Bill and Joe have committed a crime for which the police are

Bill and Joe have committed a crime for which the police are questioning them separately. The police tell each of them that if one person confesses and the other does not, the “confessor” will go free in return for cooperating, but the one who doesn’t confess will receive a 20-year sentence. If they both confess, then they will both get 5-year sentences. If neither of them confesses, the police only have evidence to convict them both for a 2-year sentence. Think of Bill and Joe as players of a game where each player has two strategies: “Confess” and “Don’t Confess.” Assume that Bill and Joe both expect to live for 30 more years and will never commit another crime, so that their payoffs are the number of years they will be able to live out of prison (30 minus their prison sentence). The payoff matrix for this “prisoners’ dilemma” game is attached

a. In the Nash equilibrium of this game, Bill will play the strategy_____________________

b. In the Nash equilibrium of this game, Joe will play the strategy _____________________.

c. Bill’s maximin strategy is _____________________

d. Joe’s maximin strategy is _____________________.

e. The cooperative outcome is when Bill\'s strategy is _____________________

f. Joe’s cooperate outcome strategy is _____________________.

e. If Bill and Joe had made an agreement to implement the cooperative outcome, then (Bill, Joe, both players, neither player) would have an incentive to cheat on that agreement.

Solution

a) In this Nash equilibrium game, Bill will play the startegy - Confess

b) In the Nash equilibrium of this game, Joe will play the startegy - Confess

c) Bill\'s maximin strategy is - Confess

Maximin startegy means maximize one’s own minimum gain. Ths, Bill\'s minimum gain is 25 if he confesses and 10 if he doesn\'t confess. Hence, his maximin strategy is Confess.

d) Joe\'s maximin startegy is - Similarly, Joe gets a minimum payoff of 10 if she doesn\'t confess and 25if she does. Thus, she will play confess.

e) The cooperative outcome is when Bill\'s startegy is - Dont\' confess

In game theory, a cooperative game is a game with no competition between the two players but rather cooperation. Both Bill and Joe will earn a higher payoff (28 years of freedom each) if they cooperate with each other and don\'t confess.

f) Joe\'s cooperative outcome startegy is - Don\'t confess

g) If Bill and Joe had made an agreement to implement the cooperative outcome, then neither player would hve an incentive to cheat on that agreement.