The anarchy of scheduling without money

Abstract

We consider the scheduling problem on n strategic unrelated machines when no payments are allowed, under the objective of minimizing the makespan. We adopt the model introduced in [Koutsoupias 2014] where a machine is bound by her declarations in the sense that if she is assigned a particular job then she will have to execute it for an amount of time at least equal to the one she reported, even if her private, true processing capabilities are actually faster. We provide a (non-truthful) randomized algorithm whose pure Price of Anarchy is arbitrarily close to 1 for the case of a single task and close to n if it is applied independently to schedule many tasks, which is asymptotically optimal for the natural class of anonymous, task-independent algorithms. Previous work considers the constraint of truthfulness and proves a tight approximation ratio of (n + 1)/2 for one task which generalizes to n(n + 1)/2 for many tasks. Furthermore, we revisit the truthfulness case and reduce the latter approximation ratio for many tasks down to n, asymptotically matching the best known lower bound. This is done via a detour to the relaxed, fractional version of the problem, for which we are also able to provide an optimal approximation ratio of 1. Finally, we mention that all our algorithms achieve optimal ratios of 1 for the social welfare objective.function that measures the quality of the outcome, the challenge is to design mechanisms which are able to elicit a desired behaviour from the players, while at the same time optimizing that objective value. A primary designer goal that has been extensively studied is that of truthfulness, under the central solution concept of dominant strategies: a player should be able to optimize her own individual utility by reporting truthfully, no matter what strategies the other players follow. However, achieving this is not always compatible with maintaining a good objective value [15,31]. The introduction of payments was suggested as a means towards achieving these goals, since a carefully designed payment scheme incentivizes the players to make truthful declarations. The goal now becomes to design such algorithms (termed mechanisms) which utilize monetary compensations in order to impose truthful behaviour while optimizing the objective function. A prominent positive result exists for utilitarian settings at which the objective function is the social welfare, where the well-known VCG mechanism [9, 16, 33] constitutes such an optimal mechanism. The study of the algorithmic aspect of mechanism design was initiated by Nisan and Ronen [26], and since then a significant body of work has been dedicated to optimization problems from the mechanism design point of view (see e.g. [4,10,21]).There are many situations, though, where the use of payments might be considered unethical [27], illegal (e.g. organ donations) or even just impractical. For this reason researchers have started turning their attention to possible ways of achieving truthfulness without the use of payments. In such a setting, in ord...

The Pareto Frontier of Inefficiency in Mechanism Design

“…In this section, we discuss some implications of our approach, as well as directions for future work. On a general level, one could follow our agenda of studying the inefficiency trade-off between the Price of Anarchy and the Price of Stability for many other problems in algorithmic mechanism design, such as auctions [30,43], machine scheduling without money [20,24], or resource allocation [11], to name a few, for which the two inefficiency notions have already been studied separately. In terms of the strategic scheduling setting, our work gives rise to a plethora of intriguing questions for future work, both on a technical and a conceptual level, which we highlight below in more detail.…”

“…t denote the set of pure Nash equilibria of mechanism M with respect to true profile t. As is standard in the literature, we focus on the case where Q M t = ∅ for all t ∈ R n×m ≥0 (see, e.g., [11,20,36]).…”

“…This approach has been adopted, among others, by central papers in the field (e.g., see [40,43] and references therein) and is by now as much a part of algorithmic mechanism design as the original framework of [36]. An interesting question that has arisen in many settings is whether non-truthful mechanisms (evaluated at the worst-case equilibrium, in terms of their PoA) can actually outperform truthful ones (evaluated at the truth-telling, dominant strategy equilibrium), for a given objective [11,20,24].While the literature that studies the concepts of PoA and PoS is long and extensive, there seems to be a lack of a systematic approach investigating the trade-off between the two notions simultaneously. More concretely, given a problem in algorithmic mechanism design, it seems quite natural to explore not only the best mechanisms in terms of the two notions independently, but also the mechanisms that achieve the best trade-off between the two.…”

Truthfulness on a budget: trading money for approximation through monitoring

“…As from the discussion above, making truthfulness less expensive requires us to depart from classical mechanism design; it turns out that our choice of alternative mechanism design paradigm goes some way towards the latter objective (i.e., obtaining a characterization of frugal payments). In detail, we look at mechanisms with monitoring, studied in [13,17,18]. 1 Mechanisms with monitoring assume that the designer is able to monitor the agents' costs so to guarantee that the agents who have exaggerated theirs end up paying the reported (higher) cost.…”

“…The terminology is introduced in[18] while[13,17] study the same model under the misnomer of mechanisms with verification. In the verification model of[20] it is effectively assumed that the designer can monitoring overbidding and punish underbidding; in[23] (and related literature) instead the designer only punishes underbidding but does not monitor overbidding -the difference between the models is furthermore studied therein.…”