PLEASE ANSWER ALL QUESTIONS WITH WORKING! THE WRITTEN WORKING IS VITAL AND NECESSARY AS IT HELPS ME UNDERSTAND.

Introduction to Game Theory Spring 2022 Problem set 4 Due: April 12 Be sure to Show all your work for partial credit. 1. (15 points) On a given week, two selsh parents simultaneously choose between spending time with their children (S) or not (N). The payoff matrix is the following. Player 2 (a) Assume that the parents repeat this game for 52 weeks. What are the Subgame Perfect Nash Equilibrium strategies? What is the SPNE outcome? (b) Now assume the parents repeat this game innitely. Consider the following strategy: \"Start by Playing S and continue playing S as long as (S,S) was played on every previous week. Otherwise, play N forever.\" Assume Payer 1's discount factor is 51 = 1/4 and Player 2's is 52 = 1/8. Is it a SPNE for both players to play the strategy above? Why or why not? 2. (12 points) Consider the innitely repeated Bertrand competition as studied in class, but assume there are N rms. (a) Show that a grim trigger strategy can sustain collusion in a Subgame Perfect Nash Equilibrium for a sufficiently high discount factor 5. (b) How does the threshold for 6 in part (a) change when N changes? Briey interpret. 3. (13 points) Consider the entry game with incomplete information studied in class. An incumbent politician's cost of campaigning can be high or low and the entrant does not know this cost (but the incumbent does). In class, we found two pure-strategy Bayesian Nash Equilibria in this game. Assume that the probability that the cost of campaigning is high is a parameter p, 0