The classic cake-cutting problem provides a model for addressing fair and efficient allocation of a divisible, heterogeneous resource (metaphorically, the cake) among agents with distinct preferences. Focusing on a standard formulation of cake cutting, in which each agent must receive a contiguous piece of the cake, this work establishes algorithmic and hardness results for multiple fairness/efficiency measures.First, we consider the well-studied notion of envy-freeness and develop an efficient algorithm that finds a cake division (with connected pieces) wherein the envy is multiplicatively within a factor of 3+o(1). The same algorithm in fact achieves an approximation ratio of 3+o(1) for the problem of finding cake divisions with as large a Nash social welfare (NSW) as possible. NSW is another standard measure of fairness and this work also establishes a connection between envy-freeness and NSW: approximately envy-free cake divisions (with connected pieces) always have near-optimal Nash social welfare. Furthermore, we develop an approximation algorithm for maximizing the ρ-mean welfarethis unifying objective, with different values of ρ, interpolates between notions of fairness (NSW) and efficiency (average social welfare). Finally, we complement these algorithmic results by proving that maximizing NSW (and, in general, the ρ-mean welfare) is APX-hard in the cake-division context.
We study the problem of fair rent division that entails splitting the rent and allocating the rooms of an apartment among roommates (agents) in a fair manner. In this setup, a distribution of the rent and an accompanying allocation is said to be fair if it is envy free, i.e., under the imposed rents, no agent has a strictly stronger preference for any other agent's room. The cardinal preferences of the agents are expressed via functions which specify the utilities of the agents for the rooms for every possible room rent/price. While envy-free solutions are guaranteed to exist under reasonably general utility functions, efficient algorithms for finding them were known only for quasilinear utilities. This work addresses this notable gap and develops approximation algorithms for fair rent division with minimal assumptions on the utility functions.Specifically, we show that if the agents have continuous, monotone decreasing, and piecewiselinear utilities, then the fair rent-division problem admits a fully polynomial-time approximation scheme (FPTAS). That is, we develop algorithms that find allocations and prices of the rooms such that for each agent a the utility of the room assigned to it is within a factor of (1 + ε) of the utility of the room most preferred by a. Here, ε > 0 is an approximation parameter, and the running time of the algorithms is polynomial in 1/ε and the input size. In addition, we show that the methods developed in this work provide efficient, truthful mechanisms for special cases of the rent-division problem. Envy-free solutions correspond to equilibria of a two-sided matching market with monetary transfers; hence, this work also provides efficient algorithms for finding approximate equilibria in such markets. We complement the algorithmic results by proving that the fair rent division problem (under continuous, monotone decreasing, and piecewise-linear utilities) lies in the intersection of the complexity classes PPAD and PLS .
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...
We study the problem of fair rent division that entails splitting the rent and allocating the rooms of an apartment among agents in a fair manner (i.e., under the imposed rents, no agent has a strictly stronger preference for any other agent’s room). The utility functions specify the cardinal preferences of the agents for the rooms for every possible room rent. Although envy-free solutions are guaranteed to exist under reasonably general utility functions, efficient algorithms for finding them were known only for quasilinear utilities. This work addresses this notable gap and develops a fully polynomial-time approximation scheme for fair rent division with minimal assumptions on the utility functions. Envy-free solutions correspond to equilibria of a two-sided matching market with monetary transfers; hence, this work also provides efficient algorithms for finding approximate equilibria in such markets. We complement the algorithmic results by proving that the fair rent division problem lies in the intersection of the complexity classes polynomial parity arguments on directed graphs and polynomial local search.
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