  We study the problem of implementing equilibria of complete information games
in settings of incomplete information, and address this problem using
"recommender mechanisms." A recommender mechanism is one that does not have the
power to enforce outcomes or to force participation, rather it only has the
power to suggestion outcomes on the basis of voluntary participation. We show
that despite these restrictions, recommender mechanisms can implement
equilibria of complete information games in settings of incomplete information
under the condition that the game is large---i.e. that there are a large number
of players, and any player's action affects any other's payoff by at most a
small amount.
  Our result follows from a novel application of differential privacy. We show
that any algorithm that computes a correlated equilibrium of a complete
information game while satisfying a variant of differential privacy---which we
call joint differential privacy---can be used as a recommender mechanism while
satisfying our desired incentive properties. Our main technical result is an
algorithm for computing a correlated equilibrium of a large game while
satisfying joint differential privacy.
  Although our recommender mechanisms are designed to satisfy game-theoretic
properties, our solution ends up satisfying a strong privacy property as well.
No group of players can learn "much" about the type of any player outside the
group from the recommendations of the mechanism, even if these players collude
in an arbitrary way. As such, our algorithm is able to implement equilibria of
complete information games, without revealing information about the realized
types.
