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

TL;DR

This paper introduces a nested evolutionary algorithm to solve complex multi-period multi-leader-follower Stackelberg games with non-linear functions and discrete variables, demonstrating its effectiveness on challenging bilevel problems.

Contribution

It presents a novel nested evolutionary approach for solving complex bilevel Stackelberg games with multiple players and non-linearities, extending beyond traditional methods.

Findings

The algorithm successfully solves difficult bilevel problems.

Player entry and exit significantly impact profits.

The method performs well on a test suite of complex problems.

Abstract

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.