## Course coordinator

## Catalog description

**MATH 104. Introduction to Mathematical Ideas**

(3-0) Cr. 3. F.S.SS.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry.*

Introduction to the use of basic mathematics to solve real-world problems in the areas of voting issues, measuring power in situations where people have different numbers of votes, apportionment, fair division, and elementary game theory. No prior background in politics or history is necessary for this course.

## Textbook

**Math 105
**Hunacek

ISU Custom Edition

ISBN: 9781269121453

## Syllabus

- The mathematics of voting; the various ways to determine a winner in preference ballot voting situations, and why none is totally satisfactory. (Chapter 1)
- Measurement of power in yes/no voting situations: Banzhaf and Shapley-Shubik power indices. (Chapter 2)
- The mathematics of fair division. (Chapter 3)
- Apportionment problems. (Chapter 4)
- Introduction to game theory. (Chapter 5)

## Objectives

### Understanding the basic methods and limitations of preference voting methods

- To be able to understand what the idea of a “preference ballot” is when voters choose among several alternatives
- To be able to understand how to tabulate a preference ballot and select a winner according to a variety of different voting methods: Borda Count, plurality, Hare, Coombs, Copeland, etc.
- To be able to understand the various criteria that have been developed to judge the adequacy of the voting methods discussed above
- To be able to understand and construct simple arguments or counter-examples illustrating that a various method does or does not violate a given criteria
- To be able to understand the statement of Arrow’s Theorem, which (loosely speaking) asserts that no “perfect” voting system exists

### Understanding the basic methods of yes-no voting and the assessment of power in such a voting system

- To be able to understand and analyze simple yes-no voting systems where different people may have a different number of votes
- To be able, in such systems, to determine who (if anybody) is a dictator, has veto power, etc.
- To understand the Banzhaf and Shapley-Shubik methods for computing power in such a voting system, and to be able to perform simple computations of voting power

### Understanding the basic ideas of apportionment and fair division

- To be able to understand what an apportionment problem is, and to understand the basic terminology concerning such problems
- To understand the various apportionment methods (Hamilton, Jefferson, Adams, Webster,Huntington-Hill) and perform simple calculations
- To understand the various methods that have been developed (Lone Divider, Divide andChoose, Method of Markers, Adjusted Winner, etc.) for solving problems in which an object or objects must be divided among several people

### Understanding the basic ideas of game theory

- To understand the meaning of a mathematical “game” and understand several classical examples: Chicken, Battle of the Bismarck Sea, Prisoner’s Dilemma, etc.
- To understand the difference between a zero-sum and nonzero-sum game
- To understand what a “saddle point” is in a zero-sum game and to be able to determine whether, for a given game, a saddle point exists
- To understand the concept of “equilibrium” for a simple zero-sum game
- To understand the mathematical techniques for determining optimal strategies for simple zero-sum games given by, at least, 2 x 2 matrices