Designing Truthful Mechanisms for Asymptotic Fair Division

We study the problem of fairly allocating a set of $m$ goods among $n$ agents in the asymptotic setting, where each item’s value for each agent is drawn from an underlying joint distribution. Prior works have shown that if this distribution is well-behaved, then an envy-free allocation exists with high probability when $m=Ω(n\log{n})$ [Dickerson et al., 2014]. Under the stronger assumption that item values are independently and identically distributed (i.i.d.) across agents, this requirement improves to $m=Ω(n\log{n}/\log{\log{n}})$, which is tight [Manurangsi and Suksompong, 2021]. However, these results rely on non-strategyproof mechanisms, such as maximum-welfare allocation or the round-robin algorithm, limiting their applicability in settings with strategic agents. In this work, we extend the theory to a broader, more realistic class of joint value distributions, allowing for correlations among agents, atomicity, and unequal probabilities of having the highest value for an item. We show that envy-free allocations continue to exist with a high probability when $m=Ω(n\log{n})$. More importantly, we give a new randomized mechanism that is truthful in expectation, efficiently implementable in polynomial time, and outputs envy-free allocations with high probability, answering an open question posed by [Manurangsi and Suksompong, 2017]. We further extend our mechanism to settings with asymptotic weighted fair division and multiple agent types and good types, proving new results in each case.

💡 Research Summary

This paper, “Designing Truthful Mechanisms for Asymptotic Fair Division,” makes significant contributions to the theory of fair resource allocation, particularly in settings with strategic agents. It addresses the problem of allocating m indivisible goods among n agents with additive valuations, where each good’s value for each agent is drawn from an underlying joint distribution.

The work first generalizes the existing theoretical framework for asymptotic fair division. Prior results by Dickerson et al. (2014) and Manurangsi and Suksompong (2021) established that envy-free (EF) allocations exist with high probability when the number of items m is sufficiently large relative to n (Ω(n log n) or Ω(n log n / log log n)), but under restrictive distributional assumptions such as independence, identical distributions (i.i.d.), or “well-behaved” distributions with equal chances of being the highest valuer. This paper relaxes these assumptions substantially, considering a much broader and more realistic class of joint value distributions. The model allows for correlations among agents’ values for the same good, atomicity (point masses) in the distributions, and unequal probabilities of having the highest value for an item. The core assumptions are minimal: each agent’s marginal distribution has a mean bounded away from zero, and for any two distinct agents, the expected absolute difference between their mean-normalized values is at least a positive constant—intuitively requiring “enough disagreement” between agents. Under this generalized model, the paper proves that EF allocations continue to exist with high probability when m = Ω(n log n).

The paper’s primary and most impactful contribution is the introduction and analysis of a new randomized mechanism: the Proportional Response with Dummy (PRD) Mechanism. This mechanism is designed to be:

1. Truthful in Expectation: No agent can increase the expected value of the bundle they receive by misreporting their true valuation.
2. Computationally Efficient: It runs in polynomial time.
3. Asymptotically Fair: It outputs an envy-free allocation with high probability for typical instances drawn from the described distributions.

The PRD Mechanism operates in two phases. In the first phase, it uses the reported valuations to compute a fractional allocation that maximizes a notion of “envy margin” between every pair of agents. A key technical innovation here is the use of the Kullback-Leibler (KL) divergence between agents’ normalized valuation vectors. The analysis shows that if the KL divergence between two agents’ preferences is large, a fractional allocation can be constructed that gives one agent a significantly better bundle than the other’s from its own perspective, creating a buffer against envy. In the second phase, the mechanism employs a randomization technique to convert this high-envy-margin fractional allocation into an integral allocation while preserving the envy-free property with high probability. The construction cleverly ensures that the bids agents submit in the first phase are optimal for them, leading to the truthfulness guarantee—a rare positive result in light of well-known impossibility theorems for truthful fair division.

Furthermore, the authors extend their results to two important settings:

• Asymptotic Weighted Fair Division: They generalize the PRD mechanism to settings where agents have different entitlements or weights, proving that their results hold even when an agent’s weight constitutes a constant fraction of the total (e.g., linear in n).
• Multiple Agent and Good Types: They adapt their framework to a model where agents belong to different types and goods have categories, deriving new asymptotic existence results for envy-freeness in this structured environment.

In summary, this paper substantially advances the theory of asymptotic fair division by both broadening the scope of distributions under which EF allocations exist and, more importantly, providing the first polynomial-time, truthful-in-expectation mechanism that guarantees envy-freeness with high probability in this general setting, thereby resolving a key open question in the field.