Michael A. Jones
Associate Professor
Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043
jonesm@mail.montclair.edu; 973-655-5448; 973-655-7686 [fax]

 

Pedagogical Materials

I try to make math for liberal arts courses more cohesive by connecting topics that are not usually connected in the curriculum; I also like taking some of the topics in math for liberal arts courses and bringing them into courses for the mathematics major.  These are the ideas behind my paper "Connecting Fair Division and Game Theory through the Optimization of Knaster's Procedure" that appeared in PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies), Vol XIII n4 (Dec. 2003) 321-336.  Here is the abstract:

Abstract for
Connecting Fair Division and Game Theory through the Optimization of Knaster's Procedure

In 1945, Bronislaw Knaster proposed a procedure to divide any number of indivisible goods between a finite number of players requiring the players to place monetary values or bids on all of the goods.  Often discussed in math for liberal arts courses that concentrate on contemporary applications of mathematics for non-major students, Knaster's procedure provides an opportunity to introduce optimization to students who will never take a course in calculus.  A simple analysis of the procedure can lead students to determine optimal monetary bids, given the bids of the other players.  More advanced students can explicitly prove these results.  The optimization problem naturally leads to pure strategy Nash equilibria of Knaster's procedure when viewed as a game, thereby providing a transition between fair division procedures and game theory that can be used in both math for liberal arts courses and upper level courses.

In the mathematics curriculum, typically, optimization is first discussed in calculus.  However, the ideas behind optimization are natural and do not require the mathematical sophistication of the calculus.  In the paper "Fairness, How to Achieve It, and How to Optimize in a Fair Division Procedure," S.F. Cohen and I show how high school students can learn about fairness and optimization without calculus by optimizing in the Adjusted Winner Procedure.  This paper is forthcoming in Mathematics Teacher.

 Richard J. Maher of Loyola University, Chicago organized a session on "Innovative Methods in Courses Beyond Calculus" at MathFest, the summer meeting of the Mathematical Association of America in 2001.  I presented the paper "Integrating Combinatorics, Geometry, and Probability through the Shapley-Shubik Power Index."  This paper is co-authored with Matthew Haines of Augsburg College in Minneapolis, MN.

Our paper is to appear in a volume of the MAA Notes series.  You can download it here: mathfest.pdf.  The paper works well as a primer on simple weighted-voting games and the Shapley-Shubik power index.  We have also included a number of exercises and examples.  These feature salient points connecting different topics and demonstrate the diverse mathematical techniques that arise in problems connected to the Shapley-Shubik power index.The references also include many citations of instructional materials and research materials that apply simple weighted-voting games to modeling of political institutions.

Jenny Dorrington (then at Colorado College, and now finishing up a degree in Landscape Architecture!) and I wrote an article about finding saddle points, the natural solution concept in zero-sum games.  Our article, "Emphasizing Saddle Points through Game Theory: A Classroom Activity," appeared in PRIMUS, Vol X n3 (2000) 206-218.  The abstract appears below:

Abstract for
Emphasizing Saddle Points through Game Theory:  A Classroom Activity

In optimization problems, often students find a single extremum of a function that is assumed to be a maximum or a minimum.  At best, saddle points are discarded when checking second order conditions en route to maxima or minima.  Game theory provides a setting where saddle points are the solution concept.  We introduce the necessary game-theoretic background and explain how game-theoretic experiments of the Matching Pennies game can be used as a classroom activity to develop intuition about saddle points.  Paralleling how students learn to find extrema, we first consider finding saddle points on the interior of a set and then consider saddle-like points that appear on the boundary of compact sets.  We conclude with a couple of examples from the game theory literature.

Diana Thomas (then at New Jersey City University; now at Montclair StateUniversity) and I had a paper published in the Proceedings of the Second Regional Conference on Quantitative Reasoning Across the Disciplines (QUAD) Conference at Richard Stockton College of New Jersey in Pomona, New Jersey.  The QUAD Conference was held on March 6, 1999 and was sponsored by the National Science Foundation.  Our paper was titled, "The Mathematical Control of Sleep/Wake Cycles," and promotes using circadian rhythms to motivate cyclic behavior in Precalculus and Differential Equations course.  The Microsoft Excel project for the Precalculus material is below.  The Differential Equations project is a Mathematica file.  It appears below, too.  (To view it however, you may need to save it first and then open it from your hard disk.)  The paper is below also.  It is a .dvi file.

Circadian Rhythm Excel  Project
Circadian Rhythm Mathematica  Project

Circadian Rhythm  DVI File
All of the graphics in the paper are not visible at the moment; I'll figure it out later.  Please e-mail me if you would like a clean copy or if you would like me to notify you when I have fixed this.

E-mail me at jonesm@mail.montclair.edu