Allocating contiguous blocks of indivisible chores fairly

Abstract

In the cake cutting problem, a cake needs to be divided among players with possibly different valuations for different pieces of the cake. We consider a variation where the cake represents chores that need to be allocated. We assume the players to be work-averse, i.e., intending to get as little of the cake as possible. We further assume that the chores are indivisible and that they are ordered along a line. We seek a fair allocation of the chores.

We study the loss in social welfare due to fairness using the price of fairness concept. Previous work has considered fair division of cakes and chores with non-connected and connected pieces as well as fair division of indivisible contiguous blocks of indivisible items (with positive utility) on a line. We complete the picture by providing price of fairness results for the allocation of contiguous blocks of indivisible chores.

Introduction

Fair division is a classic problem in economics, that deals with the question of how we can divide a set of resources between a number of different people with varying likes and dislikes. This is a very relevant problem, which occurs often in everyday social interaction and it has motivated a lot of research on different variations of the problem.

A classical problem in this area is the so called cake cutting problem [25]. In this problem we need to divide a cake with different toppings between a number of players who value these toppings differently. Our goal here is to maximize the happiness of each player, which is determined by the fraction of cake that she receives.

In this problem the players get a positive utility from their fraction of cake, so the more cake a player receives the happier she is. We can also consider a minimization problem, where the cake represents work that needs to be performed by the agents. This leads to the problem of chore division [21] where the goal is to minimize the discontent every player gets from his share of the work.

In order to evaluate the allocations we use two different kinds of welfare, utilitarian welfare and egalitarian welfare. To calculate the utilitarian welfare of an allocation, we add the utilities of all players in that allocation together. To calculate the egalitarian welfare we take the lowest utility or in the case of chore division the highest disutility as our welfare.

The resource that has to be allocated can be divisible or indivisible. In the first case, we can divide the resource into arbitrarily many pieces before allocating them to the agents. In the second case resources are divided into a set of distinct items, which cannot be divided further. Each item has to be fully allocated to one agent. When talking about allocating indivisible items, we assume these items to be lying in some order on a line.

When considering these problems of fair division, it is necessary to define some notion of fairness. In many earlier papers [1], [16], [28] the criteria for fairness are proportionality, equitability and envy-freeness. These are also used in this paper.

An allocation is proportional if every agent receives at least a fraction of $1/n$ of the utility that she would get if all of the resource gets allocated only to her (at most $1/n$ for chores). Here n is the number of players. An allocation is equitable if all agents get the same utility from their allocated share of the resource. Finally, an allocation is envy-free if every agent values the share she receives at least as much as the share any other agent receives.

As stated before, our goal is then to optimize the welfare of the players, while maintaining some notion of fairness. Since these goals do not necessarily go hand in hand, the natural question arises what the trade-off between these goals is. To this end, Caragiannis et al. [16] and independently Bertsimas et al. [9] initiated the concept of the price of fairness, which measures this trade-off, similarly to the related and well-known concepts of price of anarchy and price of stability. Caragiannis et al. [16] found upper and lower bounds for the problem of cake cutting as well as chore division in the divisible and indivisible case. However, their solutions did not limit the amount of pieces each player gets, leading to situations in which players might receive a high number of small pieces or cake crumbs. This situation can be undesirable in practice. This prompted Aumann and Dombb [1] to consider the case in which each player must be assigned a connected piece. We speak of a contiguous allocation if each player gets a contiguous block of items on the line in the case of indivisible items or a connected piece in the case of a divisible resource.

To calculate the price of fairness, we divide the welfare of an optimal allocation by the welfare of an optimal fair allocation. For chore division we use the inverse of that value to allow for easy comparisons (the optimal welfare is now never more than that of an optimal fair allocation). The price of fairness indicates by what factor the total disutility of the players must increase if we insist on a fair allocation.

Following the work of Aumann and Dombb [1], two different variations of this problem have been considered. This paper fills the gap between those works by considering both at once. Since Aumann and Dombb [1] only considered the problem of cake cutting, we can instead look at the allocation of indivisible goods, while still holding onto the requirement of the allocations being contiguous. This problem of fair division of contiguous blocks of indivisible items was studied by Suksompong [29].

The other variation keeps the goods divisible and instead considers again chores instead of goods. This problem of dividing connected chores fairly was studied by Heydrich and van Stee [22].

This paper now combines these variations and fills the gap between the papers from Suksompong and Heydrich and van Stee, considering the price of fairness for the fair allocation of contiguous blocks of indivisible chores. The optimal allocation also uses only contiguous blocks in this case. An example where this type of problem occurs could be the allocation of work shifts, which can not be further divided. A more specific example might be the division of routes between train drivers. This paper is influenced by both works; in fact many of the employed techniques are adaptations of techniques used in the one paper or the other. This is also mirrored in the results.

The price of proportionality is very similar between all three problems, being 1 in all cases for egalitarian welfare and $n,n$ and $n-1+\frac{1}{n}$ for the combination of indivisibility and chores, divisible chores and indivisible goods respectively in the case of utilitarian welfare.

Furthermore, if we make the change from goods to chores the price of envy-freeness changed from being bounded to being unbounded. Also, if we change to indivisible items, the price of equitability suddenly becomes unbounded. In this paper both changes are combined and we can observe that now both prices of fairness are unbounded (Table 1).

It should be mentioned that [29] also deals with the existence of allocations of indivisible goods, which satisfy approximate versions of the three notions of fairness. While this is not part of this paper it is possible to obtain analogous results for indivisible chores with only slight modifications to the proofs in [29].

In the case of equitability we do not change the definition when switching to chores, therefore the same proof that applies to goods also applies to chores. The proof for the existence of an allocation of indivisible goods that satisfies the approximate version of envy-freeness relies on the existence of an envy-free allocation for divisible goods. Su showed that such an allocation also always exists for divisible chores [27]. This means the proof also holds for chores. In the case of proportionality the proof needs to be altered slightly. However the structure stays the same.

Modern mathematicians started working on the topic of fair division in the 1940's with Banach, Steinhaus and Knaster giving the “Last Diminisher” mechanism for proportional divisions with n players [25]. In the following years researchers focussed on finding algorithms to achieve fair divisions [14], [15], [20], [26]. In particular Dubins and Spanier [20] gave an algorithm that guarantees the existence of a contiguous proportional allocation, Cechlárová et al. [17] showed the existence of certain contiguous equitable allocations and Stromquist [26] proved the existence of a contiguous envy-free allocation.

The problem of fair division of chores was first mentioned by Gardner [21]. The result of Steinhaus [25] can be applied to chores, thus a proportional division always exists. Su [27] proves the existence of envy-free divisions of chores, while Heydrich and van Stee [22] show the existence of equitable divisions. All these results assume that only connected pieces are allocated.

The problem of the efficiency of fair divisions was first addressed by Caragiannis et al. [16]. They gave bounds for the price of fairness for utilitarian welfare, considering the three notions of fairness as well as divisible and indivisible cakes and chores. Their analysis allows dividing the cake into any number of pieces, and giving players a collection of pieces, instead of a single one. Aumann and Dombb [1] then restricted the problem, only allowing connected pieces. This way players do not get unions of very small pieces (crumbs) assigned. They considered the problem with cakes, Heydrich and van Stee [22] then extended this work to chores.

Bouveret et al. [12] considered the allocation of contiguous blocks of indivisible items on a line and showed that determining whether a contiguous fair allocation exists is NP-hard for proportionality and envy-freeness. They also consider a more general model where each item is a vertex in a graph and the allocated bundles of goods are connected on this graph. Later Bouveret et al. [13] extend this work by considering the allocation of connected bundles of chores on a graph. Bilò et al. [10] also work in this model and study the allocations of indivisible items that are envy-free up to one good. Bei et al. [7] propose the price of connectivity which quantifies the loss of fairness when connected bundles are allocated. They derive bounds for large classes of graphs while considering envy-freeness and maximin share fairness.

Aumann et al. [2] considered the problem of finding a contiguous allocation that maximizes welfare for both divisible and indivisible items, showing that it is NP-hard to find the optimal solution, but there exists an efficient constant factor approximation algorithm. Bei et al. [5] and Cohler et al. [19] also considered this problem of maximizing welfare with additionally the constraint of respectively proportionality and envy-freeness. Suksompong [29] considered the problem of finding contiguous allocations of indivisible items on a line with respect to fairness constraints and shows the existence of certain contiguous allocations with respect to approximate versions of the usual notions of fairness, as well as giving bounds on the price of fairness with respect to exact proportionality, equitability and envy-freeness.

Recently, more work on the division of indivisible items and chores has been done. Kurokawa et al. [23] consider an alternative notion of fairness for indivisible items, the maximin share guarantee, and show that this fairness notion cannot be guaranteed even if the number of goods is small. Aziz et al. [4] add to this by considering indivisible chores with a weighted generalisation of maximin share. They provide a polynomial-time-constant-approximation algorithm for their problem.

Other methods of dividing goods are the egalitarian equivalent and the competitive equilibrium with equal incomes rules, which have been compared by Bogolmonaia et al. [11]. For goods the competitive rule fares better, however for chores this rule shows some disadvantages that the egalitarian rule does not have.

Bei et al. [6] give a truthful envy-free mechanism for cake cutting and chore division for two agents with piecewise uniform valuations without free disposal. This means the whole cake has to get allocated. Such a mechanism does not exist with some additional assumption, including the assumption that connected pieces are allocated. For more agents the class of valuations has to be restricted. They also give bounds on the efficiency ratio of mechanisms satisfying various properties. Previously Chen et al. [18] considered truthful fair division, but their results rely on the free disposal assumption.

A new approach is to consider mixed cakes, where there are both good and bad pieces. Segal-Halevi [24] considers the case of mixed cakes for envy freeness and connected pieces and shows that a division exists for 3 agents. Cariagannis et al. [3] consider this generalization for indivisible goods and chores. Bei et al. [8] continue by proposing a generalization of both envy-freeness and envy-freeness up to one good to mixed goods. They give algorithms for divisible and indivisible goods.