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Register a new userFair Resource Sharing with Externailities
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Yingkai Li
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We study a fair resource sharing problem, where a set of resources are to be shared among a set of agents. Each agent demands one resource and each resource can serve a limited number of agents. An agent cares about what resource they get as well as the externalities imposed by their mates, whom they share the same resource with. Apparently, the strong notion of envy-freeness, where no agent envies another for their resource or mates, cannot always be achieved and we show that even to decide the existence of such a strongly envy-free assignment is an intractable problem. Thus, a more interesting question is whether (and in what situations) a relaxed notion of envy-freeness, the Pareto envy-freeness, can be achieved: an agent i envies another agent j only when i envies both the resource and the mates of j. In particular, we are interested in a dorm assignment problem, where students are to be assigned to dorms with the same capacity and they have dichotomous preference over their dorm-mates. We show that when the capacity of the dorms is 2, a Pareto envy-free assignment always exists and we present a polynomial-time algorithm to compute such an assignment; nevertheless, the result fails to hold immediately when the capacities increase to 3, in which case even Pareto envy-freeness cannot be guaranteed. In addition to the existential results, we also investigate the implications of envy-freeness on proportionality in our model and show that envy-freeness in general implies approximations of proportionality.
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It is often beneficial for agents to pool their resources in order to better accommodate fluctuations in individual demand. Many multi-round resource allocation mechanisms operate in an online manner: in each round, the agents specify their demands for that round, and the mechanism determines a corresponding allocation. In this paper, we focus instead on the offline setting in which the agents specify their demand for each round at the outset. We formulate a specific resource allocation problem in this setting, and design and analyze an associated mechanism based on the solution concept of lexicographic maximin fairness. We present an efficient implementation of our mechanism, and prove that it is Pareto-efficient, envy-free, non-wasteful, resource monotonic, population monotonic, and group strategyproof. We also prove that our mechanism guarantees each agent at least half of the utility that they can obtain by not sharing their resources. We complement these positive results by proving that no maximin fair mechanism can improve on the aforementioned factor of one-half.
We consider the problem of fair allocation of indivisible goods to $n$ agents, with no transfers. When agents have equal entitlements, the well established notion of the maximin share (MMS) serves as an attractive fairness criterion, where to qualify as fair, an allocation needs to give every agent at least a substantial fraction of her MMS. In this paper we consider the case of arbitrary (unequal) entitlements. We explain shortcomings in previous attempts that extend the MMS to unequal entitlements. Our conceptual contribution is the introduction of a new notion of a share, the AnyPrice share (APS), that is appropriate for settings with arbitrary entitlements. Even for the equal entitlements case, this notion is new, and satisfies $APS ge MMS$, where the inequality is sometimes strict. We present two equivalent definitions for the APS (one as a minimization problem, the other as a maximization problem), and provide comparisons between the APS and previous notions of fairness. Our main result concerns additive valuations and arbitrary entitlements, for which we provide a polynomial-time algorithm that gives every agent at least a $frac{3}{5}$-fraction of her APS. This algorithm can also be viewed as providing strategies in a certain natural bidding game, and these strategies secure each agent at least a $frac{3}{5}$-fraction of her APS.
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 the allocative challenges that governmental and nonprofit organizations face when tasked with equitable and efficient rationing of a social good among agents whose needs (demands) realize sequentially and are possibly correlated. To better achieve their dual aims of equity and efficiency in such contexts, social planners intend to maximize the minimum fill rate across agents, where each agents fill rate must be irrevocably decided upon its arrival. For an arbitrarily correlated sequence of demands, we establish upper bounds on both the expected minimum fill rate (ex-post fairness) and the minimum expected fill rate (ex-ante fairness) achievable by any policy. Our bounds are parameterized by the number of agents and the expected demand-to-supply ratio, and they shed light on the limits of attaining equity in dynamic rationing. Further, we show that for any set of parameters, a simple adaptive policy of projected proportional allocation achieves the best possible fairness guarantee, ex post as well as ex ante. Our policy is transparent and easy to implement; yet despite its simplicity, we demonstrate that this policy provides significant improvement over the class of non-adaptive target-fill-rate policies. We obtain the performance guarantees of (i) our proposed adaptive policy by inductively designing lower-bound functions on its corresponding value-to-go, and (ii) the optimal target-fill-rate policy by establishing an intriguing connection to a monopoly-pricing optimization problem. We complement our theoretical developments with a numerical study motivated by the rationing of COVID-19 medical supplies based on a projected-demand model used by the White House. In such a setting, our simple adaptive policy significantly outperforms its theoretical guarantee as well as the optimal target-fill-rate policy.
We introduce a combinatorial variant of the cost sharing problem: several services can be provided to each player and each player values every combination of services differently. A publicly known cost function specifies the cost of providing every possible combination of services. A combinatorial cost sharing mechanism is a protocol that decides which services each player gets and at what price. We look for dominant strategy mechanisms that are (economically) efficient and cover the cost, ideally without overcharging (i.e., budget balanced). Note that unlike the standard cost sharing setting, combinatorial cost sharing is a multi-parameter domain. This makes designing dominant strategy mechanisms with good guarantees a challenging task. We present the Potential Mechanism -- a combination of the VCG mechanism and a well-known tool from the theory of cooperative games: Hart and Mas-Colells potential function. The potential mechanism is a dominant strategy mechanism that always covers the incurred cost. When the cost function is subadditive the same mechanism is also approximately efficient. Our main technical contribution shows that when the cost function is submodular the potential mechanism is approximately budget balanced in three settings: supermodular valuations, symmetric cost function and general symmetric valuations, and two players with general valuations.
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