Envy-Free and Pareto-Optimal Allocations for Agents with Asymmetric Random Valuations

TL;DR

The paper studies fair and efficient allocations of $m$ indivisible items among $n$ agents with additive utilities under an asymmetric random-valuation model. It introduces equalizing multipliers $\beta_i$ to balance each agent's chance of receiving the top-weighted item, proves existence via Sperner's lemma, and provides a polynomial-time approximation algorithm that yields a multiplier allocation which is fractionally PO and, with a positive gap in conditional expectations, envy-free with high probability when $m=\Omega(n\log n)$. The main theoretical result shows EF and (fractional) PO exist and can be found efficiently under the assumptions of interval-support and $(p,q)$-PDF-bounded utilities; this matches the known symmetric-model thresholds up to a $\log\log n$ gap. Empirically, the multiplier approach behaves robustly for larger $m$, while round-robin is EF earlier but not PO, and Maximum Nash Welfare can yield EF+PO for small instances but becomes intractable as size grows. Overall, the work provides a principled, scalable framework to achieve fair and efficient allocations in heterogeneous, randomly drawn valuations and suggests potential extensions to correlated or divisible settings.

Abstract

We study the problem of allocating $m$ indivisible items to $n$ agents with additive utilities. It is desirable for the allocation to be both fair and efficient, which we formalize through the notions of envy-freeness and Pareto-optimality. While envy-free and Pareto-optimal allocations may not exist for arbitrary utility profiles, previous work has shown that such allocations exist with high probability assuming that all agents' values for all items are independently drawn from a common distribution. In this paper, we consider a generalization of this model where each agent's utilities are drawn independently from a distribution specific to the agent. We show that envy-free and Pareto-optimal allocations are likely to exist in this asymmetric model when $m=Ω\left(n\log n\right)$, which is tight up to a log log gap that also remains open in the symmetric subsetting. Furthermore, these guarantees can be achieved by a polynomial-time algorithm.

Figures (6)

• Figure 1: The top panel shows probability density functions of five agents' utility distributions. The bottom panel shows densities after scaling distributions by the given multipliers. When drawing an independent sample from each scaled distribution, each sample is the largest with probability $1/5$.
• Figure 2: Existing and new results on when EF and EF+PO allocations are guaranteed to exist in both models. Bold results are new.
• Figure 3: Probability of different algorithms satisfying EF and PO for $n=10$ distributions. The multiplier and rounded MNW algorithms are always PO and therefore not shown. Each datapoint corresponds to 1 000 random instances, and flies indicate 95% confidence intervals. PO and MNW data not available for large $m$ due to computational cost.
• Figure 4: Illustrations of the example in the text. Dots represent utilities of 5 000 random items. Shaded regions delineate items allocated to agent A, for the maximum-percentile algorithm (left) and for the multiplier allocation (right). Three marked items (two have near-identical utilities) show that the maximum-percentile allocation is not PO.
• Figure 5: Version of the experiment in \ref{['fig:experiments']}, but with $m \equiv 3 \pmod{10}$.

Theorems & Definitions (35)

• Proposition 1: MS19
• Proposition 1
• Theorem 2
• Theorem 3
• proof : Proof sketch
• Theorem 4
• proof
• Proposition 4
• Proposition 4
• proof