Question 1 There are two players called 1 and 2. Player I can be of two types t E {0,1} with Pr (t=l) = n: E (0,1). [Here, it is a symbol] The actions and payoffs of the game are given by: where the row player is player 1. We will use the following notation: o 51(t): probability that player 1 plays up if she is of type t; o 62! probability that player 2 plays le. Part I {2 marks) Suppose :1: = 0.5. Is (61(0), 01(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 prole. To prove something is not a BNE, you need to check just one of the three type-D player 1, type-l 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 = D, i.e. 61(0) E (0,1). We therefore proceed to construct such an equilibrium and then verify for which values of II: this equilibrium exists. At the end of the exercise, you should complete the following \"Proposition" Proposition 1. If at 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. 01(0) E (0,1). In this equilibrium 01 (0) = ...; 61(1) = ...; 62 = . l. (1 mark) If type-0 player 1 is mixing, what condition must be satised 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 52. What is it? 3. (2 marks} Using your answers to parts 1 and 2, we can immediately conclude that in this equilibrium type-l player 1 must play...'? (Hint: remember to state your answer as a value for 51(1)) 4. (3 marks) Now you should be able to nd 51(0). What is it? [Hint the answer is a formula containing TE. Notice that it is easy to mess up signs when calculating 01(0), so be careful and double-check your math. 5. (1 mark) You now have a complete prole of strategies given by 01(0), 51(1), 02. But you can notice that for some values of 1|: it is not true that 01(0) E (0,1). Find the values of 11: for which 01(0) E (0,1)