Abstract
In this paper, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to matroid rank functions. This is a versatile valuation class, with several desirable properties (monotonicity, submodularity) which naturally models several real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e. utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination, by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time.
This research was funded by an MOE Grant (no. R-252-000-625-133) and a Singapore NRF Research Fellowship (no. R-252- 000-750-733). Most of this work was done when Chakraborty and Zick were employed at National University of Singapore (NUS), and Igarashi was a research visitor at NUS.
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Notes
- 1.
There is an instance of two agents with monotone supermodular/subadditive valuations where no allocation is PO and EF1 [13].
- 2.
see, e.g. https://www.ed.gov/diversity-opportunity.
- 3.
Roughly speaking, a leximin allocation is one that maximizes the realized valuation of the worst-off agent and, subject to that, maximizes that of the second worst-off agent, and so on.
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Benabbou, N., Chakraborty, M., Igarashi, A., Zick, Y. (2020). Finding Fair and Efficient Allocations When Valuations Don’t Add Up. In: Harks, T., Klimm, M. (eds) Algorithmic Game Theory. SAGT 2020. Lecture Notes in Computer Science(), vol 12283. Springer, Cham. https://doi.org/10.1007/978-3-030-57980-7_3
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