In Tom Clancy's novel Hunt for Red October, the Soviet Union has developed a submarine

named Red October that can run "silently" and thereby escape detection. On its maiden

voyage, the ship's captain, Marko Ramius, has decided to defect, because he believes that

this technology risks war by destroying the balance of power between the United States

and the U.S.S.R. He has put together a set of officers who are loyal to him and have

agreed to defect as well. However, the captain is concerned that an officer may change

his mind during the voyage and, furthermore, that an officer may be more inclined to

change his mind if he thinks that other officers will do so. The captain is then considering

writing a letterto be delivered to the Soviet government after the submarine has departed

from its basestating his plan to defect. The extensive form of this game is shown

below. The captain initially decides whether or not to send the letter. After revealing

his decision to his officers (once they are all out to sea), the officers, which, for the

sake of parsimony, are limited to Captain Ramius, Second Rank Borodin and Lieutenant

Melekhin, simultaneously decide between continuing with the plan to defect or reneging

on the plan and insisting that the submarine return to the Soviet Union. The payoffs are

such that all three players would like to defect and would prefer that it be done without the

letter being sent (which results in the Soviet government sending out another submarine

to sink Red October).

Derive all subgame perfect Nash equilibria. Show your work.

Explain why the captain would send the letter.