Economics 409: Game Theory, Fall 2021
Final Exam
Date: Friday, December 17th.
Instructions
1. There are 3 questions, worth 20, 25, and 30 points, respectively.
2. Use the space on the front and back of each question to write down your answer.
3. The last page is blank in case you need more paper.
4. Write down YOUR NAME on EVERY PAGE. We won’t grade a page without a name.
5. Write down YOUR SECTION NUMBER on EVERY PAGE. We won’t grade a page without a number.
Question 1 [20 total points].
State and explain whether each of the following claims is true or false. Each claim is worth 10 points, of which 8 points are for the explanation (of a correct statement.)
(a) Every pure BNE of a game with incomplete information induces some NE in every interim game.
(b) Consider a two-player static game in which player 1 can be of two diferent types, t and t、(and player 2 has only one type), and let p be the probability Nature puts on t. If, for some type, no strategy is interim dominated for player 1, then there exist some p such that no strategy is ex-ante dominated for player 1.
Question 2 [25 total points].
Consider a static game with incomplete information in which player 2 can be one of two possible types, t and t、. Below, the top matrix describes both players’ payofs when player is of type t, and the bottom matrix describes both players’ payofs when player is of type t、:
Suppose that nature chooses type t with probability p.
(a) [5 points] Are there strategies that are interim dominated? Explain.
(b) [5 points] Write-down the normal form of the extended (i.e., ex-ante) game.
(c) [5 points] Is some strategy (ex-ante) dominated for player 1, for some value of p? Explain.
Suppose that 0 < p < 1/2.
(d) [10 points] What are the pure Bayes Nash equilibria? Explain.
Question 3 [30 total points].
Consider the following two dynamic games:
Suppose that the top game describes the payofs of both players when the entrant is of type C, and the bottom game describes the payofs of both players when the entrant is of type W (the incumbent, I, has only one type.) Nature chooses type C with probability 1/4.
(a) [5 points] What’s the normal-form of the extended (i.e., ex-ante) game?
(b) [10 points] Find all mixed BNE.
(c) [10 points] Are all pure BNE part of some WPBE?
(d) [5 points] Are all mixed BNE part of some WPBE?