Fair Division with a Secretive Agent

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 epsilon-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.