indsigt - Algorithms and Data Structures - # Existence of MMS Allocations in Mixed Manna Settings

Existence of Maximin Share (MMS) Allocations in Mixed Manna Settings

Q: What are the implications of the existence of MMS allocations in mixed manna settings for practical applications, such as resource allocation or fair division problems

The existence of MMS allocations in mixed manna settings has significant implications for practical applications, especially in resource allocation and fair division problems. MMS allocations provide a fair and efficient way to divide resources among agents with different preferences and utility functions. By ensuring that each agent receives their maximin share guarantee, MMS allocations promote fairness and reduce envy among the agents. This can lead to more equitable outcomes and improved satisfaction among the participants in resource allocation scenarios. In practical applications, such as dividing assets in a divorce settlement, allocating budget resources among departments, or sharing household chores among roommates, MMS allocations can help ensure a fair and satisfactory division. By considering the preferences and utility functions of each agent, MMS allocations can lead to outcomes that are perceived as just and equitable by all parties involved. This can help reduce conflicts, promote cooperation, and enhance overall efficiency in resource allocation processes. The existence of MMS allocations in mixed manna settings provides a valuable tool for addressing complex allocation problems where indivisible items need to be divided among agents with diverse preferences. By understanding the conditions under which MMS allocations exist, decision-makers can make more informed choices and implement fair and efficient allocation mechanisms in various real-world scenarios.

Q: How can the techniques introduced in this paper be extended or generalized to handle the remaining cases, particularly instances with only negative mixed agents

To handle the remaining cases, particularly instances with only negative mixed agents, the techniques introduced in the paper can be extended or generalized in several ways: Exploring Different Reduction Rules: One approach could be to develop new reduction rules that are specifically tailored to handle instances with negative mixed agents. These rules could be designed to address the unique challenges posed by negative utility values and the absence of singletons in MMS allocations for such agents. Adapting Existing Techniques: The existing techniques used in the paper, such as mimicking instances and applying reduction rules, can be adapted to accommodate instances with negative mixed agents. By modifying these techniques to account for the specific characteristics of negative utility functions, it may be possible to find MMS allocations in these cases. Considering Alternative Fairness Notions: In cases where MMS allocations are not feasible for instances with negative mixed agents, exploring alternative fairness notions or solution concepts in game theory and economics could provide insights into potential allocation mechanisms that satisfy different fairness criteria. By extending and generalizing the techniques introduced in the paper, researchers and practitioners can work towards finding solutions for instances with negative mixed agents and further advance the understanding of fair division in mixed manna settings.

Q: Are there any connections between the existence of MMS allocations in mixed manna settings and other fairness notions or solution concepts in game theory and economics

The existence of MMS allocations in mixed manna settings is closely related to other fairness notions and solution concepts in game theory and economics. Some connections include: Envy-Freeness: MMS allocations are a relaxation of envy-free allocations, where each agent receives their maximin share guarantee. Envy-freeness is a fundamental fairness notion in resource allocation problems, and MMS allocations provide a way to achieve a similar level of fairness while allowing for indivisible items and diverse preferences among agents. Proportionality: MMS allocations ensure that each agent receives a share of the resources that is proportional to their maximin share guarantee. This concept of proportionality in resource allocation aligns with the principles of fairness and equity in economic decision-making. Efficiency: MMS allocations aim to maximize the minimum utility that each agent receives, leading to efficient outcomes where resources are allocated in a way that maximizes overall satisfaction among the agents. This efficiency criterion is essential in game theory and economics to ensure optimal resource utilization and allocation. By considering the connections between MMS allocations and other fairness notions or solution concepts, researchers can gain a deeper understanding of the implications of MMS allocations in mixed manna settings and their broader implications for resource allocation and fair division problems.

Kernekoncepter

An MMS allocation exists as long as the number of items m is at most n + 5, and either (i) every agent is a chores agent, or (ii) there exists a non-negative agent. Additionally, for n ≤ 3 agents, an MMS allocation exists as long as m ≤ n + 5, regardless of the types of agents.

Resumé

The paper addresses the question of whether similar results on the existence of maximin share (MMS) allocations hold for the general mixed manna setting, where items can be either goods or chores.

The key insights are:

For n ≤ 3 agents, an MMS allocation exists as long as m ≤ n + 5, regardless of the types of agents (goods, chores, or mixed).
For instances with m ≤ n + 5 and either (i) only chores agents or (ii) at least one non-negative agent, an MMS allocation exists.
The techniques used to obtain the m ≤ n + 5 bound for goods do not apply to the mixed manna setting. The paper introduces new techniques, including modifying the utility functions of agents, to handle these settings.
The main challenge is dealing with instances that contain only negative mixed agents (agents that consider some items as goods and others as chores). Handling these instances remains an open problem.

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Existence of MMS Allocations with Mixed Manna

by Kevin Hsu kl. arxiv.org 05-01-2024

https://arxiv.org/pdf/2401.07490.pdf

Existence of MMS Allocations with Mixed Manna

Dybere Forespørgsler

What are the implications of the existence of MMS allocations in mixed manna settings for practical applications, such as resource allocation or fair division problems

The existence of MMS allocations in mixed manna settings has significant implications for practical applications, especially in resource allocation and fair division problems. MMS allocations provide a fair and efficient way to divide resources among agents with different preferences and utility functions. By ensuring that each agent receives their maximin share guarantee, MMS allocations promote fairness and reduce envy among the agents. This can lead to more equitable outcomes and improved satisfaction among the participants in resource allocation scenarios.
In practical applications, such as dividing assets in a divorce settlement, allocating budget resources among departments, or sharing household chores among roommates, MMS allocations can help ensure a fair and satisfactory division. By considering the preferences and utility functions of each agent, MMS allocations can lead to outcomes that are perceived as just and equitable by all parties involved. This can help reduce conflicts, promote cooperation, and enhance overall efficiency in resource allocation processes.
The existence of MMS allocations in mixed manna settings provides a valuable tool for addressing complex allocation problems where indivisible items need to be divided among agents with diverse preferences. By understanding the conditions under which MMS allocations exist, decision-makers can make more informed choices and implement fair and efficient allocation mechanisms in various real-world scenarios.

How can the techniques introduced in this paper be extended or generalized to handle the remaining cases, particularly instances with only negative mixed agents

To handle the remaining cases, particularly instances with only negative mixed agents, the techniques introduced in the paper can be extended or generalized in several ways:

Exploring Different Reduction Rules: One approach could be to develop new reduction rules that are specifically tailored to handle instances with negative mixed agents. These rules could be designed to address the unique challenges posed by negative utility values and the absence of singletons in MMS allocations for such agents.

Adapting Existing Techniques: The existing techniques used in the paper, such as mimicking instances and applying reduction rules, can be adapted to accommodate instances with negative mixed agents. By modifying these techniques to account for the specific characteristics of negative utility functions, it may be possible to find MMS allocations in these cases.

Considering Alternative Fairness Notions: In cases where MMS allocations are not feasible for instances with negative mixed agents, exploring alternative fairness notions or solution concepts in game theory and economics could provide insights into potential allocation mechanisms that satisfy different fairness criteria.

By extending and generalizing the techniques introduced in the paper, researchers and practitioners can work towards finding solutions for instances with negative mixed agents and further advance the understanding of fair division in mixed manna settings.

Are there any connections between the existence of MMS allocations in mixed manna settings and other fairness notions or solution concepts in game theory and economics

The existence of MMS allocations in mixed manna settings is closely related to other fairness notions and solution concepts in game theory and economics. Some connections include:

Envy-Freeness: MMS allocations are a relaxation of envy-free allocations, where each agent receives their maximin share guarantee. Envy-freeness is a fundamental fairness notion in resource allocation problems, and MMS allocations provide a way to achieve a similar level of fairness while allowing for indivisible items and diverse preferences among agents.

Proportionality: MMS allocations ensure that each agent receives a share of the resources that is proportional to their maximin share guarantee. This concept of proportionality in resource allocation aligns with the principles of fairness and equity in economic decision-making.

Efficiency: MMS allocations aim to maximize the minimum utility that each agent receives, leading to efficient outcomes where resources are allocated in a way that maximizes overall satisfaction among the agents. This efficiency criterion is essential in game theory and economics to ensure optimal resource utilization and allocation.

By considering the connections between MMS allocations and other fairness notions or solution concepts, researchers can gain a deeper understanding of the implications of MMS allocations in mixed manna settings and their broader implications for resource allocation and fair division problems.