Mathematical modelling is a subtle art. Typically, the situation we would like to understand seems hopelessly complicated, and it is difficult to isolate the essential features and put them into a mathematical framework. Formulating the model is far from the final step: then we analyze it and compare the results with actual data. Does the model reflect the phenomena we care about, qualitatively? How closely does it match the data? Can we understand it conceptually? How could we improve it?
It's important to contrast this point of view with that in introductory physics courses. There, we view mathematical models as literal descriptions of physical reality. Of course, we know the descriptions aren't complete, and really aren't even true, since they neglect quantum and relativistic effects. However, the naïve point of view is appropriate, since the models are at least "nearly true." When we're dealing with such simple, fundamental principles, it's hard to get worked up about subtleties.
Undergraduate math majors often graduate without ever confronting these issues. In an ideal world, mathematical modelling would feature prominently in introductory courses in subjects that use mathematics, but that is rarely the case. For example, introductory economics would be an excellent place to treat mathematical modelling. Economic models make no pretense of being literally true, but that doesn't mean they aren't useful. Unfortunately, many students in introductory economics courses are mathematically unsophisticated, which makes it difficult to study modelling in any depth.
Mesterton-Gibbons' book deals with mathematical modelling, not by an abstract discussion of how modelling should be done, but rather by presenting many concrete examples. His theme is game theory, and particularly how it can be applied to modelling the strategic behavior that animals engage in. The book assumes relatively little background: a solid grasp of calculus, and some familiarity with probability, matrices, and multivariable calculus. It would be appropriate for use with junior math majors or advanced sophomores, and would also interest economics majors, as well as biology majors with strong mathematics backgrounds.
The mathematics described in Mesterton-Gibbons' book is fascinating, and well worth studying for its own sake even if one doesn't care about mathematical modelling. Unfortunately, I found the book frustrating to read, because much of it consists of extremely detailed calculations. It's not much fun to read through an eight-page computation line by line, and I rapidly gave up checking the details. I believe that few undergraduates would study this book with the care it deserves.
I do not want to give the impression that I consider the detailed calculations a flaw in Mesterton-Gibbons' writing. One of the book's strengths is that it analyzes interesting examples, rather than artificial examples chosen to take only one page. Because the analysis is necessarily lengthy, it would be dishonest to hide it by omitting the details. This book's examples fill the sad gap between the single-step problems one solves in calculus textbooks and the multi-step problems one faces in real life.
It's not clear to me how best to use this book. The examples are great, and the author has clearly put enormous effort into building this collection. They could form the basis of a wonderful course. However, to be honest, I would hesitate to choose this book as a textbook. If I didn't follow it closely, I doubt many students would study it. If I did use it, I might have trouble keeping the class engaged throughout the calculations. Of course, the difficulty is that it is much more interesting to do calculations oneself than to watch someone else do them. In light of this, I think the book would be perfect as a source of problems for a Moore method course. The arrangement exactly fits what would be required: all the concepts are introduced when they arise naturally in examples, and the examples grow gradually in sophistication. All it would take would be to disassemble them into manageable chunks that could be assigned to the students.
The outline of Mesterton-Gibbons' book is as follows:
- Chapter One introduces non-cooperative games, and investigates what should be considered a solution of a game, in particular Nash equilibria.
- Chapter Two discusses criteria for singling out one Nash equilibrium as the solution of a game. Population games and evolutionary stable strategies appear.
- Chapter Three introduces cooperative games.
- Chapter Four covers "characteristic function games," cooperative games that deal with dividing some benefits among several players.
- Chapter Five studies the Prisoner's Dilemma and the evolution of cooperation.
- Chapter Six analyzes a number of population games that illustrate various sorts of animal behavior.
- Chapter Seven takes a look back and considers the examples in a broader, more philosophical context.
A number of topics from a standard introduction to game theory are covered, but not all of them. For example, the minimax theorem for zero-sum games arises only tangentially, and is never formally stated, let alone proved. Of course, that's not a criticism — Mesterton-Gibbons simply has other goals, and is not trying to write a standard introduction to game theory. My only reason for mentioning it is so that nobody mistakes this book for an ordinary game theory book with a slight modelling flavor.
Each chapter contains a large number of well chosen exercises. Most of them involve verifying parts of the analysis in the text or extending it in natural ways, but several of them are quite striking (such as Exercise 4.24, which looks at the weighted voting scheme used by the Nassau county board of supervisors in 1937). An appendix contains solutions for most of the exercises.
The bibliography contains 244 items. It will be a wonderful resource for students who would like to explore these topics in more detail, for example to write senior theses.
To sum up, this book is a valuable contribution to the literature on game-theoretic modelling. It is not written in the usual textbook style, and it may require some thought to use it in a course. However, I know of nothing like it as a collection of illuminating examples. Everyone interested in game theory or mathematical modelling should take a look at it.
Henry Cohn is a postdoc in the theory group at Microsoft Research, and holds a five-year fellowship from the American Institute of Mathematics. His primary mathematical interests are number theory, combinatorics, and the theory of computation.