需求未知的主从竞争企业的定价和生产决策

Pricing and Production Decision in Leader-Follower Competiti on with Unknown Demand

研究主从竞争市场中领导企业和追随企业的定价和生产决策问题。不再传统地假设需求确定已知或服从某种已知的随机分布,而是考虑需求未知,采用基于最小二乘法的需求学习方法来预测未知的需求参数。分别对追随企业和领导企业的最优定价和生产问题进行建模,得出所有追随企业的优化问题形成一个广义微分Nash均衡问题,领导企业的优化问题是一个具有均衡约束的数学规划。证明了追随企业间广义微分Nash均衡解的存在,采用固定点算法来求解追随企业的优化子问题,采用模拟退火算法来求解领导企业的优化问题。数值实验首先展示了所有企业形成的均衡,发现相比于一个企业在寡头竞争市场中获得的利润,该企业成为主从竞争市场中的领导者将获得更高的利润,而成为追随者将获得更低的利润。

This paper considers the pricing and production decision problem for firms in a leader-follower competitive market.We do not assume demand is deterministic and known nor demand follows a known random distribution in the traditional way,but consider unknown demand and use a demand learning approach based on the least square method to forecast the unknown demand.We study the optimal pricing and production problems for the follower firms and the leader firm,respectively,and show that all follower firms＇ optimal control problem reduces to a generalized differential Nash equilibrium problem,and the leader firm＇s optimal control problem can be formulated as a mathematical program with equilibrium constraints. We prove that the generalized differential Nash equilibrium exists for the followers,and apply the fixed point algorithm to solve the followers＇ optimal sub-problem and the simulated annealing algorithm to solve the leader＇s optimal problem.Using a numerical example,we show the equilibrium of all the firms,and find that compared to the profit that a firm obtains in an oligopolistic competitive market,the firm obtaining higher profit becomes the leader in the leader-follower competitive market,and the firm obtaining lower profit becomes the follower.