**Syllabus: Mathematical Modeling (Behavioral Sciences)**

Prepared by:

Joseph Malkevitch

Department of Mathematics and Computer Studies

York College (CUNY)

Jamaica, New York

email:

__malkevitch@york.cuny.edu__

web page:

__http://york.cuny.edu/~malk__

Brief description

Mathematical modeling is the branch of mathematics which helps individuals, governments, businesses and organizations to get insight into problem situations which arise outside of mathematics. This course will focus on mathematical modeling in the behavioral sciences. Examples of models of interest in political science, economics, psychology, and sociology will be examined. Mathematical tools that will be looked at include zero-sum games, non-zero sum games, graph theory (distance, center, eccentricity), Gale-Shapley deferred acceptance algorithm, measuring power, etc. The course will be self-contained and will provide many instances where traditional mathematical skills (arithmetic, algebra, geometry) can be put to use in K-12 classrooms in a way that supports the Common Core Standards in Mathematics.

Syllabus:

1. A sampler of behavioral science problems that can be modeled using mathematics.

2. Discrete and continuous fair division problems (cut and choose, Crawford cut and choose, adjusted winner, auctions).

3. 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.)

4. 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.)

5. 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.)

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

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.)

There will be no official text but students may find it of use to look at:

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