[Paper Review] Random Serial Dictatorship versus Probabilistic Serial Rule: A Tale of Two Random Mechanisms

Hadi Hosseini, Kate Larson+1 more|arXiv (Cornell University)|Mar 4, 2015

Game Theory and Voting Systems30 references3 citations

TL;DR

This paper compares Random Serial Dictatorship (RSD) and Probabilistic Serial (PS) rules in one-sided matching with indivisible objects and ordinal preferences. It shows RSD is envy-free under lexicographic preferences, while PS is manipulable when objects exceed agents. Empirically, PS dominates RSD stochastically in fewer than 1% of cases as n/m → 1, suggesting RSD may be preferable in practice despite lacking efficiency guarantees.

ABSTRACT

For assignment problems where agents, specifying ordinal preferences, are allocated indivisible objects, two widely studied randomized mechanisms are the Random Serial Dictatorship (RSD) and Probabilistic Serial Rule (PS). These two mechanisms both have desirable economic and computational properties, but the outcomes they induce can be incomparable in many instances, thus creating challenges in deciding which mechanism to adopt in practice. In this paper we first look at the space of lexicographic preferences and show that, as opposed to the general preference domain, RSD satisfies envyfreeness. Moreover, we show that although under lexicographic preferences PS is strategyproof when the number of objects is less than or equal agents, it is strictly manipulable when there are more objects than agents. In the space of general preferences, we provide empirical results on the (in)comparability of RSD and PS, analyze economic properties, and provide further insights on the applicability of each mechanism in different application domains.

Motivation & Objective

• To analyze the comparative performance of RSD and PS in one-sided matching with indivisible objects and ordinal preferences.
• To investigate fairness and incentive properties of RSD and PS under lexicographic preferences, where agents rank objects in strict order.
• To empirically assess the frequency of dominance between RSD and PS outcomes across various agent-object ratios.
• To evaluate the practical implications of mechanism choice when object supply is limited (one copy per object).
• To determine under what conditions PS is manipulable and RSD is envy-free, especially when n ≤ m or n > m.

Proposed method

• Analyzes theoretical properties of RSD and PS under lexicographic preferences using stochastic dominance (sd) and lexicographic dominance (ld) relations.
• Empirically evaluates all possible preference profiles for varying n (agents) and m (objects), measuring dominance and manipulability.
• Uses Monte Carlo sampling across preference profile spaces to estimate the fraction of profiles where PS stochastically or lexicographically dominates RSD.
• Measures the fraction of preference profiles where PS is manipulable under sd and ld dominance, particularly for n < m.
• Compares outcomes of RSD and PS on identical preference profiles, including cases where assignments are equal but PS remains manipulable.
• Applies game-theoretic analysis to show that PS is not weakly strategyproof when m > n, even under lexicographic preferences.

Experimental results

Research questions

• RQ1 Under lexicographic preferences, does RSD satisfy envyfreeness, and how does this compare to general preferences?
• RQ2 Is PS strategyproof when m > n, and what is the extent of manipulability in such settings?
• RQ3 How frequently does PS stochastically dominate RSD across different n/m ratios, and does this frequency decline as n/m → 1?
• RQ4 In cases where RSD and PS produce identical assignments, can PS still be manipulated while RSD remains strategyproof?
• RQ5 Can a randomized mechanism be both strategyproof and fair (e.g., proportionality) when m > n?

Key findings

• RSD is envy-free under lexicographic preferences, a property it does not generally satisfy in the general preference domain.
• PS is strictly manipulable when m > n, with the fraction of sd-manipulable profiles approaching 1 as m − n increases.
• The fraction of preference profiles where PS stochastically dominates RSD drops below 1% as n/m → 1, indicating RSD often yields better expected outcomes.
• For n > 5 and m > 5, PS is nearly 100% manipulable under stochastic dominance, especially when m > n.
• Under lexicographic preferences, the fraction of ld-manipulable PS profiles converges to 1 even faster than under general preferences.
• Even when RSD and PS produce identical assignments, PS can still be manipulated—e.g., in a 3-agent, 3-object profile, agent 1 can improve expected utility via misreporting.