Expanding the duopoly Stackelberg game with marginal costs into a multipoly game with lowering the burden of mathematical calculations: a numerical analysis

In game theory, the classic Stackelberg game is a dynamic model of a leader and a follower firm contending in a market. The time evolution with the marginal cost and the dynamical traits of the duopoly and tripoly Stackelberg game can be acquired in the literature. The backward induction procedure, an analytical technique for attaining the Nash equilibrium, becomes grueling as the number of followers augments. To conquer this matter and untangle a multipoly Stackelberg game, this paper designs a numerical process based on seeking the parameter space to gain the maximum total profit. Although, due to the curse of dimensionality, a comprehensive quest in the parameter space fails as the number of followers extends. Therefore, the proposed search algorithm explores the parameter space sequentially to detect the optimum values that assure the Nash equilibrium. The significant concept of this iterative procedure is optimizing the parameters sequentially and utilizing the derived parameters for initializing the subsequent stride. The consequences of the developed numerical strategy represent that the proposed mechanism can conduce to a reasonable and even more acceptable total profit than the analytical attitude. The leader scheduled vintage is invariably greater or similar to the followers so that the leader can inevitably produce more goods and make the maximum profit in the market. The proposed algorithm can conduct a reasonable profit compared to the computational cost. In addition, solving the multipoly Stackelberg game with more than three follower firms and one leader becomes conceivable.