Random Assignment of Indivisible Goods under Constraints

TL;DR

This work advances the theory of random assignment of indivisible goods under feasibility constraints by analyzing when sd-efficient and sd-envy-free lotteries exist. It extends the Probabilistic Serial framework with constrained environments, proves that such lottery assignments may not always exist even under partition matroids, and identifies special tractable cases with constructive polynomial-time mechanisms. The authors develop matroid-specific properties, propose a round-based eating algorithm, and introduce sd-proportionality as a stronger fairness notion, while also establishing hardness results and providing an exponential-time algorithm for the general case. The results illuminate when efficient and envy-free randomized allocations can be achieved in practice and offer algorithmic tools for the tractable settings, informing applications in scheduling, course allocations, and resource sharing under constraints.

Abstract

We investigate the problem of random assignment of indivisible goods, in which each agent has an ordinal preference and a constraint. Our goal is to characterize the conditions under which there always exists a random assignment that simultaneously satisfies efficiency and envy-freeness. The probabilistic serial mechanism ensures the existence of such an assignment for the unconstrained setting. In this paper, we consider a more general setting in which each agent can consume a set of items only if the set satisfies her feasibility constraint. Such constraints must be taken into account in student course placements, employee shift assignments, and so on. We demonstrate that an efficient and envy-free assignment may not exist even for the simple case of partition matroid constraints, where the items are categorized, and each agent demands one item from each category. We then identify special cases in which an efficient and envy-free assignment always exists. For these cases, the probabilistic serial cannot be naturally extended; therefore, we provide mechanisms to find the desired assignment using various approaches.

Figures (1)

• Figure 1: Summary of our results on the existence of an sd-efficient and sd-envy-free assignment. Each of the $16$ cases is identified by four characters, such as "$2$,m,i,i." The first, second, third, and fourth characters, respectively, indicate whether there are $2$ or an arbitrary $n$ number of agents, whether the constraints are matroids or general, whether the constraints are identical or heterogeneous, and whether the preferences are identical or heterogeneous. For each case, the box is painted green if such a lottery assignment always exists and red otherwise.

Theorems & Definitions (38)

• Definition 1: sd-efficiency
• Example 1: ex post efficiency does not imply sd-efficiency
• Definition 2: sd-envy-freeness
• Proposition 1
• Example 2
• Example 3: generalized PS is not sd-envy-free
• Lemma 1
• Proposition 2: Fujishige Fujibook
• proof : Proof of Lemma \ref{['lem:lexopt=sdopt']}
• Lemma 2