## Abstract

In a Stackelberg pricing game a distinguished player, the leader, chooses prices for a set of items, and the other player, the follower, seeks to buy a minimal cost feasible subset of the items. The goal of the leader is to maximize her revenue, which is determined by the sold items and their prices. Typically, the follower is given by a combinatorial covering problem, e.g., his feasible subsets are the edges of a spanning tree or the edges of an s-t-path in a network. We initiate the study of Stackelberg pricing games where the follower solves a maximization problem. In this model, the leader offers a payment to include her items in the follower's solution. Our motivation stems from the following situation: assume the leader has a set of jobs 1, . . ., k to complete. A job i may either (a) be executed for a given cost b(i) using her own resources or (b) offered to the follower at a variable price p(i) to complete it for her. The objective function to be maximized by the leader is the sum of the margins b(i) - p(i) over those jobs i that are completed by the follower. Informally, the question is which jobs should be outsourced and what profit the leader has to offer. Our main result says that the problem can be solved to optimality in polynomial-time when the jobs have fixed starting and terminating times and the follower solves a maximum weight scheduling on a single machine. To show that the situation changes when the follower is given by other optimization problems, we prove APX-hardness for a scheduling problem that can be modeled as a bipartite maximum weight matching problem. Moreover, we show APX-hardness in the case of the maximum weight spanning tree problem. On a more general note, we prove S^{p}_{2}-completeness if the follower has a general combinatorial optimization problem given in the form of a finite ground set and a feasibility oracle. This shows that while the follower's problem is NP-complete, the leader's problem is hard even if she has an NP-oracle at hand.

Original language | English |
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Pages | 193-197 |

Number of pages | 5 |

State | Published - 2019 |

Externally published | Yes |

Event | 16th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, CTW 2018 - Paris, France Duration: 18 Jun 2018 → 20 Jun 2018 |

### Conference

Conference | 16th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, CTW 2018 |
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Country/Territory | France |

City | Paris |

Period | 18/06/18 → 20/06/18 |

### Bibliographical note

Publisher Copyright:© 2019 16th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, CTW 2018 - Proceedings of the Workshop.

## Keywords

- Algorithmic pricing
- Revenue maximization
- Stackelberg games