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Discrete Math: Social Choice

Social choice is an area of discrete mathematics that includes apportionment and voting methods. Apportionment refers to assigning representatives based on the relative sizes of a larger group. Consider this example:

There are ten student council representatives and 30 freshmen, 20 sophomores, 20 juniors, and 30 seniors. When representatives are assigned, larger groups should have more representation. Here, the freshmen and seniors would have three representatives each and the sophomores and juniors would have two representatives each, with one representative the equivalent of 10 students.

Very rarely do the numbers work out so nicely in real apportionment situations. The problems below are more realistic and look at a variety of apportionment algorithms.

Voting methods are also included here as a social choice topic. One person, one vote is a common voting method, but there are other methods such as the Borda count, sequential runoff, and Condorcet methods. Several of the following problems are based on various voting methods.

For background information elsewhere on our site, explore the High School Discrete Math area of the Ask Dr. Math archives. To find relevant sites on the Web, browse and search Discrete Mathematics in our Internet Mathematics Library.

Access to these problems requires a Membership.

The 2000 Olympics - Leigh Nataro
Discrete Math, difficulty level 4. Use the Borda count method of voting to see which country will host the Olympics. ... more>>

An Apportionment Problem - James Greene
Discrete Math, difficulty level 4. Students apportion the representatives in their class using the Hamilton method of apportionment. ... more>>

The Council of Nations - Leigh Nataro
Discrete Math, difficulty level 4. Use the Webster method of apportionment to divide up the seats for a council of nations. ... more>>

First Presidential Veto - William Bowdish
Discrete Math, difficulty level 4. This problem involves the Jefferson, Adams, and Webster apportionment methods. ... more>>

Homework and Candy - Leigh Nataro
Discrete Math, difficulty level 1. Use the Hamilton method to apportion candy among three children. ... more>>

Let's Vote - Leigh Nataro
Discrete Math, difficulty level 3. Create preference schedules that have different winners using the plurality and Borda count methods of voting. ... more>>

Voting on Survivor - Leigh Nataro
Discrete Math, difficulty level 2. Using a modified Borda count method, figure out how many people voted for each preference schedule to produce a tie. ... more>>

A Voting Problem - Leslie Johnson Nielsen
Discrete Math, difficulty level 2. Election theory: investigate the concept of plurality in voting and discuss its fairness. ... more>>

Will It Be Pumpkin Pie? - Leigh Nataro
Discrete Math, difficulty level 4. Based on voting coalitions and power indices, who has more power in deciding what will be served for dessert? ... more>>

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