Dynamic order allocation in a duopoly hybrid workforce of competition: A machine learning approach

We develop a continuous -time stochastic differential game model that aims to capture market demand and stochastic cross -network effects, and we seek to find equilibrium order allocation strategies between the firm and the platform. By solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation system, we obtain the feedback equilibrium. For a simple scenario, we derive the analytical solution which indicates that the equilibrium expenditures depend only on the marginal market thickness, and that market thickness is consistently advantageous for the value function. For the complex scenario, we propose a machine learning approach based on the Deep Galerkin Method to solve high -dimensional nonlinear HJB systems, and we demonstrate its good convergence properties. Based on reliable parameter values, our simulation results show that: (1) For higher market thickness, value functions exhibit greater sensitivity to changes in cross -network effects. (2) For a given cross -network effect, the equilibrium acquisition and retention expenditures display significant sensitivity to market thickness and time, respectively. To show the interaction between the two platforms, we present the following two results. One is that for the platform adopting a high pricing strategy, its acquisition expenditures exceed those of its competitor, while retention expenditures are the opposite. The other is that as the cross -network effect of the platform increases, its maximum profit initially rises and then declines, and the time of occurrence of the maximum profit monotonically decreases. In contrast, the competitor's maximum profit initially declines and then rises, and the time of occurrence of the maximum profit monotonically increases.