Loyalty in Cardinal Hedonic Games

Abstract

A common theme of decision making in multi-agent systems is to assign utilities to alternatives, which individuals seek to maximize. This rationale is questionable in coalition formation where agents are affected by other members of their coalition. Based on the assumption that agents are benevolent towards other agents they like to form coalitions with, we propose loyalty in hedonic games, a binary relation dependent on agents' utilities. Given a hedonic game, we define a loyal variant where agents' utilities are defined by taking the minimum of their utility and the utilities of agents towards which they are loyal. This process can be iterated to obtain various degrees of loyalty, terminating in a locally egalitarian variant of the original game. We investigate axioms of group stability and efficiency for different degrees of loyalty. Specifically, we consider the problem of finding coalition structures in the core and of computing best coalitions, obtaining both positive and intractability results. In particular, the limit game possesses Pareto optimal coalition structures in the core.

Altruistic Hedonic Games

“…Recently, Bullinger and Kober (2021) introduced loyalty in cardinal hedonic games. In their model, each player has a loyalty set that contains all players for which she has positive utility when being together with this player in a coalition of size two.…”

“…More specifically, the utility of a player for a coalition structure is derived by taking the minimum of her own original utility and the original utilities of all players in her loyalty set who are also in her coalition. Bullinger and Kober (2021) also define a k-fold loyal variant where the loyal variant is applied k times. Note that our minbased altruistic hedonic games with equal-treatment preferences (min-based EQ AHGs) are equivalent to their loyal variant of symmetric friend-oriented hedonic games.…”

The Three-Dimensional Stable Roommates Problem with Additively Separable Preferences

“…Given an instance (N, V ) of 3D-SR-SAS-BIN and matching M , let the utilitarian welfare [4,11] of a set S ⊆ N , denoted u S (M ), be αi∈S u αi (M ). Let u(M ) be short for u N (M ).…”

“…We also consider the notion of utility based on agents' valuations of their partners in a given matching. This leads us to the notion of utilitarian welfare [4,11] which is the sum of the utilities of all agents in a given matching. We consider the problem of finding a stable matching with maximum utilitarian welfare given an instance in which valuations are binary and symmetric.…”

Hedonic Games and Treewidth Revisited

“…The most well-studied case of coalition formation are hedonic games, which have the distinguishing characteristic that each agent's utility only depends on the coalition on which she is placed (and not on the coalitions of other players). Hedonic games have recently been an object of intense study also from the computer science perspective [1,2,6,7,9,10,12,19,27,34,41], due in part to their numerous applications in, among others, social network analysis [35], scheduling group activities [15], and allocating tasks to wireless agents [40]. For more information we refer the reader to [13] and the relevant chapters of standard computational social choice texbooks [4].…”