# Fair Division of Mixed Divisible and Indivisible Goods

national conference on artificial intelligence, 2020.

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Abstract:

We study the problem of fair division when the resources contain both divisible and indivisible goods. Classic fairness notions such as envy-freeness (EF) and envy-freeness up to one good (EF1) cannot be directly applied to the mixed goods setting. In this work, we propose a new fairness notion envy-freeness for mixed goods (EFM), which...More

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Introduction

- Fair division studies the allocation of scarce resources among interested agents, with the objective of finding an allocation that is fair to all participants involved.
- The literature of fair division can be divided into two classes, categorized by the type of the resources to be allocated.
- An envy-free allocation with divisible resources always exists [Alon, 1987; Su, 1999] and can be found via a discrete and bounded protocol [Aziz and Mackenzie, 2016a]

Highlights

- Fair division studies the allocation of scarce resources among interested agents, with the objective of finding an allocation that is fair to all participants involved
- The algorithm requires a perfect allocation for cake cutting oracle and can compute an envy-freeness for mixed goods (EFM) allocation in polynomial number of steps
- We introduce the envy-freeness for mixed goods (EFM) fairness notion, which generalizes both EF and EF1 to the mixed setting
- We show that an EFM allocation always exists for any number of agents
- We provide bounded protocols to compute an EFM allocation in special cases, or an ǫ-EFM allocation in general setting in time poly(n, m, 1/ǫ)
- With regard to ǫ-EFM, our algorithm runs in time poly(n, m, 1/ǫ), it remains an open question to design an algorithm that runs in time poly(n, m, log(1/ǫ))

Results

- The authors initiate the study of fair division with mixed types of resources. the authors propose a new fairness notion, denoted as envy-freeness for mixed goods (short for EFM ), that naturally combines EF and EF1 together and works for the setting where the resource may contain both divisible and indivisible goods.
- EFM requires that for each agent, if her allocation consists of only indivisible items, others will compare their bundles to hers using EF1 criterion; but if this agent’s bundle contains any positive amount of divisible resources, others will compare their bundles to hers using the more strict EF condition
- This definition generalizes both EF and EF1 to the mixed goods setting and strikes a natural balance between the two fairness notions.
- In Section 4, the authors present two algorithms that could compute an EFM allocation for two special cases without using the perfect allocation oracle: (1) two agents with general valuations in the Robertson-Webb model, and (2) any number of agents with piecewise linear valuation functions

Conclusion

- This work is concerned with fair division of a mixture of divisible and indivsible goods
- To this end, the authors introduce the envy-freeness for mixed goods (EFM) fairness notion, which generalizes both EF and EF1 to the mixed setting.
- The authors provide bounded protocols to compute an EFM allocation in special cases, or an ǫ-EFM allocation in general setting in time poly(n, m, 1/ǫ)
- It remains an important open question of whether there exists a bounded protocol in the RW model that computes an EFM allocation in the general setting for any number of players.
- With regard to ǫ-EFM, the algorithm runs in time poly(n, m, 1/ǫ), it remains an open question to design an algorithm that runs in time poly(n, m, log(1/ǫ))

Summary

## Introduction:

Fair division studies the allocation of scarce resources among interested agents, with the objective of finding an allocation that is fair to all participants involved.- The literature of fair division can be divided into two classes, categorized by the type of the resources to be allocated.
- An envy-free allocation with divisible resources always exists [Alon, 1987; Su, 1999] and can be found via a discrete and bounded protocol [Aziz and Mackenzie, 2016a]
## Results:

The authors initiate the study of fair division with mixed types of resources. the authors propose a new fairness notion, denoted as envy-freeness for mixed goods (short for EFM ), that naturally combines EF and EF1 together and works for the setting where the resource may contain both divisible and indivisible goods.- EFM requires that for each agent, if her allocation consists of only indivisible items, others will compare their bundles to hers using EF1 criterion; but if this agent’s bundle contains any positive amount of divisible resources, others will compare their bundles to hers using the more strict EF condition
- This definition generalizes both EF and EF1 to the mixed goods setting and strikes a natural balance between the two fairness notions.
- In Section 4, the authors present two algorithms that could compute an EFM allocation for two special cases without using the perfect allocation oracle: (1) two agents with general valuations in the Robertson-Webb model, and (2) any number of agents with piecewise linear valuation functions
## Conclusion:

This work is concerned with fair division of a mixture of divisible and indivsible goods- To this end, the authors introduce the envy-freeness for mixed goods (EFM) fairness notion, which generalizes both EF and EF1 to the mixed setting.
- The authors provide bounded protocols to compute an EFM allocation in special cases, or an ǫ-EFM allocation in general setting in time poly(n, m, 1/ǫ)
- It remains an important open question of whether there exists a bounded protocol in the RW model that computes an EFM allocation in the general setting for any number of players.
- With regard to ǫ-EFM, the algorithm runs in time poly(n, m, 1/ǫ), it remains an open question to design an algorithm that runs in time poly(n, m, log(1/ǫ))

Related work

- As we mentioned, most previous works in fair division are from two categories based on whether the resources to be allocated are divisible or indivisible.

When the resources are divisible, the existence of an envy-free allocation is guaranteed [Alon, 1987], even with n−1 cuts [Su, 1999]. Brams and Taylor [1995] gave the first finite (but unbounded) envy-free protocol for any number of agents. Recently, Aziz and Mackenzie [2016b] gave the first bounded protocol for computing an envy-free allocation with four agents and their follow-up work extended the result to any number of agents [Aziz and Mackenzie, 2016a]. Besides envy-freeness, other classic fairness notions include proportionality and equitability, both of which have been studied extensively [Dubins and Spanier, 1961; Even and Paz, 1984; Edmonds and Pruhs, 2006; Cechlarova and Pillarova, 2012; Procaccia and Wang, 2017].

When the resources are indivisible, none of the aforementioned fairness notions guarantees to exist, thus relaxations are considered. Among other notions, these include envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin share (MMS), etc. [Lipton et al, 2004; Budish, 2011; Caragiannis et al, 2019]. An EF1 allocation always exists and can be efficiently computed [Lipton et al, 2004; Caragiannis et al, 2019]. However, the existence for EFX is still open [Caragiannis et al, 2019] except for special cases [Plaut and Roughgarden, 2018]. As for MMS, there exist instances where no allocation satisfies MMS. However, an approximation of MMS always exists and can be efficiently computed [Kurokawa et al, 2018; Amanatidis et al, 2017; Ghodsi et al, 2018; Garg and Taki, 2019]. Several other works studied the allocation of both indivisible goods and money, with the goal of finding envy-free allocations [Maskin, 1987; Alkan et al, 1991; Klijn, 2000; Meertens et al, 2002; Halpern and Shah, 2019]. Money can be viewed as a homogeneous divisible good which is valued the same across all agents. In our work, we consider a more general setting with heterogeneous divisible goods. Moreover, these works focused on finding envy-free allocations with the help of sufficient amount of money, which is again different from our goal.

Funding

- This work is supported in part by an RGC grant (HKU 17203717E)

Reference

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