Representing the Outcome Matching in Serial Dictatorship Mechanisms

Core Concepts

The outcome-effect complexity of the serial dictatorship (SD) matching mechanism is Θ̃(n), where n is the number of applicants. This means that the function mapping one applicant's preferences to the resulting matching can be represented using a data structure of size Θ̃(n) bits, which is as efficient as possible.

Abstract

The paper studies the complexity of how one applicant can affect the outcome matching in various matching mechanisms, including serial dictatorship (SD), Deferred Acceptance (DA), and Top Trading Cycles (TTC).
For the simple SD mechanism, the authors show the following:

The outcome-effect complexity of SD is Θ̃(n), meaning the function mapping one applicant's preferences to the resulting matching can be represented using Θ̃(n) bits.

The authors prove this by constructing a novel data structure that can represent all possible matchings resulting from different preference reports by one applicant, while only requiring Θ̃(n) bits.

The key insights are: (1) there is a "filtered" preference profile Pfilt that produces the same matching as the original profile P-1, and (2) each applicant's match in SD depends only on their top 2 preferences in Pfilt.

This shows that the complexity of how one applicant can affect the SD matching is as low as possible, despite the fact that changing one applicant's preferences can drastically change the overall matching.

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Key Insights Distilled From

Structural Complexities of Matching Mechanisms

by Yannai A. Go... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2212.08709.pdf

Deeper Inquiries

How do the complexities of representing the outcome matching in SD, DA, and TTC relate to the economic properties of these mechanisms, such as strategyproofness and stability?

The complexities of representing the outcome matching in SD, DA, and TTC are closely related to the economic properties of these mechanisms, such as strategyproofness and stability.

Serial Dictatorship (SD): Despite its simplicity, SD's outcome-effect complexity is relatively low. This is because SD is a straightforward mechanism where each applicant is matched to their top choice sequentially. SD is not strategyproof in general, but it is nonbossy and produces stable matchings. The low complexity of representing the outcome matching in SD reflects its straightforward and stable nature.

Deferred Acceptance (DA): DA, both in its applicant-proposing (APDA) and institution-proposing (IPDA) variants, is strategyproof and produces stable matchings. The outcome-effect complexity of DA is higher than SD, indicating that changes in applicants' preferences can have a more structured impact on the outcome matching. This complexity aligns with the strategic nature of DA, where applicants have incentives to truthfully report their preferences.

Top Trading Cycles (TTC): TTC is strategyproof and produces Pareto-optimal matchings. The outcome-effect complexity of TTC is higher than both SD and DA, indicating a more complex relationship between applicants' preferences and the final matching. This complexity reflects the intricate nature of TTC, where cycles of trades determine the final matching in a strategic and efficient manner.
In summary, the complexities of representing the outcome matching in SD, DA, and TTC are reflective of the strategic considerations and stability properties of these mechanisms. Higher complexity often corresponds to mechanisms with stronger strategic properties and more intricate matching processes.

How can the techniques used to analyze the outcome-effect complexity of SD be extended to more complex matching mechanisms in a principled way?

The techniques used to analyze the outcome-effect complexity of SD, particularly the construction of a structured representation of the outcome matching, can be extended to more complex matching mechanisms in a principled way. Here's how:

Characterizing the Impact: Just as in SD, the impact of one applicant's preferences on the outcome matching can be systematically analyzed in more complex mechanisms. By identifying the structured changes that occur in the matching based on individual preferences, a similar approach can be applied to understand the outcome-effect complexity.

Recursive Construction: The recursive process used in SD to create filtered preference profiles can be adapted to more complex mechanisms. By iteratively modifying preference lists based on the choices of other applicants, a structured representation of the outcome matching can be constructed.

Inductive Reasoning: Utilizing inductive reasoning to prove the equivalence of different preference profiles in complex mechanisms can help establish the relationship between preferences and the final matching. This approach ensures a principled extension of the analysis to more intricate mechanisms.

By applying these techniques with appropriate modifications and adjustments to account for the complexity of the mechanisms, the analysis of outcome-effect complexity can be extended in a principled and systematic manner.

What are the implications of these complexity results for the practical implementation and transparency of matching mechanisms in real-world applications?

The complexity results obtained for matching mechanisms like SD, DA, and TTC have significant implications for their practical implementation and transparency in real-world applications:

Implementation Efficiency: Understanding the complexity of representing the outcome matching can guide the implementation of matching mechanisms. Mechanisms with lower complexity may be more computationally efficient to implement and execute, making them more practical for real-world applications.

Transparency and Explainability: Higher complexity in representing the outcome matching can impact the transparency and explainability of the mechanisms. Mechanisms with complex outcomes may be harder to communicate and understand for participants, potentially leading to confusion and mistrust.

Policy Decisions: The complexity results can inform policy decisions regarding the selection of matching mechanisms in various applications. Mechanisms with lower complexity may be preferred for their simplicity and ease of explanation, aligning with the goals of transparency and trust in real-world settings.

Structural Insights: The structured analysis of outcome-effect complexity provides valuable insights into how individual preferences influence the overall matching outcome. This understanding can enhance the design and optimization of matching mechanisms for practical applications.

Overall, the complexity results offer valuable guidance for the practical implementation, transparency, and effectiveness of matching mechanisms in real-world scenarios, helping to ensure efficiency, fairness, and trustworthiness in various applications.