2. [30 points] The Cuban missile crisis of 1962 started with the United States' discovery of Soviet nuclear missiles in Cuba, after which the United States escalated the crisis by quarantining Cuba. The USSR then backed down, agreeing to remove its missiles from Cuba, which suggests that the United States had a credible threat along the lines of "if you don't back off we both pay dearly," implying that the US would choose the nuclear option if USSR does not back down. Could this indeed be a credible threat? To answer this question, we need to analyze the crisis as a game. Consider the two superpowers, the United States and the USSR, playing the following game. The game starts with the United States' choice to either "ignore the incident" (1), resulting in maintenance of the status quo with payoffs (0, 0), or "escalate the situation" (E). Following escalation by the US, the USSR can "back down" (B), causing it to lose face and resulting in payoffs of (10, - 10), or it can choose to "proceed to a nuclear confrontation" (). If the USSR chooses N, the players play a simultaneous-move game in which they can either retreat (denote this as R for the US and r for the USSR) or choose Doomsday (denote this as D for the US and d for the USSR), in which the world is destroyed. If both call things off and retreat then they suffer a small loss due to the mobilization process and payoffs are (- 5, - 5), while if either party chooses Doomsday then the payoffs are (- 100, - 100). [Note: The first number in a payoff pair denotes the payoff to the US and the second number denotes the payoff to the USSR.] a. [10 points] Draw the game tree that represents the whole game. b. [15 points] Find the Subgame Perfect Equilibria. c. [5 points] What does your analysis suggest regarding credibility of the United States' threat? Hint: A threat is credible only if following the threat through is a best-response strategy for the player who is making the threat against some strategy of the opponent(s). In other words, the strategy corresponding to the threat has to be part of a Subgame-perfect Equilibrium