Approximation Guarantee of OSP Mechanisms: The Case of Machine Scheduling and Facility Location

Algorithmica https://doi.org/10.1007/s00453-020-00771-x Approximation Guarantee of OSP Mechanisms: The Case of Machine Scheduling and Facility Location Diodato Ferraioli1 · Carmine Ventre2 Received: 28 February 2018 / Accepted: 17 September 2020 © The Author(s) 2020 Abstract Obvious strategyproofness (OSP) is an appealing concept as it allows to maintain incentive compatibility even in the presence of agents that are not fully rational, i.e., those who struggle with contingent reasoning (Li in Am Econ Rev 107(11):3257– 3287, 2017). However, it has been shown to impose some limitations, e.g., no OSP mechanism can return a stable matching (Ashlagi and Gonczarowski in J Econ Theory 177:405–425, 2018). We here deepen the study of the limitations of OSP mechanisms by looking at their approximation guarantees for basic optimization problems paradigmatic of the area, i.e., machine scheduling and facility location. We prove a number of bounds on the approximation guarantee of OSP mechanisms, which show that OSP can come at a significant cost. However, rather surprisingly, we prove that OSP mechanisms can return optimal solutions when they use monitoring—a novel mechanism design paradigm that introduces a mild level of scrutiny on agents’ declarations (Kovács et al. in WINE 9470:398–412, 2015). Keywords Mechanism design · Obvious strategyproofness · Approximation ratio · Monitoring 1 Introduction Algorithmic mechanism design (AMD) is by now an established research area in computer science that aims at conceiving algorithms resistant to selfish manipulations. As the number of parties (a.k.a., agents) involved in the computation increases, there is, in A preliminary version of this paper appeared as [13]. * Diodato Ferraioli Carmine Ventre 1 DIEM, Università degli Studi di Salerno, Fisciano, Italy 2 King’s College London, London, UK 13 Vol.:(0123456789) Algorithmica fact, the need to realign their individual interests with the designer’s. Truthfulness is the chief concept to achieve that: in a truthful mechanism, no selfish and rational agent has an interest to misguide the mechanism. A valid question of recent interest is, however, how easy it is for the selfish agents to understand that it is useless (and possibly costly) to attempt to strategize against the truthful mechanism at hand. Recent research has come up with different approaches to deal with this question. Some authors [1, 5, 9, 39] suggest to focus on “simple” mechanisms; e.g., in postedprice mechanisms one’s own bid is immaterial for the price paid to get some goods of interest—this should immediately suggest that trying to play the mechanism is worthless no matter the cognitive abilities of the agents. However, in such a body of work, this property remains unsatisfactorily vague. An orthogonal approach is that of verifiably truthful mechanisms [7], wherein agents can run some algorithm to effectively check that the mechanism is incentive compatible. Nevertheless, these verification algorithms can run for long (i.e., time exponential in the input size) and are so far known only for quite limited scenarios. Importantly, moreover, they seem to transfer the question from the mechanism itself to the verification algorithm. Li [32] has recently formalized the aforementioned idea of simple mechanisms, by introducing the concept of Obviously Strategy-Proof (OSP) mechanisms. This notion stems from the observation that the practical evidence of truthfulness depends on implementation details. For example, in lab experiments, people facing Vickrey’s famous second-price mechanism tend to lie more when this is implemented via a sealed-bid auction than when run via an ascending auction. The quite technical definition of OSP formally captures how implementation details matter by looking at a mechanism as an extensive-form game; roughly speaking, OSP demands that strategyproofness holds among some gross-grained aggregations of strategy profiles and not only among pairs of strategies profiles (see below for a formal definition). An important validation for the ‘obviousness’ is further provided by Li [32] via a characterization of these mechanisms in terms of agents with limited cognitive abilities (i.e., agents with limited skills in contingent reasoning). Specifically, Li shows that a strategy is obviously dominant if and only if these “limited” agents can recognize it as dominant. OSP is consequently a very appealing notion as in many cases rationality has been seen as the main obstacle to concrete applications of mechanism design paradigms, cf., e.g., Ferraioli et al. [16]; such a relaxation might be a panacea in these cases. Since its introduction the concept of obviously strategyproofness has been adopted both for allocation problems [3, 6, 18, 32] and for preference aggregation problems [6]. Nevertheless, for all its significant aspects, there appear to be hints that the notion of OSP mechanisms might be too restrictive. Ashlagi and Gonczarowski [3] prove, for example, that no OSP mechanism can return a stable matching—thus implying that the Gale–Shapley matching algorithm is not OSP. 1.1 Our Contribution We investigate the power of OSP mechanisms in more detail from a theoretical computer science perspective. In particular, we want to understand the quality of approximate solutions that can be output by OSP mechanisms. To answer this question, we 13 Algorithmica focus on two fundamental optimization problems, machine scheduling [2] and facility location [34], arguably (among) the paradigmatic problems in AMD. For the former problem, we want to compute a schedule of jobs on selfish related machines (i.e., machines with job-independent speeds) so to minimize the makespan. For this single-dimensional problem, it is known that a truthful PTAS is possible [10]. In contrast, we show that there is no better than 2-approximate OSP mechanism for this problem independently from the running time of the mechanism. This result highlights a stark contrast between machine scheduling and auctions. Specifically, [32] characterizes obviously strategyproof mechanisms for every single-dimensional problem where the output is binary (e.g., item won or not); it is not too hard to see that certain optimal algorithms fall within the characterization. Our result, instead, shows that when the output is more general than binary, even simple single-parameter problems become hopeless if obvious strategyproofness is required. For the facility location problem, we want to determine the location of a facility on the real line given the preferred locations of n agents. The objective is to minimize the social cost, defined as the sum over the individual agents of the distances between their preferred location and the facility’s. Moulin [34] proves that the optimal mechanism, that places the facility on the median of the reported locations, is truthful without money (i.e., the mechanism does not pay or charge the agents). OSP mechanisms wit (...truncated)