This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 6.254 Game Theory with Engineering Applications Midterm April 8, 2008 Problem 1 : (35 points) Consider a game with two players, where the pure strategy of each player is given by x i ∈ [0 , 1]. Assume that the payoff function u i of player i is given by u i ( x 1 ,x 2 ) = a i x i + I{ x i < x j } , where 0 < a i ≤ 1 and I is the indicator function given by I{ x i < x j } = ( 1 if x i < x j 0 otherwise . (a) Does this game have a pure strategy Nash equilibrium? Verify your answer. (b) Show that there cannot be an atom at any x ∈ (0 , 1] in either player’s mixed strategy at an equilibrium. (c) Find all mixed strategy Nash equilibria. Problem 2 : (30 points) Consider the following oligopoly competition model: There are two firms. Each firm i simultaneously chooses price p i ∈ [0 , 1]. Assume that the demand function for firm i , given by d i ( p 1 ,p 2 ), is twice differentiable. Each firm is interested in maximizing his profits (assume for simplicity that there is no cost)....
View
Full
Document
This note was uploaded on 05/08/2010 for the course CS 6.254 taught by Professor Asuozdaglar during the Spring '10 term at MIT.
 Spring '10
 AsuOzdaglar

Click to edit the document details