People have gambled and played games for thousands of years. Yet, only in the 17th century we see a serious attempt for a scientific approach to the subject. The combinatorial foundations of probability theory were developed by various mathematicians such as J. BERNOULLI1 [4] as a means to understand games of chance (mostly involving rolls of dice) and to make conjectures according to mathematical principles.
Since then, game theory has grown into a wide field and appears at times quite removed from its combinatorial roots. The notion of a game has been broadened to encompass all kinds of human behavior and interactions of individuals or of groups and societies (see, e.g., BERNE [3]). Much of current research studies humans in economic and social contexts and seeks to discover behavioral laws in analogy to physical laws.
The role of mathematics in this endeavor, however, has been quite limited so far. One major reason lies certainly in the fact that players in real life often behave differently than a simple mathematical model would predict. So seemingly paradoxical situations exist where people appear to contradict the straightforward analysis of the mathematical model builder. A famous such example is the chain store paradox of SELTEN2 [41].
This is not withstanding the ground breaking work of VON NEUMANN and MORGENSTERN3 [34], who have proposed an axiomatic approach to notions of utilities and rational behavior of the players of a game.
As interesting and worthwhile as research into laws that govern psychological, social or economic behavior of humans may be, the present Mathematical Game Theory is not about these aspects of game theory. In the center of our attention are mathematical models that may be useful for the analysis of game-theoretic situations. We are concerned with the mathematics of game-theoretic models but leave the question aside whether a particular model describes a particular situation in real life appropriately.
The mathematical analysis of a game-theoretic model treats objects neutrally. Elements and sets have no feelings per se and show no psychological behavior. They are neither generous nor cost conscious unless such features are built into the model as clearly formulated mathematical properties. The advantage of mathematical neutrality is substantial, however, because it allows us to embed the mathematical analysis into a much wider framework.
This introduction into mathematical game theory sees games being played on (possibly quite general) systems. A move of a game then correspond to a transition of a system from one state to another. Such an approach reveals a close connection with fundamental physical systems via the same underlying mathematics. Indeed, it is hoped that mathematical game theory may eventually play a role for real world games akin to the role of theoretical physics to real world physical systems.
The reader of this introductory text is expected to have knowledge in mathematics, perhaps at the level of a first course in linear algebra and real analysis. Nevertheless, the text will review relevant mathematical notions and properties and point to the literature for further details.
The reader is furthermore expected to read the text “actively”. “Ex.” marks not only an “example” but also an “exercise” that might deepen the understanding of the mathematical development.
The book is based on a one-term course on the subject the author has presented repeatedly at the University of Cologne to pre-master and master level students with an interest in applied mathematics, operations research and mathematical modelling.
It is dedicated to the memory of WALTER KERN.4
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1 J. BERNOULLI (1654–1705)
2 R. SELTEN (1930–2016)
3 J. VON NEUMANN (1903–1953), O. MORGENSTERN (1902–1977)
4 W. KERN (1957–2021)