Jun Tanimoto - Fundamentals of Evolutionary Game Theory and its Applications

Graduate School of Engineering Sciences, Kyushu University Interdisciplinary, Fukuoka, Fukuoka, Japan

We define the word system as a collection of elements, all of which are connected organically to form an aggregate of elements that collectively possess an overall function. We know that most real systems are not time constant but time variable, i.e., they are dynamical system s. According to the common sense of the fields of science and engineering, a dynamical system can be described by space and time variables, i.e., x and t . Therefore, a dynamical system has a spatiotemporal structure.

Any system in the real world looks very complex. An environmental system is a typical example. If an environmental system is interpreted literally, considering every system involved with the environment, we can see there is a lot of variety within it.

This variety arises from interactions between different environments (e.g., natural, human, and social) and differences in spatial scale (i.e., from the microscopic world weaved by microorganisms to the global environment as a whole, see Fig. ). Accordingly, we have coined the phrase human environmental social system to encompass all these diverse phenomena.

One important aspect that is revealed when you shed some light on the humanenvironmentsocial system is that human intention and behavior, either supported by rational decision making, in some cases, or irrational decision making, in others, has a crucial impact on its dynamics. In fact, what is called global warming, as one example of a global environmental problem, can be understood because of human overconsumption of fossil fuels over the course of the past couple of centuries, which seems rational for people only concerned with current comfort but seems irrational for people who are carefully considering long-term consequences. Hence, in seeking to establish a certain provision to improve environmental problems, one needs to consider complex interactions between physical environmental systems and humans as well as social systems as a holistic system of individuals. In general, the modeling of the human decision-making process or actual human behavior is harder than that of the transparent physical systems dealt by traditional science and engineering, because the governing mathematical models are usually unknown. What we can guess concerning these processes is not expressed as a set of transparent, deterministic, and explicit equations but black box-like models or, in some cases, stochastic models. At any rate, in order to solve those problems in the real world, we must build a holistic model that covers not only environment as physical systems but also human beings and society as complex systems. Although this may be a difficult job, we can see some possibility of progress in the field of applied mathematical theory, which can help to model complex systems such as human decision-making processes and social dynamics. Even if it is almost impossible to obtain an all-in-one model to perfectly deal with the three spheres, i.e., environmental, human, and societal, which have different spatiotemporal scales as well as different mechanisms, it might be possible to establish bridges to connect the three. One effective tool to do this is evolutionary game theory .

Why do we cooperate? Why do we observe many animals cooperating? The mysterious labyrinth surrounding how cooperative behavior can emerge in the real world has attracted much attention. The classical metaphor for investigating this social problem is the prisoners dilemma (PD ) game, which has been thought most appropriate, and is most frequently used as a template for social dilemma.

Evolutionary game theory (e.g. Weibull ) has evolved from game theory by merging it with the basic concept of Darwinism so as to compensate for the idea of time evolution, which is partially lacking in the original game theory that primarily deals with equilibrium.

Game theory was established in the mid-twentieth century by a novel contribution by von Neumann and Morgenstern (von Neumann and Morgenstern ).

Since these developments, thousands of papers have been produced on research performed by means of computer simulations. Most of them follow the same pattern, in which each of the new models they build a priori is shown with numerical results indicating more enhanced cooperation than what the theory predicts. Those are meaningful from the constructivism viewpoint, but still less persuasive in answering the question: What is the substantial mechanism that causes mutual cooperation to emerge instead of defection?

Nowak successfully made progress in understanding this problem, to some extent, with his ground-breaking research (Nowak ) finally solved the puzzle, which was originally posed by Charles Darwins bookThe Origin of Species (1859)of why sterile social insects, such as honey bees, leave reproduction to their sisters by arguing that a selection benefit to related organisms would allow the evolution of a trait that confers the benefit but destroys the individual at the same time. Hamilton clearly deduced that kin selection favors cooperative behavior as long as the inclusive fitness surge due to the concept of relatedness is larger than the dilemma strength. This finding by Nowak, though he assumed several premises in his analytical procedure, elucidates that all the reciprocity mechanisms ever discussed can be explained with a simple mathematical formula, very similar to the Hamilton Rule, implying that Nature is controlled by a simple rule. The Nowak classificationskin selection, direct reciprocity , indirect reciprocity , network reciprocity , and group selection successfully presented a new level to the controversy, but there have still been a lot of papers reporting how much cooperation thrives if you rely on our particular model-type stories, because Nowaks deduction is based on several limitations, and thus the real reciprocity mechanism may differ from it. In fact, among the five mechanisms, network reciprocity has been very well received, since people believe complex social networks may relate to emerging mutual cooperation in social system.

This is why this book primarily focuses network reciprocity in Chap..

This book does not try to cover all the developments concerning evolutionary games, not even all the most important ones. In fact, it strives to describe several fundamental issues, a selected set of core elements of both evolutionary games and network reciprocity , and self-contained applications, which are drawn from our studies over the last decade.