Finding Optimal Strategies in Multi-period Stackelberg Games Using an Evolutionary Framework

Stackelberg games have been widely studied in the literature and are a perfect real-world example of a bilevel optimization problem. The hierarchical nature of the problem makes it difficult to arrive at the optimal solution using the existing methodologies. Approximate solution techniques are commonly employed to handle such models with simplifying assumptions like smoothness, linearity or convexity. In this paper, we apply an evolutionary bilevel programming framework to solve a multi-period Stackelberg competition model with discrete production variables and non-linear cost and price functions. The evolutionary framework allows us to relax the simplifying assumptions and consider a dynamic duopoly competition model where one firm acts as a leader and the other as a follower. The leader firm tries to maximize its total profit over multiple periods based on complete knowledge of the follower's response to any of its actions. The solution to the model leads to optimal production, investment and marketing decisions to be made in each of the time periods. Using a heuristic approach to such problems allows us to overcome some of the challenges posed by complex bilevel problems, which otherwise would be difficult to solve using conventional optimization techniques.