254 N. Brooks J. of Economic Behavior Org. 44 2001 249–267
Section 4. A brief discussion of the theoretical results and the policy implications is given in Section 5. The procedure for empirically testing the model is discussed in Section 6 along
with a discussion of the difficulties in empirically testing for neighborhood effects. The data are described and the results of the estimation are evaluated in Section 7. A discussion of
the empirical results and the conclusion is given in Section 8.
2. Model
The structure for this study is a repeated random matching prisoner’s dilemma game augmented with a sanctioning action for each player. There is a continuum of players
divided into two groups. In each stage, each player is matched randomly with a player in the other group. There are an infinite number of stages in the supergame and an individual’s
total payoffs are the expected sum of the payoffs in each stage discounted by a common discount factor δ0, 1.
In each stage game, individuals have a choice of three actions, A
i
= { cooperate, deviate,
ostracize}. Individuals have the same linear utility functions, but each has a different return to behaving cooperatively in the social trade. These heterogenous returns to cooperation are
an important feature of the game. The returns to cooperative behavior are assumed to be distributed uniformly across the community’s population. The payoff matrix for the stage
game is shown in Table 1.
If both individuals behave cooperatively each receives the return w
i
. If either player decides to be opportunistic when the other player is cooperating then his return is w
i
+ b
d
, where b
d
is the additional gain to the opportunistic behavior. The other player incurs a cost c
d
. If c
d
b
d
then there is a net social loss due to the opportunistic behavior. If both players are opportunistic then each gets the same payoff that is less than if both players
cooperate, and there is a definite social loss. Thus the community could possibly benefit from a prescribed social norm against opportunistic behavior if the aggregate costs of enforcing
the norm in equilibrium are less than the costs to society of the opportunistic behavior which would occur without the norm.
A player who is ostracized does not get to trade and thus gets a return of zero. The punishing player also does not get to benefit from the trade since no trade occurs and he faces
the cost of sanctioning. The additional returns and costs from opportunistic behavior b
d
, c
d
, and l
d
and the cost of punishing f f ≤ 0 are assumed to be the same for each individual.
Table 1 Payoff matrix for the stage game
a
Player I Player II
C D
O C
w
1
, w
2
w
1
− c
d
, w
2
+ b
d
0, f D
w
1
+ b
d
, w
2
− c
d
w
1
− l
d
, w
2
− l
d
0, f O
f, 0 f, 0
f, f
a
w
i
is the returns to trade i = 1, 2 w
i
∼ uniformly on the support [w, ¯
w], b
d
the benefit from deviating when the other player cooperates, c
d
the cost of cooperating when the other player is deviating, l
d
the loss when both players deviate, and f the cost of ostracizing f ≤ 0.
N. Brooks J. of Economic Behavior Org. 44 2001 249–267 255
Moreover, the costs of sanctioning f are the same regardless of whether you are punishing either an opportunistic individual or a non-punisher. Similarly, the costs of being punished
0 are the same whether you are punished for either opportunism or for non-punishment of another. This payoff matrix is used because it is, perhaps, the simplest and most tractable
one to investigate a community’s ability to generate adherence to prescribed social norms. The model should be used to give insights about many types of prisoner’s dilemma social
situations, but to clarify the structure of the payoff matrix you might reconsider the example of the shopkeepers and customers given in Section 1.
2.1. Brief comments on the structure of the model The repeated matching game is useful for analyzing dynamic economic relationships
because repeating a prisoner’s dilemma game for many periods can provide the opportunity for individuals to cooperate and thus potentially adhere to a prescribed behavioral norm.
10
There are two other features of the game that are important. The first is the potential third party sanctions. If an individual deviates he or she will not only be potentially sanctioned
by the individual he or she deviated against but possibly by others too. The second is that the players are randomly matched. This is important since sanctioning is costly to the
individuals in each match. A player may not be able to credibly sustain the threat to punish an opportunistic individual in every period, but if there is random matching then the player
can credibly threaten to punish in one period because he knows he might not have to punish in the next period.
2.2. Status assignments At the beginning of each period every individual has a status assignment which is derived
from the history of his previous actions. After each stage game statuses are updated. Fol- lowing the notation of Okuno-Fujiwara and Postlewaite 1995, a status is an assignment in
each period t of an element X from the set of statuses X
i
, where X
i
= { mainstream, deviant}.
The two possible status labels used in this model are “mainstream” and “deviant”. A “mainstream” individual is someone who has been cooperative when matched with an-
other mainstream individual and has punished when matched with a deviant. Everyone else is a “deviant”. This model could include more status assignments. For example, in
some real-world situations, it may be that individuals who are only non-punishers but follow all other “mainstream” behavior might be sanctioned to a lesser degree than other
opportunistic individuals in the community.
11
I have chosen to restrict my model to two status assignments since two is sufficient to address the point that both opportunists and
non-punishers might expect to face costs from being sanctioned in future interactions as
10
See Fudenberg and Tirole 1991 for a summary of folk theorem results.
11
It could possibly also be true that the stigma or punishment from non-punishing is greater than the punishment received for being opportunistic. Heckathorn 1989, p. 97 discusses what he calls “hypocritical cooperation” in
which individuals act opportunistically but then punish others who follow similar behavior. His analysis suggests that “hypocritical cooperation may play a hitherto unacknowledged role in collective action”. This example would
imply that the perceived stigma from not punishing would be greater than that received for opportunistic behavior.
256 N. Brooks J. of Economic Behavior Org. 44 2001 249–267
well as the benefits they receive from their immediate actions. In future research, it would be interesting and important to compare the results that are generated with additional status
assignments with the results derived in this paper since the results from this paper cannot necessarily be generalized to the case with more status assignments.
An individual’s status is then updated in each period according to the transition map- ping τ
i
: XA
i
= X
i
X
j
A
i
→ X
i
. The transition mapping for player i specifies his next period’s status as a function of his current status X
i
, his match’s current status X
j
and his action A
i
, where A
i
= C, D, O is the set of action choices “cooperate”, “deviate”
or “ostracize”, respectively. The players know the payoff structure of the stage game and they also know the sta-
tus label of the player with whom they are matched i.e. whether he is “mainstream” or “deviant”. This information structure is assumed. We do not model how the information is
conveyed.
12
Additionally, an individual’s return w
i
is not publicly observable, although, in equilibrium players can deduce something about each other’s returns from observing
hisher status. It does not matter in this model whether they know precisely their partners’ returns to trade or not.
3. The social norm and the candidate equilibrium
