Syllabus for Mathematical Game Theory

0. What is Game Theory? (Mathematics has provided insight into two types of games - situations which involve "conflict" or the making of decisions: i. combinatorial games include chess, checkers, Nim, etc. and ii. political and economic games include Prisoner's Dilemma and Chicken. We will deal primarily with the latter but will also deal with related topics such as elections and voting, two-sided markets, bankruptcy, and apportionment. Often the issue is to understand what it means to be fair and/or behave in a rational fashion in a wide variety of contexts.)

1. Taxonomy of games (Number of players, zero-sum games, non-zero-sum games, cooperative and non-cooperative games. Utility: transferable and non-transferable utility. Values for games: equilibrium concepts, Shapley value.)

2. 2-person zero-sum games (Matrix games, pure strategies, using spinners to find optimal mixed strategies; fair games; connections with linear programming. Applications in economics and political science.)

3. 2-person non-zero sum games (Prisoner's Dilemma, Chicken, Nash equilibria, congestion games; Braess's Paradox, price of anarchy. Rationality - connections to behavioral and experimental economics.)

4. Two-sided markets (Gale/Shapley models) (Matchings. Stable marriage. Gale/Shapley models, deferred acceptance algorithm; male optimal - female optimal stable solutions; school choice and other applications, kidney exchange.)

5. Elections and voting (Plurality, run-off, IRV, Borda count, Condorcet methods, approval voting. Arrow's Theorem.)

6. Bankruptcy (Proportionality, Maimonides gain and loss, the contest-garment rule and the talmudic method.)

7. Apportionment (Hamilton's, Jefferson's, Webster's and Huntington-Hill's methods. House and population monotonicity. Balinski-Young Theorem.)

8. Combinatorial Games (Nim, graph games. Nim addition and hackenbush.)

Note: Game Theory is not a standard topic in the K-12 curriculum. This course will show how game theory topics can be infused into the usual curriculum (including the CCSS-M) as ways of illustrating results in arithmetic, algebra, geometry, probability and statistics.

Text: (Recommended but not required)

Gura, Ein-Ya and M. Maschler, Insights into Game Theory, Cambridge U. Press, New York, 2008.