Question 1 There are two players called 1 and 2. Player 1 can be of two types t 6 {0,1} with Pr (t=1) = 71: 6 (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 01(t): probability that player 1 plays up if she is of type t; o 02: probability that player 2 plays left. Part I (2 marks) Suppose fl: = 0.5. Is (61(0), 01(1), 0'2) = (0,1,0) a Bayes-Nash equilibrium? [Hintz 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-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. (31(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 a 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) 6 (0,1). In this equilibrium 01 (0) = ...; 01(1) = ...; oz = . 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 nd 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 61(1)) 4. (3 marks) Now you should be able to nd 61(0). What is it? [Hint the answer is a formula containing 1:. 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 61(0), (31(1), (32. But you can notice that for some values of at it is not true that 01(0) E (0,1). Find the values of it for which 61(0) 6 (0,1)