[Paper Review] Obvious Manipulations in Matching without and with Contracts

R. Pablo Arribillaga, Eliana Pepa Risma|arXiv (Cornell University)|Jun 30, 2023

Game Theory and Voting Systems4 citations

TL;DR

This paper analyzes obvious manipulations in two-sided many-to-one matching markets with and without contracts, focusing on doctors' strategic incentives when hospitals' preferences are public and substitutable. It proves that only the doctor-optimal matching rule is non-obviously manipulable (NOM) in both settings, while the hospital-optimal rule is obviously manipulable with contracts—even in the one-to-one case—contrasting with its NOM status without contracts.

ABSTRACT

This paper explores many-to-one matching models, both with and without contracts, where doctors' preferences are private and hospitals' preferences are public and substitutable. It is known that any stable-dominating mechanism --which is either stable or individually rational and Pareto-dominates (from the doctors' perspective) a stable mechanism--, is susceptible to manipulation by doctors. Our study focuses on extit{obvious manipulations} and identifies stable-dominating mechanisms that prevent them. Without contracts, we show that more efficient mechanisms are less likely to be obviously manipulable and that any stable-dominating mechanism is not obviously manipulable. However, with contracts, none of these results hold. While we demonstrate that the Doctor-Proposing Deferred Acceptance (DA) Mechanism remains not obviously manipulable, we show that the Hospital-Proposing DA Mechanism and any efficient mechanism that Pareto-dominates the Doctor-Proposing DA Mechanism become (very) obviously manipulable, in the model with contracts.

Motivation & Objective

• To examine whether stable matching rules can avoid obvious manipulations by doctors when hospitals’ preferences are public and substitutable.
• To compare strategic incentives in matching models with and without contracts, particularly focusing on the hospital-optimal and doctor-optimal rules.
• To determine whether quantile stable rules—interpolating between doctor- and hospital-optimal outcomes—can be non-obviously manipulable.
• To identify the conditions under which stable matching rules are robust to obvious manipulations, especially in the presence of substitutable preferences.

Proposed method

• Uses the concept of obvious manipulation from Troyan and Morrill (2020), defined as a strategy where the best outcome under manipulation strictly dominates the best outcome under truth-telling, or the worst outcome under manipulation strictly dominates the worst under truth-telling.
• Applies the deferred acceptance algorithm generalized by Hatfield and Milgrom (2005) to construct stable allocations under substitutable hospital preferences.
• Defines q-quantile stable rules as interpolating between the doctor-optimal (q=0) and hospital-optimal (q=1) rules, selecting the ⌈kq⌉-th best stable allocation for each doctor.
• Employs counterexample constructions in one-to-one and many-to-one settings to demonstrate obvious manipulability of non-doctor-optimal rules.
• Proves that for any q∈(0,1], there exists a market where a doctor can obviously manipulate the q-quantile rule by misreporting preferences.
• Analyzes the impact of contracts by comparing models with and without multiple contracts per doctor-hospital pair, showing that contracts increase strategic vulnerability.

Experimental results

Research questions

• RQ1 Is the doctor-optimal matching rule non-obviously manipulable (NOM) in both matching models with and without contracts?
• RQ2 Is the hospital-optimal matching rule NOM in the presence of contracts, even in the one-to-one case?
• RQ3 Are there any q-quantile stable rules (for q∈(0,1)) that are NOM in the many-to-one matching model with contracts?
• RQ4 How do substitutable preferences and the law of aggregate demand affect the existence of NOM stable rules?
• RQ5 What is the strategic difference between matching models with and without contracts in terms of obvious manipulability?

Key findings

• The doctor-optimal matching rule is non-obviously manipulable (NOM) in both the many-to-one matching model with and without contracts.
• The hospital-optimal matching rule is obviously manipulable in the many-to-one matching model with contracts, even in the one-to-one case.
• In contrast, the hospital-optimal rule is NOM in the many-to-one matching model without contracts, highlighting a key difference between the two settings.
• For q-quantile stable rules, only the doctor-optimal rule (q=0) is NOM; all other rules with q∈(0,1] are obviously manipulable.
• A counterexample is constructed where a doctor can strictly improve their best possible outcome by misreporting preferences under any q-quantile rule with q>0.
• The presence of contracts fundamentally alters strategic incentives, making the hospital-optimal rule vulnerable to obvious manipulation where it was previously robust.