Characterizing the Top Trading Cycles Rule in Housing Markets with Lexicographic Preferences and Limited Externalities

Q: How would the dynamics of the housing market change if agents could express the intensity of their preferences, moving beyond a simple lexicographic structure?

Allowing agents to express the intensity of their preferences, rather than just a lexicographic ordering, would significantly impact the housing market dynamics by adding a layer of complexity and realism: More nuanced preferences: Instead of a strict hierarchy (e.g., house first, recipient second), agents could express how much they prefer one house over another or one recipient over another. This could be represented through numerical values, utility functions, or preference intensities within a matching mechanism. Increased bargaining power: Agents with strong preferences for specific houses or recipients would have increased bargaining power. For instance, an agent willing to "pay" a high price in terms of preference concessions for a particular house might influence the allocation. Emergence of new solutions: The Top Trading Cycles (TTC) rule, designed for lexicographic preferences, might not be suitable or efficient for capturing these richer preferences. New mechanisms and solution concepts would be needed, potentially incorporating concepts like stability under transfers, matching with contracts, or preference intensity-aware algorithms. Challenges in preference elicitation: Gathering and processing this more complex preference information from agents would be challenging. Designing mechanisms that are easy for agents to understand and use while effectively eliciting their intensity of preferences would be crucial. Overall, moving beyond lexicographic preferences would make the housing market model more realistic but also more complex. It would necessitate developing new mechanisms and solution concepts that can handle preference intensities and potentially side payments or compensations to achieve efficient and stable allocations.

Q: Could a mechanism involving side payments or compensations between agents overcome the impossibility result encountered when mixing demand and supply lexicographic preferences?

Yes, introducing side payments or compensations between agents could potentially overcome the impossibility result encountered when mixing demand and supply lexicographic preferences. Here's why: Addressing conflicting preferences: The impossibility result arises from the difficulty of simultaneously satisfying individual rationality, pair efficiency, and strategy-proofness when some agents prioritize the house they receive (demand lexicographic) while others prioritize the recipient of their house (supply lexicographic). Side payments can help align these conflicting preferences by compensating agents who might otherwise be worse off. Enabling mutually beneficial trades: Consider a scenario where agent A highly values receiving a specific house, while agent B strongly prefers a particular agent to receive their house. Without side payments, a trade might not occur, leaving both unsatisfied. However, if agent A can compensate agent B, a mutually beneficial agreement becomes possible. Mechanism design considerations: Designing a mechanism with side payments that achieves desirable properties like efficiency, stability, and strategy-proofness is complex. Key considerations include: Determining the payment structure: How are payments calculated and transferred between agents? Ensuring budget balance: Are total payments received equal to total payments made? Preventing strategic manipulation: Can agents manipulate the mechanism through their payments or reported preferences? Examples of mechanisms that incorporate side payments in related contexts include the Vickrey-Clarke-Groves (VCG) mechanism and various auction formats. Adapting these concepts to the housing market with mixed lexicographic preferences could be a promising avenue for future research.

核心概念

In housing markets where agents have lexicographic preferences, prioritizing either the house they receive or the recipient of their endowment, the Top Trading Cycles (TTC) rule is the only mechanism that guarantees individual rationality, pair efficiency, and strategy-proofness.

摘要

Bibliographic Information: Klaus, B. (2024). Characterizing the top trading cycles rule for housing markets with lexicographic preferences when externalities are limited. arXiv preprint arXiv:2410.16745v1.
Research Objective: This paper aims to characterize the Top Trading Cycles (TTC) rule in housing markets with limited externalities, where agents have lexicographic preferences over receiving a house and the recipient of their endowment.
Methodology: The paper employs game-theoretic modeling and analysis, focusing on properties like individual rationality, pair efficiency, strategy-proofness, Pareto efficiency, and stability in the context of housing markets with externalities.
Key Findings: The study finds that when agents prioritize the house they receive (demand lexicographic preferences), the TTC rule uniquely satisfies individual rationality, pair efficiency, and strategy-proofness. This result extends to scenarios where agents prioritize the recipient of their endowment (supply lexicographic preferences). However, no rule can simultaneously satisfy these properties when the market includes both demand and supply lexicographic preferences.
Main Conclusions: The TTC rule is a compelling mechanism for fairly and efficiently allocating houses in markets with limited externalities, particularly when agents have a clear preference hierarchy. However, the impossibility result highlights the challenges of designing mechanisms for markets with mixed lexicographic preferences.
Significance: This research contributes to the understanding of housing markets with externalities, a setting with practical relevance. It provides theoretical justification for using the TTC rule in specific scenarios and sheds light on the limitations of mechanism design in complex preference environments.
Limitations and Future Research: The study focuses on limited externalities, where agents only care about the recipient of their endowment. Future research could explore more general externality structures. Additionally, investigating the compatibility of other desirable properties with the TTC rule in this context would be valuable.

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Characterizing the top trading cycles rule for housing markets with lexicographic preferences

by Bettina Klau... 在 arxiv.org 10-23-2024

https://arxiv.org/pdf/2410.16745.pdf

Characterizing the top trading cycles rule for housing markets with lexicographic preferences

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How would the dynamics of the housing market change if agents could express the intensity of their preferences, moving beyond a simple lexicographic structure?

Allowing agents to express the intensity of their preferences, rather than just a lexicographic ordering, would significantly impact the housing market dynamics by adding a layer of complexity and realism:

More nuanced preferences:  Instead of a strict hierarchy (e.g., house first, recipient second), agents could express how much they prefer one house over another or one recipient over another. This could be represented through numerical values, utility functions, or preference intensities within a matching mechanism.
Increased bargaining power: Agents with strong preferences for specific houses or recipients would have increased bargaining power. For instance, an agent willing to "pay" a high price in terms of preference concessions for a particular house might influence the allocation.
Emergence of new solutions: The Top Trading Cycles (TTC) rule, designed for lexicographic preferences, might not be suitable or efficient for capturing these richer preferences. New mechanisms and solution concepts would be needed, potentially incorporating concepts like stability under transfers, matching with contracts, or preference intensity-aware algorithms.
Challenges in preference elicitation:  Gathering and processing this more complex preference information from agents would be challenging. Designing mechanisms that are easy for agents to understand and use while effectively eliciting their intensity of preferences would be crucial.
Overall, moving beyond lexicographic preferences would make the housing market model more realistic but also more complex. It would necessitate developing new mechanisms and solution concepts that can handle preference intensities and potentially side payments or compensations to achieve efficient and stable allocations.

Could a mechanism involving side payments or compensations between agents overcome the impossibility result encountered when mixing demand and supply lexicographic preferences?

Yes, introducing side payments or compensations between agents could potentially overcome the impossibility result encountered when mixing demand and supply lexicographic preferences. Here's why:

Addressing conflicting preferences: The impossibility result arises from the difficulty of simultaneously satisfying individual rationality, pair efficiency, and strategy-proofness when some agents prioritize the house they receive (demand lexicographic) while others prioritize the recipient of their house (supply lexicographic). Side payments can help align these conflicting preferences by compensating agents who might otherwise be worse off.
Enabling mutually beneficial trades:  Consider a scenario where agent A highly values receiving a specific house, while agent B strongly prefers a particular agent to receive their house. Without side payments, a trade might not occur, leaving both unsatisfied. However, if agent A can compensate agent B, a mutually beneficial agreement becomes possible.
Mechanism design considerations: Designing a mechanism with side payments that achieves desirable properties like efficiency, stability, and strategy-proofness is complex.  Key considerations include:

Determining the payment structure: How are payments calculated and transferred between agents?
Ensuring budget balance: Are total payments received equal to total payments made?
Preventing strategic manipulation: Can agents manipulate the mechanism through their payments or reported preferences?
Examples of mechanisms that incorporate side payments in related contexts include the Vickrey-Clarke-Groves (VCG) mechanism and various auction formats. Adapting these concepts to the housing market with mixed lexicographic preferences could be a promising avenue for future research.

If we view the housing market as a network, with agents and houses as nodes, how does the TTC rule influence the structure and properties of this network, and what insights can network theory offer in understanding such markets?

Viewing the housing market as a network, with agents and houses as nodes connected by preference relations, provides valuable insights into the TTC rule's impact:
TTC and Network Structure:

Cycle Decomposition: The TTC algorithm essentially decomposes the housing market network into a set of directed cycles. Each cycle represents a set of agents who will exchange houses within their group, with each agent's most preferred house (within the cycle) belonging to the next agent in the cycle.
Strongly Connected Components: The TTC rule prioritizes cycles involving agents with mutually achievable top preferences. This can lead to a network structure where strongly connected components (groups where every node can reach every other node) emerge, representing clusters of agents with a higher likelihood of trading with each other.
Network Theory Insights:

Centrality Measures: Analyzing centrality measures like degree centrality (number of connections) or betweenness centrality (number of shortest paths passing through a node) can identify influential agents or houses in the market. Agents or houses with high centrality might have more bargaining power or play a critical role in facilitating trades.
Network Stability and Robustness: Network theory can help assess the stability and robustness of the housing market under the TTC rule. For instance, analyzing the network's connectivity, clustering coefficient (a measure of the degree to which nodes in a graph tend to cluster together), or vulnerability to node or edge removals can provide insights into the market's resilience to shocks or disruptions.
Market Segmentation: The TTC rule, by prioritizing cycles, might lead to market segmentation, where different groups of agents with distinct preferences trade primarily amongst themselves. Network analysis can help identify these segments and understand their characteristics.
Overall, a network perspective offers a powerful framework for analyzing the housing market and the impact of the TTC rule. It allows for a deeper understanding of market structure, agent influence, stability, and potential segmentation, ultimately contributing to the design of more efficient and equitable housing allocation mechanisms.