The Distortion of Prior-Independent b-Matching Mechanisms

In a setting where $m$ items need to be partitioned among $n$ agents, we evaluate the performance of mechanisms that take as input each agent’s \emph{ordinal preferences}, i.e., their ranking of the items from most- to least-preferred. The standard measure for evaluating ordinal mechanisms is the \emph{distortion}, and the vast majority of the literature on distortion has focused on worst-case analysis, leading to some overly pessimistic results. We instead evaluate the distortion of mechanisms with respect to their expected performance when the agents’ preferences are generated stochastically. We first show that no ordinal mechanism can achieve a distortion better than $e/(e-1)\approx 1.582$, even if each agent needs to receive exactly one item (i.e., $m=n$) and every agent’s values for different items are drawn i.i.d.\ from the same known distribution. We then complement this negative result by proposing an ordinal mechanism that achieves the optimal distortion of $e/(e-1)$ even if each agent’s values are drawn from an agent-specific distribution that is unknown to the mechanism. To further refine our analysis, we also optimize the \emph{distortion gap}, i.e., the extent to which an ordinal mechanism approximates the optimal distortion possible for the instance at hand, and we propose a mechanism with a near-optimal distortion gap of $1.076$. Finally, we also evaluate the distortion and distortion gap of simple mechanisms that have a one-pass structure.

💡 Research Summary

This paper studies the problem of allocating m items to n agents under a b‑matching constraint, where each agent i has a quota b_i and the total quota equals the number of items (∑_i b_i = m). Agents’ true valuations for items are private, but the mechanism only observes their ordinal preferences (rankings). The quality of an ordinal mechanism is measured by its distortion: the worst‑case ratio, over all possible value distributions, between the expected optimal social welfare and the expected welfare achieved by the mechanism.

Main contributions

1. A universal lower bound. Even in the simplest one‑to‑one setting (b_i = 1 for all i) where every agent’s values are drawn i.i.d. from a known Bernoulli distribution, any ordinal mechanism must have distortion at least e/(e‑1) ≈ 1.582. This shows that the classic worst‑case lower bounds (Θ(n²) for deterministic, Θ(√n) for randomized) are overly pessimistic when the values are stochastic.

2. Random Survivors (RS) mechanism. To match the lower bound, the authors propose RS, a prior‑independent mechanism that works for arbitrary quota vectors and for heterogeneous, possibly correlated, agent‑specific value distributions satisfying the “uniform‑favorites” (UF) condition (every bundle of size b_i is equally likely to be an agent’s most‑valued bundle). Each agent reports only the unordered set of its top b_i items. The mechanism randomly selects a set of “survivors” with probability inversely proportional to b_i, giving smaller‑quota agents higher priority. For each item, it assigns the item uniformly at random to one of the survivors that listed it. This guarantees that every reported favorite item is received with probability at least 1 − 1/e, yielding an expected social welfare within a factor e/(e‑1) of optimal. RS requires far less communication than full rankings.

3. Distortion gap and the RSBS mechanism. Distortion gap is defined as the worst‑case ratio, over all quota vectors b, between a mechanism’s distortion and the best possible distortion achievable on instances induced by b. While RS is optimal for b_i = 1, its gap can be as large as 1.5 for other b. The authors introduce RSBS (Random Survivors with Burning and Stealing). RSBS first runs RS on all agents except the one with the largest quota, then “burns” (removes) each agent’s allocated items with a certain probability, and finally lets the largest‑quota agent “steal” all of its favorite items. RSBS retains the e/(e‑1) distortion guarantee and reduces the distortion gap to about 1.076, which is near‑optimal.

4. Sequential one‑pass mechanisms. In many practical settings the mechanism must make irrevocable decisions as agents arrive. Assuming the agent with the largest quota is approached last, the authors design the HQL (Highest‑Quota‑Last) mechanism. HQL achieves distortion 2 and a distortion gap of 2(1 − 1/e) ≈ 1.264. They also prove that no sequential mechanism can beat distortion 2 or gap 2(1 − 1/e), even with full control over the arrival order, establishing HQL’s optimality in the online setting.

5. Secretary model. When the arrival order is uniformly random, the RS mechanism can be adapted to retain the e/(e‑1) distortion, but the distortion gap cannot be better than 4/3.

6. Incentives and extensions. All proposed mechanisms satisfy Bayesian incentive compatibility (BIC) for the considered UF distributions, encouraging truthful reporting of preferences. Moreover, the analysis extends from additive valuations to submodular valuations, provided the UF condition holds (each size‑b_i bundle is equally likely to be the maximizer of the random submodular function).

Technical approach
The lower bound leverages a construction where each agent’s value for each item is a Bernoulli random variable; the optimal allocation is to give each agent the single item where its Bernoulli realized as 1, but any ordinal mechanism cannot identify these items with probability better than 1/e. The RS mechanism’s survivor‑selection probabilities are carefully calibrated to balance the competition among agents with different quotas, ensuring the 1 − 1/e guarantee. The RSBS analysis uses a probabilistic “burn” step to decouple the large‑quota agent’s allocation from the rest, allowing it to capture most of its favorite items without harming the overall distortion. The sequential lower bounds are proved via adversarial distributions that force any online algorithm to either waste items on early agents or leave too many high‑value items for later agents.

Implications
The results demonstrate that prior‑independent ordinal mechanisms can be both simple and near‑optimal in expectation, even under heterogeneous and correlated value distributions. This bridges the gap between worst‑case distortion theory and practical applications such as course registration, reviewer assignment, and cloud resource allocation, where only ordinal information is feasible to collect. The mechanisms are also communication‑efficient (RS needs only top‑b_i sets) and incentive‑compatible, making them attractive for real‑world deployment.

Overall, the paper establishes a tight bound of e/(e‑1) for the expected distortion of any ordinal, prior‑independent b‑matching mechanism, provides concrete algorithms that achieve this bound, refines the analysis with the distortion‑gap concept, and extends the findings to online and secretary settings, all while preserving incentive compatibility and supporting submodular valuations.