Fractional Hedonic Games

Abstract

The work we present in this paper initiated the formal study of fractional hedonic games, coalition formation games in which the utility of a player is the average value he ascribes to the members of his coalition. Among other settings, this covers situations in which players only distinguish between friends and non-friends and desire to be in a coalition in which the fraction of friends is maximal. Fractional hedonic games thus not only constitute a natural class of succinctly representable coalition formation games, but also provide an interesting framework for network clustering. We propose a number of conditions under which the core of fractional hedonic games is non-empty and provide algorithms for computing a core stable outcome. By contrast, we show that the core may be empty in other cases, and that it is computationally hard in general to decide non-emptiness of the core. the partitions in the strict core, and give a polynomial time algorithm to compute a unique finest partition in the strict core.• We discuss how computing desirable outcomes in fractional hedonic games provides an interesting game-theoretic perspective to community detection [see, e.g., Fortunato, 2010, Newman, 2004 and network clustering. 2 RELATED WORKFractional hedonic games are related to additively separable hedonic games [see, e.g., , Olsen, 2009, Sung and Dimitrov, 2010. In both fractional hedonic games and additively separable hedonic games, each player ascribes a cardinal value to every other player. In additively separable hedonic games, utility in a coalition is derived by adding the values for the other players. By contrast, in fractional hedonic games, utility in a coalition is derived by adding the values for the other players and then dividing the sum by the total number of players in the coalition. Although conceptually, additively separable and fractional hedonic games are similar, their formal properties are quite different. As neither of the two models is obviously superior, this shows how slight modeling decisions may affect the formal analysis. For example, in unweighted and undirected graphs, the grand coalition is trivially core stable for additively separable hedonic games.On the other hand, this is not the case for fractional hedonic games. 3 A fractional hedonic game approach to social networks with only non-negative weights may help detect like-minded and densely connected communities. In comparison, when the network only has non-negative weights for the edges, any reasonable solution for the corresponding additively separable hedonic game returns the grand coalition, which is not informative. The difference between additively separable and fractional hedonic games is reminiscent of some issues in population ethics (see, e.g., Arrhenius et al., 2017), which concerns the evaluation of states of the world with different numbers of individuals alive. Two prominent principles in population ethics are total utilitarianism and average utilitarianism. The former claims that a state of the world is better than another...

Robustness against Agent Failure in Hedonic Games

“…For instance, one might want to consider sudden failure of agents' friendship relations, due to individual or political con icts. ere are also further classes subclasses of additively separable games that we have not considered in this paper, most notably fractional hedonic games [Aziz et al, 2017[Aziz et al, , 2014, which one can study from both existence and complexity aspects; in particular, it would be interesting to investigate whether a similar graph-theoretic characterization of friendship graphs that ensure the existence of stable outcomes can be obtained.…”

Stable outcomes in modified fractional hedonic games

“…Fractional hedonic games have been introduced by Aziz et al [2]. They prove that the core can be empty for games played on general graphs and that it is not empty for games played on some classes of undirected and unweighted graphs (that is, graphs with degree at most 2, multipartite complete graphs, bipartite graphs admitting a perfect matching and regular bipartite graphs).…”

“…Finally, Aziz et al [4] consider the computational complexity of computing welfare maximizing partitions (not necessarily Nash stable) for fractional hedonic games. We point out that fractional hedonic games played on unweighted undirected graphs model realistic economic scenarios referred to in [2,10] as Bakers and Millers.…”

Strategyproof Mechanisms for Additively Separable Hedonic Games and Fractional Hedonic Games

“…Additive separability satisfies a number of desirable axiomatic properties [3] and ASHGs are the non-transferable utility generalization of graph games studied by Deng and Papadimitriou [16]. We further consider fractional hedonic games (FHGs), introduced in [2], which are similar to ASHGs, with the difference that the utility of each agent is divided by the size of her cluster. This allows to model behavioral dynamics in social environments that are not captured by ASHGs: one usually prefers having a couple of good friends in a cluster composed by few other people rather than being part of a crowded cluster populated by uninteresting agents.…”

“…Finally, hedonic games have also been considered in [6,8,14,15,16] from a classical optimization point of view, i.e., where solutions are not necessarily stable. Concerning FHGs, Aziz et al [2], give some properties guaranteeing the (non-)existence of the core. Moreover, Brandl et al [13], study the computational complexity of understanding the existence of core and individual stable outcomes.…”