  Peer-prediction is a mechanism which elicits privately-held, non-variable
information from self-interested agents---formally, truth-telling is a strict
Bayes Nash equilibrium of the mechanism. The original Peer-prediction mechanism
suffers from two main limitations: (1) the mechanism must know the "common
prior" of agents' signals; (2) additional undesirable and non-truthful
equilibria exist which often have a greater expected payoff than the
truth-telling equilibrium. A series of results has successfully weakened the
known common prior assumption. However, the equilibrium multiplicity issue
remains a challenge.
  In this paper, we address the above two problems. In the setting where a
common prior exists but is not known to the mechanism we show (1) a general
negative result applying to a large class of mechanisms showing truth-telling
can never pay strictly more in expectation than a particular set of equilibria
where agents collude to "relabel" the signals and tell the truth after
relabeling signals; (2) provide a mechanism that has no information about the
common prior but where truth-telling pays as much in expectation as any
relabeling equilibrium and pays strictly more than any other symmetric
equilibrium; (3) moreover in our mechanism, if the number of agents is
sufficiently large, truth-telling pays similarly to any equilibrium close to a
"relabeling" equilibrium and pays strictly more than any equilibrium that is
not close to a relabeling equilibrium.
