Fair division under ordinal preferences: computing envy-free allocations of indivisible goods

We study the problem of fairly dividing a set of goods amongst a group of agents, when those agents have preferences that are ordinal relations over alternative bundles of goods (rather than utility functions) and when our knowledge of those preferences is incomplete. The incompleteness of the preferences stems from the fact that each agent reports their preferences by means of an expression of bounded size in a compact preference representation language. Specifically, we assume that each agent only provides a ranking of individual goods (rather than of bundles). In this context, we consider the algorithmic problem of deciding whether there exists an allocation that is possibly (or necessarily) envy-free, given the incomplete preference information available, if in addition some mild economic efficiency criteria need to be satisfied. We provide simple characterisations, giving rise to simple algorithms, for some instances of the problem, and computational complexity results, establishing the intractability of the problem, for others.