Finding Optimal Strategies in a Multi-Period Multi-Leader-Follower Stackelberg Game Using an Evolutionary Algorithm

Stackelberg games are a classic example of bilevel optimization problems, which are often encountered in game theory and economics. These are complex problems with a hierarchical structure, where one optimization task is nested within the other. Despite a number of studies on handling bilevel optimization problems, these problems still remain a challenging territory, and existing methodologies are able to handle only simple problems with few variables under assumptions of continuity and differentiability. In this paper, we consider a special case of a multi-period multi-leader-follower Stackelberg competition model with non-linear cost and demand functions and discrete production variables. The model has potential applications, for instance in aircraft manufacturing industry, which is an oligopoly where a few giant firms enjoy a tremendous commitment power over the other smaller players. We solve cases with different number of leaders and followers, and show how the entrance or exit of a player affects the profits of the other players. In the presence of various model complexities, we use a computationally intensive nested evolutionary strategy to find an optimal solution for the model. The strategy is evaluated on a test-suite of bilevel problems, and it has been shown that the method is successful in handling difficult bilevel problems.Comment: To be published in Computers and Operations Researc

A nested Stackelberg game between the government, a mining firm, and a travel resort: an application of bilevel programming

The classic example of a negative externality is that of an upstream paper mill polluting the water used by a downstream fishery. Though this example is simplistic, similar situations abound in real life. One real-world example of such a negative externality is the case where a mining facility negatively affects the operations of a nearby vacation resort. This particular situation is the object of analysis of this thesis. I consider the situation from the government's perspective, which faces a multi-criteria decision- making problem because it would like to keep both the mine and the resort operating simultaneously in order to maximize social welfare while at the same time trying to keep environmental damages to a minimum. This approach effectively makes the considered problem into a Stackelberg competition model, whereby the government acts as the leader by subjecting the firms to a certain level of pollution taxation, and the two firms - the mine and the resort - assume the roles of followers and adjust their operations accordingly. Furthermore, since the resort is directly and adversely affected by the mine's operations, the relationship between them can also be viewed as a Stackelberg model, and the problem becomes a tri-level optimization problem compared to the classic bilevel Stackelberg problems encountered in the literature. The closed-form analytical solution to the problem is presented, which includes the formulations of the optimal taxation structures for the leader, the optimal operating levels of the followers, and the optimal profits of all parties involved. The relationships between the players are clearly illustrated by various graphs, and some shortcoming and worthwhile extensions of the model are also discussed