We investigate the fair division of a sequence of time slots among n agents when each agent is sufficiently patient. For the case of identical preferences, we construct perfectly equitable and efficient allocations. For n = 2, we construct envy-free allocations. For n = 3, we construct proportional allocations using two procedures. For the general case, we construct approximately fair allocations with two procedures
from the literature.

This works deals with an apportionment problem involving n agents and a common budget B. Each agent submits some demands which are indivisible portions of the budget, and a central authority has to decide which demands to accept. The utility of an agent corresponds to the total amount of her accepted demands. In this context, it is desirable to be fair among the agents and efficient by not wasting the budget. An ideal solution would be to spend exactly B/n for every agent but this is rarely possible because of the indivisibility of the demands. Several alternative notions of fairness exist such as envy-freeness, and the maximization of the minimum agent utility. However, combining efficiency with fairness is often impossible, and a trade-off has to be made. To do so, we use relaxed notions of fairness and efficiency. In this work, we first study the computation of almost fair and approximately efficient solutions, and we determine when these two goals can be met. Afterwards, we characterize the price of fairness which bounds the loss of efficiency caused by imposing fairness or one of its relaxations.

We propose a fair and efficient solution for assigning agents to m posts subject to congestion, when agents care about both their post and its congestion. Examples include assigning jobs to busy servers, students to crowded schools or crowded classes, commuters to congested routes, workers to crowded office spaces or to team projects etc...
Congestion is anonymous (it only depends on the number n of agents in a given post). A canonical interpretation of ex ante fairness allows each agent to choose m post-specific caps on the congestion they tolerate: these requests are mutually feasible if and only if the sum of the caps is n.
For ex post fairness we impose a competitive requirement close to envy freeness: taking the congestion profile as given each agent is assigned to one of her best posts. If a competitive assignment exists, it delivers unique congestion and welfare profiles and is also efficient and ex ante fair.
In a fractional (randomised or time sharing) version of our model, a unique competitive congestion profile always exists. It is approximately implemented by a mixture of ex post deterministic assignments: with an approxination factor equal to the largest utility loss from one more unit of congestion, the latter deliver identical welfare profiles and are weakly efficient.
Our approach to ex ante fairness generalises to the model where each agent's congestion is weighted. Now the caps on posts depend only upon own weight and total congestion, not on the number