Chore division on a graph

The paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent's share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies

Social Choice and Just Institutions: New Perspectives

It has become accepted that social choice is impossible in absence of interpersonal comparisons of well-being. This view is challenged here. Arrow obtained an impossibility theorem only by making unreasonable demands on social choice functions. With reasonable requirements, one can get very attractive possibilities and derive social preferences on the basis of non-comparable individual preferences. This new approach makes it possible to design optimal second-best institutions inspired by principles of fairness, while traditionally the analysis of optimal second-best institutions was thought to require interpersonal comparisons of well-being. In particular, this new approach turns out to be especially suitable for the application of recent philosophical theories of justice formulated in terms of fairness, such as equality of resources.social welfare, social choice, fairness, egalitarian-equivalence

Randomized and Deterministic Maximin-share Approximations for Fractionally Subadditive Valuations

We consider the problem of guaranteeing maximin-share (MMS) when allocating a set of indivisible items to a set of agents with fractionally subadditive (XOS) valuations. For XOS valuations, it has been previously shown that for some instances no allocation can guarantee a fraction better than $1/2$ of maximin-share to all the agents. Also, a deterministic allocation exists that guarantees $0.219225$ of the maximin-share of each agent. Our results involve both deterministic and randomized allocations. On the deterministic side, we improve the best approximation guarantee for fractionally subadditive valuations to $3/13 = 0.230769$. We develop new ideas on allocating large items in our allocation algorithm which might be of independent interest. Furthermore, we investigate randomized algorithms and the Best-of-both-worlds fairness guarantees. We propose a randomized allocation that is $1/4$-MMS ex-ante and $1/8$-MMS ex-post for XOS valuations. Moreover, we prove an upper bound of $3/4$ on the ex-ante guarantee for this class of valuations

Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results

Fair resource allocation is an important problem in many real-world scenarios, where resources such as goods and chores must be allocated among agents. In this survey, we delve into the intricacies of fair allocation, focusing specifically on the challenges associated with indivisible resources. We define fairness and efficiency within this context and thoroughly survey existential results, algorithms, and approximations that satisfy various fairness criteria, including envyfreeness, proportionality, MMS, and their relaxations. Additionally, we discuss algorithms that achieve fairness and efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study the computational complexity of these algorithms, the likelihood of finding fair allocations, and the price of fairness for each fairness notion. We also cover mixed instances of indivisible and divisible items and investigate different valuation and allocation settings. By summarizing the state-of-the-art research, this survey provides valuable insights into fair resource allocation of indivisible goods and chores, highlighting computational complexities, fairness guarantees, and trade-offs between fairness and efficiency. It serves as a foundation for future advancements in this vital field

Intergenerational Justice in the Hobbesian State of Nature

We analyse the issue of justice in the allocation of resources across generations. Our starting point is that if all generations have a claim to natural resources, then each generation should be entitled to exercise veto power on the unpalatable choices of the other generations. We analyse this situation as one of bargaining à la Rubinstein, Safra and Thomson [15], which incorporates a notion of justice as mutual advantage, rather than justice as impartiality, as in the Kantian-Rawlsian tradition. Our framework captures some key aspects of the interaction between isolated agents in a Hobbesian state of nature, in which agents are not placed behind a veil of ignorance, but none of them is sufficiently strong to impose their will against all others (state of war of all against all). We analyse some new social welfare relations emerging from this Hobbesian framework. JEL Categories: D63, Q01Intergenerational justice; bargaining; Hobbes; social choice.

Rethinking Resource Allocation: Fairness and Computability

In the field of multiagent systems, one important problem is fairly allocating items among a set of agents. The gold standard fairness property is envy-freeness whereby each agent prefers the bundle allocated to them over any other bundle. For indivisible goods, envy-freeness cannot be guaranteed: consider two agents and one item. In the context of indivisible goods, one key fairness notion which has gained significant attention in recent years is the maximin share guarantee (MMS). MMS extends the cut-and-choose protocol to multiple agents by guaranteeing each agent as much value as if they divided items into bundles but chose last. However, MMS is not guaranteed to exist, and even it exists, computing an MMS allocation is computationally hard. Consequently, several approximation techniques were proposed to ensure all agents receive a fraction of their MMS. We propose an orthogonal approximation which aims to guarantee MMS for a fraction of the agents. We construct instances where any optimal approximation algorithm fails to guarantee MMS for most agents. We show how to interpolate between the fraction of agents and the fraction of MMS guaranteed to agents. We prove the existence of allocations which satisfy 2/3 of the agents. Our algorithmic technique immediately implies a polynomial-time algorithm for any number of items when there are less than nine agents. Our results significantly reduce the existence gap of MMS when agents divide the items into 3n/2 bundles. We empirically demonstrate that our algorithm outperforms its worst-case bounds in practice on both synthetic and real-world data. We extend our discussion of maximin share approximations to chores. We extend many of the techniques used for MMS approximations to the chores setting. Using these new techniques, we prove the existence of allocations which satisfy all agents when they expect to divide the workload among 2n/3 agents