International audienceWe investigate five different fairness criteria in a simple model of fair resource allocation of indivisible goods based on additive preferences. We show how these criteria are connected to each other, forming an ordered scale that can be used to characterize how conflicting the agents' preferences are: for a given instance of a resource allocation problem, the less conflicting the agents' preferences are, the more demanding criterion this instance is able to satisfy, and the more satisfactory the allocation can be. We analyze the computational properties of the five criteria, give some experimental results about them, and further investigate a slightly richer model with k-additive preferences
We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases where the underlying graph has simple structure, and/or the number of agents-or, less restrictively, the number of agent types-is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.
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.
Distributed mechanisms for allocating indivisible goods are mechanisms lacking central control, in which agents can locally agree on deals to exchange some of the goods in their possession. We study convergence properties for such distributed mechanisms when used as fair division procedures. Specifically, we identify sets of assumptions under which any sequence of deals meeting certain conditions can be shown to converge to a proportionally fair allocation and to an envy-free allocation, respectively. We also introduce an extension of the basic framework where agents are vertices of a graph limiting which agents can interact with each other and prove a similar convergence result for envy-freeness in this context. Finally, when not all assumptions guaranteeing envy-freeness are satisfied, we may want to minimise the degree of envy exhibited by an outcome. To this end, we introduce a generic framework for measuring the degree of envy in a society and establish the computational complexity of checking whether a given scenario allows for a deal that is beneficial to every agent involved and that will reduce envy.
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