Worst-Case Welfare of Item Pricing in the Tollbooth Problem

TL;DR

This paper analyzes the worst-case welfare performance of item pricing in the tollbooth problem, revealing tight bounds for paths and trees, and establishing lower bounds for general graphs under adversarial buyer order.

Contribution

It provides tight bounds for the competitive ratio in paths and trees, and demonstrates inherent limitations for general graphs in the tollbooth problem.

Findings

Tight competitive ratio of 3/2 for paths (highway problem).

Item-pricing can achieve optimal welfare with proper tie-breaking.

Lower bounds of (m^{1/8}) for general graphs, even with capacity augmentation.

Abstract

We study the worst-case welfare of item pricing in the \emph{tollbooth problem}. The problem was first introduced by Guruswami et al, and is a special case of the combinatorial auction in which (i) each of the $m$ items in the auction is an edge of some underlying graph; and (ii) each of the $n$ buyers is single-minded and only interested in buying all edges of a single path. We consider the competitive ratio between the hindsight optimal welfare and the optimal worst-case welfare among all item-pricing mechanisms, when the order of the arriving buyers is adversarial. We assume that buyers own the \emph{tie-breaking} power, i.e. they can choose whether or not to buy the demand path at 0 utility. We prove a tight competitive ratio of $3/2$ when the underlying graph is a single path (also known as the \emph{highway} problem), whereas item-pricing can achieve the hindsight optimal if the seller is allowed to choose a proper tie-breaking rule to maximize the welfare. Moreover, we prove an $O(1)$ upper bound of competitive ratio when the underlying graph is a tree. For general graphs, we prove an $\Omega(m^{1/8})$ lower bound of the competitive ratio. We show that an $m^{\Omega(1)}$ competitive ratio is unavoidable even if the graph is a grid, or if the capacity of every edge is augmented by a constant factor $c$. The results hold even if the seller has tie-breaking power.