A demand-based optimization approach to find market equilibria in oligopolies

Oligopolistic competition occurs often in transportation as well as in other markets due to reasons such as barriers to entry, limited capacity of the infrastructure and external regulations. In transport oligopolies, suppliers are profit maximizers and take decisions that are influenced both by the preferences of the customers, who want to purchase one of the services on the market, and by the strategies of the competitors. In our work, the preferences of the customers are modelled at a disaggregate level using discrete choice models and are embedded in each operator's optimization problem. Using a disaggregate approach that accounts for heterogeneous demand allows to better model supply-demand interactions. Competition among market players is modelled explicitly as a non-cooperative game. The result is a multi-leader-follower game. We present a MIP model inspired by the fixed-point iteration algorithm that find Nash equilibrium solutions of finite games. Numerical experiments show that the computational performance of the model depends on the type of decision variables used to model supply strategies and on the discrete choice model used to describe demand. The nonlinear formulation is non-convex and becomes intractable when many discrete variables are introduced, while the linearized formulation is convex but combinatorial due to the nature of the proposed simulation framework. We propose an algorithmic framework in which candidate equilibrium solutions are first found by means of heuristic blocks and constitute the initial restricted sets of strategies in the fixed-point MIP model, which is used to find subgame equilibrium solutions. Subgame equilibria are checked against the original game by solving best-response problems, and new candidate strategies can be added to the next iterations of the fixed-point MIP model using a column-generation-like approach.