Dynamic Analysis Of Generalized Tullock Contest

Because of contest theory’s wide application in the real world,it has gradually become a research hotspot in academia since Tullock proposed.With Tullock contest theory as the background,this paper assumes that the players in the game are homogeneous and have bounded rationality,combines game theory and nonlinear complex system theory,comprehensively considers the influence of various factors among the players in the game,and establishes two duopoly game models with effort level as the decision variable based on gradient adjustment mechanism.In order to analyze the complex behavior of the models,The following work is done by means of parametric bifurcation diagram,stability curve,maximum Lyapunov exponent diagram,phase diagram,basin of attraction,particle swarm optimization and other tools:1.By introducing the spillover effect and penalty mechanism,a generalized dynamic Tulloch competition model is established.The existence of equilibrium points of the model is analyzed theoretically,and the condition for the existence of Nash equilibrium is given.The stability domain of the system is found by means of the parameter bifurcation diagram,and the influence of parameters on the stability of Nash equilibrium is discussed.It is found that the system is more sensitive to the changes of spillover parameters and cost parameters,and the parameter range of the stable operation of the system is determined.It is found that there are two types of bifurcation in the system,namely,period-doubling bifurcation and Neimark-Sacker bifurcation.The system will eventually fall into chaos through these two bifurcations;Using the tool of the basin of attraction,it is found that the attractor will contact with the boundary of its domain of attraction,resulting in a "boundary crisis".With the change of parameters,the number and structure of the attractor as well as the overall outline of the basin of attraction will change.In addition,the change of various parameters will also cause the coexistence of locally stable basic steady state and stable(quasi-)periodic orbit,thus showing path dependence.A set of optimal spillover parameters are solved by particle swarm optimization algorithm to ensure the smooth operation of the system and maximize the market benefits.2.This paper studies the dynamic Tullock contest model with nonlinear variable cost and does not consider spillover,gives the unique Nash equilibrium of the model and its local stability condition,analyzes the dynamic evolution behavior of the parametric equilibrium point after instability,and finds that there are abundant isoperiodic structures in the two-dimensional bifurcation diagram,namely,Anoral’d tongue.They are adjacent to the Neimark-Sacker bifurcation line,which indicates that the system will show strong periodic oscillation with the increase of the parameter.The existence of global bifurcation and the coexistence of multiple attractors are proved by using the basin of attraction,which means the historical dependence of the nonlinear economic system on the initial conditions.Finally,from the point of view of the designer of the competition,through the particle swarm optimization algorithm,a set of optimal cost parameters are given under the condition of ensuring the stability of the dynamic system to achieve the maximum market benefits and create a benign competitive environment.