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Truthfulness on a budget: trading money for approximation through monitoring

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Abstract

Albeit a pervasive desideratum when computing in the presence of selfish agents, truthfulness typically imposes severe limitations to what can be implemented. The price of these limitations is typically paid either economically, in terms of the financial resources needed to enforce truthfulness, or algorithmically, in terms of restricting the set of implementable objective functions, which often leads to renouncing optimality and resorting to approximate allocations. In this paper, with regards to utilitarian problems, we ask two fundamental questions: (i) what is the minimum sufficient budget needed by optimal truthful mechanisms, and (ii) whether it is possible to sacrifice optimality in order to achieve truthfulness with a lower budget. To answer these questions, we connect two streams of work on mechanism design and look at monitoring—a paradigm wherein agents’ actual costs are bound to their declarations. In this setting, we prove that the social cost is always a sufficient budget, even for collusion-resistant mechanisms, and, under mild conditions, also a necessary budget for a large class of utilitarian problems that encompass set system problems. Furthermore, for two well-studied problems outside of this class, namely facility location and obnoxious facility location, we draw a novel picture about the relationship between (additive) approximation and frugality. While for optimal mechanisms we prove that the social cost is always a sufficient and necessary budget for both problems, for approximate mechanisms we do have a dichotomy: for the facility location problem (i.e., agents want to be close to the facilities) we show that “good” approximations still need a budget equal to the social cost; on the contrary, for the obnoxious facility location problem (i.e. agents want to be as far away from the facilities as possible) we show that it is possible to trade approximation for frugality, thus obtaining truthfulness with a lower budget.

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Notes

  1. The terminology is introduced in [33] while [23, 32] study the same model under the misnomer of mechanisms with verification. In the verification model of [39] it is effectively assumed that the designer can monitoring overbidding and punish underbidding; in [41] (and related literature) instead the designer only punishes underbidding but does not monitor overbidding—the difference between the models is furthermore studied therein. We here only monitor overbidding as in [23, 32, 33].

  2. A set system \((E, \mathcal {F})\) is an r-out-of-k-system if there exists a partition of E into k disjoint sets \(S_1,\ldots ,S_k\), such that every set \(F \in \mathcal {F}\) contains exactly r out of these k sets of elements.

  3. One might wonder whether relaxing the upper bound for some agents, and allowing a surplus for them, would allow to save more on the remaining agents. Our results of necessity cover also this case since they prove that no single agent can have a negative discount.

  4. For the sake of brevity, unless otherwise stated, in the remainder we will always refer to bid-independent discounted first-price mechanisms simply as discounted first-price mechanisms, whereas we will explicitly write single-bid discounted first-price mechanisms to refer to payment functions that employ single-bid discounts.

  5. See [37] for the similarities among facility location and clustering/classification with strategic data sources.

  6. A simpler proof works when agents have unrestricted domains (i.e., simply set \(b_i\) big enough with respect to \({\mathbf {b}}_{-i}\) so to force \(f^*\) to locate one facility on \(b_i\)). Our argument applies also to more restricted settings as considered in, e.g., [6].

  7. We justify our focus on \(f_\varepsilon ^*\) by highlighting how useful a tool it is to study the tradeoff between frugality and approximation in this setting, as it allows us to both: (i) easily predict the location of the facility; and (ii) compute the approximation ratio achievable given the particular configuration of the instance at hand (see Theorem 8). If we were to focus, for instance, on the class of mechanisms having a given approximation guarantee, we would certainly retain (ii) but we would have to restrict ourselves to specific algorithms within this class to also retain (i).

  8. We note that unless the length of the interval is specified, the problem is not well-defined, as a solution with lower social cost can always be obtained by moving the facility farther away.

  9. Note that we are assuming a fixed tie-breaking rule when there are multiple optimal allocations (i.e., n is even and n / 2 agents are at 0 and n / 2 agents at \(\ell \)). We assume that the algorithm will allocate consistently the facility at one extreme of the interval.

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Authors and Affiliations

  1. Gran Sasso Science Institute, L’Aquila, Italy

    Paolo Serafino

  2. King’s College London, London, UK

    Carmine Ventre

  3. University of Athens, Athens, Greece

    Angelina Vidali

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  1. Paolo Serafino
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Serafino, P., Ventre, C. & Vidali, A. Truthfulness on a budget: trading money for approximation through monitoring. Auton Agent Multi-Agent Syst 34, 5 (2020). https://doi.org/10.1007/s10458-019-09435-9

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