Question

5. Textbook 4.6: Two roommates each need to choose to clean their apartment, and each can choose any amount of time ti 2 0 to clean. If their choices are t and tj, then player i's payoff is given by (10-t)t-7. (This payoff function implies that the more one roommate cleans, the less valuable is cleaning for the other roommate) (a) What is the best response correspondence for each player i 1,2? (b) Which choices survive one round of IESDS? (c) Which choices survive IESDS?

Answer 1

Given that Two roommates each need to choose to clean their apartment, and each can choose any amount of time t_i greaterthanorequalto 0 to clean. (a) Player i maximizes (10 - t_i) t_i - t_i^2 The first - order optimally condition is 10 - t_i - 2t_i = 0 Implying the best response is 2t_i = 10 - t_i t_i = 10 - t_i/2 (b) The most player i could choose is t_i = 5 which is a BR to t_i = 0 So, any t_i > 5 is dominated by t_i = 5 Hence t_i elementof [0, 5] are the choices that survive one round of RESDS (c) In the second round of elimination t_2 lessthanorequalto 5. The best response t_i = 10 - t_i/2, it implies that Roommate 1 will choose t_1 greaterthanorequalto 2.5 The symmetric argument applies to Roommate 2. The second round of elimination implies,