Approaching Utopia: Strong Truthfulness and Externality-Resistant Mechanisms

We introduce and study strongly truthful mechanisms and their applications. We use strongly truthful mechanisms as a tool for implementation in undominated strategies for several problems,including the design of externality resistant auctions and a variant of multi-dimensional scheduling

A characterization of 2-player mechanisms for scheduling

We study the mechanism design problem of scheduling unrelated machines and we completely characterize the decisive truthful mechanisms for two players when the domain contains both positive and negative values. We show that the class of truthful mechanisms is very limited: A decisive truthful mechanism partitions the tasks into groups so that the tasks in each group are allocated independently of the other groups. Tasks in a group of size at least two are allocated by an affine minimizer and tasks in singleton groups by a task-independent mechanism. This characterization is about all truthful mechanisms, including those with unbounded approximation ratio. A direct consequence of this approach is that the approximation ratio of mechanisms for two players is 2, even for two tasks. In fact, it follows that for two players, VCG is the unique algorithm with optimal approximation 2. This characterization provides some support that any decisive truthful mechanism (for 3 or more players) partitions the tasks into groups some of which are allocated by affine minimizers, while the rest are allocated by a threshold mechanism (in which a task is allocated to a player when it is below a threshold value which depends only on the values of the other players). We also show here that the class of threshold mechanisms is identical to the class of additive mechanisms.Comment: 20 pages, 4 figures, ESA'0

The Geometry of Truthfulness.

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 geometrical considerations can lead to lower bound proofs.

A Characterization of n-Player Strongly Monotone Scheduling Mechanisms

Our work deals with the important problem of globally characterizing truthful mechanisms where players have multi-parameter, additive valuations, like scheduling unrelated machines or additive combinatorial auctions. Very few mechanisms are known for these settings and the question is: Can we prove that no other truthful mechanisms exist? We characterize truthful mechanisms for n players and 2 tasks or items, as either task-independent, or a player-grouping minimizer, a new class of mechanisms we discover, which generalizes affine minimizers. We assume decisiveness, strong monotonicity and that the truthful payments1 are continuous functions of players’ bids

Game-theoretic analysis of networks

Algorithmic mechanism design is an important area between computer science and economics. One of the most fundamental problems in this area is the problem of scheduling unrelated machines to minimize the makespan. The machines behave like selfish players: they have to get paid in order to process the tasks, and would lie about their processing times if they could increase their utility in this way. The problem was proposed and studied in the seminal paper of Nisan and Ronen, where it was shown that the approximation ratio of mechanisms is between 2 and n. In this thesis, we present some recent improvements of the lower bound to 1 + ?2 for three or more machines and to 1 + φ for many machines. Since the gap between the lower bound of 2.618 and the upper bound of n is huge, we also propose an alternative approach to the problem, which first attempts to characterize all truthful mechanisms and then study their approximation ratio. Towards this goal, we show that the class of truthful mechanisms for two players (regardless of approximation ratio) is very limited: tasks can be partitioned in groups allocated by affine minimizers (a natural generalization of the wellknown VCG mechanism) and groups allocated by threshold mechanisms. Finally we generalize a tool we have used in the proof of the 1 + ?2 lower bound: we give a geometrical characterization of truthfulness for the case of three tasks, which we believe that might be useful for proving improved lower bounds and which provides a more complete understanding of truthfulness.Η Αλγοριθμική Σχεδίαση Μηχανισμών είναι μια σημαντική περιοχή μεταξύ της επιστήμης των υπολογιστών και της οικονομικής θεωρίας. ΄Ενα από τα σημαν- τικότερα προβλήματα της περιοχής αυτής είναι το πρόβλημα του χρονοπρογραμ- ματισμού ασυσχέτιστων μηχανών με σκοπό την ελαχιστοποίηση του χρόνου που δουλεύει η μηχανή που τελειώνει τελευταία. Οι μηχανές συμπεριφέρονται σαν εγ- ωιστές παίκτες : θέλουν να πληρωθούν για να εκτελέσουν τις εργασίες και είναι δι- ατεθημένες να δηλώσουν ψευδείς χρόνους εκτέλεσης εάν με αυτό τον τρόπο μπορούν να αυξήσουν την ωφέλειά τους. Το πρόβλημα αυτό προτάθηκε και μελετήθηκε στην πολύ σημαντική δημοσίευση των Nisan και Ronen, που θεμελίωσε τον κλάδο της Αλγοριθμικής Σχεδίασης Μηχανισμών, όπου έδειξαν ότι ο λόγος προσέγγισης για το πρόβλημα είναι μεταξύ 2 και n. Στην παρούσα διατριβή παρουσιάζουμε βελτιώσεις του κάτω φράγματος σε 1+?2 για την περίπτωση των τριών μηχανών και σε 1 + φ για πολλές μηχανές. Dεδομένου ότι το χάσμα ανάμεσα στο κάτω φράγμα 2.618 και το άνω φράγμα n είναι τεράστιο προτείνουμε μία εναλλακτική προσέγγιση στο πρόβλημα, να χαρακ- τηρίσουμε πρώτα όλους τους δυνατούς μηχανισμούς και κατόπιν να μελετήσουμε τον λόγο προσέγγισής τους. Προς αυτό τον στόχο δείχνουμε ότι η κλάση των φιλαλήθων μηχανισμών για την περίπτωση των δύο παικτών είναι πολύ περιορισμένη: οι εργασίες μπορούν να χωριστούν σε ομάδες που ανατείθονται από αφφινικούς μεγιστοποιητές (που αποτελούν γενίκευση των γνωστών μηχανισμών VCG) και ομάδες που ανατείθονται από μηχανισμούς κατωφλίου. Τέλος γενικεύουμε ένα από τα εργαλεία που χρησιμοποιήσαμε για την απόδειξη του κάτω φράγματος 1+?2 δίνοντας ένα γεωμετρικό χαρακτηρισμό της φιλαλήθειας για την περίπτωση των τριών εργασιών, ο οποίος πιστεύουμε ότι θα είναι χρήσιμος για την απόδειξη νέων κάτω φραγμάτων και που σαφώς οδηγεί σε μια πιο πλήρη κατανόηση της φιλαλήθειας

A

lower bound for scheduling mechanisms