# Introduction

The phenomenon of cooperative interactions among animals has puzzled biologists since Charles Darwin. Nevertheless, theoretical concepts to study cooperation appeared only a century later and originated in economics and political sciences rather than biology. John von Neumann and Oskar Morgenstern developed a mathematical framework termed Game Theory to describe interactions between individuals. This theory emerged in the wake of World War II and was mainly intended to provide a strategic basis to prevent a nuclear holocaust. John Nash, working at the post-WWII US military think tank, the RAND Corporation, augmented the theory by developing and introducing the concept of equilibria, the so-called Nash equilibrium:

- An equilibrium is reached as soon as no party can increase its profit by
*unilaterally*deciding differently.

Another generation later, John Maynard Smith and George R. Price ingeniously related the economic concept of *payoff functions* with *evolutionary fitness* as the only relevant currency in evolution. Furthermore, Maynard Smith refined the concept of Nash equilibria in an evolutionary context and introduced the notion of evolutionary stable strategies (ESS). All ESS represent a subset of the Nash equilibria because an ESS applies only at the population level and adds stability requirements.

- A strategy is called evolutionarily stable if a population of individuals homogenously playing this strategy is able to outperform and eliminate a small amount of any mutant strategy introduced into the population.

These achievements mark the advent of an entirely new, approach to behavioral ecology where theoretical models and predictions inspired and continue to inspire numerous experiments and field studies.

## Prisoner's Dilemma

The Prisoner's Dilemma is probably the most famous mathematical methaphor for modelling the evolution of cooperation. This section gives a brief introduction into the Prisoner's Dilemma and provides examples of the game dynamics in well-mixed populations with random interactions as well as structured populations with limited local interactions.

## Snowdrift Game

A closely related game, which is also addressing the problem of cooperation but under slightly relaxed conditions, is called the Snowdrift game. This section briefly introduces the Snowdrift game and then, similarly to the Prisoner's dilemma, exemplifies the game dynamics in well-mixed populations as well as structured populations.

## Rock-Scissors-Paper Game

The fascination of the Rock-Scissors-Paper game is not restricted to children but is equally thrilling for evolutionary biologists. The cyclic dominance of the three strategies can lead to very interesting dynamics both in well-mixed as well as structured populations.

## References

Selected publications on game theory and evolutionary game theory:

- von Neumann, J. & Morgenstern, O. (1944)
*Theory of Games and Economic Behaviour*, Princeton: Princeton University Press. - Nash, J. (1950)
*The bargaining problem*, Econometrica**18**155-162. - Maynard-Smith, J. & Price, G. R. (1973)
*The logic of animal conflict*, Nature**246**15-18. - Axelrod, R. (1984)
*The Evolution of Cooperation*, New York: Basic Books. - Hofbauer, J. & Sigmund, K. (1998)
*Evolutionary Games and Population Dynamics*, Cambridge: Cambridge University Press. - Sigmund, K. (1995)
*Games of Life*, Harmondsworth, UK: Penguin. - Nowak, M. A. (2006)
*Evolutionary Dynamics*, Harvard University Press, Cambridge MA, USA.

## Acknowledgments

The development of these pages would not have been possible without the encouragement and the insightful advice of Karl Sigmund. Financial support for the first version of these pages of the Swiss National Science Foundation is gratefully acknowledged.