The Geometry of Truthfulness. Angelina Vidali ∗ September 30, 2009 Abstract We study the geometrical shape of the partitions of the input space created by the allocation rule of a truthful mechanism for multi-unit auctions with multidimensional types and additive quasilinear utilities. We introduce a new method for describing the allocation graph and the geometry of truthful mechanisms for an arbitrary number of items(/tasks). Applying this method we characterize all possible mechanisms for the case of three items. Previous work shows that Monotonicity is a necessary and sufficient condition for truthfulness in convex domains. If there is only one item, monotonicity is the most practical description of truthfulness we could hope for, however for the case of more than two items and additive valuations (like in the scheduling domain) we would need a global and more intuitive description, hopefully also practical for proving lower bounds. We replace Monotonicity by a geometrical and global characterization of truthfulness. Our results apply directly to the scheduling unrelated machines problem. Until now such a characterization was only known for the case of two tasks. It was one of the tools used for proving a lower bound of 1 + √ 2 for the case of 3 players. This makes our work potentially useful for obtaining improved lower bounds for this very important problem. Finally we show lower bounds of 1 + √ n and n respectively for two special classes of scheduling mechanisms, defined in terms of their geometry, demonstrating how geomet- rical considerations can lead to lower bound proofs. 1 Introduction Mechanism design is the branch of game theory that tries to implement social goals taking into account the selfish nature of the individuals involved. Mechanism design constructs allocation algorithms that together with appropriate payments elicit from the players their secret values or preferences. In this paper we give a characterization result that reveals the exact geometry of truthful mechanisms. The goal of this paper is to understand and visualize truthful mechanisms better. We realized the need for such a result while trying to improve the lower bound for the scheduling selfish unrelated machines problem [15, 6, 11], however the result is more broadly applicable and interesting from itself continuing a line of research attempting to grasp truthfulness better [18, 10, 13, 1, 4]. What differentiates our work from this line of research is that we fully exploit the linearity in the geometry of additive valuations. There exists a simple necessary and sufficient condition for truthfulness in convex domains and a finite number of outcomes, the Monotonicity Property. In single parameter domains, like for example in an auction where there is only one item, monotonicity is exactly the monotonicity we know from calculus and the most practical description of truthfulness we could hope for. The allocation should be a monotone (for the case of auctions an increasing, ∗ Department of Informatics, University of Athens, Email: avidali@di.uoa.gr 1