Abstract
In this paper, we extend the Rent Sharing problem to the case that every room must be allocated to a group of agents. In the classic Rent Sharing problem, there are n agents and a house with n rooms. The goal is to allocate one room to each agent and assign a rent to each room in a way that no agent envies any other option. Our setting deviates from the classic Rent Sharing problem in a sense that the rent charged to each room must be divided among the members of the resident group.
We define three notions to evaluate fairness, namely, weak envy-freeness, aggregate envy-freeness and strong envy-freeness. We also define three different policies to divide the cost among the group members, namely, equal, proportional, and free cost-sharing policies.
We present several positive and negative results for different combinations of the fairness criteria and rent-division policies. Specifically, when the groups are pre-determined, we propose a strong envy-free solution that allocates the rooms to the agents, with free cost-sharing policy. In addition, for the case that the groups are not pre-determined, we propose a strong envy-free allocation algorithm with equal cost-sharing policy. We leverage our results to obtain an algorithm that determines the maximum total rent along with the proper allocation and rent-division method.
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- 1.
[n] refers to the set \(\{1,2,\ldots ,n\}\).
- 2.
\(\mathcal N=\bigcup _{i,j} a_{i,j}\).
- 3.
Note that if \(V_{i,k}=0\), by individual rationality, \(\mathcal{R}(k)=0\) and no agent has to pay any cost.
- 4.
We refer the reader to the full version of the paper for this proof.
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Ghodsi, M., Latifian, M., Mohammadi, A., Moradian, S., Seddighin, M. (2018). Rent Division Among Groups. In: Kim, D., Uma, R., Zelikovsky, A. (eds) Combinatorial Optimization and Applications. COCOA 2018. Lecture Notes in Computer Science(), vol 11346. Springer, Cham. https://doi.org/10.1007/978-3-030-04651-4_39
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