6. (Prisoner's Dilemmas) An important class of games is called the Prisoner's Dilemma

which gets itself name from a book by William Poundstone, where they explain it as1:

Two members of a criminal gang are arrested and imprisoned. Each pris-

oner is in solitary confinement with no means of communicating with the other.

The prosecutors lack sufficient evidence to convict the pair on the principal

charge, but they have enough to convict both on a lesser charge. Simultane-

ously, the prosecutors offer each prisoner a bargain. Each prisoner is given the

opportunity either to betray the other by testifying that the other committed

the crime, or to cooperate with the other by remaining silent. The offer is:

If A and B each betray the other, each of them serves two years in prison

If A betrays B but B remains silent, A will be set free and B will serve

three years in prison (and vice versa)

If A and B both remain silent, both of them will serve only one year in

prison (on the lesser charge).

We often phrase "staying silent" as cooperating (with the other prisoner) and "betraying"

as defecting, since this language is more common.

(a) (5 points) Model this situation as a strategic game, being careful to explain your

choices.

(b) (3 points) What is the ordering of the payoffs implied by the story above?

(c) (5 points) Draw the matrix representation of this game.

(d) (2 points) What is the Nash equilibrium of this game?

(e) (5 points) Recall, from microeconomics, that we define an outcome as Pareto effi-

cient if you cannot find another outcome with makes both players at least as well

off, and one player strictly better off. Which outcomes in the Prisoner's Dilemma

are Pareto efficient? Is the Nash equilibrium outcome Pareto efficient?

(f) (5 points) Does the behaviour in this game depend on the payoffs being the same

across Player 1 and Player 2? Explain your answer.