Resource allocation under uncertainty: an algebraic and qualitative treatment

TL;DR

This paper introduces a matrix-based algebraic framework for resource allocation under uncertainty, using qualitative data like plausibility, hierarchy, and preferences to identify optimal allocations and simple deals.

Contribution

It presents a novel qualitative approach with a matrix framework to determine and reach optimal resource allocations under uncertainty.

Findings

Existence of maximal allocations under the qualitative relation

Optimal allocations can be reached through sequences of simple deals

Mechanisms for discriminating optimal allocations are proposed

Abstract

We use an algebraic viewpoint, namely a matrix framework to deal with the problem of resource allocation under uncertainty in the context of a qualitative approach. Our basic qualitative data are a plausibility relation over the resources, a hierarchical relation over the agents and of course the preference that the agents have over the resources. With this data we propose a qualitative binary relation $\unrhd$ between allocations such that $\mathcal{F}\unrhd \mathcal{G}$ has the following intended meaning: the allocation $\mathcal{F}$ produces more or equal social welfare than the allocation $\mathcal{G}$. We prove that there is a family of allocations which are maximal with respect to $\unrhd$. We prove also that there is a notion of simple deal such that optimal allocations can be reached by sequences of simple deals. Finally, we introduce some mechanism for discriminating {optimal} allocations.