A Game Theory problem set number four for economics classes

6. Learning to cooperate: Imagine a parent with two kids, Abel (A) and Carlos (C), each ar01md 6 to 10 years old. After an afternoon of playing the house is a mess with all the toys of the two kids scattered around. The parent wants to take this opportunity to teach A and C to cooperate, the parent is more interested in getting the kids to cooperate than to actually getting them to clean the house. For this purpose the parent designs the following scenario. She/He tell the kids that if they clean, they will go and get ice cream afterwards. That is, regardless of what kid decides to clean, if the house is cleaned both get ice cream. Each kid can choose to clean e,- : 1 or shirk ei : 0. The payo' function for each is given byDr n,(q,e_,) = 8 x 1{e; = 1 or e; = 1} e; (a) Assume that kids play this game simultaneously. Find the Nash Equilibria of this game (don't forget to nd the mixed strategies NE, if any) (b) Now imagine that A plays rst, C observes; and then choose. Find the subgameperfect Nash equilibria. (c) Assume that the kids will be playing each weekend over a month (4 times). In each stage they choose simultaneously. Let 5 E (0, 1) be the disccnmt factor- Consider the following strategies and determine if they are part of the subgameperfect nash equilibria set. (Hint: At least one is not, you may need to make some assumptions for the value of 5) a They both choose to clean in the rst period. In each subsequent period, both kids choose to clean if in the previous period both kids also choose to clean. 0 I clean rst, then you clean. That is, kids alternate between 6,- = 1 and e,- = 0. a Only A clean in each period