Fair Division with a Secretive Agent

Abstract

We study classic fair-division problems in a partial information setting. This paper respectively addresses fair division of rent, cake, and indivisible goods among agents with cardinal preferences. We will show that, for all of these settings and under appropriate valuations, a fair (or an approximately fair) division among n agents can be efficiently computed using only the valuations of n − 1 agents. The nth (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.For the rent-division setting we prove that the (well-behaved) utilities of n−1 agents suffice to find a rent division among n rooms such that, for every possible room selection of the secretive agent, there exists an allocation (of the remaining n − 1 rooms among the n − 1 agents) which ensures overall envy freeness (fairness). We complement this existential result by developing a polynomial-time algorithm that finds such a fair rent division under quasilinear utilities.In this partial information setting, we also develop efficient algorithms to compute allocations that are envy-free up to one good (EF1) and ε-approximate envy free. These two notions of fairness are applicable in the context of indivisible goods and divisible goods (cake cutting), respectively. This work also addresses fairness in terms of proportionality and maximin shares. Our key result here is an efficient algorithm that, even with a secretive agent, finds a 1/19-approximate maximin fair allocation (of indivisible goods) under submodular valuations of the non-secretive agents.One of the main technical contributions of this paper is the development of novel connections between different fair-division paradigms, e.g., we use our existential results for envy-free rent-division to develop an efficient EF1 algorithm. * Indian Institute of Science. eshwarram.arunachaleswaran@gmail.com † Indian Institute of Science. barman@iisc.ac.in ‡ Indian Institute of Science. nidhirathi@iisc.ac.in 1 http://www.spliddit.org/ 2 http://www.nyu.edu/projects/adjustedwinner/ 1 agent) and, then, the second agent can select her most preferred piece. This protocol leads to an envy-free cake division, i.e., in the resulting allocation no agent has a strictly stronger preference for the other agent's piece. Envy freeness is a standard notion of fairness [Fol67, Var74, Str80] and the divide-andchoose method shows a fair division with respect to this notion can be found even if the second agent is secretive and just selects a piece after we partition the cake.Asada et al. [AFP + 18] showed that this result extends to higher number of agents who are endowed with ordinal preferences: there exists a division of the cake into n parts, which depends on the (subjective) preferences of only n − 1 agents, such that the nth (secretive) agent can select an arbitrary piece and still we would be able to allocate the remaining n − 1 pieces in an envy-free manner. In other words, independent of the cho...

How to Cut a Discrete Cake Fairly

“…More generally, our result shows that the valuations of n − 1 agents suffice to find an EF1 outer connected division among n agents. Note that without connectivity constraints, a secretive EF1 division is known to exist and can be computed in polynomial time when the agents have monotone valuations (Arunachaleswaran, Barman, and Rathi 2019). However, our result is the first to show that the existential result holds in conjunction with connectivity requirements.…”

Mind the gap: Cake cutting with separation

“…Cake cutting has long been studied by mathematicians and economists, and more recently attracted substantial interest from computer scientists, as it suggests a plethora of computational challenges. In particular, a long line of work in the artificial intelligence community in recent years has focused on cake cutting and its variants [1,3,8,12,20,38,42,44].…”