Perfect Bayesian Equilibrium

Problem 1 There is a sandbox in the school playground. Ann is enjoying the sandbox by herself. Bob would like to also play in the sandbox. Ann could be one of two types: friendly (99% chance) or violent (1% chance). When Bob shows interest in joining the sandbox, she can do one of two things: play or start a fight. If she is violent, she can only start a fight. After observing Ann's behavior, Bob can choose either to enter the sandbox or leave. If he enters, Ann's behavior continues as she had started before he entered. The payoffs are as shown in the figure. They reflect two important things: Ann prefers to play alone even if she is friendly; and while Bob does not mind if a friendly Ann fights with him, he will suffer greatly if he ends up in a fight with a violent Ann. Nature Friendly Violent Ann 0.99 0.01 Ann Play Fight Fight Bob Bob Enter Leave Enter Leave Enter Leave 10 10 10 N a NO 0 200 0