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Question There are two players called 1 and 2. Player 1 can be of two types te {0,1} with Pr (t=1) =x E (0,1). The actions and payoffs of the game are given by: left right 0.4 1,1 down 1,2 t,4 where the row player is player 1. We will use the following notation: 1(t): probability that player 1 plays up if she is of type t; 62: probability that player 2 plays left. Part I (2 marks) Suppose It = 0.5. Is (G1(0), 1(1), 62) = (0,1,0) a Bayes-Nash equilibrium? [Hint: To prove something is a BNE you have to check no player-type has an incentive to deviate from the proposed strategy profile. To prove something is not a BNE, you need to check just one of the three - type-0 player 1, type-1 player 1, or player 2 has an incentive to deviate] Part II (8 marks) We want to know whether and when it is possible that in a Bayes Nash equilibrium player 1 mixes between up and down whenever she is of type t = 0, i.e. G(0) E (0,1). We therefore proceed to construct such an equilibrium and then verify for which values of a this equilibrium exists. At the end of the exercise, you should complete the following "Proposition" Proposition 1. If E (..., ...), then there exists a Bayes Nash equilibrium in which player 1 mixes between up and down whenever she is of type t=0 , i.e. G1(0) E (0,1). In this equilibrium of (0) = ...; 61(1)=..; 61 =... ... in. . 1. (1 mark) If type-0 player 1 is mixing, what condition must be satisfied in this equilibrium? (Hint: if I am mixing then it means that I am ...)2. (1 mark) Using the condition derived in part 1, you should be able to find player 2 's equilibrium strategy 62. What is it? 3. (2 marks) Using your answers to parts 1 and 2, we can immediately conclude that in this equilibrium type-1 player 1 must play...? (Hint: remember to state your answer as a value for of(1)) 4. (3 marks) Now you should be able to find o(0). What is it? [Hint: the answer is a formula containing a. Notice that it is easy to mess up signs when calculating 1 (0), so be careful and double-check your math. 5. (1 mark) You now have a complete profile of strategies given by ci(0), o(1), 62. But you can notice that for some values of x it is not true that of (0) 6 (0,1). Find the values of a for which GI(0) E (0,1)