# Negative Prices in Network Pricing Games

## TL;DR

This paper investigates how negative pricing in network pricing games can paradoxically increase a leader's profit, revealing conditions and network structures where this effect is significant or absent.

## Contribution

It quantifies the profit increase possible with negative prices and identifies network topologies immune to this paradox, advancing understanding of strategic pricing in network games.

## Key findings

- Profit with negative prices can be a factor of Θ(log(m·k̄)) larger.
- Single follower case cannot bound the profit increase.
- Series-parallel graphs with one follower are immune to negative price paradox.

## Abstract

In a Stackelberg network pricing game, a leader sets prices for a given subset of edges so as to maximize profit, after which one or multiple followers choose a shortest path from their source to sink. We study the counter-intuitive phenomenon that the use of negative prices by the leader may in fact increase his profit. In doing so, we answer the following two questions. First, how much more profit can the leader earn by setting negative prices? Second, for which network topologies can the profit be increased by using negative prices? Our main result shows that the profit with negative prices can be a factor $\Theta(\log (m\cdot\bar k))$ larger than the maximum profit with positive prices, where $m$ is the number of priceable edges in the graph and $\bar k \leq 2^m$ the number of followers. In particular, this factor cannot be bounded for the single follower case, and can even grow linearly in $m$ if the number of followers is large. Our second result shows that series-parallel graphs with a single follower and Stackelberg games with matroid followers are immune to the negative price paradox.