Efficiency and Envy-freeness in Fair Division of Indivisible Goods: Logical Representation and Complexity

Abstract

We consider the problem of allocating fairly a set of indivisible goods among agents from the point of view of compact representation and computational complexity. We start by assuming that agents have dichotomous preferences expressed by propositional formulae. We express efficiency and envy-freeness in a logical setting, which reveals unexpected connections to nonmonotonic reasoning. Then we identify the complexity of determining whether there exists an efficient and envy-free allocation, for several notions of efficiency, when preferences are represented in a succinct way (as well as restrictions of this problem). We first study the problem under the assumption that preferences are dichotomous, and then in the general case.

On Low-Envy Truthful Allocations

“…They also present algorithms that compute allocations that approximate the minimum envy-ratio; the envy ratio of a player p for a player q is the utility of player p for the items allocated to player q over p's utility for the items allocated to her. Complexity considerations about envy-freeness for indivisible items and non-additive utilities are presented in [4]. The papers [10,11] study the problem of achieving envy-free and efficient allocations in distributed settings and when the allocation of items is accompanied by monetary side payments (in this case, envy-freeness is always a feasible goal).…”

Fair Allocation of Indivisible Goods

“…envy-free allocation exists has been shown to be computationally intractable [3]. Fortunately, in the presence of money, when envy-freeness is defined in terms of both bundles of goods and payments received, the situation is more favourable and envy-free solutions do exist [2].…”

“…While fair division is a problem originating in Economics and Political Science, with a number of important contribution by mathematicians, it has recently begun to also attract the attention of researchers in Artificial Intelligence, Multiagent Systems, and Theoretical Computer Science [7,23,14,3,6,10]. The reason for this trend is twofold: on the one hand, concepts from fair division are immediately relevant to these disciplines (e.g., finding acceptable agreements in a multiagent systems) and the tools and techniques of these disciplines can shed new light on previously unexplored aspects of fair division (e.g., by applying ideas from complexity theory).…”

Nash Social Welfare in Multiagent Resource Allocation

“…We have implemented the presented heuristic algorithm and run some experiments to test its performance. 2 For generating the test cases we have used a simple random algorithm. Note that it would have been more desirable to use more "realistic" test data, of the kind generated by the CATS software [11], a standard benchmark for generating test cases for combinatorial auctions.…”

“…Both criteria have been widely used in Artificial Intelligence, Multiagent Systems and Electronic Commerce. Some recent work in these disciplines has also recognized the fact that the desideratum of finding efficient allocations needs to be balanced with appropriate fairness considerations [2,8,12], a dilemma that has long been discussed in Economics, Political Science, and Philosophy [13]. Fairness criteria include egalitarian social welfare (measuring quality in terms of the utility experienced by the poorest agent) and envy-freeness (an allocation is envy-free if no agent would want to change bundle with any of the others).…”