Question A: Monty Hall’s three doors

Suppose you're on a game show, and you're given the choice of three doors: behind one door is a car; behind the others are goats. You pick a door, say Door 1, and the host, Monty, who knows what's      behind the doors, opens another door, say Door 3, which has a goat. He then says to you, "Do you     want to pick Door 2?" Is it to your advantage to switch your choice? Explain your answer.

Question B: A guessing game

A large number of players have to state simultaneously a number in the closed interval [0, 100]. The winner is the person whose chosen number is closest to 2/3 of the average. If you want to win the     game, which number would you pick?

Question C: Chicken

The classic toy game called Chicken derives from the James Dean movie Rebel without a Cause, in which two teenage boys drive cars toward a cliff edge to see who chickens out first. The same game is played by middle-aged drivers who approach each other in streets too narrow for them to pass      without someone slowing down. If both of them speed up, each gets a utility of -1. If both of them   slow down, each gets a utility of 2. If one speeds up and the other slows down, the former gets a      utility of 3 and the latter 0.  Show the payoff matrix and find the Nash equilibria of this game.

Question D: Prisoners’ Dilemma

Two accomplices in crime have been caught by the police. The police have enough evidence to put them in prison for 5 years. They also know that the pair have committed a more serious crime, one that carries a 30-year sentence. Unfortunately, the police don’t have enough evidence to convict the men on that charge. To do so, they would need each of the prisoners to implicate the other in the second crime.