We here address the problem of fairly allocating indivisible goods or chores to n agents with weights that define their entitlement to the set of indivisible resources. Stemming from well-studied fairness concepts such as envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) for agents with equal entitlements, we present, in this study, the first set of impossibility results alongside algorithmic guarantees for fairness among agents with unequal entitlements. Within this paper, we expand the concept of envy-freeness up to any good or chore to the weighted context (WEFX and XWEF respectively), demonstrating that these allocations are not guaranteed to exist for two or three agents. Despite these negative results, we develop a WEFX procedure for two agents with integer weights, and furthermore, we devise an approximate WEFX procedure for two agents with normalized weights. We further present a polynomial-time algorithm that guarantees a weighted envy-free allocation up to one chore (1WEF) for any number of agents with additive cost functions. Our work underscores the heightened complexity of the weighted fair division problem when compared to its unweighted counterpart.

Envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents and additive valuations. In addition, the best approximation guarantee for EFX is (φ − 1) ≃ 0.61 by Amanitidis et al.. In order to find a middle ground to bridge this gap, in this paper we suggest another fairness criterion, namely envy-freeness up to a random good or EFR, which is weaker than EFX, yet stronger than EF1. For this notion, we provide a polynomial-time 0.73-approximation allocation algorithm. For our algorithm we introduce Nash Social Welfare Matching which makes a connection between Nash Social Welfare and envy freeness.

Search ads have evolved in recent years from simple text formats to rich ads that allow deep site links, rating, images and videos. In this paper, we consider a model where several slots are available on the search results page, as in the classic generalized second-price auction (GSP), but now a bidder can be allocated several consecutive slots, which are interpreted as a rich ad. As in the GSP, each bidder submits a bid-per-click, but the click-through rate (CTR) function is generalized from a simple CTR for each slot to a general CTR function over sets of consecutive slots. We study allocation and pricing in this model under subadditive and fractionally subadditive CTRs. We design and analyze a constant-factor approximation algorithm for the efficient allocation problem under fractionally subadditive CTRs, and a log-approximation algorithm for the subadditive case. Building on these results, we show that approximate competitive equilibrium prices exist and can be computed for subadditive and fractionally subadditive CTRs, with the same guarantees as for allocation.

We study fair allocation of indivisible goods to agents with unequal entitlements. Fair allocation has been the subject of many studies in both divisible and indivisible settings. Our emphasis is on the case where the goods are indivisible and agents have unequal entitlements. This problem is a generalization of the work by Procaccia and Wang (2014) wherein the agents are assumed to be symmetric with respect to their entitlements. Although Procaccia and Wang show an almost fair (constant approximation) allocation exists in their setting, our main result is in sharp contrast to their observation. We show that, in some cases with n agents, no allocation can guarantee better than 1/n approximation of a fair allocation when the entitlements are not necessarily equal. Furthermore, we devise a simple algorithm that ensures a 1/n approximation guarantee. Our second result is for a restricted version of the problem where the valuation of every agent for each good is bounded by the total value he wishes to receive in a fair allocation. Although this assumption might seem without loss of generality, we show it enables us to find a 1/2 approximation fair allocation via a greedy algorithm. Finally, we run some experiments on real-world data and show that, in practice, a fair allocation is likely to exist. We also support our experiments by showing positive results for two stochastic variants of the problem, namely stochastic agents and stochastic items.

We study the problem of fair allocation for indivisible goods. We use the maxmin share paradigm introduced by Budish~\citeBudish:first as a measure for fairness. \procacciafirst ~\citeProcaccia:first were the first to investigate this fundamental problem in the additive setting. They show that a maxmin guarantee (1-$\MMS$ allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a $1/2$-$\MMS$ allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. \sprocacciafirst ~\citeProcaccia:first provide a proof for the existence of a $2/3$-$\MMS$ allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of \sprocacciafirst~to a $3/4$ factor in the additive setting. The main idea for our $3/4$-$\MMS$ allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time. The reader can find a summary of our results in Table \refresultstable.

Chore division, introduced by Gardner in 1970s [10], is the problem of fairly dividing a chore among n different agents. In particular, in an envy-free chore division, we would like to divide a negatively valued heterogeneous object among a number of agents who have different valuations for different parts of the object, such that no agent envies another agent. It is the dual variant of the celebrated cake cutting problem, in which we would like to divide a desirable object among agents. There has been an extensive amount of study and effort to design bounded and envy-free protocols/algorithms for fair division of chores and goods, such that envy-free cake cutting became one of the most important open problems in 20-th century mathematics according to Garfunkel [11]. However, despite persistent efforts, due to delicate nature of the problem, there was no bounded protocol known for cake cutting even among four agents, until the breakthrough of Aziz and Mackenzie [2], which provided the first discrete and bounded envy-free protocol for cake cutting for four agents. Afterward, Aziz and Mackenzie [3], generalized their work and provided an envy-free cake cutting protocol for any number of agents to settle a significant and longstanding open problem. However, there is much less known for chore division. Unfortunately, there is no general method known to apply cake cutting techniques to chore division. Thus, it remained an open problem to find a discrete and bounded envy-free chore division protocol even for four agents. In this paper, we provide the first discrete and bounded envy-free protocol for chore division for an arbitrary number of agents. We produce major and powerful tools for designing protocols for the fair division of negatively valued objects. These tools are based on structural results and important observations. In general, we believe these structures and techniques may be useful not only in chore division but also in other fairness problems. Interestingly, we show that applying these techniques simplifies Core Protocol provided in Aziz and Mackenzie [3].

Internet-enabled marketplaces such as Amazon deal with huge datasets registering transaction of merchandises between lots of buyers and sellers. It is important that algorithms become more time and space efficient as the size of datasets increase. An algorithm that runs in polynomial time may not have a reasonable running time for such large datasets. Here, we study the development of pricing algorithms that are appropriate for use with massive datasets. We especially focus on the streaming setting, the common model for big data analysis. We present an envy-free mechanism for social welfare maximization problem in the streaming setting using O(k2 l) space, where k is the number of different goods and l is the number of available items of each good. We also provide an α-approximation mechanism for revenue maximization in this setting given an α-approximation mechanism for the corresponding offline problem exists. Moreover, we provide mechanisms to approximate the optimum social welfare (or revenue) within 1 – ε factor, in space independent of l which would be favorable in case l is large compared to k. Finally, we present hardness results showing approximation of optimal prices that maximize social welfare (or revenue) in the streaming setting needs Ω(l) space. We achieve our results by developing a powerful sampling technique for bipartite networks. The simplicity of our sampling technique empowers us to maintain the sample over the input sequence. Indeed, one can construct this sample in the distributed setting (a.k.a, MapReduce) and get the same results in two rounds of computations, or one may simply apply this sampling technique to provide faster offline algorithms.

In this paper we consider the maximum matching problem in large bipartite graphs. We present a new algorithm that finds the maximum matching in a few iterations of a novel edge sampling technique. This algorithm can be implemented in big data settings such as streaming setting and MapReduce setting, where each iteration of the algorithm maps to one pass over the stream, or one MapReduce round of computation, respectively. We prove that our algorithm provides a 1-\eps approximate solution to the maximum matching in 1/\eps rounds which improves the prior work in terms of the number of passes/rounds. Our algorithm works even better when we run it on real datasets and finds the exact maximum matching in 4 to 8 rounds while sampling only about %1 of the total edges.