The Price of Stability of Simple Symmetric Fractional Hedonic Games

We consider simple symmetric fractional hedonic games, in which a group of utility maximizing players have hedonic preferences over the players' set, and wish to be partitioned into clusters so that they are grouped together with players they prefer. Each player either wishes to be in the same cluster with another player (and, hence, values this agent at 1) or is indifferent (and values this player at 0). Given a cluster, the utility of each player is defined as the number of players inside the cluster that are valued at 1 divided by the cluster size, and a player will deviate to another cluster if this leads to higher utility. We are interested in Nash equilibria of such games, where no player has an incentive to unilaterally deviate to another cluster, and we focus on the notion of the price of stability. We present new and improved bounds on the price of stability both for the normal utility function and for a slightly modified one.