Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Complexity of Stackelberg Pricing Games

Christoph Grüne, Dorothee Henke, Eva Rotenberg, Lasse Wulf

TL;DR

This paper shows that Stackelberg pricing games, where a leader sets prices and a follower solves a combinatorial optimization, are $Σ^p_2$-complete for natural pricing variants. The authors develop an SSP-based meta-framework and prove $Σ^p_2$-hardness for the knapsack pricing problem and the independent set pricing problem, then extend the result to a broad class of SSP-NP-complete problems via a general meta-reduction. The approach starts with a SAT-pricing construction and lifts hardness to any pricing problem $Pricing$-$Π$ where $Π$ is SSP-NP-complete, using large constant separations and SSP reductions. The results imply intrinsic higher-level complexity for bilevel pricing, even when the follower’s problem is classical NP-complete, and they illuminate limits of compact MILP formulations for these problems, while covering at least 50 known NP-complete problems.

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., $Σ^p_2$-complete. The same conjecture was also made by Böhnlein, 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 $Σ^p_2$-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 Grüne and Wulf.

The Complexity of Stackelberg Pricing Games

TL;DR

This paper shows that Stackelberg pricing games, where a leader sets prices and a follower solves a combinatorial optimization, are -complete for natural pricing variants. The authors develop an SSP-based meta-framework and prove -hardness for the knapsack pricing problem and the independent set pricing problem, then extend the result to a broad class of SSP-NP-complete problems via a general meta-reduction. The approach starts with a SAT-pricing construction and lifts hardness to any pricing problem - where is SSP-NP-complete, using large constant separations and SSP reductions. The results imply intrinsic higher-level complexity for bilevel pricing, even when the follower’s problem is classical NP-complete, and they illuminate limits of compact MILP formulations for these problems, while covering at least 50 known NP-complete problems.

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., -complete. The same conjecture was also made by Böhnlein, 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 -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 Grüne and Wulf.

Paper Structure

This paper contains 18 sections, 17 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Example.
  3. The polynomial hierarchy.
  4. Our contribution.
  5. Outline.
  6. Preliminaries
  7. SSP Framework
  8. Example.
  9. Problem Definition
  10. Difference between the domains $\mathbb{R}$ and $\mathbb{R}_{\geq 0}$.
  11. Example.
  12. Feasibility pricing.
  13. Feasibility-Pricing-Sat is $\Sigma^p_2$-complete
  14. "If" direction.
  15. "Only if" direction.
  16. ...and 3 more sections

Key Result

theorem 1

For all LOP problems $\Pi$, we have $\textsc{Pricing-$\Pi$} \in \Sigma^p_2$. Also for all $\textsf{SSP-NP}$ problems $\Pi$, we have $\textsc{Feas-Pricing-$\Pi$} \in \Sigma^p_2$.

Figures (1)

  • Figure 1: The relation between the universes when applying the meta-reduction from Feas-Pricing-Sat and Pricing-$\Pi$. The function $f$ maintains a one-to-one correspondence between the elements of $\mathcal{U}_L$ and $f(\mathcal{U}_L)$ as well as $\mathcal{U}_F$ and $f(\mathcal{U}_F)$.

Theorems & Definitions (35)

  • definition 1: Linear Optimization Problem, from grune2025completeness
  • definition 2: Subset Search Problem (SSP), from grune2025completeness
  • definition 3: SSP Reduction, from grune2025completeness
  • theorem 1
  • proof
  • theorem 2
  • proof
  • corollary 1
  • proof
  • theorem 3
  • ...and 25 more