We study a class of non-cooperative aggregative games—referred to as social purpose games—in which the payoffs depend separately on a player’s own strategy (individual benefits) and on a function of the strategy profile which is common to all players (social benefits) weighted by an individual benefit parameter. This structure allows for an asymmetric assessment of a common social benefit across players. We show that these games have a weighted potential, and we investigate its properties. We investigate the payoff structure and the uniqueness of Nash equilibria and social optima. Furthermore, following the literature on partial cooperation, we investigate the leadership of a single coalition of cooperators, while the rest of players act as non-cooperative followers. In particular, we show that social purpose games admit the emergence of a stable coalition of cooperators for the subclass of strict social purpose games. As a particular application, we study a standard formulation of the tragedy of the commons. We show that there emerges a single stable coalition of cooperators that curbs the over-exploitation of the common resource.