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CHAPTER 10
The Prisoners’ Dilemma and Repeated Games
Teaching Suggestions
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 not familiar with the concepts of compound interest,
discounted present values, and summing infinite geometric sums. Be prepared to spend a lot of time on
this in such cases, repeating the exposition and clarifying difficulties.
You know that a given outcome with tacit cooperation can often be sustained using multiple
combinations of the players’ strategies. At this elementary level, we have focused on just two, namely, the
grim trigger strategy of reversion to playing the Nash equilibrium of the one-shot game Forever, and
Tit-for-tat (TFT), where it is possible to resume cooperation after a single defection. Many students will
have heard of TFT from popular discussions of the prisoners’ dilemma, so it may be best to start with that.
As usual, it helps to have a specific game with which students are familiar or which they have
been required to play. (We describe a simple prisoners’ dilemma game in Game 2, Leadership in an Oil
Production, 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 option first, showing the gain from cheating and the loss that must be
incurred one period later before cooperation can be resumed. (Of course, the 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 possibility of placing money in the bank where it can earn interest,
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 resolution of the dilemma based on rewards for cooperating in which the
credibility of promises to reward is questionable but 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 provided at the end of the chapter or from your own experience. You may want
to ask 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 prisoners’ 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, Paired Prisoners’ Dilemma, below, or against a computer. One Web-based
interactive version of the prisoners’ dilemma is available at
http://serendip.brynmawr.edu/playground/pd.html. 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 explanation of the game that
tells you the computer is playing TFT. 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 discussion during class. They will most
certainly come up with a variety of different stories about how they tried to take advantage 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 sometimes predictions are wrong, not because the
theory is wrong but because the theorist misunderstands the incentive structures or payoff functions of the
players whose behavior she is predicting.
If your class has some analytical sophistication, you may want to discuss the multiplicity of
strategies that can sustain a given cooperative outcome. The conditions on the payoffs and discount
factors that make cooperation an equilibrium of the repeated game are similar for the different strategy
combinations, but not usually exactly the same. Therefore, some dilemmas may be resolved more easily
using one approach (say, grim trigger strategies) than another (say, TFT). We provide an analysis of this
type here; you could develop this specific example in class or in a more-specialized section, or you could
even use it as an extra exercise.
Consider a prisoners’ dilemma where the Row player’s payoffs are
Column
Defect Cooperate
Row Defect c – b a
Cooperate –b0
and where a, b, and c are all positive numbers with c < b, so c – b < 0. The payoff c is the benefit to Row
from choosing Defect instead of Cooperate when Column is also choosing Defect; call this the defensive
aspect of Row’s dilemma. The payoff a is the benefit to Row from choosing Defect instead of Cooperate
when Column is choosing Cooperate; call this the offensive aspect of Row’s dilemma.
Now consider infinite repetition of this game with discount factor d and two methods of
sustaining the (Cooperate, Cooperate) outcome.
1. Grim Trigger Strategy (GTS). Here the first-period gain to the cheater is a, and the
subsequent loss is 0 – (c – b) = b – c every period. The present discounted value of the subsequent loss is
found by multiplying by the capitalization factor:
(d + d2 + d3 + … ) = d/(1 – d).
Therefore, the condition to sustain Row’s cooperative action is
a < [d/(1 – d)](b – c), or a < d(b + a – c).
2. Tit-for-Tat (TFT). Here the cheater gains a in the first period, and in the second period plays
Cooperate while the other TFT player is playing Defect, thereby suffering a loss of b compared with what
he would have gotten by sticking to the equilibrium. Then cooperation is restored from the third period
onward. The first-period present value of the second-period loss is just db. Therefore, the condition to rule
out Row’s one-time defection (then suffering the bad payoff in the second period to restore cooperation
from the third period on) is
a < db.
So the condition for GTS to work is stricter than that for TFT to work if and only if
b + a – c < b, or a < c;
that is, the dilemma is worse in its defensive aspect.
The separate offensive and defensive aspects of the prisoners’ dilemma are also a new interesting
feature worth pointing out to students.
