Prisoners’ dilemma game, Game Theory

The Prisoners’ Dilemma Game

The idea that tacit cooperation can be sustained in an ongoing relationship is very simple and students easily accept it. The formal analysis in a repeated game is much harder if your students do not have an economics or business background and are unused to compound interest, discounted present values, and summing infinite geometric sums. Be prepared to spend a lot of time on this, repeating the exposition and clarifying difficulties.

As usual, it helps to have a specific game with which they are familiar or which they have been required to play. (We describe a simple prisoners' dilemma game in Game 2 below that can be used to motivate the concept of leadership. A symmetric version of that game can be used as an example elsewhere in the presentation of material from this chapter.) Build up the intuitive nature of the cheat once op- tion first, showing the gain from cheating and the loss that must be incurred one period later before cooperation can be resumed. (Of course, resumption of cooperation is possible only with an opponent who is playing TFT.) Those students who have had little experience with present value should be amenable to the argument that time is valuable and that money now is better than money later. Remind them about the pos- sibility of placing money in the bank where it can earn inter- est, even over the span of a month or two. If your example has a gain from cheating that is exactly equal to the loss during the punishment phase, it is easier to see that cheating would be worthwhile in such a situation, at any positive rate of interest. Once students grasp this idea, you can move on to the cheat forever option and a more complicated present discounted value calculation. Don't forget to remind them that the conditions you derive relating to the interest rate at which cheating becomes worthwhile are example-specific; just because you need r > 0.5 in one game does not mean that r > 0.5 is a general condition that determines whether players will cheat.

By contrast, the solutions to the prisoners' dilemma based on penalties are much simpler and can largely be left for the students to read. You can use different illustrations of penalty systems depending on your audience. For example, to convey the idea that if the active players in the dilemma solve it and sustain cooperation, this can be bad for the rest of society, you can use the example of a cartel for economics students or of logrolling among a group of incumbent legislators for political science students. You can also introduce the resolu- tion of the dilemma based on rewards for cooperating in which the credibility of promises to reward is questionable and can be achieved if some third party can hold a player's promise in an "escrow account." If you chose to have your students play the game of Zenda (in the Game Playing in Class section of this chapter) early in the term, they have already had an example of this approach. Now that they have seen a thorough analysis of the prisoners' dilemma, they may have additional insights into the game and their choices; this would be a good time to encourage discussion of their thoughts.

The possibility of solving the prisoners' dilemma through leadership is often less intuitive to students. Having them play an in-class game like Game 2 below can help them to appreciate the differing incentives faced by large and small players in a dilemma game. The game's framework can also be used to show how incentives change as the size disparity between players grows.

There are many, many examples of the prisoners' dilemma that you can point out to your students, from the stories pro- vided at the end of the chapter or from your own experience. You may want to ask the students to try to find their own examples of dilemma situations; they can look for strategic situations that have the three identifying characteristics of a prisoner's dilemma. Students could suggest situations in class that might fit the criteria, and the class could debate whether the various situations qualify as prisoners' dilemmas.

Another good way to encourage discussion about the pri- soners' dilemma is to get students thinking and talking about how the actual actions of players in such dilemmas often diverge from the predictions of the theory. This is easiest if they have been forced to play, either against each other, as suggested in Game 1 below, or against a computer. Each game played against the computer (Serendip) is of finite but unknown length (usually between 10 and 20 rounds), and the computer tracks the total as well as the average gain for each player during the game; there is a link to an expla- nation of the game that tells you the computer is playing tit- for-tat. Although most students don't read this first, virtually all of them figure it out. You can ask students to play against the computer before class and to keep track of their choices and their outcomes so that they can participate in a discus- sion during class. They will most certainly come up with a variety of different stories about how they tried to take ad- vantage of the computer's forgiving play. You may also find that they play more to beat the computer than to maximize their own gains; this is your chance to suggest that some- times predictions are wrong, not because the theory is wrong but because the theorist misunderstands the incentive struc- tures or payoff functions of the players whose behavior she is predicting.

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