Problem 5

Consider a game with two players, who both have two options: they can either attempt to Cooperate with the other, or Defect, which they both decide in advance in secret. The choices are then revealed. If both players chose to Cooperate, they both receive 3 points. If one Defected while the other Cooperated, the defector gets 5 points while the cooperator gets nothing. If both chose to Defect, they both get 1 point.

This game is called the Prisoner's Dilemma, and has been intensely studied in game theory. It is intriguing because no matter what your opponent chooses to do, Defecting will give you a higher score than Cooperating. However, if both players Defect, they end up with less points than if they had both chosen to Cooperate. So the strategy that is optimal for both players yields an overall suboptimal outcome.

This paradoxical nature of the Prisoner's Dilemma goes away if one considers an iterated version, where the game is played over and over and the scores are added up. In this setting, Defecting in one round might cause the opponent to retaliate in future rounds, which incentivizes cooperation.

Suppose that you play 25 rounds of Iterated Prisoner's Dilemma, and you know that your opponent has the following strategy. On the first round, they randomly choose between Cooperate and Defect with equal probability. On each subsequent round, they Cooperate with probability $p$, where $p$ is the proportion of your Cooperations up until that round. (E.g., if you played C, D, C in the first three rounds, then on the fourth round, your opponent will choose to Cooperate with probability 2/3 and Defect with probability 1/3.)

In expected value, what is the highest overall score you can achieve against this player? Give your answer rounded to 4 decimal points.

(No further information is needed to solve the question, but if you are interested, there is a great video about the Prisoner's Dilemma here.)

To deter guessing without thinking, we ask that you also solve the following simple arithmetic problem before checking your answer:

If x = 3, y = 3 and z = 5, what is x + y + z?

This problem was first solved on Wed 25th February at 4:48:16pm