Resource allocation under uncertainty: an algebraic and qualitative treatment

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.