Dynamics in Games

Dynamics in Games
Eizo Akiyama, SFI Postdoctoral Fellow

The main topics of game theory have traditionally been equilibrium analysis and not dynamics. These days, however, dynamical phenomena among interacting agents are getting more and more interest of many researchers, and models that deal with open ended evolution of strategies are becoming popular, e.g., the evolutionary models using the Iterated Prisoners' Dilemma. This situation may be compared to the history of development of physics; that is, though physics originally analyzed the phenomena that, in most cases, converge on fixed point or limit cycles, recently physics has been enlarging its sphere of influence, with the development of non-linear science, into chaotic phenomena.

SFI Postdoctoral Fellow Eizo Akiyama is interested in two aspects of game theory: the dynamics produced in static (traditional) games, and the dynamics of games themselves in which the payoff matrix is influenced by the players.

Dynamics in static games

Akiyama is investigating in detail the relation between the rules of games and the dynamics emerged from (fixed or static) games; namely, dynamics of the strategies' evolution, or the dynamics that can observed in the players' communication patterns. Thus, by analyzing various types of games with various parameters, I would like to know what types of games produce the static phenomena, such as the end of the strategies' evolution, and especially, on the contrary, what conditions of games can cause the dynamical phenomena.

As a part of this work, he has already completed a work focusing on the qualitative difference between the games of two players and those of more than two players (3, 4, 5, ...). A non-zero-sum 3-person coalition game is presented in order to study the evolution of complexity and diversity in communication and strategies, where the population dynamics of players with strategies is given according to their scores in the iterated game and mutations. Communication rules are self-organized in a society through evolution. The co-evolution of diversity and complexity of strategies and communications are found at later stages of the simulation.

By extending this work to more number of players or by exploring other setting of games, such as the number of feasible strategies for a player, symmetric or asymmetric pay-off matrices for players, and so on, he would like to study the mechanism of the dynamical phenomena observed in the various types communities in our world from the game theoretical point of view; that is, the mathematical formulation of the relationship among plural agents, such as the coalition structures, the nature of dilemma, and so on. As the next step of this work, I would study four and five person coalition games by conducting computer simulations in which only the number of players is changed from the previous research. (The fundamental difference between the number of players, four and five, is already found out, which difference is caused by the conflictive structure between coalitions in more-than-five-person games. However, what type of effect such a difference actually has on the social and evolutionary phenomena is not known.)

Problems in Game Dynamics

In traditional game theory payoff matrix is given as fixed one. There are two issues that Akiyama is considering. The first one is an effect that a player's actions can have on the game environment. Players' decision-making and their consequent behaviors sometimes change their game environment itself. For example, if a cow has most of the grasses in the pastureland where she lives and gets large utility, the pastureland will go to ruin and she will not get so much utility by the next day's meal. In addition to this, He also wants to consider the second issue of the connection between a player's payoff function and that player's 'state'. (Use of the word 'state' here means any general properties of a player that may change, such as the actual condition of the player.) For example, consider a player continuing a contest with the same opponents in a game environment that does not change with time. Will the utility of the player's possible actions always continue to be the same? Won't the player's assessment of the possible actions vary accompanied by changes in the player's state? For example, suppose that a person eats an apple and gets enough satisfaction from it, he or she may not get so large utility from the same action, , to eat an (additional) apple.

In game theory this situation is sometimes represented by one (large) game. That is, from the past into the future, all possible actions of all players at all points of time are taken into account. Thus all possible bifurcation patterns of the game are derived, with this situation as a whole depicted as one huge game-tree. In this way, we can project the course of time into a static game and analyze its solution in the form of a game-tree or a game matrix. "Strategy" here means the action plan for all points in time, and the analysis of the rational solution for a game is possible only when we know all the possible actions from the past to the future. However, do we always make our decisions in this way? Moreover, admitting that we do so, is it possible for us to make such a decision for the first place? In order to deal with these types of problems, in which the game itself is affected and changed by the players' behaviors or states,, Akiyama is considering a model where the game itself is described as a "Dynamical Systems." He calls this model "the Dynamical Systems Game model."

Application of the Dynamical Systems Game Up To Now

He have applied the DS Game model to a concrete example. The particular example he studied is the "Lumberjack's Dilemma (LD) Game model," which deals with a problem of social dilemmas (specifically, the problem of forming and maintaining cooperation between a relatively large set of individuals). "Static" N-person dilemma games have usually been adopted to study the basis for research into social dilemmas. He plans to consider social dilemmas from a perspective of the DS Game model. In particular, he will investigate a model that explicitly takes into consideration the element of "time," focusing on its effects.

One of the important perspectives is:


A social norm based on dynamics can be established between players. For social dilemmas, stable orbits inherent in the game dynamics (with regard to variations in the players' choices of strategy) make possible the emergence and persistence of cooperative social communities.

Further, the following features have also been highlighted from a DS Game viewpoint for the Lumberjack's Dilemma.


As well as a direct influence to an opponent, a mechanism to affect an opponent through managing the game environment itself is important in the DS Game model. Constructing a diagram that we call an "SPG Diagram" can represent this mechanism).


Depending on how a strategy refers to its own or to others' states, completely different game dynamics can be observed.


The social structure of players' group intermittently evolves through the mutual interaction between players' actions and game dynamics. That is, once one of the stable and cooperative game dynamics is constructed by players, the observable dynamics of the society lasts for a period, while the players' strategies are continuously evolving. However, when the players' strategies evolve enough to realize another norm of cooperation, the dynamics of the players' game environment shifts to another state almost all at once.


Attempting to represent some games with "dynamical" definitions sometimes produces only one logically equivalent "static game", and by applying this type of abstraction about dynamics, traditional game theory have succeeded in formulating the common structure among several game-like situations. In this DS Game framework, however, keeping the features of dynamics intact in the model often brings to light wholly different phenomena among games that have been supposed logically equivalent, which differences at the level of description of dynamics cannot have been detected by traditional game theory.

Future Developments

There are still many conditions that have not been studied in the previous research of the LD Game. For example, a game with other types of dynamics (such as chaos), a larger number of players (the number of players is essential in case of "social dilemma'). In the first place, Akiyama would like to investigate the LD Game with regard to these conditions by extending my previous research.

Furthermore, there are many game-like situations that are appropriate to be represented by the DS Game model. For example, in the situation of the "chicken game," such as the arms race with nuclear weapons, the dilemma of both players becomes heightened with time, and so, the chicken game can be naturally expressed as a DS Game where cooperation may become more urgent with time. Other social dilemmas such as the problem of consumption of resources in a common pastureland and of petroleum are different problems if they are considered from the viewpoint of the DS Game. Petroleum does not increase actually in the time scale of our human lives, and so, the formation of cooperation should be treated as another problem.

Thus, using the DS Game framework, Akiyama is focusing on decision-making problems of multiple agents whose payoff matrix and behavior rules change in time.