Expected payoffs in a variant of BoS with imperfect information:-
Construct tables like the one in Figure 1 for type n1 of player 1, and for types y2 and n2 of player 2. I claim that ((B, B), (B, S)) and ((S, B), (S, S)) are Nash equilibria of the game, where in each case the first component gives the actions of the two types of player 1
And the second component gives the actions of the two types of player 2. Using Figure 1 you may verify that B is a best response of type y1 of player 1 to the pair (B, S) of actions of player 2, and S is a best response to the pair of actions (S, S).
You may use your answer to verify that in each of the claimed Nash equilibria the action of type n1 of player 1 and the action of each type of player 2 is a best response to the other players' actions. In each of these examples a Nash equilibrium is a list of actions, one for each type of each player, such that the action of each type of each player is a best response to the actions of all the types of the other player, given the player's beliefs about the state after she observes her signal.
The actions planned by the various types of player i are not relevant to the decision problem of any type of player i, but there is no harm in taking them, as well as the actions of the types of the other player, as given when player i is choosing an action. Thus we may define a Nash equilibrium in each example to be a Nash equilibrium of the strategic game in which the set of players is the set of all types of all players in the original situation.
In the next section I define the general notion of a Bayesian game, and the notion of Nash equilibrium in such a game. These definitions require significant theoretical development. If you find the theory in the next section heavy-going, you may be able to skim the section and then study the subsequent illustrations, relying on the intuition developed in the examples in this section, and returning to the theory only as necessary for clarification.