Computationally-feasible truthful auctions for convex bundles

In many economic settings, convex figures on the plane are for sale. For example, one might want to sell advertising space on a newspaper page. Selfish agents must be motivated to report their true values for the figures as well as to report the true figures. Moreover, an approximation algorithm should be used for guaranteeing a reasonable solution for the underlying NP-complete problem. We present truthful mechanisms that guarantee a certain fraction of the social welfare, as a function of a measure on the geometric diversity of the shapes. We give the first approximation algorithm for packing arbitrary weighted compact convex figures. We use this algorithm, and variants of existing algorithms, to create polynomial-time truthful mechanisms that approximate the social welfare. We show that each mechanism achieves the best approximation over all the mechanisms of its kind. We also study different models of information and a discrete model, where players bid for sets of predefined building blocks.