text: Fair division has long been an important problem in the economics literature. In this note, we consider the existence of proportionally fair allocations of indivisible goods, i.e., allocations of indivisible goods in which every agent gets at least her proportionally fair share according to her own utility function. We show that when utilities are additive and utilities for individual goods are drawn independently at random from a distribution, proportionally fair allocations exist with high probability if the number of goods is a multiple of the number of agents or if the number of goods grows asymptotically faster than the number of agents.
  @cite_14: We study the problem of computing maximin share allocations, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of an agent is the best she can guarantee to herself, if she is allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then is to find a partition, where each agent is guaranteed her maximin share. Such allocations do not always 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 and goods. This improves upon the algorithm of Procaccia and Wang (2014), which is also 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 Procaccia and Wang (2014), exploiting the construction of carefully selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty in establishing lower bounds, we undertake a probabilistic analysis. We prove that in randomly generated instances, maximin share allocations exist with high probability. This can be seen as a justification of previously reported experimental evidence. Finally, we provide further positive results for two special cases arising from previous works. The first is the intriguing case of three agents, where we provide an improved 7 8-approximation. The second case is when all item values belong to 0, 1, 2 , where we obtain an exact algorithm.
  @cite_7: 
  @cite_8: The fairness notion of maximin share (MMS) guarantee underlies a deployed algorithm for allocating indivisible goods under additive valuations. Our goal is to understand when we can expect to be able to give each player his MMS guarantee. Previous work has shown that such an MMS allocation may not exist, but the counterexample requires a number of goods that is exponential in the number of players; we give a new construction that uses only a linear number of goods. On the positive side, we formalize the intuition that these counterexamples are very delicate by designing an algorithm that provably finds an MMS allocation with high probability when valuations are drawn at random.
  @cite_3: 
  @cite_5: A number of labor markets and student placement systems can be modeled as many-to-one matching markets. We analyze the scope for manipulation in many-to-one matching markets under the student-optimal stable mechanism when the number of participants is large. Under some regularity conditions, we show that the fraction of participants with incentives to misrepresent their preferences when others are truthful approaches zero as the market becomes large. With an additional condition, truthful reporting by every participant is an approximate equilibrium under the student-optimal stable mechanism in large markets. (JEL C78)
  @cite_11: The fair division of indivisible goods has long been an important topic in economics and, more recently, computer science. We investigate the existence of envyfree allocations of indivisible goods, that is, allocations where each player values her own allocated set of goods at least as highly as any other player's allocated set of goods. Under additive valuations, we show that even when the number of goods is larger than the number of agents by a linear fraction, envy-free allocations are unlikely to exist.We then show that when the number of goods is larger by a logarithmic factor, such allocations exist with high probability. We support these results experimentally and show that the asymptotic behavior of the theory holds even when the number of goods and agents is quite small. We demonstrate that there is a sharp phase transition from nonexistence to existence of envy-free allocations, and that on average the computational problem is hardest at that transition.
summary: @cite_11 considered asymptotic existence and nonexistence of envy-free allocations. They showed that under additive utilities, envy-free allocations are unlikely to exist even when the number of goods is larger than the number of agents by a linear fraction. On the other hand, they proved that when the number of goods is larger than the number of agents by a logarithmic factor, such allocations are likely to exist under certain technical conditions on the probability distribution. @cite_14 and @cite_8 considered allocations that gives each agent a maximin share guarantee and showed that such allocations exist with high probability when utilities are additive. Asymptotic statements have been considered in other areas of economics as well. For instance, Manea @cite_3 established that the allocation obtained by the random serial dictatorship mechanism is ordinally inefficient with high probability when agents' preference profiles are drawn at random. Incentives and stability in large matching markets have also been considered in the literature @cite_7 @cite_5 .