Decision-making: Game Theory and its Applications
Alan Sauter
Department of Developmental Math
Background
Game theory can be described rather simply as the study of how ‘rational’ decisions
are made by individuals with different goals. These ‘individuals’ may be people,
companies, countries, animals, genes, or even computers, to name a few.
The origin of mathematical approaches to decision-making and the study of games are
often dated to Blaise Pascal and the development of probability in the 17th century. The
foundations of modern game theory as a subject in its own right are generally seen as
beginning with John von Neumann. In 1944 he published Theory of Games and
Economic Behavior (with Oskar Morgenstern) in which they presented a mathematical
approach to game theory and its use in explaining economic behavior. From that point on,
the study of game theory and its applications in creating models for a variety of
circumstances in economics, the natural and behavioral sciences, mathematics, etc., grew
exponentially. Some of the most important contributions to game theory were made by
John Nash (well known to everyone from the movie ‘A Beautiful Mind’) in the 1950’s,
when he published several papers on game theory, including his discovery of what
became known as the ‘Nash equilibrium’. Many later contributions are generalizations of
his ideas.
Rationale
For the last twelve years my research has been focused mainly on Riemannian
geometry. I originally became familiar with the work of John Nash through his
contributions in this area. (He obtained a rather important result known as the ‘Nash
embedding theorem’.) I then became aware of his contributions in game theory. I have
had a long-standing interest in the use of mathematics to model real-world situations.
(Previous to my studies in Riemannian geometry I had focused on chaos theory and
fractal geometry. These mathematical subjects are useful in explaining certain natural
phenomena, such as models involving ‘strange attractors’.)
I am interested in learning more about game theory and how it can be used in
applications. This study grant would allow me the opportunity to pursue this interest. I
also wish to explore whether the ideas of game theory can be adapted to a more
geometric approach for modeling purposes. That is, by studying the basic mathematical
ideas of game theory and its applications, I aim to look for ways in which it can be
connected to the geometric approaches I am familiar with from my previous studies.
Mathematical topics are often divided up into ‘discrete mathematics’ and ‘continuous
mathematics’. Most of game theory is considered to be part of the ‘discrete’ branch and
most of geometry falls under the ‘continuous’ branch. I’ve always found it interesting
that Nash’s most important contributions came in these two seemingly dichotomous areas.
Plan of Study
My studies would begin by with applications to politics and elections. The game
theory models used here are among the simplest and it would present a good opportunity
to master the basic (though still powerful) mathematical tools while studying some rather
important applications. (I will also admit to a slight bias in choice of topics – I began my
college career as a philosophy and political science student.) These topics lead naturally
into a study of international relations and defense strategies.
The next focus would be on game theory used in economic modeling and decisionmaking. This is probably the area that has been impacted the most by the development of
game theory. (Nash received the Nobel Prize for Economics, the influence of his ideas
was so extensive.) My goal is to look at examples from both macroeconomics and
microeconomics.
This would be followed by a study of the use of game theory to explain certain
features of the process of evolution. There are a number of examples of the application of
game theory in the sciences. I chose this because of my long-standing interest in
evolution (and genetics) and the mathematical models used to describe how it ‘works’. I
was profoundly influenced twenty years ago by reading Richard Dawkins’ The Selfish
Gene (the one book on my list I have already read) - which I’ve always maintained was a
mathematical book without mathematics!
Although I intentionally chose readings throughout that emphasize a mathematical
approach (with a couple of exceptions), I wanted to set aside some time for studying the
tools of game theory from a strictly mathematical perspective. Thus, I would spend
several weeks learning some more advanced techniques of mathematical game theory,
while still keeping an eye on their usability for modeling. (This would also give me a
chance to read some of the key papers in the foundations of game theory by Nash, Kuhn,
Shapley, Aumann, etc.)
Finally, a few brave souls have attempted to introduce a game-theoretic framework
into philosophical discussions and I would conclude by studying a couple of those works
in the areas of religion and political philosophy.
Decision-Making: Game Theory and its Applications
Study Plan
Weeks 1 – 3: Game Theory and Politics
First, study the application of game theory to voting systems, election strategies and
social decision-making. Second, examine a specific example of a political process (union
organizing) from a game theoretic perspective.
The Mathematics of Voting and Elections: A Hands-on Approach
Hodge, Jonathan and Kilma, Richard; American Mathematical Society (2005)
Mathematics and Politics: Strategy, Voting, Power and Proof
(selected chapters on voting systems, political power and social choice)
Taylor, Alan; Springer-Verlag (1995)
Racial Competition and Class Solidarity
Boswell, Terry et al; SUNY Press (2006)
Weeks 4 – 5:
Game Theory and International Relations
Study the use of game theory in analyzing international conflict, arms races, deterrence,
and issues of national security.
Game Theory and National Security (selected chapters)
Brams, Steven and Kilgour, D. Marc; Basil Blackwell (1988)
‘Arms control, game theory and the twenty-first century’ by Kilgour, D. Marc
in Terrorism and Weapons of Mass Destruction: Responding to the Challenge
Bellany, Ian (ed.); Routledge (2007)
Mathematics and Politics: Strategy, Voting, Power and Proof
(selected chapters on models of escalation and international conflict)
Taylor, Alan; Springer-Verlag (1995)
Weeks 6 – 8: Game Theory and Economics
Study how game theory is used to create economic models of competition, trade,
bargaining, etc.
Game Theory with Applications to Economics (selected chapters)
Friedman, James; Oxford University Press (1986)
Game-Theoretic Models of Bargaining (selected chapters)
Roth, Alvin (ed.); Cambridge University Press (1985)
The Economics of Sports Broadcasting
Ch. 6 – ‘Competition between channels: a game theory approach’
Gratton, Chris and Solberg, Harry; Routledge (2002)
Week 9: Game Theory and Science
Study the use of game theory in explaining evolutionary processes.
The Survival Game: How Game Theory Explains the Biology of Cooperation and
Competition
Barash, David; Times Books (2003)
The Selfish Gene (3rd ed.)
Richard Dawkins; Oxford University Press (1976)
Weeks 10 – 13: Mathematics of Game Theory
Study more advanced mathematical treatments of game theory and strategy. Also review
some of the seminal papers in the development of game theory.
An Introduction to Game-Theoretic Modelling (2nd ed)
Mesterton-Gibbons, Michael; American Mathematical Society (2001)
Playing for Real: A Text on Game Theory (selected chapters)
Binmore, Ken; Oxford University Press (2007)
Classics in Game Theory (selected chapters)
Kuhn, Harold (ed.); Princeton University Press (1997)
Weeks 14 – 15: Game Theory and Philosophy
Examine approaches to philosophical discussions within a game-theoretic framework.
Superior Beings: If They Exist, How Would We Know (2nd ed.)
Brams, Steven; Springer (2007)
The Game of Justice: A Theory of Individual Self-Government
Lane, Ruth; SUNY Press (2007)
Secondary Sources (time permitting)
A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature
Siegfried, Tom; Joseph Henry Press (2006)
(A layman’s introduction to game theory with discussions of Von Neumann, Nash,
Pascal and Freud.)
Competition: The Birth of a New Science
Case, James; Hill and Wang (2007)
(Another layman’s introduction.)
The Game of Language: Studies in Game Theoretical Semantics and Its Applications
Hintikka, Jaako and Kulas, Jack; D. Reidel Publishing Company (1983)
Game Theory and Policymaking in Natural Resources and the Environment
Dinar, Ariel et al (ed.); Routledge (2008)
(Includes game theory as applied to water allocation, fisheries, etc.)