0_Problem_Set_1

Premium Support This page features MathJax technology to render mathematical formulae. If you are using a screen reader, please visit MathPlayer to download the plugin for your browser. Please note that this is an Internet Explorer-only plugin at this time. Game Theory Need technical help? Visit our new Learner Help Center! Top Navigation Bar • Courses • Anonymous Student About UsCareersPeopleContact Us My CoursesProfileCourse RecordsSettingsCourse Preferences Sign Out Game Theory by Matthew O. Jackson, Kevin Leyton-Brown, Yoav Shoham

Search this course Search Side Navigation Bar • Quick Questions15 Please help us improve this course by answering some quick questions! • Home • Syllabus • Video Lectures • In-Video Quizzes • Problem Sets • Final Exam • Online Lab Exercises • Discussion Forums • YouTube Channel opens in new browser tab • Surveys • Coursera Demographic Survey opens in new browser tab • Zip File of All Slides - LARGE FILE (20M) opens in new browser tab • Help Center Problem Set 1 Help Center Learn more Warning: The hard deadline has passed. You can attempt it, but you will not get credit for it . You are welcome to try it as a learning exercise. In accordance with the Coursera Honor Code, I (Anonymous Student) certify that the answers here are my own work. Thank you! Question 1 Dominance 1\ 2 x y z

a 1 ,2 2 ,2 5 ,1 b 4 ,1 3 ,5 3 ,3 c 5 ,2 4 ,4 7 ,0 d 2 ,3 0 ,4 3 ,0 When player 1 plays d , what is player 2's best response: a) Only x b) Only y c) Only z d) Both y and z Question 4 Dominance 1\ 2 x y z a 1 ,2 2 ,2 5 ,1 b 4 ,1 3 ,5 3 ,3 c 5 ,2 4 ,4 7 ,0 d 2 ,3 0 ,4 3 ,0 Find all strategy profiles that form pure strategy Nash equilibria (there may be more than one, or none): 1) (a, x); 2) (b, x); 3) (c, x); 4) (d, x); 5) (a, y); 6) (b, y); 7) (c, y); 8) (d, y); 9) (a, z); 10) (b, z); 11) (c, z); 12) (d, z). Question 5 Nash Equilibrium - Bargaining

There are 2 players that have to decide how to split one dollar. The bargaining process works as follows. Players simultaneously announce the share they would like to receive s 1 and s 2, with 0≤ s 1, s 2≤1. If s 1+ s 2≤1, then the players receive the shares they named and if s 1+ s 2>1, then both players fail to achieve an agreement and receive zero. Which of the following is a strictly dominant strategy? a)1; b) 0.5; c) 0; d) None of the above. Question 6 Nash Equilibrium - Bargaining There are 2 players that have to decide how to split one dollar. The bargaining process works as follows. Players simultaneously announce the share they would like to receive s 1 and s 2, with 0≤ s 1, s 2≤1. If s 1+ s 2≤1, then the players receive the shares they named and if s 1+ s 2>1, then both players fail to achieve an agreement and receive zero. Which of the following strategy profiles is a pure strategy Nash equilibrium? a) (0.3, 0.7); b) (0.5, 0.5); c) (1.0, 1.0); d) All of the above Question 7 Bertrand Duopoly • Two firms produce identical goods, with a production cost of c >0 per unit. • Each firm sets a nonnegative price ( p 1 and p 2). • All consumers buy from the firm with the lower price, if pi ≠ pj . Half of the consumers buy from each firm if pi = pj . • D is the total demand. • Profit of firm i is: • 0 if pi > pj (no one buys from firm i); • D ( pi − c )/2 if pi = pj (Half of customers buy from firm i); • D ( pi − c ) if pi < pj (All customers buy from firm i); Find the pure strategy Nash equilibrium: a) Both firms set p =0. b) Firm 1 sets p =0, and firm 2 sets p = c . c) Both firms set p = c . d) No pure strategy Nash equilibrium exists.