Question is as follows:

Recall that an assessment a = (o, #) (where o is a profile of behavioral strategies and u is a profile of beliefs) constitutes a Perfect Bayesian Equilibrium (PBE) iff for every i E N: 1. Sequential Rationality: of is optimal at every information set of player i given o_, and #, and 2. Weak Consistency: #, is obtained via Bayes' rule at all information sets on the equilibrium path, i.e. at all information sets belonging to player i that is reached with positive probability given o. Consider the simplified game of poker that we have learned in class. Here is the timing of the game. First, Nature assigns player 1 a High card (t, = H) with probability 1/2 and a Low card (t, = [) with probability 1/2. After observing his/her card, player 1 chooses to Raise (R) or See (S). Player 2 does not observe player 1's card, but observes whether player 1 has played R or S. After observing this, player 2 either Calls (C) or Folds (F). The extensive form of the game is as follows: Nature L H 1 R S R S -1,1 -1,1 ha 2 C F C F -2,2 1,-1 2,-2 1,-1(a) Show that in any PBE, we must have 1 (R|H) = 1. (5 points) (b) Can there be a Pooling Equilibrium in which both types of player 1 plays R -i.e. a PBE in which 1(RH) = 01(R|Z) = 1? Fully explain your answers. (15 points) (c) Can there be a Separating Equilibrium in which the H type of player 1 plays R and the L type of player 1 plays S - i.e. a PBE in which of (R|H) = 1 and of (R|L) = 0? Fully explain your answers. (15 points) (d) Given o, Calculate player 2's beliefs #2 in information set I1 -i.e. what is #2(h3) and #2(h4)? (5 points) (e) Using your answer to (d), calculate the value of of (R|L) = x that would make it optimal for player 2 to use a mixed strategy. (5 points) (f) Calculate the value of 02 (C|/1) = y that would make it optimal for the L type of player 1 to use a mixed strategy. (5 points) (g) Based on your answers from (d) to (f), characterize the PBE in which the L type of player 1 and player 2 uses mixed strategies. (5 points)