We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, hence, we resort to approximation algorithms. Our main result is a 2/3-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang [14], which also produces a 2/3-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of the algorithm in [14], exploiting the construction of certain, appropriately selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances there exists a maximin share allocation with high probability. This can be seen as a justification of the experimental evidence reported in [5,14], that maximin share allocations exist almost always.Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of 3 agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of 2 agents). We provide a 7/8-approximation algorithm, improving the previously known result of 3/4 [14]. The second case is when all item values belong to {0, 1, 2}, extending the {0, 1} setting studied in [5]. We obtain an exact algorithm for any number of agents in this case.

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We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, hence, we resort to approximation algorithms. Our main result is a 2/3-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang [14], which also produces a 2/3-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of the algorithm in [14], exploiting the construction of certain, appropriately selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances there exists a maximin share allocation with high probability. This can be seen as a justification of the experimental evidence reported in [5,14], that maximin share allocations exist almost always.Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of 3 agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of 2 agents). We provide a 7/8-approximation algorithm, improving the previously known result of 3/4 [14]. The second case is when all item values belong to {0, 1, 2}, extending the {0, 1} setting studied in [5]. We obtain an exact algorithm for any number of agents in this case.

In this paper we consider a mechanism design problem in the context of large-scale crowdsourcing markets such as Amazon's Mechanical Turk (MTrk), ClickWorker (ClkWrkr), CrowdFlower (CrdFlwr). In these markets, there is a requester who wants to hire workers to accomplish some tasks. Each worker is assumed to give some utility to the requester on getting hired. Moreover each worker has a minimum cost that he wants to get paid for getting hired. This minimum cost is assumed to be private information of the workers. The question then is -if the requester has a limited budget, how to design a direct revelation mechanism that picks the right set of workers to hire in order to maximize the requester's utility?We note that although the previous work (Singer (2010); Chen et al. (2011)) has studied this problem, a crucial difference in which we deviate from earlier work is the notion of large-scale markets that we introduce in our model. The notion of a large-scale market that we consider is a natural one which states that the (private) cost of each worker is small compared to the budget of the requester. Without the large market assumption, it is known that no mechanism can achieve a competitive ratio better than 0.414 and 0.5 for deterministic and randomized mechanisms respectively (while the best known deterministic and randomized mechanisms achieve an approximation ratio of 0.292 and 0.33 respectively). In this paper, we design a budget-feasible mechanism for large markets that achieves a competitive ratio of 1−1/e 0.63. Our mechanism can be seen as a generalization of an alternate way to look at the proportional share mechanism, which is used in all the previous works so far on this problem. Interestingly, we can also show that our mechanism is optimal by showing that no truthful mechanism can achieve a factor better than 1 − 1/e; thus, fully resolving this setting. Finally we consider the more general case of submodular utility functions and give new and improved mechanisms for the case when the market is large.

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