Developments in Multi-Agent Fair Allocation

Fairness is becoming an increasingly important concern when designing markets, allocation procedures, and computer systems. I survey some recent developments in the field of multi-agent fair allocation

Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results

Fair resource allocation is an important problem in many real-world scenarios, where resources such as goods and chores must be allocated among agents. In this survey, we delve into the intricacies of fair allocation, focusing specifically on the challenges associated with indivisible resources. We define fairness and efficiency within this context and thoroughly survey existential results, algorithms, and approximations that satisfy various fairness criteria, including envyfreeness, proportionality, MMS, and their relaxations. Additionally, we discuss algorithms that achieve fairness and efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study the computational complexity of these algorithms, the likelihood of finding fair allocations, and the price of fairness for each fairness notion. We also cover mixed instances of indivisible and divisible items and investigate different valuation and allocation settings. By summarizing the state-of-the-art research, this survey provides valuable insights into fair resource allocation of indivisible goods and chores, highlighting computational complexities, fairness guarantees, and trade-offs between fairness and efficiency. It serves as a foundation for future advancements in this vital field

Maximin Fairness with Mixed Divisible and Indivisible Goods

We study fair resource allocation when the resources contain a mixture of divisible and indivisible goods, focusing on the well-studied fairness notion of maximin share fairness (MMS). With only indivisible goods, a full MMS allocation may not exist, but a constant multiplicative approximate allocation always does. We analyze how the MMS approximation guarantee would be affected when the resources to be allocated also contain divisible goods. In particular, we show that the worst-case MMS approximation guarantee with mixed goods is no worse than that with only indivisible goods. However, there exist problem instances to which adding some divisible resources would strictly decrease the MMS approximation ratio of the instance. On the algorithmic front, we propose a constructive algorithm that will always produce an $\alpha$-MMS allocation for any number of agents, where $\alpha$ takes values between $1/2$ and $1$ and is a monotone increasing function determined by how agents value the divisible goods relative to their MMS values.Comment: Appears in the 35th AAAI Conference on Artificial Intelligence (AAAI), 202

Simplification and Improvement of MMS Approximation

We consider the problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The current best approximation factor, for which the existence is known, is $(\frac{3}{4} + \frac{1}{12n})$ [Garg and Taki, 2021]. Most of these results are based on complicated analyses, especially those providing better than $2/3$ factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of $(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$. For small $n$, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques

Fairness Guarantees in Allocation Problems

Fair division problems have been vastly studied in the past 60 years. This line of research was initiated by the work of Steinhaus in 1948 in which the author introduced the cake cutting problem as follows: given a heterogeneous cake and a set of agents with different valuation functions, the goal is to find a fair allocation of the cake to the agents. In order to study this problem, several notions of fairness are proposed, the most famous of which are proportionality and envy-freeness, introduced by Steinhaus in 1948 and Foley in 1967. The fair allocation problems have been studied in both divisible and indivisible settings. For the divisible setting, we explore the "Chore Division Problem". The chore division problem is the problem of fairly dividing an object deemed undesirable among a number of agents. The object is possibly heterogeneous, and hence agents may have different valuations for different parts of the object. Chore division is the dual problem of the celebrated cake cutting problem. We give the first discrete and bounded envy-free chore division protocol for any number of agents. For the indivisible setting, we use the maximin share paradigm introduced by Budish as a measure of fairness. We improve previous results on this measure of fairness in the additive setting and generalize our results for submodular, fractionally subadditive, as well as subadditive settings. We also model the maxmin share fairness paradigm for indivisible goods with different entitlements. For the indivisible setting, we also consider the most studied notion of fairness, envy-freeness. It is known that envy-freeness cannot be always guaranteed in the allocation of indivisible items. We suggest envy-freeness up to a random item (EFR) property which is a relaxation of envy-freeness up to any item (EFX) and give an approximation guarantee. For this notion, we provide a polynomial-time 0.72-approximation allocation algorithm