Instead, we seek to learn a fair distribution over the arms. Drawing on a long line of research in economics and computer science, we use the Nash social welfare as our notion of fairness.

The appendix is organized as follows: Appendix A: In this section, we prove IS functions are XOS. Appendix B: In this section, we formally prove our main results in Section 3. Appendix C: In this section, we formally prove our main results in Section 4. Appendix D: In this section, we introduce a milder efficiency requirement IO and study its compatibility with EF1. Appendix E: In this section, we discuss our experiments in more details. Regarding agents' utilities, FISP contains three cases, from the most special to the most general: Unweighted: u J, i.e., agents have unary utility for jobs. J, i.e., all jobs have unit processing time.

A, B to denote allocations that are exclusively integral and R for randomized allocations. On a high level, the PS-Lottery algorithm uses Birkhoff's We begin by proving a lemma that highlights a connection between not obvious manipulability and randomized mechanisms that output ex-ante proportional allocations. Lemma 5. Inequality (1) (the worst-case guarantee) is satisfied for every randomized mechanism Note that multiple randomized allocations may have the same expected fractional allocation. Recall that, Birkhoff's algorithm, given a square bistochastic matrix, decomposes it into a convex combination (or a lottery) over permutation matrices. Using Lemma 5 we can prove the following theorem.

It is well-understood that, in the absence of monetary transfers, fairness, efficiency and truthfulness cannot be reconciled, in a very strong sense.

Data is the new oil of the 21st century. The growing trend of trading data for greater welfare has led to the emergence of data markets. A data market is any mechanism whereby the exchange of data products including datasets and data derivatives takes place as a result of data buyers and data sellers being in contact with one another, either directly or through mediating agents. It serves as a coordinating mechanism by which several functions, including the pricing and the distribution of data as the most important ones, interact to make the value of data fully exploited and enhanced. In this article, we present a comprehensive survey of this important and emerging direction from the aspects of data search, data productization, data transaction, data pricing, revenue allocation as well as privacy, security, and trust issues. We also investigate the government policies and industry status of data markets across different countries and different domains. Finally, we identify the unresolved challenges and discuss possible future directions for the development of data markets.

We study learning-based design of fair allocation mechanisms for divisible resources, using proportional fairness (PF) as a benchmark. The learning setting is a significant departure from the classic mechanism design literature, in that, we need to learn fair mechanisms solely from data. In particular, we consider the challenging problem of learning one-shot allocation mechanisms -- without the use of money -- that incentivize strategic agents to be truthful when reporting their valuations. It is well-known that the mechanism that directly seeks to optimize PF is not incentive compatible, meaning that the agents can potentially misreport their preferences to gain increased allocations. We introduce the notion of "exploitability" of a mechanism to measure the relative gain in utility from misreport, and make the following important contributions in the paper: (i) Using sophisticated techniques inspired by differentiable convex programming literature, we design a numerically efficient approach for computing the exploitability of the PF mechanism. This novel contribution enables us to quantify the gap that needs to be bridged to approximate PF via incentive compatible mechanisms. (ii) Next, we modify the PF mechanism to introduce a trade-off between fairness and exploitability. By properly controlling this trade-off using data, we show that our proposed mechanism, ExPF-Net, provides a strong approximation to the PF mechanism while maintaining low exploitability. This mechanism, however, comes with a high computational cost. (iii) To address the computational challenges, we propose another mechanism ExS-Net, which is end-to-end parameterized by a neural network. ExS-Net enjoys similar (slightly inferior) performance and significantly accelerated training and inference time performance. (iv) Extensive numerical simulations demonstrate the robustness and efficacy of the proposed mechanisms.