Question

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There are two players called 1 and 2. Player 1 can be of two types t E 0,1) with Pr (t 1)E (0,1). The actions and payoffs of the game are given by rt up 0,41.1 own 1,2t,4 where the row player is player 1 We will use the following notation: ơ1(t) is the probability that player 1 plays up if she is of type t; σ2 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 doun whenever she is of type t = 0, i.e. σι (0) E (0,1). We therefore proceed to construct such an equilibrium and then verify for which values of π this equilibrium exists. At the end of the exercise, you s hould complete the following "Proposition" Proposition 1. Ifm... . . . .. then there erists a Bayes Nash equilibrium in which player 1 mixes between up and down whenever she is of type t = 0, 1.1 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...) 1.2 Using the condition you derived in part 1.1, you should be able to find player 2's equilibrium strategy σ2-what is it? 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 σ1(1)) 1.3 1.4 Now you should be able to find σ1(0). What is it? (Hint: the answer is a formula containing π. Notice that it is easy to mess up signs when calculating σ| (0), so be careful and double-check your math) 1.5 You now have a complete profile of strategies given by σι (0), σι (1), σ2. But you can notice that for some values of π it is not true that n(0) E (0,1). Find the values of π for which ơi (0) (0.1).

Answer 1

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