If a game admits a unique Nash equilibirum, does common knowledge of rationality implies Nash equilibirum? In a highly controversial paper by Robert Aumann(see here), it is stated as a theorem:


  In PI games, common knowledge of rationality implies backward induction.


If we stick to the strong and controversial definition of rationality in the paper


  Rationality of a player means that he is a habitual payoff maximizer:
  that no matter where he finds himself,
  at which vertex,he will
  not knowingly continue with a strategy that yields him less than he could
  have gotten with a different strategy. 


Can we have some other implication, like, If a game admits a unique Nash equilibirum, does common knowledge of rationality implies Nash equilibirum?
No. In static games, common knowledge of rationality is equivalent to rationalizability. Bernheim, in "Rationalizable Strategic Behavior" (Econometrica, July 1984) gives an example on page 1012 in Figure 1 of a normalform game in which there is a unique Nash equilibrium, yet multiple strategies are rationalizable.