Question There are two players called 1 and 2. Player 1 can be of two types t E {0,1} with Pr (t=l] = 11' E (0,1). The actions and payoffs of the game are given by: where the row player is player 1. We will use the following notation: e elm: probability that player 1 plays a. if she is of type t; e :32: probability that player 2 plays ls. Part I {2 marks) Suppose I = 0.5. Is (mm), on: I}, or} = (, 1,13} a BayeseNash equilibrium? [H.iot: To prove something is a ENE you have to check no player-type has an inoentiye to deviate from the proposed strategy prole. To prove something is not a E'J'JE1 you need to eheekjust one of the three type- player 1, type-l player 1, or player 2 has an inoentiye to deyiate] 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 it this equilibrium exists. At the end of the exercise, you should complete the following "Proposition" Proposition 1. If n 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 61 (0) = ...; 61(1) = ...; 61 = ... .. .. 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 ...). {1 mark) Using the condition derived in part I, you should be able to nd player 2 's equilibrium strategy as. What is it? . {2 marks} Using your answers to parts 1 and 2, we can immediately,r conclude that in this equilibrium type-l player 1 must play...'? (Hint: remember to state your answer as a value for o.{l]) . {3 marks} Now you should be able to nd lm). What is it? [I-Iint: the answer is a formula containing it. Notice that it is easy to mess up signs when calculating mt'), so be careful and double-check your math. . {1 mark) You now have a complete prole of strategies given by mm}, 61\"], 52. But you can notice that for some values of it it is not true that o.{} E {0,1}. Find the values of it for which mm) E (U, l}