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Truthful and Fair Mechanisms for Matroid-Rank Valuations
arXiv - CS - Computer Science and Game Theory Pub Date : 2021-09-13 , DOI: arxiv-2109.05810
Siddharth Barman, Paritosh Verma

We study the problem of allocating indivisible goods among strategic agents. We focus on settings wherein monetary transfers are not available and each agent's private valuation is a submodular function with binary marginals, i.e., the agents' valuations are matroid-rank functions. In this setup, we establish a notable dichotomy between two of the most well-studied fairness notions in discrete fair division; specifically, between envy-freeness up to one good (EF1) and maximin shares (MMS). First, we show that a Pareto-efficient mechanism of Babaioff et al. (2021) is group strategy-proof for finding EF1 allocations, under matroid-rank valuations. The group strategy-proofness guarantee strengthens the result of Babaioff et al. (2021), that establishes truthfulness (individually for each agent) in the same context. Our result also generalizes a work of Halpern et al. (2020), from binary additive valuations to the matroid-rank case. Next, we establish that an analogous positive result cannot be achieved for MMS, even when considering truthfulness on an individual level. Specifically, we prove that, for matroid-rank valuations, there does not exist a truthful mechanism that is index oblivious, Pareto efficient, and maximin fair. For establishing our results, we develop a characterization of truthful mechanisms for matroid-rank functions. This characterization in fact holds for a broader class of valuations (specifically, holds for binary XOS functions) and might be of independent interest.

中文翻译:

拟阵秩估价的真实公平机制

我们研究了在战略代理人之间分配不可分割的物品的问题。我们专注于货币转移不可用且每个代理的私人估值是具有二元边际的子模函数的设置,即代理的估值是拟阵秩函数。在这种设置中,我们在离散公平划分中两个最深入研究的公平概念之间建立了显着的二分法;具体来说,在达到一种商品的无嫉妒(EF1)和最大份额(MMS)之间。首先,我们展示了 Babaioff 等人的帕累托有效机制。(2021) 是在拟阵等级估值下寻找 EF1 分配的组策略证明。群体策略证明性保证加强了 Babaioff 等人的结果。(2021),在相同的上下文中建立真实性(每个代理单独)。我们的结果还概括了 Halpern 等人的工作。(2020),从二元加法估值到拟阵秩案例。接下来,我们确定 MMS 无法获得类似的积极结果,即使在个人层面考虑真实性也是如此。具体来说,我们证明,对于拟阵秩估值,不存在指数遗忘、帕累托有效和最大化公平的真实机制。为了建立我们的结果,我们开发了拟阵秩函数的真实机制的表征。这种特征实际上适用于更广泛的估值类别(特别是,适用于二元 XOS 函数)并且可能具有独立的兴趣。即使在个人层面考虑真实性。具体来说,我们证明,对于拟阵秩估值,不存在指数遗忘、帕累托有效和最大化公平的真实机制。为了建立我们的结果,我们开发了拟阵秩函数的真实机制的表征。这种特征实际上适用于更广泛的估值类别(特别是,适用于二元 XOS 函数)并且可能具有独立的兴趣。即使在个人层面考虑真实性。具体来说,我们证明,对于拟阵秩估值,不存在指数遗忘、帕累托有效和最大化公平的真实机制。为了建立我们的结果,我们开发了拟阵秩函数的真实机制的表征。这种特征实际上适用于更广泛的估值类别(特别是,适用于二元 XOS 函数)并且可能具有独立的兴趣。
更新日期:2021-09-14
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