  Studying games in the complete information model makes them analytically
tractable. However, large $n$ player interactions are more realistically
modeled as games of incomplete information, where players may know little to
nothing about the types of other players. Unfortunately, games in incomplete
information settings lose many of the nice properties of complete information
games: the quality of equilibria can become worse, the equilibria lose their
ex-post properties, and coordinating on an equilibrium becomes even more
difficult. Because of these problems, we would like to study games of
incomplete information, but still implement equilibria of the complete
information game induced by the (unknown) realized player types.
  This problem was recently studied by Kearns et al. and solved in large games
by means of introducing a weak mediator: their mediator took as input reported
types of players, and output suggested actions which formed a correlated
equilibrium of the underlying game. Players had the option to play
independently of the mediator, or ignore its suggestions, but crucially, if
they decided to opt-in to the mediator, they did not have the power to lie
about their type. In this paper, we rectify this deficiency in the setting of
large congestion games. We give, in a sense, the weakest possible mediator: it
cannot enforce participation, verify types, or enforce its suggestions.
Moreover, our mediator implements a Nash equilibrium of the complete
information game. We show that it is an (asymptotic) ex-post equilibrium of the
incomplete information game for all players to use the mediator honestly, and
that when they do so, they end up playing an approximate Nash equilibrium of
the induced complete information game. In particular, truthful use of the
mediator is a Bayes-Nash equilibrium in any Bayesian game for any prior.
