teachers can use modeling situations as a tool for reviewing or teaching topics that deal with mathematical techniques.
For example, given the ballot in Figure 2, one can ask students to address the following questions:
How many distinct ballots can one voter produce when there are 4 candidates to be considered and a voter cannot be indifferent between two candidates?
How many distinct ballots can one voter produce when there are 3 candidates to be considered and a voter cannot be indifferent between two candidates?
How many distinct ballots can one voter produce when there are n (a positive integer) candidates to be considered and a voter cannot be indifferent between two candidates?
How many distinct ballots can one voter produce when there are 3 candidates to be considered and a voter can be indifferent between two candidates?
In this enumeration, for example, there is a ballot in which all 3 candidates are ranked at the same level, that is, the voter is indifferent between the choice of any of the three candidates. One would also count, for the candidates (choices) A, B, and C, that A was ranked above B and C and the voter was indifferent between B and C.
How many distinct ballots can one voter produce when there are 4 candidates to be considered and a voter cannot be indifferent between two candidates and cannot "truncate" the ballot?
Truncation refers to the option when voting, not considered in the prior problems, that a voter might not rank all of the available candidates but only rank a proper subset of the candidates. Thus, with the candidates A, B, C, and D, a voter might vote for D above A and C, and be indifferent between A and C who are listed on the ballot but do not list B on the ballot.
If truncation of ballots is allowed, under what circumstances might a voter choose to not rank all of the candidates?
If truncation of ballots is allowed, what might explain the difference between voter's choosing the ballots shown in Figure 3.
Invent a new election system of your own design. What features of your system appeal to you that are not reflected in other election systems you are aware of?
Börgers, C. Mathematics of Social Choice, SIAM, Philadelphia, 2010
Saari, D., Chaotic Elections!, American Mathematical Society, Providence, 2001.
Saari, D., Decisions and Elections, Cambridge U. Press, New York, 2001.
Saari, D., Disposing Dictators, Dymstifying Voting Paradoxes, Cambridge U. Press, New York, 2008.
Taylor, A., Social Choice and the Mathematics of Manipulation, Cambridge U. Press, New York, 2005