Dynamic game of incomplete information:

Now  we  introduce yet  another  concept  of equilibrium viz. perfect Bayesian Nash  equilibrium. Perfect Bayesian Nash equilibrium was invented  in order to refine  (i.e., strengthen  the requirements of) Bayesian Nash  equilibrium  in  the same way  that  sub-game-perfect Nash equilibrium refines Nash  equilibrium. A Perfect Bayesian Nash equilibrium must satisfy two conditions viz.,  first the equilibrium must  be  a Bayesian Nash  equilibrium and secondly,  it must  also constitute a Bayesian Nash  equilibrium  in every continuation  game. Later we formally define a perfect Bayesian Nash equilibrium.  

Before we give the formal definition of perfect Bayesian Nash equilibrium we must  first define four  requirements needed to define a perfect Bayesian Nash equilibrium. The four requirements are stated below:

Requirement 1: At each information set, the player with  the move must have a belief about which node in the information set has been  reached  by  the play of  the game.  For a  nonsingleton  information  set,  a belief  is  a  probability distribution over the nodes  in  the  information set; for a singleton information set, the player's belief puts probability one on the single decision node.

Requirement  2:  Given their  beliefs,  the  players' strategies  must be sequentially rational. That  is,  at each  information  set the action taken by  the player with  the move (and the players' subsequent strategy) must be optimal given  the  player's belief  at  that  information set  and  the  other  players' subsequent strategies (where a  "subsequent strategy"  is  a complete  plan  of action covering every contingency that might arise after the given information set has been reached).

Definition:  For  a.  given  equilibrium  in  a  given  extensive  form  game,  an information  set  is on the equilibrium  path  if  it will be reached  with  positive probability and  even  when  the  game  played  according  to  the  equilibrium strategies.  It  is off the equilibriunt  path if  it is certain not to be reached if the game  is  played  according to the  equilibrium  strategies (where "equilibrium" can  mean  Nash,  sub-game-perfect, Bayesian,  or  perfect Bayesian equilibrium).

Requirement  3:  At  information  sets  on  the  equilibrium  path, beliefs  are determined by Bayes'  rule and the players' equilibrium strategies.

Requirement  4;  At  information  sets off  the  equilibrium path,  beliefs  are determined  by  the  Bayes'  rule  and  players' equilibrium strategies  where possible.

Definition: A  perfect Bayesian Nash  equilibrium  consists of  strategies and beliefs satisfying Requirements 1 through 4.