# Almost Envy-freeness, Envy-rank, and Nash Social Welfare Matchings

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Abstract:

Envy-free up to one good (EF1) and envy-free up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents an...More

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Introduction

- Fair division is a fundamental and interdisciplinary problem that has been extensively studied in economics, mathematics, political science, and computer science [21, 30, 24, 20, 4, 25, 28, 26, 27, 5, 18, 2, 14].
- To shed light on their usefulness, assume that after a Nash Social Welfare Matching, agent i envies agent j with a ratio α > 1, meaning that he thinks the value of the good allocated to agent j is α times more than the value of his item.
- In the first step of the algorithm, the authors allocate one item to each agent such that the Nash social welfare of the agents is maximized.

Highlights

- Fair division is a fundamental and interdisciplinary problem that has been extensively studied in economics, mathematics, political science, and computer science [21, 30, 24, 20, 4, 25, 28, 26, 27, 5, 18, 2, 14]
- evny-free up to a random good (EFR) is a weaker notion than EFX, yet stronger than EF1
- The intuition behind EFR is to use randomness to reduce the severe impact of small items
- As the number of items allocated to an agent grows larger, we expect EFX and EFR to be more and more aligned
- Similar to EFX, we provide a counter example which shows that a Nash Social Welfare allocation is not necessarily EFR
- In Example 4, we show one structural difference between EF1 and EFR: in contrast to EF1, EFR is not implied by an allocation that maximizes Nash social welfare

Results

- This allocation is not envy-free, it is both EFX and EFR since each agent receives only one item.
- Suppose that the authors allocate one item to each agent using a Nash social welfare matching.
- The authors allocate each agent one item using a Nash social welfare matching and divide the agents into three groups based on their envy-rank.
- In the second step the authors allocate a set of goods to the agents in each group, and in the third step the authors allocate the rest of the items using the classic envy-cycle elimination method.
- The authors use the envy-graph to allocate the remaining unallocated items.
- It follows from the observation above that for every agent i the valuation of every remaining item is at most vi(Ai)/φ after the second step of the algorithm.
- EFR allocation, our (φ − 1)-EFX algorithm consists of 3 steps, namely NSW matching, allocation refinement, and envy-graph based allocation.

Conclusion

- // Step 1 Allocate NSW matching; Let ri be envy-rank of an agent i.
- Eliminate all directed cycles in the envy-graph; Let s be an arbitrary source in the envy-graph; Ask agent s to pick his most valuable remaining item; end return the allocation; 4.2 Step 2.
- Suppose that the authors are given a partial α-EFX allocation A such that for every agent i and every remaining item b, the authors have vi(b) ≤ α · (Ai) for some constant α ≤ 1.

Summary

- Fair division is a fundamental and interdisciplinary problem that has been extensively studied in economics, mathematics, political science, and computer science [21, 30, 24, 20, 4, 25, 28, 26, 27, 5, 18, 2, 14].
- To shed light on their usefulness, assume that after a Nash Social Welfare Matching, agent i envies agent j with a ratio α > 1, meaning that he thinks the value of the good allocated to agent j is α times more than the value of his item.
- In the first step of the algorithm, the authors allocate one item to each agent such that the Nash social welfare of the agents is maximized.
- This allocation is not envy-free, it is both EFX and EFR since each agent receives only one item.
- Suppose that the authors allocate one item to each agent using a Nash social welfare matching.
- The authors allocate each agent one item using a Nash social welfare matching and divide the agents into three groups based on their envy-rank.
- In the second step the authors allocate a set of goods to the agents in each group, and in the third step the authors allocate the rest of the items using the classic envy-cycle elimination method.
- The authors use the envy-graph to allocate the remaining unallocated items.
- It follows from the observation above that for every agent i the valuation of every remaining item is at most vi(Ai)/φ after the second step of the algorithm.
- EFR allocation, our (φ − 1)-EFX algorithm consists of 3 steps, namely NSW matching, allocation refinement, and envy-graph based allocation.
- // Step 1 Allocate NSW matching; Let ri be envy-rank of an agent i.
- Eliminate all directed cycles in the envy-graph; Let s be an arbitrary source in the envy-graph; Ask agent s to pick his most valuable remaining item; end return the allocation; 4.2 Step 2.
- Suppose that the authors are given a partial α-EFX allocation A such that for every agent i and every remaining item b, the authors have vi(b) ≤ α · (Ai) for some constant α ≤ 1.

Related work

- Fair allocation of a divisible resource (known as cake cutting) was first introduced by Steinhaus[31] in 1948, and since then has been the subject of intensive studies. We refer the reader to [11] and [29] for an overview of different fairness notions and their related results. Proportionality and Envy-freeness are among the most well-established notions for cake cutting. As mentioned, the literature of cake cutting admits strong positive results for these two notions (see [31] for details).

Since neither EF nor proportionality or any approximation of these notions can be guaranteed for indivisible goods, several relaxations are introduced for these two notions in the past decade. These relaxations include EF1 and EFX for envy-freeness and maximin-share [12] for proportionality. Nash Social Welfare (NSW) is also another important notion in allocation of indivisible goods which is somewhat a trade off between fairness and optimality.

Reference

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