Investigations For Financial Models Based On Evolutionary Games Of A Single Population

Evolutionary game theory which is applied to investigate some phenomena in biological evolution is originated from the field of biology.Economists have modified the form of evolutionary game theory so as to explore the phenomena of economics.Assuming that players are rational and the adjustment process of the population is dynamic,the relationship between individual behavior and the behavior of the whole population is described.Based on evolutionary game theory,this paper studies a competition model of commercial banks,a manager's income model in financial market and a financial bubble model.Friedman and Ostrove describe the evolutionary process of a single population by using the first-order partial differential equation(PDE).We give an evolutionary dynamic equation which contains the effect of technology and obtain the solution based on the work in Friedman and Ostrove .We consider an evolutionary game model of commercial banks with technological effects.Technology factor has a certain impact on the cost of commercial banks.Scholars have found that advantageous technology results in the decreases of the cost.On the contrary,backward technology leads to the increase of the cost.As time increases,we find that the income of commercial banks increases in the case of advantageous technology,the income of commercial banks decreases in the case of backward technology.We also find that the evolutionary dynamic process tends to be a steady state.As the solution of the gradient dynamics converges to a demonstrative function,each of the bank in the population will choose the same strategy or the same amount of credit supply.When the technology goes backward,the equilibrium output of credit is smaller than that of the advantageous technology.We verify our conclusions by numerical method.From the perspective of price competition and based on the model of Bertrand's price games,we study the evolutionary game model of commercial banks with technological progress and substituting parameters.The equilibrium price of the model is found.It depends not only on the substitution parameters but also on the marginal cost.We classify the results into five categories and verify the conclusions by numerical methods.Friedman and Ostrove study the evolution dynamic model of income of managers in financial markets.Unlike Friedman and Ostrove ,we establish the model with managers' absolute returns and use the balance equation to represent the evolutionary dynamic process of managers' population.Making use of the characteristic method,we obtain the explicit solution of the balance equation.We use the linear function and quadratic function to describe the impact of ability level of managers,respectively.We give the evolutionary dynamic equation of managers' income.It is found that if the excess rate of return is positive for a long period of time,the players in the population will choose the same and higher leverage because of herd behavior.At this time,the evolutionary dynamic process of population reaches a steady state at the equilibrium point.If the excess rate of return is negative for a long period of time and the market does not allow short selling,the players will exit the market until the excess rate of return becomes positive.When short selling is allowed,the solution of the evolutionary dynamic equation converges to an Heaviside step function,that is,the players will choose the same negative leverage.The evolutionary dynamic process reaches steady state at equilibrium point.When the difference level of managers is described by the linear function and the quadratic function,respectively,the evolutionary game equilibrium points of the population are different and the convergence speed is different.Some scholars have analyzed the causes of financial bubbles and discussed the method to prevent the crash of the financial bubbles.We investigate an evolutionary dynamic model of financial bubbles with a constant parameter in which the model of Friedman and Abraham does not contain.The model of the financial bubbles and crashes has a stable equilibrium with classic property.The bubbles and crashes occur in our extended models.Our results are similar to that in .