# Maximin Fairness with Mixed Divisible and Indivisible Goods

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Abstract:

We study fair resource allocation when the resources contain a mixture of divisible and indivisible goods, focusing on the well-studied fairness notion of maximin share fairness (MMS). With only indivisible goods, a full MMS allocation may not exist, but a constant multiplicative approximate allocation always does. We analyze how the MM...More

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Introduction

- Fair division concerns the problem of allocating a set of goods among interested agents in a way that is fair to all participants involved.
- As one of the most classic and widely known fairness notions, Steinhaus [1948] proposed that in an allocation that involves n participating agents, each agent should receive a bundle which is worth at least 1/n of her value for the entire set of goods.
- An allocation satisfying such property is known as a proportional allocation.
- An allocation is said to be an MMS allocation if every agent receives a bundle which is worth at least her maximin share

Highlights

- Fair division concerns the problem of allocating a set of goods among interested agents in a way that is fair to all participants involved
- In Section 3, we first show that any problem instance of mixed goods can be converted into another problem instance with only indivisible goods, such that the two instances have the same maximin share (MMS) value for every agent, and any allocation of the indivisible instance can be converted to an allocation in the mixed instance
- It is reasonable to think that adding divisible goods to the set of indivisible goods can only help with the MMS approximation guarantee
- We provide a problem instance with only indivisible goods, such that when a small amount of divisible goods is added to the instance, the MMS approximation guarantee of the instance strictly decreases, i.e., while an α-approximate maximin share fairness (α-MMS) allocation exists in the original instance, no α-MMS allocation exists after adding cake
- MMS approximation guarantees between mixed goods instances and indivisible goods instances
- We present an algorithm to produce an α-MMS allocation for any number of agents, where α monotonically increases in terms of the ratio between agents’ values for the divisible goods and their MMS values

Results

- In Section 3, the authors first show that any problem instance of mixed goods can be converted into another problem instance with only indivisible goods, such that the two instances have the same MMS value for every agent, and any allocation of the indivisible instance can be converted to an allocation in the mixed instance
- This reduction directly implies that the worst-case MMS approximation guarantee across all mixed goods instances is the same as that across all indivisible goods instances.
- The authors provide a problem instance with only indivisible goods, such that when a small amount of divisible goods is added to the instance, the MMS approximation guarantee of the instance strictly decreases, i.e., while an α-MMS allocation exists in the original instance, no α-MMS allocation exists after adding cake

Conclusion

- The authors study the extent to which the authors can find approximate MMS allocations when the resources contain both divisible and indivisible goods.
- MMS approximation guarantees between mixed goods instances and indivisible goods instances.
- It would be interesting to improve the MMS approximation guarantee with mixed goods.
- Another working direction is to study fair allocations in the mixed goods setting in conjunction with economic efficiency notions such as Pareto optimality

Summary

## Introduction:

Fair division concerns the problem of allocating a set of goods among interested agents in a way that is fair to all participants involved.- As one of the most classic and widely known fairness notions, Steinhaus [1948] proposed that in an allocation that involves n participating agents, each agent should receive a bundle which is worth at least 1/n of her value for the entire set of goods.
- An allocation satisfying such property is known as a proportional allocation.
- An allocation is said to be an MMS allocation if every agent receives a bundle which is worth at least her maximin share
## Objectives:

The authors aim to provide such a comparison. The authors' goal is to design algorithms that could compute allocations with good MMS approximation ratios in a mixed goods problem instance.## Results:

In Section 3, the authors first show that any problem instance of mixed goods can be converted into another problem instance with only indivisible goods, such that the two instances have the same MMS value for every agent, and any allocation of the indivisible instance can be converted to an allocation in the mixed instance- This reduction directly implies that the worst-case MMS approximation guarantee across all mixed goods instances is the same as that across all indivisible goods instances.
- The authors provide a problem instance with only indivisible goods, such that when a small amount of divisible goods is added to the instance, the MMS approximation guarantee of the instance strictly decreases, i.e., while an α-MMS allocation exists in the original instance, no α-MMS allocation exists after adding cake
## Conclusion:

The authors study the extent to which the authors can find approximate MMS allocations when the resources contain both divisible and indivisible goods.- MMS approximation guarantees between mixed goods instances and indivisible goods instances.
- It would be interesting to improve the MMS approximation guarantee with mixed goods.
- Another working direction is to study fair allocations in the mixed goods setting in conjunction with economic efficiency notions such as Pareto optimality

Related work

- Maximin share (MMS) fairness was first introduced by Budish [2011]. In addition to the works we mentioned above, MMS allocations of indivisible resources have also been extensively studied in several other settings [Bouveret et al, 2017; Farhadi et al, 2019; Gourves and Monnot, 2019; Igarashi and Peters, 2019; Lonc and Truszczynski, 2020; Suksompong, 2018].

A related line of research incorporates money into the fair division of indivisible goods, with the consideration of finding envy-free allocations [Alkan et al, 1991; Aziz, 2021; Brustle et al, 2020; Caragiannis and Ioannidis, 2020; Halpern and Shah, 2019; Klijn, 2000; Maskin, 1987; Meertens et al, 2002]. Another closely related problem is called rent division (see [Abdulkadiroglu et al, 2004; Arunachaleswaran et al, 2019; Brams, 2008; Gal et al, 2017; Haake et al, 2002; Su, 1999]). Its cardinal utility version can be viewed as a special case of the mixed setting where one wants to allocate (indivisible) rooms and the (divisible) rent among agents. However, in the mixed setting of fair division, the divisible goods (the rent) must be allocated and the agents are not allowed to use additional money to achieve more strict fairness condition.

Reference

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- Our counterexample will utilize the following lemma from [Kurokawa et al., 2018], which is also used for showing the nonexistence of full MMS allocation with indivisible goods.
- Lemma A.1 (Base of counterexample [Kurokawa et al., 2018]). For any n ≥ 6, there exists an n × n matrix M, satisfying the following properties: 1. All entries are non-negative (i.e., ∀i, j: Mi,j ≥ 0).
- 2. All entries of the last row and column, and the first entry in the first row, are positive (i.e., ∀i: Mi,n, Mn,i > 0 and M1,1 > 0).
- 3. All rows and columns sum to 1 (i.e., M 1 = M ⊤1 = 1).
- 4. Define M + as the set of all positive entries in M. Then if we wish to partition M + into n subsets that sum to exactly 1, then our partition must correspond to either the rows of M or the columns of M.

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