Production game of multifirms with different decision-making patterns

In production market, the decision-making patterns of firms producing a homogeneous kind of products are different. Some are under the control of a dominant firm called centralized pattern. In this case, there is only one decision maker who takes charge of all the firms and attempts to maximize the total utility of the system. Another is decentralized pattern, in which firms plan by themselves. In reality, there always exists a third pattern, where the centralized pattern and decentralized pattern firms are in the same market. This article deals with the last case and studies the effects of decentralized pattern firms' decisions on centralized pattern firms in the same market, which is described as a Stackelberg game. It is supposed that centralized pattern firms are the leaders who affect the whole market by adjusting the production of each firm. Decentralized pattern firms are the followers who determine their productions in a manner of competitive equilibrium against the given centralized firm's strategy. This game in this article is formulated as a bilevel programming model. Solution algorithm based on the sensitivity analysis is adopted to solve the model, and a numerical example is given to illustrate the model and algorithm.