# Combinatorial Cost Sharing

## TL;DR

This paper introduces a combinatorial cost sharing mechanism that ensures cost coverage and approximate efficiency, leveraging the potential function to handle complex multi-parameter valuation scenarios.

## Contribution

It presents the Potential Mechanism combining VCG and the potential function, achieving cost coverage and approximate efficiency in complex combinatorial settings.

## Key findings

- Mechanism always covers the cost.
- Achieves approximate efficiency with subadditive costs.
- Ensures approximate budget balance under specific conditions.

## Abstract

We introduce a combinatorial variant of the cost sharing problem: several services can be provided to each player and each player values every combination of services differently. A publicly known cost function specifies the cost of providing every possible combination of services. A combinatorial cost sharing mechanism is a protocol that decides which services each player gets and at what price. We look for dominant strategy mechanisms that are (economically) efficient and cover the cost, ideally without overcharging (i.e., budget balanced). Note that unlike the standard cost sharing setting, combinatorial cost sharing is a multi-parameter domain. This makes designing dominant strategy mechanisms with good guarantees a challenging task. We present the Potential Mechanism -- a combination of the VCG mechanism and a well-known tool from the theory of cooperative games: Hart and Mas-Colell's potential function. The potential mechanism is a dominant strategy mechanism that always covers the incurred cost. When the cost function is subadditive the same mechanism is also approximately efficient. Our main technical contribution shows that when the cost function is submodular the potential mechanism is approximately budget balanced in three settings: supermodular valuations, symmetric cost function and general symmetric valuations, and two players with general valuations.