  Grim-trigger strategies are a fundamental mechanism for sustaining equilibria
in iterated games: the players cooperate along an agreed path, and as soon as
one player deviates, the others form a coalition to play him down to his minmax
level. A precondition to triggering such a strategy is that the identity of the
deviating player becomes common knowledge among the other players. This can be
difficult or impossible to attain in games where the information structure
allows only imperfect monitoring of the played actions or of the global state.
  We study the problem of synthesising finite-state strategies for detecting
the deviator from an agreed strategy profile in games played on finite graphs
with different information structures. We show that the problem is undecidable
in the general case where the global state cannot be monitored. On the other
hand, we prove that under perfect monitoring of the global state and imperfect
monitoring of actions, the problem becomes decidable, and we present an
effective synthesis procedure that covers infinitely repeated games with
private monitoring.
