Super Altruistic Hedonic Games Jacob Schlueter and Judy Goldsmith University of Kentucky Lexington, Kentucky [email protected], [email protected] Abstract have its packets relayed. There are many other applications in which agents care not only about immediate connections, Hedonic games are coalition formation games in which but also those farther away. We introduce a family of hedonic agents’ utility depends only on their own coalition. The intro- games that model such broad evaluations of coalitions: the duction of Altruistic Hedonic Games increased the expressive potential of Hedonic Games by considering the utility of each Super Altruistic Hedonic Games. of the agent’s friends within the coalition. We introduce Su- per Altruistic Hedonic Games (SAHGs), in which an agent’s Related Work utility may depend on the utility of all other agents in the coalition, weighted according to their distance in the friend- SAHGs are a natural extension of Altruistic Hedonic Games ship graph. We establish the framework for this new model (AHGs) wherein agents consider the preferences of other and investigate the complexity of multiple notions of stabil- agents (Nguyen et al. 2016). In AHGs, agents only consider ity. We show that SAHGs generalize Friend-oriented Hedo- the preferences of their friends. In SAHGS, agents consider nic Games, Enemy-oriented Hedonic Games, and selfish-first the preferences of all agents in their coalition. In AHGs, Altruistic Hedonic Games, inheriting the hardness results of friends are assigned fixed weights. In SAHGs, the weights these games as minimum upper complexity bounds. We also assigned to friends and enemies are not fixed, and the prefer- give SAHGs that have neither Nash stable nor strictly core ences of all agents in a coalition are considered, often taking stable partitions. advantage of indirect relationships such as friends of friends to adjust weights. (Note that friendship is not transitive: a Introduction friend of a friend could be our enemy.) Social Distance Games (SDGs) are a class of coalition Consider the process of choosing where to live. Much has formation games wherein an agent’s utility is a measure of been written (in the RecSys literature, preferences, etc.) their closeness, or social distance, from the other members about how to choose the right house or apartment, even the of their coalition (Branzeiˆ and Larson 2011). SDGs have cer- right roommates for a stable configuration. Let us consider tain similarities to SAHGs, but we believe that SAHGs can the choice of neighbors, perhaps in a setting where students better model realistic human interactions by combining the are choosing their dormitories/hostels. We can see the par- notion of social distance with the consideration of others’ titioning of students into living units (floors, buildings, etc.) preferences proposed in AHGs. as an hedonic game. It is clear that we value our friends’ happiness with the living situation, as we will hear about it As we demonstrate later, SAHGs generalize Friends and from them; our enemies’ happiness could be assumed to also Enemies-oriented Hedonic Games (FHGs and EHGs) (Dim- affect how they treat us. (If we stopped there, we would be itrov et al. 2006). In the former, agents seek to find coalitions modeling evaluation as a Altruistic Hedonic Game.) More that maximize the number of friends with a secondary goal generally we can also argue that our friends’ friends’ hap- of minimizing the number of enemies. In the latter, mini- piness will affect our friends’, and thus indirectly, our own, mizing the number of enemies is the primary goal, while and that this continues out friendship chains, with decreasing maximizing the number of friends becomes secondary. Re- (or at least, non-increasing) effect as we increase the social cent work has investigated the impact that neutral agents distance from ourselves. have on these games, defining a neutral agent as one that If we were building intranets, a node could evaluate the is neither friend nor enemy (Ohta et al. 2017). It was shown quality of the local network in terms of the bandwidth to that permitting neutral agents in EHGs allows for games that reachable nodes. However, it would also need to take into have no core stable partition (Ohta et al. 2017). Core stable account the quality of more distant connections, if it hopes to partitions are still guaranteed to exist in FHGs with neutral agents; however, strict-core stable partitions are not (Ohta Copyright c 2020, Association for the Advancement of Artificial et al. 2017). The proofs of these findings cannot be read- Intelligence (www.aaai.org). All rights reserved. ily translated to SAHGs, because SAHGs do not allow neu- tral agents. Neutral agents could be modeled as graph-based Enemy-oriented Hedonic Games (EHGs) are based on games by labeling appropriate edges as neutral, but SAHGs the same principles as FHGs, but friends are instead as- are focused on simple graph-based models, so the addition signed a value of 1, while enemies are assigned a value of of neutral edges is beyond the scope of this paper. −n. There are graph-related hedonic games that depend on Altruistic hedonic games are another sub-category that edge-weighted graphs. For instance, B and W games are a expands on the ideas in FHGs and EHGs and is a major in- category of hedonic games in which an agent’s utility is de- spiration for the work done in this paper. fined by the agents in their coalition that they rate as the Definition 3. (Nguyen et al. 2016) An altruistic hedonic best or the worst, respectively (Cechlarov´ a´ and Romero- game (AHG) is a hedonic game in which agents derive util- Medina 2001). While these games fall into the category of ity from both their own basic preferences and those of any hedonic games, we don’t believe SAHGs can generalize B friends in the same coalition. or W games. Similarly, we do not believe that either B or Let each agent i 2 N have utility u , and let i parti- W games can generalize SAHGs. This is due to the differ- i tion other agents into friends and enemies, given by F ;E . ences between B and W games and SAHGS, such as the for- i i Three levels of altruism are considered in AHGs: selfish- mer two categories assuming each agent can assign a unique first, equal treatment, and altruistic first. The function used value to each other agent, while SAHGs restrict agents to to determine an agent’s utility depends on their altruism placing others into one of two categories. Additionally, B level and on pre-utility preference values calculated as the and W games do not consider the preferences of others as utility agents would have in a friends-oriented hedonic game SAHGs do. based on the same graph (njC \Fij−jC \Eij). Two of these functions utilize a weight parameter of M = n5 to ensure Preliminaries that one of the terms in the equation dominates the other. Below, we outline three types of cooperative games with This weight value is the smallest whole number exponent of non-transferable utility, specifically coalition formation n which guarantees this for both equations that make use of games. In each type, a game G consists of M. Definitions for each altruism level and their utility func- tions are outlined below: 1. N, a finite set of n agents, with 1. Selfish-First: agents prioritize their own preferences, but 2. preference set P = fPi : i 2 Ng, where Pi is the prefer- use the preferences of others to break ties. ence of each agent i over partitions of N into coalitions. u = M(njC \ F j − jC \ E j) Depending on the type of game, P may exhaustively list i i i each individual’s preferences or provide a succinct represen- X njC \ Faj − jC \ Eaj + tation from which preferences are derived. jC \ Fij a2C\Fi Definition 1. (Banerjee, Konishi, and Sonmez¨ 2001; Bogo- 2. Equal Treatment: all preferences are treated equally. molnaia and Jackson 2002) Hedonic games are coalition formation games with nontransferable utility wherein play- X njC \ Faj − jC \ Eaj ers are concerned only with their own coalition. This inher- ui = jC \ (Fi [ fig)j ently self-interested means of determining utility makes such a2C\(Fi[fig) games hedonic in nature. 3. Altruistic First: agents prioritize the preferences of oth- Let Ni be the set of possible coalitions containing agent ers, but use their own preferences to break ties. i 2 N. A preference ordering of Ni is derived from the pref- ui = njC \ Fij − jC \ Eij erence set Pi 2 P . A solution for a game is a partition π, which is contained in the set of all distinct partitions Γ. Each X njC \ Faj − jC \ Eaj player i 2 N ranks each partition π 2 Γ based on the coali- +M · jC \ Fij tion to which they belong. a2C\Fi We assume familiarity with the complexity classes P and Hedonic games are a broad category, so it can be use- NP, but also reference two lesser-known complexity classes, ful to define sub-categories that exhibit certain interesting or p DP and Θ . useful properties. Friend and enemy-oriented hedonic games 2 are two categories. Definition 4. (Papadimitriou and Yannakakis 1982) The complexity class DP contains languages defined as the dif- Definition 2. A Friend-oriented Hedonic Game (FHG) ference between two languages in NP. (Dimitrov et al. 2006) is characterized by agents assigned For example, let C be an NP-complete language, and let values to each other based on whether they view each other ∗ L = fhc1; c2i : c1 2 C ^ c2 2= Cg.