The Complexity of Stackelberg Pricing Games

TL;DR

This paper proves that Stackelberg pricing games are computationally very complex, specifically $ ext{Sigma}_2^p$-complete, for a wide range of classic NP-complete problems, highlighting the inherent difficulty of these bilevel pricing problems.

Contribution

It establishes the $ ext{Sigma}_2^p$-completeness of Stackelberg pricing games across many NP-complete problems, confirming a longstanding conjecture and broadening understanding of their complexity.

Findings

Stackelberg pricing game is $ ext{Sigma}_2^p$-complete for the knapsack problem.

The $ ext{Sigma}_2^p$-completeness extends to over 50 NP-complete problems.

This result aligns with recent meta-theorems on higher complexity in the polynomial hierarchy.

Abstract

We consider Stackelberg pricing games, which are also known as bilevel pricing problems, or combinatorial price-setting problems. This family of problems consists of games between two players: the leader and the follower. There is a market that is partitioned into two parts: the part of the leader and the part of the leader's competitors. The leader controls one part of the market and can freely set the prices for products. By contrast, the prices of the competitors' products are fixed and known in advance. The follower, then, needs to solve a combinatorial optimization problem in order to satisfy their own demands, while comparing the leader's offers to the offers of the competitors. Therefore, the leader has to hit the intricate balance of making an attractive offer to the follower, while at the same time ensuring that their own profit is maximized. Pferschy, Nicosia, Pacifici, and Schauer considered the Stackelberg pricing game where the follower solves a knapsack problem. They raised the question whether this problem is complete for the second level of the polynomial hierarchy, i.e., $\Sigma^p_2$-complete. The same conjecture was also made by B\"ohnlein, Schaudt, and Schauer. In this paper, we positively settle this conjecture. Moreover, we show that this result holds actually in a much broader context: The Stackelberg pricing game is $\Sigma^p_2$-complete for over 50 NP-complete problems, including most classics such as TSP, vertex cover, clique, subset sum, etc. This result falls in line of recent meta-theorems about higher complexity in the polynomial hierarchy by Gr\"une and Wulf.