Title
Fair is Fair (Greta Lawlor)
Goals
1) Students will explore several voting methods and decide which is the "most" fair using Arrow's Conditions.
2) Students will be able to use and justify a method for determining a group ranking and apply Arrow's Method to justify that their method is fair.
3) Students will work in small groups to come to a consensual solution and be able to justify that solution.
Abstract
This activity is set in the context of choosing a site for a recreation center geared at teenagers. Students are to determine a group ranking based on a set of already collected preferences and to apply several voting methods to decide on the site for the recreation center. They will use Arrow's conditions to justify the "fairness" of their solution.
Problem Statement
Discuss with students what advantages the recreation center might have for a community. Also, discuss the disadvantages. Let students know that they will be looking at the problem in the context that the recreation center is sought after by all communities under consideration. The students will be examining the voter information and examining ways to some up with a fair way to decide the location of the center.
Instructor Suggestions
1) Set the stage with the discussions of the advantages and disadvantages of the center.
2) Distribute the activity "Fair is Fair" and after the students have read the problem, discuss any questions they may have.
3) Have the students work in groups and use at least two methods to determine a winner for the recreation center.
4) Have the students present their solutions and justify their solution using Arrow's conditions.
5) Discuss the students work in relation to "fairness."
Materials
"Fair is Fair" activity sheet, transparencies, markers.
Time
Introduction to the problem (5 minutes), group work (25 minutes), presentations and discussion (25 minutes.)
Mathematics Concepts
Discrete Mathematics Concepts
Group ranking, plurality, majority, Borda Method, runoff, sequential runoff, Condorcet method, fairness, approval, Arrow's conditions, weighted voting, winning coalitions, power index
Related Mathematics Concepts
Matrices, permutation, and combinations.
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Algebra, Geometry, Discrete Mathematics
Colorado Model Content Standards
Number Sense (1), Algebraic Techniques (2), Geometric Techniques (4), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into Algebra or Geometry class as the topic of matrices and their related operations are discussed. Borda's method would bridge the integration. This activity could also be used when permutations and combinations are discussed particularly when determining power indices and winning coalitions.
Further Investigation
This activity could be extended by changing the demographic data or by changing the premise that the recreation center was not wanted by the communities. Students idea of fairness must fit with the premise.
Variations/Comments
This activity could be used as an assessment if enough time is given. More restriction might be used to limit the extent of the problem. It would also be a valuable problem of the week.
References/Resources
Colorado Model Mathematics Standards Task Force. (1995) Colorado Model Content Standards for Mathematics.
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete Mathematics Through Applications. New York: W. H. Freeman and Company.
National Council of Teachers of Mathematics. (1989) Curriculum and Evaluation Standards for Mathematics.