The Prisoner's Dilemma is a well-known problem in game theory. Two thieves are arrested and held in custody separately. The police offer each the same deal.
Inform on your partner and we will reduce your sentence. The following outcomes and costs are possible:
1. If both you and your partner stay quiet, you will both be convicted of misdemeanour offences (lesser charges). The cost of this is 10.
2. If you turn state's evidence (cooperate with the police), you will be convicted of a misdemeanour and fined. The cost of this is 50.
3. If you do not cooperate, but your partner does, you will be convicted of a felony (a major crime). The cost of this is 100.
The dilemma is that the best course of action is for both of you to stay quiet, but since there is no honour among thieves, you do not thrust one another. Then you both will turn state's evidence in order to avoid being convicted of the major crime (which happens if your partner turns state's evidence and you do not).
Consider this twist. Before you are hauled away, you and your partner swear to keep quiet. You believe that there is a 60% chance that he will keep his word.
(i) Draw a decision tree that represents your decision (to keep quiet or to cooperate) and the possible outcomes.
(ii) What decision has the highest expected value?
(iii) If x represents the probability that your partner will keep quiet, for what value of x is the value of keeping quiet to the value of cooperating?