Mathematics in Social Choice and Decision Making
Revised: November 2006 (Joe Klerlein)
This course covers topics from voting methods, weighted voting, fair division, apportionment, and game theory.
1. Analyze and interpret a preference schedule and the possible results using different voting methods.
2. Analyze a given weighted voting system. Calculate the Banzhaf power index and the Shapley-Shubik index for a given weighted voting system.
3. Describe the goal of a fair-division problem and calculate a discrete fair division
for a small number of players and objects when:
1. each player has an equal share;
2. the players all have different shares.
4. Calculate the apportionment of seats in a representative body when the individual population sizes and number of seats are given, using the methods of Hamilton, Jefferson, Webster, and Hill-Huntington.
5. Analyze a two-dimensional game matrix, to determine if saddle points exist and produce the players. best strategies.
Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.
Attendance policy is left to the discretion of individual instructors, subject to general university policy.
- We discuss the important problem of social choice. How does a group of individuals, each with his or her own set of values, select one outcome from a list of possibilities? This problem arises frequently in a democratic society and even in authoritarian institutions where decisions are made by more than one person. While "majority rule" is a good system for deciding an election involving just two candidates, there is no perfect way of deciding an election when three or more candidates are running. Group decision making is often a strategic encounter, and citizens need to be aware of the difficulties that can arise when some participants have an incentive to manipulate the outcome.
- We consider decision-making bodies in which the individual voters or parties do not have equal power. In particular, we will look at weighted voting systems such as the electoral college, stockholders in a corporation, or political parties in a national assembly, in which the voters cast different numbers of votes. While the notion of power is central in political science, it is typically difficulty to quantify. We will find that a voter's power in such a system may not be proportional to the number of votes that he or she is entitled to cast. We describe two well-known indices for measuring power in weighted voting systems that will enable us to assess the fairness of weighted voting systems.
- The idea of fairness in decision making becomes most explicit. Here we describe some fair-decision schemes in which a group of individuals with different values can be assured of each receiving what he or she views as a fair share when dividing up objects like cakes or the goods in an estate. An important theme of this chapter is finding procedures that produce "envy-free" allocations, in which each person gets a largest portion (as he or she values the cake or other goods) and hence does not envy anybody else.
- We discuss the apportionment problem, which is to round a set of fractions to whole numbers while preserving their sum; of course, the sum of the original fractions must be a whole number to start. Apportionment problems occur when resources must be allocated in integer quantities--for instance, when college administrators allocate faculty positions to each department.
- We discuss the mathematical field called game theory, which describes situations involving two or more decision makers having different goals. Game theory provides a collection of models to assist in the analysis of conflict and cooperation. It prescribes optimal strategies for games of total conflict in which one player's gain is equal to the other player's loss. It also provides insights into more cooperative situations in which players are trying to coordinate their choices, as well as encounters of partial conflict that involve aspects of both competition and cooperation.