Abstract
In hedonic games, coalitions are created as a result of the strategic interaction of independent players. In particular, in additively separable hedonic games, every player has valuations for all other ones, and the utility for belonging to a coalition is given by the sum of the valuations for all other players belonging to it. So far, non-cooperative hedonic games have been considered in the literature only with respect to totally selfish players. Starting from the fundamental class of additively separable hedonic games, we define and study a new model in which, given a social graph, players also care about the happiness of their friends: we call this class of games social context additively separable hedonic games (SCASHGs). We focus on the fundamental stability notion of Nash equilibrium, and study the existence, convergence and performance of stable outcomes (with respect to the classical notions of price of anarchy and price of stability) in SCASHGs. In particular, we show that SCASHGs are potential games, and therefore Nash equilibria always exist and can be reached after a sequence of Nash moves of the players. Finally, we provide tight or asymptotically tight bounds on the price of anarchy and the price of stability of SCASHGs.
Highlights
We provide an exact potential function for social context additively separable hedonic games (SCASHGs), proving that these games always possess a pure Nash equilibrium and that the convergence to Nash equilibria is guaranteed
In order to evaluate the performance of SCASHGs, we consider two social welfare functions
The first social function, SW, is given by the summation, for each player, of the values she assigns to the members of her coalition, while the second social function, denoted by SW, is the summation of the players’ utilities. We evaluate, for both of them, the performance of the Nash outcomes by means of the notions of price of anarchy and price of stability (PoA and PoA denote the price of anarchy with respect to SW and SW, respectively; analogously PoS and PoS denote the price of stability with respect to SW and SW, respectively)
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Hedonic games constitute a framework for formally studying the stability and the evolution of the process of forming player coalitions Given that they model natural behavioral dynamics of real-life situations, this class of games has received great interest in the literature: in economic, social and political environments, individuals perform activities in groups rather than by themselves. Separable hedonic games constitute a natural and succinctly representable class of hedonic games In these games, each player has a value for any other player, and the utility of a coalition to a particular player is the sum of the values she assigns to the members of her coalition. Games 2021, 12, 71 is defined as the ratio between the social optimum value and the social value of the best stable outcome
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