In Mario Puzo’s The Godfather, Michael Corleone (played by Al Pacino in the movie version of the book) would like to meet with Virgil “The Turk” Sollozzo. Michael was concerned that if he meets with Sollozzo, Sollozzo will kill him. We can think of their problem as a game. First, Michael decides whether or not to meet. If they do not meet, suppose Sollozzo and the Corleones each get a payoff of zero. If they do agree to meet then Sollozzo will decide whether or not to kill Michael. If he decides to kill him, then Sollozzo gets a payoff of 20 and the Corleones get a payoff of -10; if he does not kill him then each gets a payoff of 10.
a. Draw the game tree.
b. Use backward induction to show that Michael will decide not to meet.
c. The Bocchicchio family had a well-deserved reputation for ruthlessness. They had a simple code of vengeance; if you were responsible for the death of a member of their family they would kill a member of yours, regardless of the cost to them. Suppose that when Michael meets with Sollozzo, he also hires a member of the Bocchicchio family to go to Michael’s house. There, the "hostage" will be guarded by Michael’s men. If Michael does not return safely, Michael’s men will kill the hostage. The Bocchicchio family, seeking revenge, will blame Sollozzo for the death, since he made the promise that Michael will not be harmed, and will eventually kill Sollozzo. If Michael and Sollozzo are both killed, the Corleone family and Sollozzo each gets a payoff of -10. Use backward induction to determine how this game will be played.