Walsh[2] studies an online variant of fair cake-cutting, in which agents arrive and depart during the division process, like in a party.
Sinclair, Jain, Bannerjee and Yu[3][4] study allocation of divisible resources when individuals arrive randomly over time.
The following extensions are known: The cake redivision problem[10] is a variant of fair cake-cutting in which the cake is already divided in an unfair way (e.g. among a subset of the agents), and it should be re-divided in a fair way (among all the agents) while letting the incumbent owners keep a substantial fraction of their present value.
The food bank problem is an online variant of fair allocation of indivisible goods.
Working with Foodbank Australia, Aleksandrov, Aziz, Gaspers and Walsh[11] have initiated the study of the food bank problem when all agents have binary valuations {0,1}, that is, for each arriving item, every agent states whether he likes the item or not.
They study two simple mechanisms for this setting: In a more general case of the food bank problem, agents can have additive valuations, which are normalized to [0,1].
Due to the online nature of the problem, it may be impossible to attain some fairness and efficiency guarantees that are possible in the offline setting.
In particular, Kahana and Hazon[12] prove that no online algorithm always finds a PROP1 (proportional up to at most one good) allocation, even for two agents with additive valuations.
Moreover, no online algorithm always finds any positive approximation of RRS (round-robin share).
Benade, Kazachkov, Procaccia and Psomas[13] study another fairness criterion - envy-freeness.
They show that: Jiang, Kulkarni and Singla[14] improve the bound for the case of n=2 agents, when the values are random (rather than adversarial).
Zeng and Psomas[15] study the trade-off between efficiency and fairness under five adversary models, from weak to strong.
In many fair division problems, such as production of energy from solar cells, the exact amount of available resource may not be known at the time the allocation is decided.
The latter criterion is weaker (since envy-freeness holds only in expectation), but it allows a higher social welfare.
Each village has a different probability distribution over storms: The government has two generators, each of which can supply electricity to a single house.
Igarashi, Lackner, Nardi and Novaro[28] study repeated fair allocation of indivisible items.
They consider two settings: the number of rounds can be either fixed in advance, or variable (can be chosen by the algorithm).