Game theory problem set 5

1 . There are two players called 1 and 2. Player 1 can be of two types 15 E {0, 1} with Pr (1: = 1) = 7r 6 (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: 01 (t) is the probability that player 1 plays up if she is of type t; 02 is the probability that player 2 plays left. 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. 0'1 (0) E (0, 1). We therefore proceed to construct such an equilibrium and then verify for which values of 7r this equilibrium exists. At the end of the exercise, you should complete the following \"Proposition\" Proposition 1. If 7r ............... , 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) = ............ ; 01 (1) = ......... ,- aud 02 = .......... 1.1 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...) 1.2 Using the condition you derived in part 1.1, you should be able to nd player 2's equilibrium strategy 02. What is it? 1.3 1.4 1.5 Using your answers to parts 1.1 and 1.2, we can imme- diately conclude that in this equilibrium type-1 player 1 must play...? (Hint: remember to state your answer as a value for 01 (1)) Now you should be able to nd 01(0). What is it? (Hint: the answer is a formula containing 71'. Notice that it is easy to mess up signs when calculating 01 (0), so be careful and double-check your math) You now have a complete prole of strategies given by 01(0),01(1),crg. But you can notice that for some values of 71' it is not true that 01(0) 6 (0,1). Find the values of 71' for which 01 (0) 6 (0,1)