There are two players called 1 and 2. Player 1 can be of two types t {0, 1} with Pr (t = 1) = (0, 1).

t,4

where the row player is player 1.

We will use the following notation: σ1 (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 down whenever she is of type t = 0, i.e. σ1 (0) ∈ (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 should complete the following "Proposition"

Proposition 1. If π..............., 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. σ1 (0) ∈ (0,1). In this equilibrium σ1 (0) = ............; σ1 (1) = .........; and σ2 = ..........

1. 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...)
2. 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?
3. 1.3 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))
4. 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 σ1 (0), so be careful and double-check your math)
5. 1.5 You now have a complete profile of strategies given by σ1 (0) , σ1 (1) , σ2. But you can notice that for some values of π it is not true that σ1 (0) ∈ (0, 1). Find the values of π for which σ1 (0) ∈ (0, 1).