Random Serial Dictatorship Rule

• Random Serial Dictatorship Rule is a mechanism that randomly orders agents to sequentially select their top available choices, ensuring fairness and strategyproofness.
• It is characterized by axioms such as equal-treatment-of-equals, ex post efficiency, and probabilistic monotonicity, which underpin its robust theoretical foundations.
• While RSD can be simulated efficiently, calculating its exact ex-ante allocation probabilities is #P-complete, posing challenges in precise computation.

Random Serial Dictatorship Rule

Random Serial Dictatorship (RSD) is a fundamental stochastic mechanism in social choice, assignment, and resource allocation problems. Under RSD, agents are ordered via a uniform random permutation, and they sequentially select their most-preferred available alternative. RSD is lauded for powerful axiomatic properties—including strategyproofness, ex post efficiency, anonymity, and equal treatment of equals—but also exhibits inherent tradeoffs, computational hardness, and quantitative vulnerability to manipulation. Its characterization, fairness, and efficiency properties anchor much of the modern theory of random assignment and stochastic social choice.

1. Definition and Structural Properties

Given a set of $n$ agents and a set of $m$ indivisible objects (or alternatives), the RSD rule first draws a permutation $\pi \in S_n$ uniformly at random. Under $\pi$, agent $\pi(1)$ selects her top choice, agent $\pi(2)$ picks her favorite among the remaining, and the process continues until all objects are assigned or the outcome is determined uniquely.

In the deterministic serial dictatorship (SD), a fixed agent order prescribes the outcome—always Pareto efficient and strategyproof. RSD is the uniform lottery over all $n!$ deterministic SD outcomes. The probability simplex over all assignments (or alternatives, in the voting case) is the output space of RSD.

If $R$ denotes the profile of agents' preferences and $f^\pi(R)$ denotes the SD outcome under permutation $\pi$, RSD is

$$\text{RSD}(R) = \frac{1}{n!} \sum_{\pi \in S_n} f^\pi(R)$$

RSD inherits ex post efficiency and strategyproofness from SD, and is anonymous and neutral. In assignment and voting domains with linear or strict preferences, RSD reduces to the uniform random dictatorship (for $m = 1$ object).

2. Characterization and Axiomatic Foundations

Several foundational results establish RSD as the unique random assignment rule (or random priority rule) satisfying certain combinations of fairness, efficiency, and monotonicity axioms. For $n \le 3$, RSD is uniquely characterized by equal-treatment-of-equals, ex post efficiency, and strategyproofness, as detailed in (Brandt et al., 2023). Equal-treatment-of-equals requires agents with identical preferences to receive identical distributions over allocations; ex post efficiency mandates the outcome is a convex combination of Pareto efficient deterministic assignments; strategyproofness (sd-strategyproofness) ensures that no agent can benefit (in the sense of stochastic dominance) from misreporting their preferences.

For general $n$, a recent explicit axiomatization states that RSD is uniquely characterized by the following three properties (Basteck, 22 Jun 2025):

Axiom	Meaning
Equal-Treatment-of-Equals	Agents with identical preferences have identical allocation probabilities
Ex Post Efficiency	Only Pareto efficient assignments are in the support of the random assignment
Probabilistic (Maskin) Monotonicity	If an outcome rises in all agents' rankings, its probability does not decrease

Probabilistic monotonicity is strictly stronger than sd-strategyproofness for random assignment but coincides with it on the universal domain of strict preferences; for deterministic rules, it collapses to conventional Maskin monotonicity.

Characterization via matrix rank (as in (Brandt et al., 2023)) for arbitrary $n$ remains a challenging open problem: it is equivalent to showing that a certain large, sparse system of linear equations, capturing the constraints above, has full rank—so that RSD is the unique solution.

3. Incentive Compatibility and Strategic Vulnerability

RSD is strategyproof: for any agent, truth-telling weakly dominates any misreport. This holds in the stochastic dominance sense—that is, for any utility consistent with the declared ranking, truthful reporting maximizes expected utility.

However, in the voting context, RSD is not a dictatorship. The quantitative version of the Gibbard–Satterthwaite theorem (Friedgut et al., 2011) proves that any non-dictatorial rule with at least three alternatives that is "far" from being dictatorial (i.e., all alternatives are selected with non-negligible probability) is susceptible to manipulation with non-negligible probability $\Omega(\varepsilon^2)$. Specifically, for each voter $i$, the manipulation power $M_i(F)$ (the probability that a random misreport is profitable) satisfies

$\max_i M_i(F) \geq C' \cdot \varepsilon^2$

If RSD is used with three or more alternatives and is far from dictatorial, this lower bound on manipulation power applies: RSD is not immune to strategic manipulation in quantitative terms, even with random manipulations.

Hybrid mechanisms (Mennle et al., 2013) use RSD as the strategyproof "base" and mix it with more efficient mechanisms (e.g., Probabilistic Serial), yielding a controlled tradeoff between incentive compatibility and ex-ante efficiency. Admissibility conditions ensure that monotonicity and partial strategyproofness can be maintained as the mixture parameter is varied.

4. Efficiency: Ex Post, Ex Ante, and Welfare Guarantees

RSD guarantees ex post efficiency: every deterministic assignment in its support is Pareto efficient. However, RSD is not SD-efficient (efficient in the sense of stochastic dominance): there exist preference profiles such that another random assignment stochastically dominates the RSD outcome for all agents (Aziz, 2016). The necessary and sufficient condition: RSD is not SD-efficient if and only if there exists an ex post assignment (i.e., a mixture over Pareto optima) that is not SD-efficient.

RSD fails robust ex post efficiency: some decompositions consistent with the support of the random assignment may include non–Pareto optimal deterministic assignments (Aziz et al., 2014).

Social welfare guarantees for RSD depend on the utility model:

• Dichotomous preferences: For $n$ agents and $n$ items, RSD always achieves at least 1/3 of the maximum possible social welfare; a refinement using the "RANKING"-like algorithm achieves 0.69-approximation under non-adversarial reporting (Adamczyk et al., 2014).
• Normalized von Neumann–Morgenstern utilities: For homogeneously normalized preferences, RSD achieves a lower bound of $\frac{1}{e}\frac{\nu(O)^2}{n}$, where $\nu(O)$ is the optimal social welfare (Adamczyk et al., 2014). No truthful mechanism can exceed the bound $\frac{\nu(O)^2}{n}$.
• Egalitarian value: In the worst case, RSD guarantees only an $O(1/n)$-fraction of the optimal egalitarian value (Aziz et al., 2015).

RSD's lack of SD-efficiency can be significant in applications requiring strong welfare guarantees or robust envy-freeness. Even so, under lexicographic preferences, RSD is envyfree (Hosseini et al., 2015, Hosseini et al., 2015).

5. Computational Complexity

Executing RSD—namely, simulating a run and producing an outcome—is efficient: generating a random permutation and having agents pick in turn is $O(n^2)$. However, computing the exact probabilities with which each assignment (or object/alternative) is chosen—the so-called RSD probabilities—is #P-complete in both the voting and assignment settings (Aziz et al., 2013).

The intractability arises because one must sum over exponentially many ($n!$) permutations, with nontrivial dependencies introduced by agent preferences and indifferences. This is critical for tasks that require expected allocations or ex-ante fairness rather than a realization.

For instances where key parameters (number of agents, objects, or agent types) are small, parametrized algorithms can compute RSD probabilities efficiently (Aziz et al., 2014). Specifically, the approach is fixed-parameter tractable with respect to these parameters.

For practical estimation of expected welfare or cost, sampling-based algorithms (e.g., Monte Carlo simulation over a modest number of permutations) suffice. With high probability, the sample mean approaches the true expectation quickly, due to concentration of measure phenomena—despite the #P-hardness of exact computation (Caragiannis et al., 23 May 2024).

6. Extensions, Modifications, and Applied Perspectives

Transfers and Aftermarket Trading

Allowing monetary or utility transfers after the initial RSD allocation fundamentally alters incentive and efficiency properties. When agents have quasilinear utilities, ex-post trading following RSD can lead to Pareto improvements and outcomes matching the VCG (optimal welfare-maximizing) assignment if post-allocation trades are unrestricted. However, RSD is no longer strategyproof, as agents have incentive to select items based on resale value rather than intrinsic preferences (Sundar et al., 2023).

Resource Augmentation

In resource allocation with facility constraints and strategic agents, RSD with augmented capacities ($g$ times the base capacity, $g \geq 3$) achieves approximation ratio $g/(g-2)$ for social cost minimization; with no augmentation, the ratio lies between $n^{0.26}$ and $n$ (Caragiannis et al., 2016). This demonstrates that mild resource augmentation can drastically improve the worst-case performance of mechanisms constrained by strategyproofness.

Robust Stability and Democratic Foundations

Self-equivalence—stability under self-application—characterizes the uniform random dictatorship (the distributional analog of RSD) as the unique democratic, anonymous, optimal, monotonic, and neutral voting rule stable under its own use (Hermida-Rivera, 18 Jun 2025). This axiomatization strengthens the rationale for using RSD-type procedures in collective decision making.

Information Aggregation and Adversarial Settings

For robust binary information aggregation with unknown and possibly adversarial correlations, the unique optimal rule (in minimax, regret, and approximation ratio paradigms) is random dictator: select a voter at random and follow her recommendation (Arieli et al., 2023), underscoring the broad optimality of the RSD/random dictator framework beyond pure assignment.

7. Open Questions and Ongoing Directions

• Establishing the unique characterization of RSD for arbitrary $n$ via equal-treatment-of-equals, ex post efficiency, and sd-strategyproofness remains an open algebraic problem, tied to the full-rank condition on constraint matrices (Brandt et al., 2023).
• Precise quantitative bounds on manipulation power for RSD and related rules, as well as constructing more finely tuned hybrid mechanisms with optimal tradeoffs, continue to be a subject of active research (Friedgut et al., 2011, Mennle et al., 2013).
• Practical algorithms for estimating welfare or assignment probabilities via sampling, and the design of mechanisms that achieve better efficiency/robustness trade-offs under specific domain or fairness constraints, are being refined and tested in applications (Caragiannis et al., 23 May 2024).

Summary Table: Central Properties of RSD

Property	Holds for RSD	Reference
Ex post efficiency	Yes	(Aziz et al., 2013, Basteck, 22 Jun 2025)
Sd-strategyproofness	Yes	(Hosseini et al., 2015, Basteck, 22 Jun 2025)
Robust ex post efficiency	No	(Aziz et al., 2014)
SD-efficiency	Not in general	(Aziz, 2016)
Egalitarian worst-case ratio	$O(1/n)$	(Aziz et al., 2015)
Anonymity, neutrality	Yes	(Hermida-Rivera, 18 Jun 2025, Basteck, 22 Jun 2025)
Computationally tractable execution	Yes (sampling)	(Aziz et al., 2013, Caragiannis et al., 23 May 2024)
Computation of ex-ante probabilities	#P-complete	(Aziz et al., 2013, Aziz et al., 2014)

RSD occupies a central and canonical place in discrete allocation and stochastic social choice theory. Its universal axiomatic underpinnings, strategic robustness, and computational pragmatics inform mechanism design, economic theory, and practical allocation in diverse domains—though it is also subject to rigorous limitations in terms of manipulability, ex-ante inefficiency, and computational complexity.