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Estimating the Expected Social Welfare and Cost of Random Serial Dictatorship

Ioannis Caragiannis, Sebastian Homrighausen

TL;DR

The paper studies estimating the expected efficiency of random serial dictatorship (RSD) in assignment problems under two settings: value-based and metric-cost. It first establishes #P-hardness for computing the exact RSD lottery entries, then demonstrates practical, sampling-based estimators. In the value setting, a simple sampling scheme yields an ε-approximation with O(n/ε^2 log(1/δ)) samples. For the metric-cost setting, a non-trivial variance bound on the social cost and a median-of-means approach reduce the required samples to O(n^3/ε^2 log(1/δ)), despite an inherent exponential lower bound for naive sampling. Together, the results provide scalable methods to compare RSD against alternatives while acknowledging the computational hardness of exact calculations.

Abstract

We consider the assignment problem, where $n$ agents have to be matched to $n$ items. Each agent has a preference order over the items. In the serial dictatorship (SD) mechanism the agents act in a particular order and pick their most preferred available item when it is their turn to act. Applying SD using a uniformly random permutation as agent ordering results in the well-known random serial dictatorship (RSD) mechanism. Accurate estimates of the (expected) efficiency of its outcome can be used to assess whether RSD is attractive compared to other mechanisms. In this paper, we explore whether such estimates are possible by sampling a (hopefully) small number of agent orderings and applying SD using them. We consider a value setting in which agents have values for the items as well as a metric cost setting where agents and items are assumed to be points in a metric space, and the cost of an agent for an item is equal to the distance of the corresponding points. We show that a (relatively) small number of samples is enough to approximate the expected social welfare of RSD in the value setting and its expected social cost in the metric cost setting despite the #P-hardness of the corresponding exact computation problems.

Estimating the Expected Social Welfare and Cost of Random Serial Dictatorship

TL;DR

The paper studies estimating the expected efficiency of random serial dictatorship (RSD) in assignment problems under two settings: value-based and metric-cost. It first establishes #P-hardness for computing the exact RSD lottery entries, then demonstrates practical, sampling-based estimators. In the value setting, a simple sampling scheme yields an ε-approximation with O(n/ε^2 log(1/δ)) samples. For the metric-cost setting, a non-trivial variance bound on the social cost and a median-of-means approach reduce the required samples to O(n^3/ε^2 log(1/δ)), despite an inherent exponential lower bound for naive sampling. Together, the results provide scalable methods to compare RSD against alternatives while acknowledging the computational hardness of exact calculations.

Abstract

We consider the assignment problem, where agents have to be matched to items. Each agent has a preference order over the items. In the serial dictatorship (SD) mechanism the agents act in a particular order and pick their most preferred available item when it is their turn to act. Applying SD using a uniformly random permutation as agent ordering results in the well-known random serial dictatorship (RSD) mechanism. Accurate estimates of the (expected) efficiency of its outcome can be used to assess whether RSD is attractive compared to other mechanisms. In this paper, we explore whether such estimates are possible by sampling a (hopefully) small number of agent orderings and applying SD using them. We consider a value setting in which agents have values for the items as well as a metric cost setting where agents and items are assumed to be points in a metric space, and the cost of an agent for an item is equal to the distance of the corresponding points. We show that a (relatively) small number of samples is enough to approximate the expected social welfare of RSD in the value setting and its expected social cost in the metric cost setting despite the #P-hardness of the corresponding exact computation problems.
Paper Structure (13 sections, 18 theorems, 39 equations, 2 algorithms)

This paper contains 13 sections, 18 theorems, 39 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Further Related Work
  4. Preliminaries
  5. #P-Hardness of Random Serial Dictatorship
  6. Approximating the Expected Social Welfare
  7. Approximating the Expected Social Cost
  8. A Lower Bound for \ref{['alg:approx_sw_of_rsd']}
  9. Bounding the Variance of Social Cost
  10. The Upper Bound for \ref{['alg:approx_sc_of_rsd']}
  11. Conclusion
  12. A Weaker Statement for \ref{['alg:approx_sw_of_rsd']} in the Value Setting
  13. Upper Bounds for \ref{['alg:approx_sw_of_rsd']} in the Metric Cost Setting

Key Result

Lemma 1

Let $X_1, X_2, \dots, X_k$ be independent random variables satisfying $| X_i | \leq \alpha$ for $i\in [k]$, with mean $0$ and variance $\sigma^2(X_i)$. Then

Theorems & Definitions (33)

  • Lemma 1: Bernstein inequality, e.g. see DP09, page 9
  • Lemma 2: Bhatia-Davis inequality, BD00
  • Lemma 3: Chebyshev-Cantelli inequality, e.g. see MR95, page 64
  • Lemma 4: Chernoff bound, e.g. see MR95, page 68
  • Lemma 5: Reverse Chernoff bound, KY15
  • Theorem 1
  • Lemma 6
  • proof
  • Theorem 2
  • proof
  • ...and 23 more