Simple combinatorial auctions with budget constraints

TL;DR

This paper analyzes the efficiency of simple combinatorial auctions with budget constraints, establishing tight bounds on their price of anarchy and stability, and comparing them to more complex auction formats.

Contribution

It provides the first tight bounds on the price of anarchy and stability for a broad class of simple auctions with budget constraints, including well-known auction types.

Findings

Price of anarchy is at most 2 for all auctions in the class.

Price of stability is exactly 2 for simultaneous first and second price auctions.

Lower bounds are established for more complex auctions and common knowledge budgets.

Abstract

We study the efficiency of simple combinatorial auctions for the allocation of a set of items to a set of agents, with private subadditive valuation functions and budget constraints. The class we consider includes all auctions that allocate each item independently to the agent that submits the highest bid for it, and requests a payment that depends on the bids of all agents only for this item. Two well-known examples of this class are the simultaneous first and second price auctions. We focus on the pure equilibria of the induced strategic games, and using the liquid welfare as our efficiency benchmark, we show an upper bound of 2 on the price of anarchy for any auction in this class, as well as a tight corresponding lower bound on the price of stability for all auctions whose payment rules are convex combinations of the bids. This implies a tight bound of 2 on the price of stability of the well-known simultaneous first and second price auctions, which are members of the class. Additionally, we show lower bounds for the whole class, for more complex auctions (like VCG), and for settings where the budgets are assumed to be common knowledge rather than private information.