The price of fairness for a small number of indivisible items

Incorporating fairness criteria in optimization problems comes at a certain cost, which is measured by the so-called price of fairness. Here we consider the allocation of indivisible goods. For envy-freeness as fairness criterion it is known from literature that the price of fairness can increase linearly in terms of the number of agents. For the constructive lower bound a quadratic number of items was used. In practice this might be inadequately large. So we introduce the price of fairness in terms of both the number of agents and items, i.e., key parameters which generally may be considered as common and available knowledge. It turned out that the price of fairness increases sublinear if the number of items is not too much larger than the number of agents. For the special case of coincide of both counts exact asymptotics could be determined. Additionally an efficient integer programming formulation is given.Comment: 5 page

Competitive Equilibrium with Indivisible Goods and Generic Budgets

Competitive equilibrium from equal incomes (CEEI) is a classic solution to the problem of fair and efficient allocation of goods [Foley'67, Varian'74]. Every agent receives an equal budget of artificial currency with which to purchase goods, and prices match demand and supply. However, a CEEI is not guaranteed to exist when the goods are indivisible, even in the simple two-agent, single-item market. Yet, it is easy to see that once the two budgets are slightly perturbed (made generic), a competitive equilibrium does exist. In this paper we aim to extend this approach beyond the single-item case, and study the existence of equilibria in markets with two agents and additive preferences over multiple items. We show that for agents with equal budgets, making the budgets generic -- by adding vanishingly small random perturbations -- ensures the existence of an equilibrium. We further consider agents with arbitrary non-equal budgets, representing non-equal entitlements for goods. We show that competitive equilibrium guarantees a new notion of fairness among non-equal agents, and that it exists in cases of interest (like when the agents have identical preferences) if budgets are perturbed. Our results open opportunities for future research on generic equilibrium existence and fair treatment of non-equals.Comment: Major revision (R&R

Competitive Equilibrium For Almost All Incomes: Existence and Fairness

Competitive equilibrium (CE) is a fundamental concept in market economics. Its efficiency and fairness properties make it particularly appealing as a rule for fair allocation of resources among agents with possibly different entitlements. However, when the resources are indivisible, a CE might not exist even when there is one resource and two agents with equal incomes. Recently, Babaioff and Nisan and Talgam-Cohen (2017) have suggested to consider the entire space of possible incomes, and check whether there exists a competitive equilibrium for almost all income-vectors --- all income-space except a subset of measure zero. They proved various existence and non-existence results, but left open the cases of four goods and three or four agents with monotonically-increasing preferences. This paper proves non-existence in both these cases, thus completing the characterization of CE existence for almost all incomes in the domain of monotonically increasing preferences. Additionally, the paper provides a complete characterization of CE existence in the domain of monotonically decreasing preferences, corresponding to allocation of chores. On the positive side, the paper proves that CE exists for almost all incomes when there are four goods and three agents with additive preferences. The proof uses a new tool for describing a CE, as a subgame-perfect equilibrium of a specific sequential game. The same tool also enables substantially simpler proofs to the cases already proved by Babaioff et al. Additionally, this paper proves several strong fairness properties that are satisfied by any CE allocation, illustrating its usefulness for fair allocation among agents with different entitlements.Comment: All the inexistence results are now stronger - they show that even CE-fairness alone (without efficiency) cannot be attaine

The Price of Fairness for Indivisible Goods

We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We also introduce the concept of strong price of fairness, which captures the efficiency loss in the worst fair allocation as opposed to that in the best fair allocation as in the price of fairness. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions, for both the price of fairness and the strong price of fairness.Comment: A preliminary version appears in the 28th International Joint Conference on Artificial Intelligence (IJCAI), 201

To Give or Not to Give: Fair Division for Single Minded Valuations

Single minded agents have strict preferences, in which a bundle is acceptable only if it meets a certain demand. Such preferences arise naturally in scenarios such as allocating computational resources among users, where the goal is to fairly serve as many requests as possible. In this paper we study the fair division problem for such agents, which is harder to handle due to discontinuity and complementarities of the preferences. Our solution concept---the competitive allocation from equal incomes (CAEI)---is inspired from market equilibria and implements fair outcomes through a pricing mechanism. We study the existence and computation of CAEI for multiple divisible goods, cake cutting, and multiple discrete goods. For the first two scenarios we show that existence of CAEI solutions is guaranteed, while for the third we give a succinct characterization of instances that admit this solution; then we give an efficient algorithm to find one in all three cases. Maximizing social welfare turns out to be NP-hard in general, however we obtain efficient algorithms for (i) divisible and discrete goods when the number of different \emph{types} of players is a constant, (ii) cake cutting with contiguous demands, for which we establish an interesting connection with interval scheduling, and (iii) cake cutting with a constant number of players with arbitrary demands. Our solution is useful more generally, when the players have a target set of desired goods, and very small positive values for any bundle not containing their target set

Fair Division Minimizing Inequality

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

Price of Fairness for Allocating a Bounded Resource

In this paper we study the problem of allocating a scarce resource among several players (or agents). A central decision maker wants to maximize the total utility of all agents. However, such a solution may be unfair for one or more agents in the sense that it can be achieved through a very unbalanced allocation of the resource. On the other hand fair/balanced allocations may be far from being optimal from a central point of view. So, in this paper we are interested in assessing the quality of fair solutions, i.e. in measuring the system efficiency loss under a fair allocation compared to the one that maximizes the sum of agents utilities. This indicator is usually called the Price of Fairness and we study it under three different definitions of fairness, namely maximin, Kalai-Smorodinski and proportional fairness. Our results are of two different types. We first formalize a number of properties holding for any general multi-agent problem without any special assumption on the agents utility sets. Then we introduce an allocation problem, where each agent can consume the resource in given discrete quantities (items), such that the maximization of the total utility is given by a Subset Sum Problem. For the resulting Fair Subset Sum Problem, in the case of two agents, we provide upper and lower bounds on the Price of Fairness as functions of an upper bound on the items size

Fair Division: The Computer Scientist's Perspective

I survey recent progress on a classic and challenging problem in social choice: the fair division of indivisible items. I discuss how a computational perspective has provided interesting insights into and understanding of how to divide items fairly and efficiently. This has involved bringing to bear tools such as those used in knowledge representation, computational complexity, approximation methods, game theory, online analysis and communication complexityComment: To appear in Proceedings of IJCAI 202

Online Fair Division: analysing a Food Bank problem

We study an online model of fair division designed to capture features of a real world charity problem. We consider two simple mechanisms for this model in which agents simply declare what items they like. We analyse several axiomatic properties of these mechanisms like strategy-proofness and envy-freeness. Finally, we perform a competitive analysis and compute the price of anarchy.Comment: 7 pages, 2 figures, 1 tabl

Satiation in Fisher Markets and Approximation of Nash Social Welfare

We study linear Fisher markets with satiation. In these markets, sellers have earning limits and buyers have utility limits. Beyond natural applications in economics, these markets arise in the context of maximizing Nash social welfare when allocating indivisible items to agents. In contrast to markets with either earning or utility limits, markets with both limits have not been studied before. They turn out to have fundamentally different properties. In general, the existence of competitive equilibria is not guaranteed. We identify a natural property of markets (termed money clearing) that implies existence. We show that the set of equilibria is not always convex, answering a question of Cole et al. [EC'17]. We design an FPTAS to compute an approximate equilibrium and prove that the problem of computing an exact equilibrium lies in the intersection of complexity classes PLS and PPAD. For a constant number of buyers or goods, we give a polynomial-time algorithm to compute an exact equilibrium. We show how (approximate) equilibria can be rounded and provide the first constant-factor approximation algorithm (with a factor of 2.404) for maximizing Nash social welfare when agents have budget-additive valuations. Finally, we significantly improve the approximation hardness for additive valuations to \sqrt{8/7} > 1.069 (over 1.00008 by Lee [IPL'17]).Comment: Restructured the paper with improved lower boun