We study a very general class of games --- multi-dimensional aggregative games --- which in particular generalize both anonymous games and weighted congestion games. For any such game that is also large (meaning that the influence that any single player's action has on the utility of others is diminishing with the number of players in the game), we solve the equilibrium selection problem in a strong sense. In particular, we give an efficient weak mediator: an algorithm or mechanism which has only the power to listen to reported types and provide non-binding suggested actions, such that (a) it is an asymptotic Nash equilibrium for every player to truthfully report their type to the mediator, and then follow its suggested action; and (b) that when players do so, they end up coordinating on a particular asymptotic pure strategy Nash equilibrium of the induced complete information game. In fact, truthful reporting is an ex-post Nash equilibrium of the mediated game, so our solution applies even in settings of incomplete information, and even when player types are arbitrary or worst-case (i.e. not drawn from a common prior). We achieve this by giving an efficient differentially private algorithm for computing a Nash equilibrium in such games. We also give similar results for a related class of one-dimensional games with weaker conditions on the aggregation function, and apply our main results to a multi-dimensional market game.

Our results can be viewed as giving, for a rich class of games, a more robust version of the Revelation Principle, in that we work with weaker informational assumptions (no common prior), yet provide a stronger solution concept (Nash versus Bayes Nash equilibrium). Previously, similar results were only known for the special case of unweighted congestion games. In the process, we derive several algorithmic results that are of independent interest.