Proportional Borda allocations.

Andreas Darmann,Christian Klamler

doi:10.1007/s00355-016-0982-z

Abstract

In this paper we study the allocation of indivisible items among a group of agents, a problem which has received increased attention in recent years, especially in areas such as computer science and economics. A major fairness property in the fair division literature is proportionality, which is satisfied whenever each of the n agents receives at least frac{1}{n} of the value attached to the whole set of items. To simplify the determination of values of (sets of) items from ordinal rankings of the items, we use the Borda rule, a concept used extensively and well-known in voting theory. Although, in general, proportionality cannot be guaranteed, we show that, under certain assumptions, proportional allocations of indivisible items are possible and finding such allocations is computationally easy.

Highlights

In recent years, the problem of fair division has received increased attention, especially in areas such as computer science and economics [see, e.g., the surveys by Bouveret et al (2016) and Thomson (2016)]
When we are concerned with indivisible items, even the satisfaction of one of the most basic fairness criterion, proportionality, which requires each of n agents to receive at least one nth of her value of the whole set of items, might be out of reach
With the exception of Proposition 1, we focus on Borda scores only and proportionality refers to Borda scores exclusively

Summary

Introduction

The problem of fair division has received increased attention, especially in areas such as computer science and economics [see, e.g., the surveys by Bouveret et al (2016) and Thomson (2016)]. Budish (2011) translates the ”I cut you choose” procedure from the cake-cutting literature to the indivisible items setting and focuses on what he calls maximin shares, i.e., the minimal share that an agent would receive if she was to divide the items into n piles and being the last to pick her pile This approach was used by Procaccia and Wang (2014) who showed that there is always an allocation that guarantees an agent 2/3 of her maximin share. Since we assume additivity, the value of a set of items depends directly on the positions in which the items are ranked by the agent [see, e.g., Brams et al (2001)].2 This seems to be a reasonable approach in case no other than ordinal information is available.

Formal framework

Proportional allocations in the general case

Proportional allocations in the two agents case

Conclusion