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Question 3. [30 points] (Bayesian game) Consider the Bayesian game represented by the following table.

There are three different states: i) Ygritte hates Jon with prior probability 1/4 ii) Ygritte loves Jon with prior probability 1/2 and iii) Ygritte is indifferent for Jon with probability 1/4 . Suppose Ygritte knows her attitude perfectly. However, Jon is only fully informed if Ygritte hates him but cannot tell whether Ygritte loves him or is indifferent.

\fQuestion 3. [30 points] (Bayesian game) Consider the Bayesian game represented by the following table. Ygritte Hate (Prob=1) Love (Prob=) Indifferent (Prob=) CB BW CB BW CB BW CB 3.-1 2.3 CB 4.2 1.-1 CB 1.0 1.0 Jon BW 0.2 1.-1 BW 0.-2 3.3 BW 3.0 There are three different states: i) Ygritte hates Jon with prior probability 1/4 ii) Ygritte loves Jon with prior probability 1/2 and iii) Ygritte is indifferent for Jon with probability 1/4 . Suppose Ygritte knows her attitude perfectly. However, Jon is only fully informed if Ygritte hates him but cannot tell whether Ygritte loves him or is indifferent. 1. [4 points] How many different signals could Jon receive and how many different signals could Ygritte receive? Please write down both players' signal set and signal function. 2. [3 points] Suppose Jon receives a signal that tells him either state Love or state Indifferent will be played. Using Bayes rule calculate Jon's belief about the probability of state Love after he receives such a signal. 3. [6 points] What are Jon and Ygritte's pure strategy sets respectively? (Hint: a player's strategy in Bayesian game is a plan of actions for each signal/type.) 4. [10 points] For each possible pure strategy taken by Ygritte, please calculate Jon's expected payoffs of each of his pure strategy when he receives a signal that tells him either state Love or state Indifferent will be played 2 5. [7 points] Find the set of all pure strategy Nash Equilibria of this game