Economics 409: Game Theory, Fall 2023
Final Exam
Date: December 12th.
Question 1 [25 total points].
State and explain whether the following claim is true or false. The claim is worth 25 points, of which 20 points are for the explanation (of a correct statement.)
Suppose P1 has possible type θ 2 f1, 2g, which they will observe but P2 will not know. After P1 realizes their type, P1 and P2 play the following static game
This game has no mixed BNE regardless of what probabilities are assigned to each type.
Question 2 [25 total points].
Consider the following static game of incomplete information with two players.
Player 1 only has 1 type, while the player 2 has two possible types t1 and t2 . When player 2 is of type t1 , both players play the game on top. If player 2 is of type t2 , both players play the game on the bottom. Nature informs player 2 of her type, but not to player 1. Hence, player 1 is never informed of what the game they play. Suppose that each type of player 2 occurs with probability 1/2.
(a) [10 points] Find all pure Bayesian Nash equilibria, or explain why none exists.
(b) [10 points] Find all mixed Bayesian Nash equilibria where player 1 plays a pure strategy. If none exist, explain why.
(c) [5 points] Find all mixed Bayesian Nash equilibria where player 2 plays a pure strategy. If none exist, explain why.
Question 3 [25 total points].
This question considers a three-person game with incomplete information in which the three players have two strategies, A and B. Suppose that players 1 (row) and 2 (column) have only one type, but player 3 (matrix) has two equally like types, high (H) and low (L); if player 3 is of type H the three-player game is:
whereas if player 3 is of type L the three-player game is:
(a) [5 points] Are there strategies that are interim strictly dominated for either of the players?
(b) [10 points] What are the pure Bayes Nash Equilibria of the game? Hint: You don’t need to write down the normal-form of the ex-ante game.
(c) [10 points] Find all the mixed BNE of the game, or explain why none exist.