Walrasian Equilibrium: Hardness, Approximations and Tractable Instances

We study the complexity issues for Walrasian equilibrium in a special case of combinatorial auction, called single-minded auction, in which every participant is interested in only one subset of commodities. Chen et al. [5] showed that it is NP-hard to decide the existence of a Walrasian equilibrium for a single-minded auction and proposed a notion of approximate Walrasian equilibrium called relaxed Walrasian equilibrium. We show that every single-minded auction has a relaxed Walrasian equilibrium that satisfies at least two-thirds of the participants, proving a conjecture posed in [5]. Motivated by practical considerations, we introduce another concept of approximate Walrasian equilibrium called weak Walrasian equilibrium. We show NP-completeness and hardness of approximation results for weak Walrasian equilibria. In search of positive results, we restrict our attention to the tollbooth problem [19], where every participant is interested in a single path in some underlying graph. We g...

• Type: Conference
• Year: 2005
• Where: WINE
• Authors: Ning Chen, Atri Rudra
• Keywords: Approximate Walrasian Equilibrium, Economy, Single-minded Auction, Walrasian Equilibrium, WINE 2005