## Abstract

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.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Editors | Klaus Jansen, Sanjeev Khanna, Jose D. P. Rolim, Dana Ron |

Publisher | Springer Verlag |

Pages | 27-38 |

Number of pages | 12 |

ISBN (Print) | 3540228942, 9783540228943 |

DOIs | |

State | Published - 2004 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 3122 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Bibliographical note

Funding 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.