  Peer-prediction is a (meta-)mechanism which, given any proper scoring rule,
produces a mechanism to elicit privately-held, non-verifiable information from
self-interested agents. Formally, truth-telling is a strict Nash equilibrium of
the mechanism. Unfortunately, there may be other equilibria as well (including
uninformative equilibria where all players simply report the same fixed signal,
regardless of their true signal) and, typically, the truth-telling equilibrium
does not have the highest expected payoff. The main result of this paper is to
show that, in the symmetric binary setting, by tweaking peer-prediction, in
part by carefully selecting the proper scoring rule it is based on, we can make
the truth-telling equilibrium focal---that is, truth-telling has higher
expected payoff than any other equilibrium.
  Along the way, we prove the following: in the setting where agents receive
binary signals we 1) classify all equilibria of the peer-prediction mechanism;
2) introduce a new technical tool for understanding scoring rules, which allows
us to make truth-telling pay better than any other informative equilibrium; 3)
leverage this tool to provide an optimal version of the previous result; that
is, we optimize the gap between the expected payoff of truth-telling and other
informative equilibria; and 4) show that with a slight modification to the peer
prediction framework, we can, in general, make the truth-telling equilibrium
focal---that is, truth-telling pays more than any other equilibrium (including
the uninformative equilibria).
