  We initiate the study of trembling-hand perfection in sequential (i.e.,
extensive-form) games with correlation. We introduce the extensive-form perfect
correlated equilibrium (EFPCE) as a refinement of the classical extensive-form
correlated equilibrium (EFCE) that amends its weaknesses off the equilibrium
path. This is achieved by accounting for the possibility that players may make
mistakes while following recommendations independently at each information set
of the game. After providing an axiomatic definition of EFPCE, we show that one
always exists since any perfect (Nash) equilibrium constitutes an EFPCE, and
that it is a refinement of EFCE, as any EFPCE is also an EFCE. Then, we prove
that, surprisingly, computing an EFPCE is not harder than finding an EFCE,
since the problem can be solved in polynomial time for general n-player
extensive-form games (also with chance). This is achieved by formulating the
problem as that of finding a limit solution (as $\epsilon \rightarrow 0$) to a
suitably defined trembling LP parametrized by $\epsilon$, featuring
exponentially many variables and polynomially many constraints. To this end, we
show how a recently developed polynomial-time algorithm for trembling LPs can
be adapted to deal with problems having an exponential number of variables.
This calls for the solution of a sequence of (non-trembling) LPs with
exponentially many variables and polynomially many constraints, which is
possible in polynomial time by applying an ellipsoid against hope approach.
