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.
We now make some comments on the relation between our results and those of @cite_11 concerning envy-free allocations. As we will later elaborate, proportional fairness is a weaker notion of fairness than envy-freeness, i.e., envy-free allocations are also proportionally fair when utilities are additive. showed that under additive utilities, envy-free allocations are unlikely to exist when the number of goods is larger than the number of agents by a linear fraction. Theorem contrasts that result by showing that proportionally fair allocations are likely to exist even when the number of goods is the same as the number of agents.
@cite_11
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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. Mechanisms for dividing a set of goods amongst a number of autonomous agents need to balance efficiency and fairness requirements. A common interpretation of fairness is envy-freeness, while efficiency is usually understood as yielding maximal overall utility. We show how to set up a distributed negotiation framework that will allow a group of agents to reach an allocation of goods that is both efficient and envy-free. In the context of fair allocation of indivisible items, fairness concepts often compare the satisfaction of an agent to the satisfaction she would have from items that are not allocated to her: in particular, envy-freeness requires that no agent prefers the share of someone else to her own share. We argue that these notions could also be defined relative to the knowledge that an agent has on how the items that she does not receive are distributed among other agents. We define a family of epistemic notions of envy-freeness, parameterized by a social graph, where an agent observes the share of her neighbours but not of her non-neighbours. We also define an intermediate notion between envy-freeness and proportionality, also parameterized by a social graph. These weaker notions of envy-freeness are useful when seeking a fair allocation, since envy-freeness is often too strong. We position these notions with respect to known ones, thus revealing new rich hierarchies of fairness concepts. Finally, we present a very general framework that covers all the existing and many new fairness concepts. Abstract Consider a heterogeneous but divisible commodity, bundles of which are represented by the (measurable) subsets of the good. One such commodity might be land. The mathematics literature has considered agents with utilities that are nonatomic measures over the commodity (and hence are additive). The existence of ‘α-fair’ allocations, in which each agent receives a utility proportional to his utility of the endowment of the entire economy, was demonstrated there. Here we extend these existence results to α-fair efficient allocations, envy-free allocations, envy-free efficient allocations, group envy-free and nicely shaped allocations of these types. We examine utilities that are not additive and relate the mathematics literature to the economics literature. We find sufficient conditions for the existence of egalitarian-equivalent efficient allocations. Finally, we consider the problem of allocating a time interval (uses of a facility). Existence of an envy-free allocation had been demonstrated in earlier literature. We show that any envy-free allocation is efficient as well as group envy-free. We extend this last result to a more general setting.