In this paper, we examine in which each player's preferences over partitions of players depend only on the members of his coalition. We present three main results in which restrictions on the preferences of the players guarantee the existence of stable partitions for various notions of stability. The preference restrictions pertain to and which model optimistic and pessimistic behavior of players respectively. The existence results apply to natural subclasses of and . It is also shown that our existence results cannot be strengthened to the case of stronger known stability concepts.
proposed a natural preference restriction called top responsiveness which is based on the idea that players value other players on how they could complement them in research teams. They showed that there exists an algorithm called the which finds a core stable partition for top responsive hedonic games. The Top Covering Algorithm can be seen as a generalization of @cite_0 . simplified the Top Covering Algorithm and proved that top responsiveness implies non-emptiness of the strict core and if mutuality is additionally satisfied, then a Nash stable partition exists.
@cite_0
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We consider the partitioning of a society into coalitions in purely hedonic settings, i.e., where each player's payoff is completely determined by the identity of other members of her coalition. We first discuss how hedonic and nonhedonic settings differ and some sufficient conditions for the existence of core stable coalition partitions in hedonic settings. We then focus on a weaker stability condition: individual stability, where no player can benefit from moving to another coalition while not hurting the members of that new coalition. We show that if coalitions can be ordered according to some characteristic over which players have single-peaked preferences, or where players have symmetric and additively separable preferences, then there exists an individually stable coalition partition. Examples show that without these conditions, individually stable coalition partitions may not exist. We also discuss some other stability concepts, and the incompatibility of stability with other normative properties. Journal of Economic Literature Classification Numbers: C71, A14, D20. Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i’s share is precisely equal to i’s expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), and give a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ±e) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ±e)-approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic. We resolve the worst-case price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest non-trivial special case of affine cost functions. We make two distinct contributions. On the upper-bound side, we define the framework of "local smoothness", which refines the standard smoothness framework for games with convex strategy sets. While standard smoothness arguments cannot establish tight bounds on the POA in atomic splittable congestion games, we prove that local smoothness arguments can. Further, we prove that every POA bound derived via local smoothness applies automatically to every correlated equilibrium of the game. Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to the coarse correlated equilibria of the game. Our second contribution is a very general lower bound: for every set L that satisfies mild technical conditions, the worst-case POA of pure Nash equilibria in atomic splittable congestion games with cost functions in L is exactly the smallest upper bound provable using local smoothness arguments. In particular, the worst-case POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games. To help their users to discover important items at a particular time, major websites like Twitter, Yelp, TripAdvisor or NYTimes provide Top-K recommendations (e.g., 10 Trending Topics, Top 5 Hotels in Paris or 10 Most Viewed News Stories), which rely on crowdsourced popularity signals to select the items. However, different sections of a crowd may have different preferences, and there is a large silent majority who do not explicitly express their opinion. Also, the crowd often consists of actors like bots, spammers, or people running orchestrated campaigns. Recommendation algorithms today largely do not consider such nuances, hence are vulnerable to strategic manipulation by small but hyper-active user groups. To fairly aggregate the preferences of all users while recommending top-K items, we borrow ideas from prior research on social choice theory, and identify a voting mechanism called Single Transferable Vote (STV) as having many of the fairness properties we desire in top-K item (s)elections. We develop an innovative mechanism to attribute preferences of silent majority which also make STV completely operational. We show the generalizability of our approach by implementing it on two different real-world datasets. Through extensive experimentation and comparison with state-of-the-art techniques, we show that our proposed approach provides maximum user satisfaction, and cuts down drastically on items disliked by most but hyper-actively promoted by a few users.