Analysis Of The Dynamics Of The Bertrand Game With Heterogeneous Players And Time Delay

Price competition among enterprises is one of the main ways in the market competition.Bertrand game is a classical game theory to discuss oligarch enterprises’ price competition.In this thesis,we consider a class of Bertrand price-making game models with heterogeneous players and time delay.Considering heterogeneous players with single delay structure,a Bertrand game is built in the paper firstly.One player with delayed bounded rationality adjusts the price strategy according to his own marginal profit with time delay,while another player adopts adaptive adjustment method to make the price strategy.A dynamics system is established,theoretical analyses and numerical simulations are done on the system stability and system complexity.The instability of the boundary equilibrium is proved,and the local stability conditions of the interior equilibrium are obtained.In addition,the results from numerical simulations reveal that either Neimark-Sacker bifurcations or period-doubling bifurcations would make the system lose stability.This paper also consider a Bertrand game played by two heterogeneous players with different delayed structures.In this model,one player adjusts the price strategy according to his own marginal profit with time delay,while another player makes the price strategy as a best response to an expectation that averages the opponent’s actions in the previous periods.The dynamics is built for this decision making process and the local stability of its equilibrium states are studied.In order to explore the impact of model parameters on the system,the numerical simulations are done to investigate the system stability and complexity.It is shown that the system may lose stability and take complex behaviors,and the stability loss may be due to either Neimark-Sacker bifurcation or period-doubling bifurcation.The research of the two systems shows that the time delay play an important role on the system stability and the loss of stability may be caused by Neimark-Sacker bifurcations or period doubling bifurcations.The numerical simulations show that both marginal profit delay and price delay affect the system’s dynamical behaviors,but the difference is that the time delay for the marginal profithas a greater effect than the time delay considered for the price variable.An intermediate level of delay on the marginal profit can not only defer the occurrence of complicated behaviors,but also enlarge the stability region,thereby improving the stability of the system.