2. Consider the following game. There are N players. Each player i has two choices: A or B. The state of the world is also either A or B, and each player gains one util from matching her action to the state, no utils from mismatching. All players share a common prior belief about the probability of A, denoted π = P(A) € (0,1). Each player i observes a private signal si, conditionally IID across players. The signal is either a or b, with probability P(s; = a/A) = P(s; = b|B) = q € (1, 1). Players take turns making their choices, and observe the choices made by their pre- decessors. The order of moves coincides with the natural ordering of players' names, thus, player 1 chooses first, player 2 second, etc. (a) Find player i's posterior beliefs and optimal choice given her signal predecessors' actions for i = 1, 2, 3. (b) Assume that, when a player is indifferent between two choices, she flips a fair coin to decide. Prove that player 3 relies on her information to make her own decision with probability q(1-q), and that, with probability 1-q(1-q), every subsequent player starting from player 3 will disregard her own signal when making a decision.