A look at the applications of mathematics to various aspects of politics and other examples of social interaction. The main applications can be grouped under the headings of Conflict, Voting, and Power.
Situations of conflict occur whenever people have opposing goals. The subject of Game Theory is the mathematical study of strategic behavior, the goal of which is to systematically identify the best possible strategies in situations of conflict. The players of these games could be individuals, corporations, or entire nations. Part of the course may include looking at specific historical episodes in International Relations from a game-theoretic viewpoint, including the Cuban Missile Crisis, the US/Soviet arms race, and the Yom Kippur War. Game Theory is introduced in Chapter 4 of the text, and covered in detail in Chapters 4 and 10. We intend to cover more Game Theory than what is in our text.
There are two different types of voting subject to mathematical analysis:
There are various schemes for Yes/No voting, ranging from simple majority voting to weighted voting systems to complicated procedures such as the US Federal government's method for passing laws with its two houses, presidential veto power, and congressional overrides of the presidential veto. We will look at some of these schemes with an eye toward seeing the relationships between them. Yes/No Voting Theory is covered in Chapters 2 and 8 of our text.
How does one measure political power? Various power indices have been proposed for assigning a numerical measure of power in Yes/No voting systems. However, the different indices tend to yield widely different measures of power. We intend to look at the more common power indices, compare and contrast them, and try to understand why they vary so much. Political Power is covered in Chapters 3 and 9 of our text.
In addition to conflict, voting and power, the mathematical applications to social science which are discussed in our text include Fair Division (see chapters 5 and 11), and Escalation (chapters 6 and 12.)
Escalation can be seen in various types of auctions, where the bidding wars can result in prices paid for items that are perhaps far more than the item is worth. The same sort of behavior can be seen in the arms races of various countries.
Fair division is just what it sounds like. How does one go about dividing up something in a fair way? Questions like this come up in all sorts of situations. One typical situation might be heirs dividing up an estate; another could be a team dividing up it's winnings from a game. And it comes up in politics in the problem of apportionment.
As stated in our introductory meeting, there is not enough time to cover all these topics in one semester, so a couple of topics will have to be skipped or only mentioned briefly. Since I covered social choice voting in detail the last time I taught this course, this time we will omit it. Therefore, we intend to begin the course with yes/no voting, and go from there directly to the related topic of measuring political power. Thus we begin with Chapters 2 and 3, with a selection of further topics from Chapters 8 and 9. Once we complete that we will cover the topic of game theory, covering Chapter 4 and perhaps topics from Chapter 10. I will supplement the material from the text with more material from other sources. If time permits at the end of the course, and if there is interest, we may come back to our text to look at the fairness chapters (5 and 11). We will not have time to cover escalation.
No mathematics beyond high school algebra is assumed. The requirements for the course are merely passing QRI, and an interest in seeing mathematics applied to social situations and politics.
