In many economic settings, like spectrum and real-estate auctions, geometric figures on the plane are for sale. Each bidder bids for his desired figure, and the auctioneer has to choose a set of disjoint figures that maximizes the social welfare. In this work, we design mechanisms that are both incentive compatible and computationally feasible for these environments. Since the underlying algorithmic problem is computationally hard, these mechanisms cannot always achieve the optimal welfare; Nevertheless, they do guarantee a fraction of the optimal solution. We differentiate between two information models-when both the desired figures and their values are unknown to the auctioneer or when only the agents' values are private data. We guarantee different fractions of the optimal welfare for each information model and for different families of figures (e.g., arbitrary convex figures or axis-aligned rectangles). We suggest using a measure on the geometric diversity of the figures for expressing the quality of the approximations that our mechanisms provide.
Bibliographical noteFunding Information:
The authors are grateful to Noam Nisan for many helpful discussions, to Ron Lavi for his comments on an earlier draft, and to anonymous referees for their helpful comments on an earlier draft. The first author was supported by the Yeshaya Horowitz Association and by the National Science Foundation grant number ANI-0331659. Both authors were supported by grants from the Israel Science Foundation and the USA–Israel Binational Science Foundation.
- Approximation algorithms
- Mechanism design