Mechanism Design for Fair Allocation

Mechanism design for a social utility being the sum of agents' utilities (SoU) is a well-studied problem. There are, however, a number of problems of theoretical and practical interest where a designer may have a different objective than maximization of the SoU. One motivation for this is the desire for more equitable allocation of resources among agents. A second, more subtle, motivation is the fact that a fairer allocation indirectly implies less variation in taxes which can be desirable in a situation where (implicit) individual agent budgetary constraints make payment of large taxes unrealistic. In this paper we study a family of social utilities that provide fair allocation (with SoU being subsumed as an extreme case) and derive conditions under which Bayesian and Dominant strategy implementation is possible. Furthermore, it is shown how a simple modification of the above mechanism can guarantee full Bayesian implementation. Through a numerical example it is shown that the proposed method can result in significant gains both in allocation fairness and tax reduction

Mechanism design for fair allocation on uniform machines

In traditional machine scheduling problems, a central decision maker, provided with all the relevant information about a system, is asked to derive an allocation scheme that optimizes some global objective, while simultaneously satisfying all the side constraints of the problem. However, since the emergence of the Internet as a computation platform, the assumption of information completeness does not hold anymore and algorithm designers are encouraged to reconsider the problem from a decentralized perspective. Most importantly, when decisions are made by independent agents, it is more likely that a rational agent will implement the strategy in such a way that maximizes their own interests, regardless of the overall system performance. Such situations require algorithm designers to not only focus on the global performance of the system, but also to take into account the strategic behaviour of the individuals involved. Algorithmic Mechanism Design (AMD), a term coined by Nisan and Ronen (1999), specifcally targets this kind of problem, where part of the input is under the control of selfish agents who do not have an incentive to tell the truth, unless truth-telling is for their own good. This type of design endeavours to merge the challenges from two classic disciplines: algorithm design in computer science, and mechanism design in game theory. The former emphasizes the importance of the computational e - ciency of an algorithm, while ignoring the elements related to incentives; the latter, instead, normally yields game theoretic outcomes with poor computations. AMD, on the other hand, aims to present good game theoretic properties and good computational properties at the same time. Guided by the idea of AMD, research has been conducted on scheduling problems with various models and various objectives. Among them, some of the most popular objectives include the minimization of the maximum completion time (also known as the makespan), and the maximization of the minimum completion time (also known as the cover). Minimizing the makespan is naturally related to effciency, as it ensures that the entire job set is completed within the shortest possible time; maximizing cover, instead, embodies the concept of fairness from the machine owner's perspective in the sense that a machine will not get exemption due to its slowness. However, it can be argued that the fairness embodied by both objectives is only to a limited extent as they both can lead to extreme situations. Fairness is an important social concept that has not been well considered in the literature of AMD. This is surprising given that \each person possesses an inviolability founded on justice that even the welfare of the society as a whole cannot override" (Rawls, 2009, pg.3). To concretely state the importance of fairness in the scheduling context, let us consider the problem faced by the U.S. Federal Aviation Administration. Billions of monetary losses are incurred as a result of unpredictable system delays each year. To improve this situation, proposals have been raised by scholars which have guaranteed more efficient schedules and claimed greater savings in costs. Unfortunately, few of those proposals have been implemented in practice, mainly because they fail to take the issue of fairness into consideration (Bertsimas and Patterson, 2000). Fairness plays a key role in resource allocation, especially in socially oriented areas, including education, medical systems, and businesses. Although it is in the interest of the central authority to achieve system efficiency when allocating resources, individual players tend to care more about their own interests. If they cannot maximize their own benefits, then they at least want to be treated fairly. As a special case of resource allocation, machine scheduling problems also face similar challenges deriving from the players' desire for fairness. To achieve higher levels of fairness, we propose a new objective called minimizing the maximum deviation, which aims to minimize the maximum deviation between the completion time of each individual machine and the average completion time of the system, calculated as the sum of all the job sizes divided by the sum of all the machine speeds. To the best of our knowledge, this objective has not been considered by others before