Mathematics and Democracy Teach Out

Ed Scheinerman

This course is aimed at anyone who has interest in the lens through which mathematicians view democracy. You will learn theories and approaches to the mathematics of voting.

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Syllabus

Module 1

This module sets the stage for the entire Teach Out. We begin by discussing the nature of mathematics (as opposed to arithmetic) and develop a workable abstract model for democracy: a function that takes individual preferences and returns a group decision. We then look closely at different methods for two-party elections, fairness criteria we want these functions to have, and conclude that only the Simple Majority method satisfies those criteria.

Module 2

This module builds on the work from Module 1 by considering elections with three (or more) candidates. We examine various decision functions (such as Plurality and Borda Counts) as well as properties we want those functions to have. We conclude with Arrow’s Theorem that shows that there are no decision functions satisfying basic fairness criteria.

Module 3

In the previous modules we assumed that each voter ranks the candidates from most to least desirable; these individual rankings are the inputs to the decision functions. In this module we question the viability of asking voters to rank both for psychological reasons (it is very difficult to rank a long list of options) and—more to the point of this Teach Out—for mathematical reasons. We model preference using a simple game played with dice that illustrates non-transitive preference: A is better than B, B is better than C, but C is better than A!

Module 4

This module considers mathematical issues arising in representative democracy in which elected officials make decisions for the larger population. In the United States House of Representatives, the number of representatives from a given state is proportional to the population of that state. However, since the number of representatives from a state must be a whole number, and the total number of representatives is 435, we need a method by which seats are allocated to states. We present the apportionment methods of Hamilton and Jefferson, and discuss problems arising with these methods. We conclude with a theorem of Balinsky and Young that shows there are no apportionment methods satisfying basic fairness conditions.

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Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Mathematics and Democracy Teach Out with these activities:

Review Basic Probability and Statistics

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Refresh and strengthen foundational skills in probability and statistics to enhance understanding of concepts related to voting methods.

Browse courses on Probability

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Review lecture notes or textbooks on probability and statistics.
Solve practice problems to test understanding.
Identify areas where further review is needed.

Linear Algebra Basics

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Review basic linear algebra skills to enhance understanding of mathematical foundations of voting systems

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Review matrix operations and properties
Refresh concepts of determinants

Organize and Review Course Materials

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Improve retention and understanding by organizing and reviewing notes, assignments, and course materials.

Browse courses on Learning Strategies

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Review and summarize lecture notes.
Organize assignments and quizzes by topic.
Highlight key concepts and make connections between different topics.

Seven other activities

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Review 'Social Choice and Welfare'

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Gain a comprehensive understanding of the foundations of social choice theory, including Arrow's Theorem and its implications for democratic decision-making.

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