Mathematics

### Material on the Final Exam

The final exam will cover voting methods (Chapters 12 of For All Practical Purposes), game theory, and weighted voting systems (Chapters 13 of For All Practical Purposes). It will NOT cover apportionment (Chapters 15 of For All Practical Purposes).

#### Voting methods

Find the winner of an election using (1) plurality voting, (2) Borda count, (3) the Hare system, (4) sequential pairwise voting with a given agenda.

Answer questions such as: Does this example violate the Condorcet winner criterion? Independence of Irrelevant Alternatives? Monotonicity? The Pareto Condition?

Some of the voting methods always satisfy certain of the fairness criteria. You should be able to explain why.

#### Game theory

Dominated strategies and Nash equilibria for zero-sum games. "Game Theory and Strategy," pp. 11 - 12, exercises 1, 2, 3. On exercise 3, omit the "movement diagrams." I suggest finding Nash equilibria by the "circle and box" method described in lecture.

Mixed strategies for zero-sum games. Problems assigned Feb. 25.

Game trees: solution by rollback. "Game Theory and Strategy," pp. 42-43, exercises 3, 5a.

Dominated strategies, Nash equilibria, and Pareto optima for non-zero-sum games. "Game Theory and Strategy," p. 72, exercises 2ab; last problem assigned March 5. I suggest finding Nash equilibria by the "circling" method described in lecture.

Commitments, promises, and threats. Problems assigned March 19, 1-5; problems assigned March 23.

Constructing and analyzing game theory models: "Game Theory and Strategy," p, 80, exercise 5; p. 92, exercises 3ab.

Three-person games: Problems assigned March 26.

#### Weighted voting

Identify dictators, players with veto power, dummies.

Find a weighted voting system that corresponds to a voting system described in words.

Find the Banzhaf power distribution of a weighted voting system.

Use the combination formula to find the Banzhaf power distribution of a weighted voting system.

Find the Shapley-Shubik power distribution of a weighted voting system.

#### Personal Help Sheets

You may bring to the test one sheet of paper for each of the three topics containing whatever you find helpful. (Three sheets total. This is to allow you to reuse sheets you made for the earlier tests, if you want.)