In this paper, we consider the problem of fair division of indivisible goods, where the allocation of goods impacts society. Specifically, we introduce a second valuation function for each agent, which determines the social impact of allocating a good to the agent. Such impact is considered desirable for the society -- the higher, the better. Our goal is to understand how to allocate goods fairly from the agents' perspective while maintaining society as happy as possible. To this end, we measure the impact on society using the utilitarian social welfare, and provide both possibility and impossibility results. Our findings reveal that achieving good approximations, better than linear in the number of agents, is not possible while ensuring fairness to the agents. These impossibility results can be attributed to the fact that agents are completely unconscious of their social impact. Consequently, we explore scenarios where agents are socially aware, by introducing related fairness notions, and demonstrate that an appropriate definition of fairness is compatible with the social objective.

In this work, we consider the design of Non-Obviously Manipulable (NOM) mechanisms, mechanisms that bounded rational agents may fail to recognize as manipulable, for two relevant classes of succinctly representable Hedonic Games: Additively Separable and Fractional Hedonic Games. In these classes, agents have cardinal scores towards other agents, and their preferences over coalitions are determined by aggregating such scores. This aggregation results in a utility function for each agent, which enables the evaluation of outcomes via the utilitarian social welfare. We first prove that, when scores can be arbitrary, every optimal mechanism is NOM; moreover, when scores are limited in a continuous interval, an optimal mechanism that is NOM exists. Given the hardness of computing optimal outcomes in these settings, we turn our attention to efficient and NOM mechanisms. To this aim, we first prove a characterization of NOM mechanisms that simplifies the class of mechanisms of interest. Then, we design a NOM mechanism returning approximations that asymptotically match the best-known approximation achievable in polynomial time. Finally, we focus on discrete scores, where the compatibility of NOM with optimality depends on the specific values. Therefore, we initiate a systematic analysis to identify which discrete values support this compatibility and which do not.

A decade ago, Gerhard Woeginger posed an open problem that became well-known as “Woeginger’s Hiking Problem”: Consider a group of n people that want to go hiking; everyone expresses preferences over the size of their hiking group in the form of an interval between 1 and n. Is it possible to efficiently assign the n people to a set of hiking subgroups so that every person approves the size of their assigned subgroup? The problem is also known as efficiently deciding if an instance of an anonymous Hedonic Game with interval approval preferences admits a wonderful partition. We resolve the open problem in the affirmative by presenting an O(n5) time algorithm for Woeginger’s Hiking Problem. Our solution is based on employing a dynamic programming approach for a specific rectangle stabbing problem from computational geometry. Moreover, we propose natural, more demanding extensions of the problem, e.g., maximizing the number of satisfied participants and variants with single-peaked preferences, and show that they are also efficiently solvable. Last but not least, we employ our solution to efficiently compute a partition that maximizes the egalitarian welfare for anonymous single-peaked Hedonic Games.

Fair division of indivisible goods is a central challenge in artificial intelligence. For many prominent fairness criteria including envy-freeness (EF) or proportionality (PROP), no allocations satisfying these criteria might exist. Two popular remedies to this problem are randomization or relaxation of fairness concepts. A timely research direction is to combine the advantages of both, commonly referred to as Best of Both Worlds (BoBW). We consider fair division with entitlements, which allows to adjust notions of fairness to heterogeneous priorities among agents. This is an important generalization to standard fair division models and is not well-understood in terms of BoBW results. Our main result is a lottery for additive valuations and different entitlements that is ex-ante weighted envy-free (WEF), as well as ex-post weighted proportional up to one good (WPROP1) and weighted transfer envy-free up to one good (WEF(1, 1)). We show that this result is tight – ex-ante WEF is incompatible with any stronger ex-post WEF relaxation. In addition, we extend BoBW results on group fairness to entitlements and explore generalizations of our results to instances with more expressive valuation functions.

We study PAC learnability and PAC stabilizability of Hedonic Games (HGs), i.e., efficiently inferring preferences or core-stable partitions from samples. We first expand the known learnability/stabilizability landscape for some of the most prominent HGs classes, providing results for Friends and Enemies Games, Bottom Responsive, and Anonymous HGs. Then, having a broader view in mind, we attempt to shed light on the structural properties leading to learnability/stabilizability, or lack thereof, for specific HGs classes. Along this path, we focus on the fully expressive Hedonic Coalition Nets representation of HGs. We identify two sets of conditions that lead to efficient learnability, and which encompass all of the known positive learnability results. On the side of stability, we reveal that, while the freedom of choosing an ad hoc adversarial distribution is the most obvious hurdle to achieving PAC stability, it is not the only one. First, we show a distribution independent necessary condition for PAC stability. Then, we focus on W-games, where players have individual preferences over other players and evaluate coalitions based on the least preferred member. We prove that these games are PAC stabilizable under the class of bounded distributions, which assign positive probability mass to all coalitions. Finally, we discuss why such a result is not easily extendable to other HGs classes even in this promising scenario. Namely, we establish a purely computational property necessary for achieving PAC stability.

For the fundamental problem of fairly dividing a set of indivisible items among agents, envy-freeness up to any item (EFX) and maximin fairness (MMS) are arguably the most compelling fairness concepts proposed till now. Unfortunately, despite significant efforts over the past few years, whether EFX allocations always exist is still an enigmatic open problem, let alone their efficient computation. Furthermore, today we know that MMS allocations are not always guaranteed to exist. These facts weaken the usefulness of both EFX and MMS, albeit their appealing conceptual characteristics. We propose two alternative fairness concepts—called epistemic EFX (EEFX) and minimum EFX value fairness (MXS)---inspired by EFX and MMS. For both, we explore their relationships to well-studied fairness notions and, more importantly, prove that EEFX and MXS allocations always exist and can be computed efficiently for additive valuations. Our results justify that the new fairness concepts are excellent alternatives to EFX and MMS.

We consider the problem of maximizing the Nash social welfare when allocating a set G of indivisible goods to a set N of agents. We study instances, in which all agents have 2-value additive valuations: The value of every agent for every good is either p or q, where p and q are integers and p2. In terms of approximation, we present positive and negative results for general p and q. We show that our algorithm obtains an approximation ratio of at most 1.0345. Moreover, we prove that the problem is APX-hard, with a lower bound of 1.000015 achieved at p/q = 4/5.

We investigate strategyproof mechanisms in the Group Activity Selection Problem with the additively separable property. Namely, agents have distinct preferences for each activity and individual weights for the other agents. We evaluate our mechanisms in terms of their approximation ratio with respect to the maximum utilitarian social welfare. We first show that, for arbitrary non-negative preferences, no deterministic mechanism can achieve a bounded approximation ratio. Thus, we provide a randomized k-approximate mechanism, where k is the number of activities, and a corresponding 2-2/(k+1) lower bound. Furthermore, we propose a tight (2 - 1/k)-approximate randomized mechanism when activities are copyable. We then turn our attention to instances where preferences can only be unitary, that is 0 or 1. In this case, we provide a k-approximate deterministic mechanism, which we show to be the best possible one within the class of strategyproof and anonymous mechanisms. We also provide a general lower bound of Ω({\sqrt{k}) when anonymity is no longer a constraint. Finally, we focus on unitary preferences and weights, and prove that, while any mechanism returning the optimum is not strategyproof, there exists a 2-approximate deterministic mechanism.

In this paper we consider Distance Hedonic Games (DHGs), a class of non-transferable utility coalition formation games that properly generalizes previously existing models, like Social Distance Games (SDGs) and unweighted Fractional Hedonic Games (FHGs). In particular, in DHGs we assume the existence of a scoring vector \(\alpha \), in which the i-th coefficient \(\alpha _i\) expresses the extent to which an agent x contributes to the utility of an agent y if they are at distance i. We focus on Nash stable outcomes in the arising games, i.e., on coalition structures in which no agent can unilaterally improve her gain by deviating.We consider two different natural scenarios for the scoring vector, with monotonically increasing and monotonically decreasing coefficients. In both cases we give NP-hardness and inapproximability results on the problems of finding a social optimum and a best Nash stable outcome. Moreover, we characterize the topologies of coalitions that provide high social welfare and consequently give suitable bounds on the Price of Anarchy and on the Price of Stability.

We investigate strategyproof mechanisms for Friends and Enemies Games, a subclass of Hedonic Games in which every agent classifies any other one as a friend or as an enemy. In this setting, we consider the two classical scenarios proposed in the literature, called Friends Appreciation (FA) and Enemies Aversion (EA). Roughly speaking, in the former each agent gives priority to the number of friends in her coalition, while in the latter to the number of enemies.We provide strategyproof mechanisms for both settings. More precisely, for FA we first present a deterministic n-approximation mechanism, and then show that a much better result can be accomplished by resorting to randomization. Namely, we provide a randomized mechanism whose expected approximation ratio is 4, and arbitrarily close to 4 with high probability. For EA, we give a simple (1+√2)n-approximation mechanism, and show that its performance is asymptotically tight by proving that it is NP-hard to approximate the optimal solution within O(n1−ɛ) for any fixed ɛ > 0.Finally, we show how to extend our results in the presence of neutrals, i.e., when agents can also be indifferent about other agents, and we discuss anonymity.