Study Of Dynamics In Complex Systems Based On Game Models

Complex behavior can occur in any system made up of large numbers of interacting constituents, such as atoms in a solid, cells in a living organism, or consumers in a national economy. Recently, the growing interest in the study of complex systems has led to novel generalizations of classical models in physics, economics and biology, e.g., the Ising model, the minority game and the prisoner's dilemma game. Since the discovery of some novel properties in real-world networks in various areas including physics, biology, economics, ecology, and computer science, researchers have been exploring suitable theories or models, and spatial evolutionary game models have become one of the important tools and attracted more and more attentions nowadays.In comparison to the work done in statistical physics, we find spatial game models are quite useful for us to study the phenomenon of self-organization and critical behavior in the physical world. Based on two game models——the minority game and the snowdrift game, we explore the dynamical processes of multi-particle interactions on different networks. The methods used in our research are naturally very similar to those used in statistical physics. The contents presented in this dissertation are mainly about our investigations on the effects of networking on the dynamical processes.The thesis consists of two parts. The first part focuses on the introduction of game theory and graph theory, especially the characteristics of game models and network topologies. In order to emphasize the relationship between statistical physics and game theory, in this part we also introduce some related models and concepts in physics, such as Ising model, potts model, phase transition, correlation and probability distribution. We present three network models, i.e. random network, small world network and scale-free network, and give a comparison between them. The geometrical properties of networks are described in detail, including degree distribution, shortest-path length and clustering coefficient. A random growing network and a network model with both random and preferential attachments are introduced at the end of this part.The second part focuses on two kinds of dynamical problems, which include the minority game and the snowdrift game. For the minority game, we study the effects of the presence of contrarians in a competing environment. These contrarians are agents who deliberately prefer to hold an opinion that is contrary to the prevailing idea of the commons or normal agents. Results of numerical simulations reveal that the average success rate among the agents depends non-monotonically on the fraction ac of contrarians. For small ac, the contrarians systematically outperform the normal agents by avoiding the crowd effect and enhance the overall success rate. For high ac, the anti-persistent nature of the MG is disturbed and the few normal agents outperform the contrarians. Qualitative discussion and analytic results for the small ac and high ac regimes are also presented, and the crossover behavior between the two regimes is discussed.For the snowdrift game, we investigate the following three effects on the extent of cooperation emerging in a competitive setting separately:the effects of networking, the effects of an additional strategy or character called loner and the effects of finite populations.The evolutionary snowdrift game, which represents a realistic alternative to the well-known Prisoner's Dilemma, is studied in the Watts-Strogatz network that spans the regular, small-world, and random networks through random re-wiring. Over a wide range of payoffs, a re-wired network is found to suppress cooperation when compared with a well-mixed or fully connected system. Two extinction payoffs, that characterize the emergence of a homogeneous steady state, are identified. It is found that, unlike in the Prisoner's Dilemma, the standard deviation of the degree distribution is the dominant network property that governs the extinction payoffs.The effects of an additional strategy or character called loner are studied in a well-mixed population or fully-connected network and in a square lattice. In a fully-connected network, it is found that either C (cooperator) lives with D (defector) or the loners take over the whole population. In a square lattice, three possible situations are found:a uniform C-population, C lives with D, and the coexistence of all three characters. The presence of loners is found to enhance cooperation in a square lattice by enhancing the payoff of cooperators. The results are discussed in terms of the effects in restricting a player to compete only with his nearest neighbors in a square lattice, as opposed to competing with all players in a fully-connected network.Based on the evolutionary snowdrift game, frequency dependent evolutionary dynamics is studied in finite populations. It is found that, in a well-mixed or fully connected system, phase transitions that characterize the emergence of a homogeneous steady state easily occur in contrast to the case in infinite populations, where no such phase transitions are identified. Analytical calculations are given in comparison to the Monte Carlo simulation results. We find that the occurrence of the homogeneous steady state is related to the synchronously updating rule and the finite population size.