Variant of BoS with imperfect information:-

Consider another variant of the situation modeled by BoS, in which neither player knows whether the other wants to go out with her. Specifically, suppose that player 1 thinks that with probability ½ player 2 wants to go out with her, and with probability ½ player 2 wants to avoid her, and player 2 thinks that with probability 2/3 player 1 wants to go out with her and with probability 1/3 player 1 wants to avoid her. As before, assume that each player knows her own preferences.

We can model this situation by introducing four states, one for each of the possible configurations of preferences. I refer to these states as yy (each player wants to go out with the other), yn (player 1 wants to go out with player 2, but player 2 wants to avoid player 1), ny, and nn. The fact that player 1 does not know player 2's preferences means that she cannot distinguish between states yy and yn, or between states ny and nn. Similarly, player 2 cannot distinguish between states yy and ny, and between states yn and nn. We can model the players' information by assuming that each player receives a signal before choosing an action.

Player 1 receives the same signal, say y₁, in states yy and yn, and a different signal, say n₁, in states ny and nn; player 2 receives the same signal, say y₂, in states yy and ny, and a different signal, say n₂, in states yn and nn. After player 1 receives the signal y₁, she is referred to as type y₁ of player 1 (who wishes to go out with player 2); after she receives the signal n₁ she is referred to as type n₁ of player 1 (who wishes to avoid player 2). Similarly, player 2 has two types, y₂ and n₂.