Dynamical Systems Game

(with K. Kaneko)

1. Framework and application to 1-person games

A theoretical framework we call dynamical systems game is presented, in which the game itself can change due to the influence of players' behaviors and states. That is, the nature of the game itself is described as a dynamical system.
The relation between game dynamics and the evolution of strategies is discussed by applying this framework. Computer experiments are carried out for simple one-person games to demonstrate the evolution of dynamical systems with the effective use of dynamical resources.

2. Application to the problem of Social Dilemma

The ``social dilemma'' is a problem inherent in forming and maintaining cooperation among selfish individuals, and is of fundamental importance in the biological and social sciences. From the viewpoint of traditional game theory, the existence of the social dilemma necessarily implies degeneration into selfish behavior as the numbers of members in a community increases, unless there exists some external power.

In the real world, however, cooperation is often formed and maintained merely through mutual interactions, without the influence of an external power. To answer questions concerning appearance and maintenance of cooperative behavior in societies, we study what we call the ``Lumberjacks' Dilemma (LD) game,'' as an application of the dynamical systems (DS) game theory, which can naturally deal with the dynamic aspects of games.

Dynamical processes that lead to the formation and maintenance of cooperation, which is often observed in the real communities, are realized in our model. The mechanism underlying this formation and maintenance is explained from the DS game point of view, by analyzing the functional dependence of the attractor of the game dynamics on a parameter characterizing the strategy. It is demonstrated that norms for cooperation are formed as strategies that are manifested as specific attractors of game dynamics. The change in the stability of this cooperative behavior as the number of members increases is also discussed. Finally, the relevance of our study to cooperation seen in the real world is discussed.