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CHAPTER 11
Collective-Action Games
Teaching Suggestions
A good way to introduce the topic of coordination games is to have the students play one. There
are two suggestions below. One can be used to lead into a general discussion of the issues surrounding
games of this type; the other can be used as a numerical example for a game with a multiperson prisoners’
dilemma structure. Another way to play such a game can be used if you give frequent quizzes; you can
prepare the class by including one of the game questions as the last question on the quiz the week before.
Another way to start the topic is to ask the class why, with so many different styles of dress
available, all students are dressed pretty much alike (these days you will find most male students and
many females wearing reversed baseball caps, jeans, and flannel shirts with shirttails hanging outside).
The answer will usually be some variant of “conformity to the group norm” or “seeking peer approval.”
You can develop these suggestions into the ideas that conformity creates a positive feedback, uniformity
is the equilibrium outcome of such behavior, or some other chance development could have led to a
different equilibrium where the common dress style was something else. It is then possible to develop a
diagram like the Windows-Unix one in the text out of this discussion and to talk about the different
equilibria that might arise in various situations.
Ask students for stories of attempts, successful or otherwise, to get volunteers in their high school
class or their community for some worthy project such as cleaning up a local beach. Relate the arguments
given for participating or shirking and the reasons for the success or failure of the project to the
conceptual ideas in this chapter. You can make connections here to the second game described below,
which is essentially a test of player willingness to contribute to a public good or to the third game, which
deals with the problem of a common resource.
There are many case studies in political science, law, and anthropology of attempts to resolve
collective-action problems. Some are mentioned in the text. You can mention others, or ask students to
read them and then lead discussions in class. This is a good way to give noneconomics students a feel for
applying game theory to their fields. Here is a small list of such readings:
Robert C. Ellickson, Order without Law: How Neighbors Settle Disputes (Cambridge, Mass.: Harvard
University Press, 1991).
Jean Ensminger, Making a Market: The Institutional Transformation of an African Society (Cambridge,
U.K.: Cambridge University Press, 1992), esp. chaps. 4–6.
Avner Greif, “Contract Enforceability and Economic Institutions in Early Trade: The Maghribi Traders’
Coalition,” American Economic Review, vol. 83 (1993), pp. 525–548.
________, “Cultural Beliefs and the Organization of Society: A Historical and Theoretical Reflection on
Collectivist and Individualist Societies,” Journal of Political Economy, vol. 102 (1994), pp.
912–950.
________, “Contracting, Enforcement, and Efficiency: Economics Beyond the Law,” in Annual World
Bank Conference on Development Economics, ed. Michael Bruno and Boris Pleskovic
(Washington, D.C.: World Bank, 1997) pp. 239–265.
Timothy W. Guinnane, “A Failed Institutional Transplant: Raiffeisen’s Credit Cooperatives in Ireland,
1894–1914,” Explorations in Economic History, vol. 31 (2003), pp. 38–61.
John McMillan and Christopher Woodruff, “Private Order under Dysfunctional Public Order,” Michigan
Law Review, vol. 98 (2000), pp. 2421–2458.
Elinor Ostrom, Governing the Commons: The Evolution of Institutions for Collective Action (Cambridge,
U.K.: Cambridge University Press, 1990).
James Q. Wilson and George L. Kelling, “Broken Windows,” Atlantic Monthly, vol. 249, no. 3 (March
1982), pp. 29–38.
At some point you should give more precise numerical values to the various payoffs in the games
you are discussing to enable students to connect the examples with their game-theoretic analytical
structures. Here is one numeric example about a group of college students and their decision to participate
in a dorm cleanup. Suppose that there are 100 residents of the dorm. If n of them choose to work on the
cleanup, then the benefit to each one, measured in dollars, is B = n – (1/200)n2. For each student the cost
of the approximately 5-hour cleanup is $40. Then each student gets a net payoff of 0 if no one works and
a net payoff of 50 – 40 = 10 if everyone works.
To analyze this collective-action game, you need to look for the equilibrium participation choices
of the students. If n other students are working, then an individual student’s payoff from choosing to work
with the group is
(n + 1) – (n + 1)2/200 – 40,
and his payoff from choosing not to work (choosing to shirk) is
n – n2/200.
Then the student will shirk if
(n + 1) – (n + 1)2/200 – 40 < n – n2/200,
or if –(2n + 1)/200 – 39 < 0, which is always true. Every student will want to shirk.
Another way to illustrate this outcome is with a diagram showing the payoffs to an individual
student from working and shirking given different numbers of other students who have chosen to work.
Such a diagram for this example is shown below:
Note that the payoff from shirking is always higher than the payoff from working, regardless of
how many other students are already working. This game fits the description of a multiperson prisoners’
dilemma.
Consider how to resolve this dilemma:
1. Use the repeated (ongoing) relationship among dorm residents and punish shirkers (through
ostracism or exclusion).
2. Establish a social norm of participation; then shirkers entail a cost—shame, and so on.
3. Reward workers with money, an office, and so on.
4. Take advantage of the existence of people who have larger benefit (or smaller cost) who take the
lead in organization, and so on.
5. Resort to external enforcement (administration or government), but the enforcing institution may be
poorly informed. The same type of analysis applies to natural resource management.
Where is the aggregate optimum in this game? You need to find the n that maximizes
100[n – (1/200)n2] – 40n = 60n – (1/2)n2.
That n solves 60 – (1/2)2n = 0; n = 60. Each worker’s payoff is 60 – 3,600/200 – 40 = 2, and each
shirker’s payoff = 42. There is an additional problem of distributive justice of this optimum; who should
be allowed to be a shirker? You can solve this problem in a repeated game using a rotation scheme. (You
can relate this to Ostrom’s case involving rotation of rights to good locations in a fishing community. See
pages 19–20 of her book cited above.) In a one-shot game, the problem can be solved with side payments
(from shirkers to workers).
The other most easily constructed examples are those involving externalities. Consider an
example based on career choice decisions of students in your class: suppose there are 1,000 students in
the class (or college). Let n represent the number who choose to become doctors, so 1,000 – n is the
number who choose to become lawyers. Each doctor’s income is a function of the number of others who
choose to be doctors: 250 – n/6 (thousands of dollars); each lawyer’s income is a constant 150 (thousand
dollars). In the equilibrium with free choice, 600 students will choose to become lawyers as can be seen in
the following diagram:
But what is the socially optimum outcome here; what number of doctors would maximize total class
income? You want to find the value of n that maximizes
n(250 – n/6) + (1,000 – n)150.
Taking the derivative with respect to n shows that the optimal n must satisfy 250 – (1/6)2n – 150 = 0, or n
= 300. Thus the optimal number of doctors is only one-half the number that arises in the free-choice
equilibrium.
You can interpret this story with additional economic jargon if you are so inclined. In this case,
the doctors’ total income is n(250 – n/6). An average doctor earns 250 – n/6, but the marginal doctor earns
250 – n/3. For lawyers, the total income is 150(1,000 – n); the average and marginal incomes for lawyers
are also 150.
You can use the diagram below to show that maximizing the total income of the group requires
equating the marginal income of a doctor to the marginal income of a lawyer. The free-choice equilibrium
is attained from equating the average incomes.
For a collection of additional examples that are different from the ones in the book, see Dixit and
Nalebuff, Thinking Strategically, Chapter 9.
