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Register a new userPicking Sequences and Monotonicity in Weighted Fair Division
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We study the problem of fairly allocating indivisible items to agents with different entitlements, which captures, for example, the distribution of ministries among political parties in a coalition government. Our focus is on picking sequences derived from common apportionment methods, including five traditional divisor methods and the quota method. We paint a complete picture of these methods in relation to known envy-freeness and proportionality relaxations for indivisible items as well as monotonicity properties with respect to the resource, population, and weights. In addition, we provide characterizations of picking sequences satisfying each of the fairness notions, and show that the well-studied maximum Nash welfare solution fails resource- and population-monotonicity even in the unweighted setting. Our results serve as an argument in favor of using picking sequences in weighted fair division problems.
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We study the problem of fairly dividing a heterogeneous resource, commonly known as cake cutting and chore division, in the presence of strategic agents. While a number of results in this setting have been established in previous works, they rely crucially on the free disposal assumption, meaning that the mechanism is allowed to throw away part of the resource at no cost. In the present work, we remove this assumption and focus on mechanisms that always allocate the entire resource. We exhibit a truthful and envy-free mechanism for cake cutting and chore division for two agents with piecewise uniform valuations, and we complement our result by showing that such a mechanism does not exist when certain additional constraints are imposed on the mechanisms. Moreover, we provide bounds on the efficiency of mechanisms satisfying various properties, and give truthful mechanisms for multiple agents with restricted classes of valuations.
Computing market equilibria is a problem of both theoretical and applied interest. Much research focuses on the static case, but in many markets items arrive sequentially and stochastically. We focus on the case of online Fisher markets: individuals have linear, additive utility and items drawn from a distribution arrive one at a time in an online setting. We define the notion of an equilibrium in such a market and provide a dynamics which converges to these equilibria asymptotically. An important use-case of market equilibria is the problem of fair division. With this in mind, we show that our dynamics can also be used as an online item-allocation rule such that the time-averaged allocations and utilities converge to those of a corresponding static Fisher market. This implies that other good properties of market equilibrium-based fair division such as no envy, Pareto optimality, and the proportional share guarantee are also attained in the online setting. An attractive part of the proposed dynamics is that the market designer does not need to know the underlying distribution from which items are drawn. We show that these convergences happen at a rate of $O(tfrac{log t}{t})$ or $O(tfrac{(log t)^2}{t})$ in theory and quickly in real datasets.
We study a new but simple model for online fair division in which indivisible items arrive one-by-one and agents have monotone utilities over bundles of the items. We consider axiomatic properties of mechanisms for this model such as strategy-proofness, envy-freeness, and Pareto efficiency. We prove a number of impossibility results that justify why we consider relaxations of the properties, as well as why we consider restricted preference domains on which good axiomatic properties can be achieved. We propose two mechanisms that have good axiomatic fairness properties on restricted but common preference domains.
Behavioural economists have shown that people are often averse to inequality and will make choices to avoid unequal outcomes. In this paper, we consider how to allocate indivisible goods fairly so as to minimize inequality. We consider how this interacts with axiomatic properties such as envy-freeness, Pareto efficiency and strategy-proofness. We also consider the computational complexity of computing allocations minimizing inequality. Unfortunately, this is computationally intractable in general so we consider several tractable greedy online mechanisms that minimize inequality. Finally, we run experiments to explore the performance of these methods.
Two simple and attractive mechanisms for the fair division of indivisible goods in an online setting are LIKE and BALANCED LIKE. We study some fundamental computational problems concerning the outcomes of these mechanisms. In particular, we consider what expected outcomes are possible, what outcomes are necessary, and how to compute their exact outcomes. In general, we show that such questions are more tractable to compute for LIKE than for BALANCED LIKE. As LIKE is strategy-proof but BALANCED LIKE is not, we also consider the computational problem of how, with BALANCED LIKE, an agent can compute a strategic bid to improve their outcome. We prove that this problem is intractable in general.
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