Microscopic Replicator Dynamics

An individual's behavior of is called purposive behavior, if it is based on the notion of having preference, pursuing goal, or maximizing his own interest. The goal or purpose of an individual often relates directly to others, and their behavior are often constrained by an environment that consists of others who are pursuing their own goals. This type of behavior that depends on what others are doing is called as contingent behavior. They often have to deal with conscious decisions or adaptations in the pursuit of goals within the limits of their information and their comprehension of how to navigate through their environment toward whatever their objectives are. The purpose of this paper is to propose a new nonlinear dynamic model for analyzing emergent properties, especially a self-organizing process of the aggregate of contingent behaviors. We consider a large population of heterogeneous agents in which each member interacts each other in a different manner. We formalize each interdependent and asymmetric and interaction between any two agents as a mixed -motivation game. There are growing literatures on the evolutionary approaches for game theory. The standard interpretation of game theory is that the game is played exactly once between fully rational individuals who know all details of the game, including each other's preferences over outcomes. Evolutionary game theory, instead, assumes that the game is repeated many times by individuals who are randomly drawn from large populations. An evolutionary selection process operates over time on the population distribution of behaviors. In most of previous literatures on evolutionary games, agents are viewed as being genetically coded with a strategy and selection pressure favors agents which are fitter, i.e., whose strategy yields a higher payoff against the average payoff of the population.In many applications, it is of interest to know which strategies survive in the long run. The dynamic evolutionary process with the assumption of uniform matching can be characterized by using replicator dynamics. In general, an evolutionary game combines two basic elements: a mutation mechanism that provides variety and a selection mechanism that favors some varieties over others. The criterion of evolutionary equilibrium highlights the role of mutations. The replicator dynamics, on the other hand, highlight the role of selection. We propose a new model of evolutionary dynamics, termed as microscopic replicator dynamics, which describe the aggregate behavior from the bottom up. We obtain each agent's contingent behavior can be obtained as a threshold model, which is a function of each individual's characteristic and the common information. The common information represent the proportion of agents who adapt the same behavior. We characterize the heterogeneity of agents by the distribution pattern of the thresholds, which provides the macroscopic property of the population.The criteria of selection are the average payoff of the population, which are the absolute criteria. However, in our model, no one needs to calculate the average payoff of the population. We discuss a self-organizing process of the collective behavior, which shifts among several equilibrium states without any central authority or the pressure from the outside. It is also shown that weakly dominated strategies need not to be eliminated, and even strongly dominated strategies can survive in certain special cases. This property is essential to maintain both variety and harmony of the population.