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 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?
- 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))
- 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)
- 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).

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