Equilibria of a variant of BoS with imperfect information:-

Show that there is no pure strategy Nash equilibrium of this game in which player 1 chooses S. If you have studied mixed strategy Nash equilibrium (Chapter 4), find the mixed strategy Nash equilibria of the game. (First check whether there is an equilibrium in which both types of player 2 use pure strategies, then look for equilibria in which one or both of these types randomize.)

We can interpret the actions of the two types of player 2 to reflect player 2's intentions in the hypothetical situation before she knows the state. We can tell the following story. Initially player 2 does not know the state; she is informed of the state by a signal that depends on the state. Before receiving this signal, she plans an action for each possible signal.

After receiving the signal she carries out her planned action for that signal. We can tell a similar story for player 1. To be consistent with her not knowing the state when she takes an action, her signal must be uninformative: it must be the same in each state. Given her signal, she is unsure of the state; when choosing an action she takes into account her belief about the likelihood of each state, given her signal. The framework of states, beliefs, and signals is unnecessarily baroque in this simple example, but comes into its own in the analysis of more complex situations.