Outcome-equivalence of self-confirming equilibrium and Nash equilibrium

We introduce a condition, Nash-equivalent self-confirming equilibrium. If beliefs are assumed to be independent and unitary, Nash-equivalent self-confirming equilibrium and Nash equilibrium are outcome-equivalent. We show that the set of Nash-equivalent self-confirming equilibria and the set of self-confirming equilibria which are outcome-equivalent to Nash equilibria coincide. Our condition identifies the collection of information sets and requires the existence of beliefs shared by (certain sets of) players regarding these information sets. If the information sets are off the equilibrium path, the beliefs regarding them do not have to be correct. Our condition is weaker than that of strongly consistent self-confirming equilibrium by Kamada (2010).