Asymmetric enforcement of cooperation in a social dilemma.
Nikiforakis, Nikos ; Normann, Hans-Theo ; Wallace, Brian 等
[I]f only a few very powerful actors want to promote a certain
pattern of behavior, their punishments alone can often be sufficient to
establish it, even if the others are not vengeful against defections.
(Axelrod 1997; p. 63)
1. Introduction
A number of daily decisions involve making a choice between the
private and the public interest. In these cases, following the private
interest reduces efficiency and imposes a negative externality on
others. A common way of dealing with negative externalities is the
imposition of sanctions on parties that deviate from a widely accepted
norm of behavior (Ostrom 1990). Given that the purpose of such sanctions
is to lower the return from acting against the social interest, the
efficacy of sanctions in enforcing cooperation will depend critically on
the punishment power of the sanctioning party. The greater the power of
the party abiding to a given norm, the less appealing a deviation from
that norm will be for other parties.
In this article, we present the results from a laboratory
experiment investigating the efficacy of decentralized sanctions in
fostering cooperation when negative externalities exist and players are
asymmetric in their punishment effectiveness. The use of a laboratory
experiment permits us to control a number of factors in a way that is
difficult to do in the field (e.g. endowments, returns to cooperation,
information), and focus on variables of interest, such as the
players' punishment power and the extent of the asymmetry in their
power.
To the best of our knowledge, this is the first study to
investigate behavior in asymmetric punishment institutions. However,
numerous social dilemma experiments have been conducted in which players
have the same punishment power (see below). Our goal is to examine how
the asymmetry in the players' punishment power affects cooperation
rates and efficiency, as well as how behavioral regularities observed in
symmetric institutions, such as the determinants of punishment, carry
over to asymmetric institutions.
For our study, we use the two-stage public-good game that was
introduced by Fehr and Gachter (2002). In the first stage, each
individual is given an endowment in experimental currency units (ECUs).
Individuals have to decide how much of their endowment they want to
contribute to a public account. The higher the contributions to the
public account, the higher the group earnings. However, every individual
has also an incentive to free ride and not contribute. In the second
stage, individuals are informed of each group member's contribution
to the public account and can reduce their earnings by assigning costly
punishment points. Each punishment point costs the punisher one ECU and
reduces the earnings of the punishment recipient by a factor larger than
one. We refer to this factor as punishment effectiveness and use the
term interchangeably with punishment and sanctioning power.
To evaluate the effect of asymmetries in the players'
punishment effectiveness, we compare behavior in symmetric and
asymmetric punishment institutions. In the symmetric case, as studied in
previous articles, group members are given the same punishment
effectiveness. That is, they are all equally effective in punishing each
other. By contrast, in asymmetric punishment institutions, players
differ in their punishment effectiveness: One player has higher
punishment effectiveness than the other three players, who all have the
same power. To ensure the robustness of our findings, we examined
behavior in two symmetric institutions that differ in the players'
average punishment effectiveness and four asymmetric punishment
institutions. The asymmetric institutions differ in the extent of the
asymmetry between strong and weak players.
Experimental economists have repeatedly used the two-stage
public-good game to analyze the resolution of the tension between social
good and self interest in symmetric punishment institutions (Fehr and
Gachter 2000, 2002; Masclet et al. 2003; Noussair and Tucker 2005; Page,
Putterman, and Unel 2005; Anderson and Putterman 2006; Bochet, Page, and
Putterman 2006; Carpenter 2007a, b; Carpenter and Matthews 2009; Sefton,
Shupp, and Walker 2007; Egas and Riedl 2008; Nikiforakis and Normann
2008; Nikiforakis, forthcoming). The two main goals of this literature
have been to understand what triggers punishment and whether the threat
of punishment can promote cooperation.
With respect to the motivation behind punishment, the experimental
results show that individuals tend to punish those who free ride by
contributing less than the group average even in one-shot interactions
(e.g., Fehr and Gachter 2002). This suggests that punishment does not
serve the exclusive purpose of increasing one's earnings by raising
future contributions to the public account (Falk, Fehr, and Fischbacher
2005). However, despite the non-strategic use of sanctions, punishment
is sensitive to economic incentives. Anderson and Putterman (2006),
Carpenter (2007a), Egas and Riedl (2008), and Nikiforakis and Normann
(2008) find that the higher the cost of punishment (which is the inverse
of punishment effectiveness), the less the punishment is inflicted on
free riders and the disciplinary power of decentralized punishment. (1)
With respect to the efficacy of punishment in fostering
cooperation, the evidence shows that when individuals interact
repeatedly with the same people throughout the experiment (fixed
matching), decentralized punishment leads to high cooperation rates as
free riders react to punishment by raising their contribution in
subsequent periods. When individuals interact with different people in
every period (random matching), cooperation rates are higher than they
are in the absence of a punishment institution, but lower than under
fixed matching. However, the efficacy of punishment depends critically
on its effectiveness. Nikiforakis and Normann (2008) find a monotonic
relation between punishment effectiveness, cooperation rates, and
efficiency. Unless punishment is sufficiently effective, cooperation
unravels. (2)
The main reason for studying behavior in asymmetric punishment
institutions is the empirical relevance of such institutions. In most
naturally occurring situations, asymmetric players are the norm rather
than the exception. (3) Individuals differ in their physical ability to
enforce cooperation just as countries differ in their military budgets
and technology. Given the documented willingness to punish free riders
even in one-shot interactions and the efficacy of symmetric punishment
institutions in promoting cooperation, we believe it is interesting to
investigate whether asymmetric institutions are as effective in
mitigating free riding.
Another reason for studying the performance of asymmetric
punishment institutions is that it seems challenging to predict the
effect these asymmetries will have on cooperation. This is because, for
our setting, arguments can be made in either direction. On the one hand,
an asymmetric institution might have a positive effect on cooperation.
First, experimental evidence suggests that higher punishment
effectiveness increases the demand for punishment (Anderson and
Putterman 2006; Carpenter 2007a; Egas and Riedl 2008; Nikiforakis and
Normann 2008). The high punishment effectiveness of strong players
suggests that they might be able to unilaterally enforce cooperation. In
contrast, in a symmetric punishment institution, more than one
individual might be required to discipline free riders (controlling for
average punishment effectiveness). This is essentially the reason
Axelrod (1997) argues that a "dominant power" will have a
positive effect on cooperation. Second, punishment of free riders is
(partly) a public good: All group members benefit from the increase in
cooperation because of punishment, but all have an incentive to let
others carry the cost of punishment. (4) By giving more power to one
player, the free rider problem at the punishment stage might be
alleviated. Similarly, endowing a single player with more punishment
power provides a natural focal point and might help alleviate any
coordination problems in punishment.
On the other hand, the asymmetric nature of the institution might
affect cooperation negatively. This can happen through two different
channels. First, strong players face a reduced threat relative to
players in symmetric institutions because we control for average
punishment effectiveness in our experiments. Therefore, they will be
more likely to free ride, ceteris paribus, if they are self-regarding.
In turn, this jeopardizes the cooperative performance of the whole group
because their behavior is likely to be imitated by reciprocal group
members (Fischbacher, Gachter, and Fehr 2001). Second, the reliance on
fewer individuals to enforce cooperation (or, in our case, the reliance
on a single strong player) might make cooperation more fragile compared
with a symmetric situation in that the outcome depends on the
preferences of the strong player who might not wish to enforce
cooperation.
The effect of asymmetries in public-good games has been studied
previously almost exclusively in the absence of punishment
opportunities. In general, there seems to be no consensus about the
effect of asymmetries on cooperation (see Varughese and Ostrom 2001;
Anderson, Mellor, and Milyo 2008). Isaac, McCue, and Plott (1985) were
the first to study behavior in a public-good game when agents have
asymmetric endowments from which to contribute. The authors argue that
the asymmetric environment gives the game-theoretic prediction of zero
contributions to the public account its best chance. Cherry, Kroll, and
Shogren (2005) provide evidence supporting this conjecture by comparing
treatments with symmetric and asymmetric endowments. Another study
finding evidence that asymmetries have a negative effect on cooperation
is Anderson, Mellor, and Milyo (2008), who examine a public-good game in
which participants receive different show-up fees. When this is made
salient through a public announcement, cooperation levels are negatively
affected. In contrast, Fisher et al. (1995) investigate the effect of
asymmetric returns to cooperation (i.e., unequal marginal per capita
returns) and find that they do not affect cooperation rates. Similarly,
Visser and Burns (2008) study a linear public-good experiment with South
African fishermen and find that heterogeneous endowments do not affect
the effectiveness of a symmetric punishment institution in promoting
cooperation.
The main results from our experiment are as follows. At the
aggregate level, asymmetric institutions are not only equally successful
in fostering cooperation as symmetric institutions, but they are also
equally efficient. At the individual level, we find that all of the
behavioral regularities observed in symmetric institutions carry over to
asymmetric institutions. Interestingly, we find that strong players do
not take advantage of their privileged position; they contribute amounts
to the public account similar to those of their weak counterparts.
However, strong players punish more than others and enjoy higher
earnings than weak players.
The rest of the article is organized as follows. Section 2 presents
the experimental design and the procedures, section 3 presents the
experimental results, and section 4 concludes.
2. The Experiment
The experiment is based on the design of Fehr and Gachter (2002),
who use the public-good game with n players. In each period, all
participants are given an endowment y. Players then decide
simultaneously and without communication how much of the endowment to
contribute to a public account, [c.sub.i], where 0 [less than or equal
to] [c.sub.i] [less than or equal to] y. The rest (y - [c.sub.i])
remains in the player's own account. In addition to the money
player i keeps, i receives a fixed proportion of the group's total
contribution to the public account, [alpha], where l/n < [alpha] <
1. The earnings of player i in the first stage of a period are
[[pi].sup.1.sub.i] = y - [c.sub.i] + [alpha] [n.summation over
(i=1)] [c.sub.i]. (1)
In the second stage of a period, participants are informed how much
the other individuals in their group contributed. They can then, if they
wish, purchase punishment points to reduce the earnings of one or more
other participants. Punishment is costly for the punisher. The price for
each punishment point is 1 ECU (experimental currency unit). Let
[p.sub.ij] denote the number of punishment points that player i assigns
to j (where i, j = 1, ..., n; j [not equal to] i), and [e.sub.i] denotes
i's punishment effectiveness, that is, the reduction in earnings
that one punishment point from player i causes to its recipient.
Punishment effectiveness is the inverse of the cost of punishment (i.e.,
the cost of reducing the earnings of player j by 1 ECU). Player i's
earnings at the end of a period are accordingly
[[pi].sup.2.sub.i] = y - [c.sub.i] + [alpha] [n.summation over
(i=1)] [c.sub.i] - [summation over (j[not equal to]i]) [p.sub.ij] -
[summation over (j[not equal to]i)] [e.sub.j][p.sub.ji]). (2)
The maximum number of points a participant can distribute to others
is equal to his earnings from the first stage; that is,
[[summation over].sub.j[not equal to]i] [p.sub.ij] [less than or
equal to] y - [c.sub.i] + [alpha] [[summation].sup.n.sub.i=1] [c.sub.i].
As in stage one, punishment decisions are made simultaneously and
without communication.
Table 1 describes the treatments in the experiment. The treatments
can be divided into two categories: symmetric treatments, wherein
[e.sub.j] is the same for all players, and asymmetric treatments, with
one "strong" and three "weak" players. In the
asymmetric treatments, the punishment effectiveness of the strong
player, [e.sub.s], is larger than the effectiveness of the weak players,
[e.sub.w]. In all treatments, it is common knowledge that y = 20, n = 4,
and [alpha] = 0.4. The treatment labels read [e.sub.s]_[e.sub.w], such
that the number on the left indicates the effectiveness of the strong
player and the number on the right the effectiveness of the weak
players. So, for example, in treatment 5_1, one punishment point from
the strong player reduces the earnings of its recipient by 5 ECU;
whereas, one punishment point from the weak players reduces the earnings
of its recipient by 1 ECU. In the instructions, strong players were
referred to as "type A" and weak players as "type
B." (5)
The treatments differ in two dimensions: First, average
effectiveness, [bar.e], is the average punishment effectiveness of the
group members, [bar.e] [equivalent to]([e.sub.s] + 3[e.sub.w])/4. We ran
treatments with [bar.e] = 2 and [bar.e] = 3. This permits us to test the
robustness of our findings with respect to the effect of asymmetric
punishment institutions. On the basis of the findings of Nikiforakis and
Normann (2008), we anticipated that the higher the level of average
punishment effectiveness the higher the level of cooperation. Second,
the asymmetry level indicates the relative strength of the strong
player's punishment and is denoted by l [equivalent to]
[e.sub.s]/[e.sub.w]. For both [bar.[epsilon]] = 2 and [bar.e] = 3, we
conducted sessions with l = 3 in addition to the symmetric control
sessions with l = 1. We also ran a treatment with [bar.[epsilon]] = 2
and l = 5 (5_1). However, we were concerned that a treatment in which
[bar.e] = 3 and l = 5 (i.e., 7.5_1.5) would be risking losses for the
weak players because of the magnitude of the strong players'
punishment effectiveness in this case. This could have caused
frustration and have led to erratic behavior. Instead, we decided to
conduct treatment 4_2.6 with [bar.e] = 3 and l = 1.5. (6)
Information feedback is as follows. At the beginning of each
experimental session, subjects are informed as to whether they are
assigned the role of a strong (type A) or a weak player (type B). These
roles remained fixed for the duration of the experiment. At the
beginning of each period, every player is randomly given a number
between 1 and 4 to distinguish their actions from those of the others in
that period. To prevent the formation of individual reputation, the
numbers are randomly reallocated in the beginning of every period.
Participants are aware of this. Such a mechanism ensures that, even
though the group members remain the same, the participants cannot link
the actions of the other subjects across the periods.
Once participants have completed stage 1, they are informed about
their group's total contribution to the public account, individual
contributions, and their earnings from the period, as given by Equation
1. At the end of each period, participants are informed about the total
number of punishment points they received from their peers, the
associated earnings reduction, and their earnings from the period, as
given by Equation 2. Subjects are not informed about the individual
punishment decisions of the other players. That is, subjects know
neither which of their peers punished them nor whether other group
members were punished. Participants only know how many points they
assigned to the other group members; thus, targeted counter-punishment,
as in Denant-Boemont, Masclet, and Noussair (2007) and Nikiforakis
(2008), is not possible. (7)
All treatments last for 10 periods. For the experiment, we use
fixed (or "partners") matching. This implies that each group
can be regarded as an independent observation. For treatments 2_2, 3_3,
and 5_1, we have six independent observations. For treatments 4_1.3,
4_2.6, and 6_2, we have five independent observations. In two cases, we
missed the target of six groups because individuals did not show up. In
the third case (4_1.3), we had to discard one group from the analysis
because of a bankruptcy problem. (8) The duration of each experiment and
the matching protocol were common knowledge. From Equations 1 and 2, and
backward induction, the finite duration of the experiment implies that
in the subgame perfect equilibrium, self-regarding players do not punish
and do not contribute to the public account in all periods and in all
treatments.
The experiments were conducted at Royal Holloway (University of
London) and University College London. The total number of participants
was 132 (not counting the discarded group). (9) The subjects were
recruited with the use of an e-mail list of voluntary potential
(student) candidates. Participants were from a variety of backgrounds.
None of the participants had participated previously in a public-good
experiment. Sessions lasted approximately 50 minutes. The rate of
exchange between the ECU and the British pound was 1 ECU = 0.04 [pounds
sterling]. The average earnings in the experiment were 10.61 [pounds
sterling] or roughly $20 at the time of the experiment. The experiments
were conducted with the use of z-Tree (Fischbacher 2007).
3. Results
We begin the analysis by taking an overview of the data and
reporting non-parametric tests of our key variables. We then proceed
with a more detailed analysis of the effect of the asymmetric punishment
institution on contributions to the public account, punishment behavior,
and earnings. All p-values reported are based on two-tailed tests.
Overview
Table 2 presents summary statistics from the experiment. Columns 2
to 4 include average contributions, columns 5 to 7 average punishment
inflicted, and columns 8 to 10 average earnings. The information is also
presented separately for strong and weak players. Looking at column 2,
one can see that, in all treatments, the punishment institution can
sustain cooperation at higher levels than those predicted by the subgame
perfect Nash equilibrium.
The results from non-parametric tests (counting one group as one
observation and using data from all periods) are as follows. At the
aggregate level, when we pool the data into two groups according to the
presence or not of asymmetries, we find that asymmetric institutions
appear to be as effective as symmetric institutions, in the sense that
contribution levels are not significantly different (Mann-Whitney
rank-sum test, p = 0.99), and as efficient, in the sense that earnings
are not significantly different (Mann-Whitney rank-sum test, p = 0.63).
Comparing the entries in columns 3 and 4, we see that the average
contribution of strong players tends to be lower than that of weak
players. However, this difference is not significant (Wilcoxon
signed-rank test, p = 0.30). Comparing columns 6 and 7, it appears that
strong players take the role of the enforcer of cooperation as they
punish more than weak players (Wilcoxon signed-rank test, p < 0.01).
In addition, columns 9 and 10 reveal that strong players have higher
earnings on average than their weak counterparts (Wilcoxon signed-rank
test, p = 0.02). Finally, contributions in treatments with [bar.e] = 3
are higher than in treatments with [bar.e] = 2 (Mann Whitney rank-sum
test, p = 0.07; column 2); whereas, earnings are not (Mann-Whitney
rank-sum test, p = 0.45; column 8).
[FIGURE 1 OMITTED]
Contributions to the Public Account
The effect of average effectiveness ([bar.e]) and asymmetry level
(l) on contribution rates can be identified clearly in Figure 1.
Contributions in treatments with [bar.e] = 3 are at a high level and,
overall, appear to be increasing over time. On the other hand,
contributions in treatments with [bar.e] = 2 are at a lower level and
remain more or less constant. The striking fact is that the evolution of
contributions is very similar amongst the three treatments with [bar.e]
= 2 (the lower three lines) and also amongst the three treatments with
[bar.e] = 3 (the upper three lines). The level of asymmetry appears to
have no effect on cooperation at the aggregate level.
Table 3 shows the results of a series of regressions investigating
the effect of different sets of explanatory variables on contributions.
To model the effect of asymmetric institutions, we use Asymmetric, a
dummy variable that takes the value of 1 for all asymmetric treatments
(i.e., whenever l = [e.sub.s]/[e.sub.w] > 1) and that equals 0
otherwise, or separate dummy variables for each realized level of
asymmetry. The other independent variables are Three, a dummy variable
that takes the value of 1 for all treatments with [bar.e] = 3 and that
equals 0 otherwise; Strong, a dummy variable that takes the value of 1
when individual i is a strong player in an asymmetric treatment and that
equals 0 otherwise; (10) Period to account for the different time paths
in Figure 1, and the interaction Period*Three. (11) All regressions
include individual random effects, to account for the fact that we have
repeated observations from the same individuals, and group random
effects, to control for the interaction within groups. (12)
The results in Table 3 are fully consistent with the non-parametric
tests and the observations made regarding Figure 1. Contributions in
asymmetric institutions are not significantly different from those in
symmetric institutions. Similarly, the level of asymmetry does not have
a significant effect on contributions. As average effectiveness
increases from [bar.e] = 2 to [bar.e] = 3, so do average contributions.
The difference increases over time, as indicated by the positive sign of
Period*Three. Strong players contribute less than weak players on
average. However, this difference fails to be significant. This implies
that strong players do not exploit their relative power to free ride. We
summarize:
RESULT l. Asymmetric punishment institutions are as successful in
fostering cooperation as symmetric institutions.
RESULT 2. A higher average effectiveness of punishment
significantly increases contributions.
RESULT 3. Strong and weak players contribute similar amounts to the
public account.
Punishment Behavior
We now turn our attention to punishment behavior. The literature
has already identified some regularities with regard to punishment
behavior (see, e.g., Carpenter and Matthews 2009). The first is that
punishment is mostly aimed toward individuals who contribute less than
their peers on average and that the severity of punishment increases as
the difference between the free rider's contribution and that of
his group members becomes larger. Second, individuals often punish group
members who contribute less than they do, irrespective of the
target's relative position in the group. We will look for these
regularities in our data set. In addition, we are interested in
providing answers to two questions that have previously not been
investigated: (i) Do strong players punish more than weak players? And
if they do, what accounts for the difference in the behavior of strong
and weak players? (ii) Does punishment differ in asymmetric
institutions? That is, is there more or less punishment in asymmetric
institutions?
To address the first question, Figure 2 compares the punishment of
strong, symmetric, and weak players as a function of the deviation of
player j's contributions from that of his peers. Figure 2 provides
clear evidence that, as in previous studies, the greater the extent of
free riding, the greater the punishment. With respect to the punishment
activity of strong and weak players, Figure 2 reveals that, for all
levels of deviation, strong players are found to punish more than
symmetric and weak players. However, Figure 2 cannot explain the cause
of this difference. This could be because of the low punishment cost for
strong players (Anderson and Putterman 2006; Carpenter 2007a; Egas and
Riedl 2008; Nikiforakis and Normann 2008), but it could also be because
of the very role of the strong player. For example, it is plausible that
some weak players abstain from punishing, knowing that the strong player
will punish free riders. As a result, strong players might adopt (or
might be pushed to adopt) the role of the cooperation enforcer and
punish more than they otherwise would.
Table 4 presents a series of regressions analyzing the determinants
of punishment inflicted. Formally, the dependent variable in these
regressions is [p.sub.ij][e.sub.i]. As before, we build our model in
steps, starting with our variables of interest and controlling for other
factors that might be affecting behavior. The first regression shows no
difference in the punishment activity in symmetric and asymmetric
institutions. The result is robust when we control for the average
punishment effectiveness in regression (2), which also shows that
punishment is higher when average effectiveness increases.
In regression (3), the negative coefficient of Asymmetric shows
that weak players punish less than players in symmetric treatments. The
positive coefficient of Strong shows that strong players punish more
than players in symmetric treatments.
To test for the robustness of these findings and to account for the
regularities in punishment behavior discussed at the beginning of the
section, regressions (4) to (7) include the independent variables
[Own_Neg_Diff.sub.j,t] [equivalent to] max{0, [c.sub.j,t] -
[c.sub.i,t]},
[Group-Neg-Diff.sub.j,t] [equivalent to]
max{0,([[summation].sub.h[not equal to]j] [c.sub.h,t])/(n-1)},
as well as
[Own_Pos_Diff.sub.j,t] [equivalent to] max{0, [c.sub.j,t] -
[c.sub.i,t]},
and
[Group-Pos-Diff.sub.j,t] [equivalent to] max{0,[c.sub.j,t] -
([[summation].sub.[not equal to]j] [c.sub.h,t])/(n-1)},
where [c.sub.j,t] is the contribution of individual j (i.e., the
target of the punishment) in period t and [c.sub.i,t] is the
contribution of the punisher, individual i, in period t. Given the
diminishing returns to punishment as the experiment approaches its end,
the variable Period is also included in these regressions. The results
show that deviating from the group average increases the extent of
punishment ([Group_Neg_Diff.sub.j,t]). Similarly, individuals on average
punish more heavily group members who contribute less than they do
([Own_Neg_Diff.sub.j,t]). The variable Period is found to be highly
significant. Given that we control for the variance in contributions in
regressions (4) to (7), the negative sign of Period suggests that
punishment is to some extent used strategically to promote cooperation;
as the experiment approaches the end, the future benefits of punishment
decrease. As a result, so does punishment. Therefore, we conclude that
we find in our data set the punishment patterns observed in previous
experiments.
To understand the reason behind the higher punishment activity of
strong players, we need to disentangle the effect of being a strong
player from that of having a low punishment cost and to evaluate their
relative effect on punishment behavior. To this end, regressions (6) and
(7) add Punishment Effectiveness (el) as an explanatory variable. Of
course, ei has to be significant by construction. However, if the
variable Strong remains significant in these regressions, it will imply
that the mere role of a strong player induces individuals to purchase
more points to punish others.
The results of the regressions indicate that the higher punishment
activity of strong players (and the lower activity of weak players) can
be attributed to their higher punishment effectiveness (or,
alternatively, their lower punishment cost). Comparing the estimates in
regressions (4) and (5) and regressions (6) and (7), we see that the way
we model asymmetric treatments has little effect on our results.
Moreover, we see that there does not appear to be a systematic relation
between punishment and the level of asymmetry.
Results 4 and 5 summarize the main findings from this section and
provide answers to questions (i) and (ii) stated above.
RESULT 4. Strong players punish more than weak players. The
difference is due to the higher punishment effectiveness of strong
players.
RESULT 5. The punishment inflicted does not differ between
asymmetric and symmetric punishment institutions.
Figure 2 and Table 4 overlook the fact that, of the 3960 possible
punishment cases (132 participants times 3 targets per period times 10
periods), punishment was carried out in only 613 cases. That is, the
modal behavior in the second stage is not to punish. It seems possible
that the asymmetric institutions have an effect on punishment that is
masked by studying the extent of punishment. For example, individuals
who would normally abstain from punishing might be more willing to
engage in punishment if they are assigned the role of the strong player.
In other words, the asymmetric nature of the institutions might affect
the decision as to whether to punish or not.
[FIGURE 3 OMITTED]
Figure 3 gives a first answer whether this is the case in the
experiment. As before, we plot the deviation of player j's
contributions from that of his peers on the horizontal axis. On the
vertical axis, we plot the likelihood of j being punished by strong,
symmetric, and weak players. For all levels of deviation, strong players
are found to be more likely to punish than weak players. (13) The
difference exceeds 20% in some cases.
To evaluate the significance of the difference in Figure 3, Table 5
presents a series of probit regressions. The dependent variable is a
dummy, taking the value of 1 if subject i punished subject j and equal
to 0 otherwise. The logic behind the different regressions is the same
as that in Table 4. The results from the regressions show that once we
control for individual and group random effects the difference between
strong and other players in the propensity to punish is not significant.
(14) Therefore, we conclude that the role of a player does not influence
significantly the decision to punish or not in our experiment. The
decision to punish appears to be a negative function of the time that
has elapsed and the extent of free riding.
Evolution of Contributions
To better understand punishment behavior, we need to take a look at
how individuals adjust their contributions across periods. As before, we
will be looking for regularities observed in previous public-good
experiments, while focusing our attention on two questions that have
previously not been investigated: (i) How do strong players adjust their
behavior across periods? (ii) Is the way in which individuals adjust
their contributions across periods different in symmetric and asymmetric
institutions?
To provide answers for these questions, Table 6 presents
regressions of the changes in individual contributions from period t to
period t + 1 (i.e., [c.sup.t.sub.i] + - [c.sup.t.sub.i]), on whether a
subject was assigned the role of a strong player, the average
effectiveness in the punishment institution (Three), the punishment
inflicted to player i in period t, the average contribution of the other
group members in period t, and a variable that controls for the number
of periods left (Period). As we did before, we present regressions in
which we pool the data across asymmetric institutions and regressions in
which we include dummies for each level of asymmetry.
Previous studies have shown that the way in which individuals
respond to punishment depends on their relative position in the group
and, in particular, whether a subject was contributing more or less than
his peers on average (e.g., Masclet et al. 2003). For this purpose, we
run separate regressions for the individuals who contributed more than
the average of the group in period t (henceforth, high contributors) and
those who contributed less than the average of the group in period t
(henceforth, low contributors).
We begin by addressing question (i). Table 6 shows that strong
players do not adjust their contributions between periods differently
from weak players. This is interesting given the reduced threat that
they face. Regarding question (ii), we find that the way in which
individuals adjust their contributions from one period to the other is
not different in asymmetric and symmetric institutions. However,
regression (4) provides some evidence that there is a positive
relationship between the level of asymmetry and the reduction in
contribution in period t + 1 for high contributors. A possible
explanation for this is that if a high contributor reduces her
contribution, she risks becoming a low contributor. In treatments with
high levels of asymmetry, low contributors might be subject to (heavy)
punishment from strong players.
The significant constant in regressions (1) and (3) in Table 6
shows that low contributors increase their contributions to the public
account in the following period on average. In contrast, the significant
constant in regressions (2) and (4) show that, as in previous studies,
high contributors lower their contributions in the following period.
These results explain why punishment reduces the variance in
contributions within groups; if we compare the standard deviation of
contributions in the first and the second half of the experiment
(excluding the final period), we find that standard deviation is smaller
in the second half of the experiment in 31 out of 33 groups. The reason
behind the observed convergence in contribution levels is presumably
because contributing less than the group average triggers punishment;
whereas, contributing more than the average is individually costly.
Regressions (1) and (3) in Table 6 show that low contributors
respond to punishment by increasing their contribution in period t + 1;
whereas, they are not influenced by the contribution of the other group
members in period t. For high contributors, the relation between the
contribution of other group members in period t and the contribution of
high contributors in period t + 1 is positive, but punishment does not
have an effect. The latter might be because of the few cases in which
high contributors were punished. These results are comparable to those
in previous studies (e.g., Carpenter 2007a). High and low contributors
become less responsive to punishment as the experiment approaches the
end. We summarize:
RESULT 6. The change in contributions across periods is similar in
symmetric and asymmetric institutions. However, the higher the level of
asymmetry, the lower the reduction in the contribution of high
contributors in the following period.
RESULT 7. Strong and weak players adjust their contributions over
time in a similar manner.
RESULT 8. High (low) contributors in period t tend to lower
(increase) their contribution in period t + 1. The increase in the
contribution of low contributors becomes greater as punishment
increases.
Earnings
Earnings are a measure of the efficiency of a punishment
institution. From Equation 2, the earnings per period is 20 ECU per
person in the subgame perfect Nash equilibrium (no punishments and no
contributions). If each member contributes the whole endowment to the
public account and abstains from punishment, each individual will earn
32 ECU. These are the benchmarks against which we measure the
performance of asymmetric and symmetric punishment institutions.
The average earnings in each of the six treatments can be found in
Table 2 (column 8). As indicated by Equation 1, higher contributions
(column 2) imply higher group earnings. However, earnings are reduced by
the punishment inflicted to individuals (column 5) and the cost of
punishment paid by the punisher. Figure 4 complements Table 2 by
illustrating the evolution of average earnings for each of the
treatments separately.
[FIGURE 4 OMITTED]
The following facts become apparent. First, earnings are somewhere
between the Pareto-optimal earnings of 32 ECU and the earnings predicted
by the subgame perfect Nash equilibrium of 20 ECU. Second, there appear
to be no pronounced differences in earnings between asymmetric and
symmetric institutions. This is not surprising given Results 1 and 5.
Columns 9 and 10 in Table 2 present the earnings of strong and weak
players. Strong players have higher earnings in all treatments except
4_1.3. Given the similar amounts contributed by strong and weak players
to the public account, the difference in earnings can be attributed to
the greater punishment inflicted by strong players (see columns 6 and 7)
and the lower cost of punishment for strong players.
Table 7 presents the results of a random-effects regression in
which the dependent variable is the earnings of individual i at the end
of a period, [[pi].sup.2.sub.i]. Earnings are not significantly
different in symmetric and asymmetric institutions. Also, the relation
between earnings and the level of asymmetry does not appear to be
systematic, as it can be seen in regression (5). Strong players have
higher earnings than weak players and players in symmetric treatments.
Regressions (2) and (3) show that, on average, the level of average
effectiveness does not have a significant effect on earnings. However,
earnings increase over time in treatments with [bar.e] = 3 relative to
earnings in treatments with [bar.e] = 2, as regressions (4) and (5)
show. This can be attributed to the increasing contributions in
treatments with [bar.e] = 3 and the falling expenditure on punishment.
In contrast, although punishment also declines in treatments with
[bar.e] = 2, contributions remain stable over time. We summarize:
RESULT 9. Earnings are not significantly different in asymmetric
and symmetric institutions.
RESULT 10. The average effectiveness of punishment does not have a
significant effect on average earnings.
RESULT 11. Strong players have significantly higher earnings than
weak players.
4. Conclusion
Most naturally occurring interactions involve players that are
asymmetric in their ability to enforce cooperation when free riding
incentives exist. In this article, we have presented the results from a
laboratory experiment to investigate the efficacy of an asymmetric
punishment institution in fostering cooperation. As such, our study
contributes to two strands of the literature on social dilemmas. The
first of them examines whether groups can sustain cooperation when free
riding incentives exist and there is no central authority to enforce
cooperation. To this end, we find that the asymmetric punishment
institution is as effective at sustaining cooperation and is as
efficient as the symmetric institution. In other words, asymmetries in
punishment effectiveness neither promote cooperation nor do they
constitute an obstacle to it. The second strand of the literature deals
with the effect of asymmetries on cooperation (see Isaac, McCue, and
Plott 1985; Fisher et al. 1995; Cherry, Kroll, and Shogren 2005;
Anderson, Mellor, and Milyo 2008; Visser and Burns 2008). The effect of
different types of asymmetries on cooperation has only mixed support for
the conjecture that asymmetries hinder cooperation (Varughese and Ostrom
2001). Our article provides further evidence against this conjecture by
showing that heterogeneity in punishment effectiveness does not harm
cooperation.
At the individual level, we find that the behavioral regularities
observed in symmetric punishment institutions are also found in
asymmetric punishment institutions. Punishment is mostly aimed toward
free riders. It increases with the extent of free riding and decreases
with the cost of punishment. Interestingly, we also find that the
percentage of punishments aimed toward cooperators is equally high in
symmetric and asymmetric institutions, even though there are strong
players in the latter. Another interesting finding is that strong
players do not exploit their privileged position and contribute similar
amounts as weak players to the public account. This might be taken as
further evidence for the importance of reciprocity in social dilemmas:
Strong players either do not wish to exploit their peers or understand
that by exploiting them they will harm cooperation. Strong players,
however, punish more than weak players. The increased punishment
activity can be attributed to the reduced cost of punishing for strong
players rather than strong players adopting the role of the enforcer.
The upshot is that strong players benefit from the asymmetry and enjoy
higher earnings than the weak players.
Our study should be seen as only a first step toward understanding
how players that differ in their punishment effectiveness can cooperate
in the presence of externalities without an external intervention. It
would be interesting to examine the effect of asymmetric punishment
institutions when punishment can lead to a cycle of punishment and
counter-punishment (e.g., Nikiforakis and Engelmann 2008). One
possibility is that the presence of strong players will reduce the cases
of counter-punishment in fear of further reprisals. This could
potentially lead to higher rates of cooperation and efficiency. It will
be also interesting to see how players in public-good games would behave
if they were given the opportunity to invest in improving their
punishment effectiveness. The difference in earnings between strong and
weak players could indicate that, given the option, weak players will
have an incentive to invest a part of their endowment in enhancing their
punishment effectiveness, and such incentives might give rise to an
"arms race" for punishment effectiveness.
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(1) Similarly, Denant-Boemont, Masclet, and Noussair (2007),
Nikiforakis (2008a), and Nikiforakis and Engelmann (2008) observe that,
if counter-punishment is possible, individuals are less willing to
punish free riders. This behavior reflects an increase in the expected
cost of enforcing cooperation.
(2) Nikiforakis (forthcoming) finds that the efficacy of
decentralized punishment also depends critically on the information
available to subjects. If subjects receive information about individual
earnings (rather than individual contributions), the threat of
punishment cannot prevent the breakdown of cooperation.
(3) Of course, in field settings, players may well differ not only
in their punishment effectiveness but also in their endowments, the
returns from cooperation, and so on. In this article, we focus on the
effect of asymmetric punishment effectiveness on behavior while
maintaining symmetry elsewhere.
(4) The fact that individuals punish even in one-shot interactions
suggests that punishment has private returns. On the notion of
punishment as a second-order public good, see Yamagishi (1986) and
Heckathorn (1989).
(5) Neutral language was used throughout the instructions.
Punishment points were termed "points that reduce another
player's income." The instructions can be downloaded at
www.economics.unimelb.edu.au/nnikiforakis/research.htm.
(6) We decided not to use treatments with [bar.e] = 1 because this
would require [e.sub.w] < 1 in the asymmetric treatments. If
[e.sub.w] < 1, a punishment carried out by the weak players increases
payoff inequality between the victim and the punisher. This would
prevent a ceteris paribus comparison with treatments in which [bar.e] =
2 and [bar.e] = 3 because, in these eases, punishment reduces
inequality. Similarly, we chose not to study treatments with [e.sub.w] =
0, (e.g., 8_0 or 12_0 such that 1 = [infinity]) because these treatments
would not only risk losses for weak players but also differ along two
dimensions relative to the other treatments: the level of asymmetry and
the fact that weak players cannot punish at all.
(7) As one of our referees points out, although targeted
counter-punishment is not possible in our experiment, it is possible
that a free rider anticipating being punished (e.g., because he has
contributed substantially less than his peers) uses preemptive
"counter-punishment." Indeed, there is evidence that
cooperators often get punished (e.g., Anderson and Putterman 2006;
Cinyabuguma, Page, and Putterman 2006; Hermann, Thoeni, and Gachter
2008). Nikiforakis (2008, pp.102-3) reports evidence suggesting that
punishment of cooperators might be more from a dislike of cooperators
and less because of pre-emptive "counter-punishment."
(8) The group had one subject contributing nothing for several
periods and the other three subjects, who contributed the maximum
amount, punished her harshly, causing the bankruptcy of the subject in
period 5. For the experiment to continue, we credited her account with 5
[pounds sterling]. Although we did continue collecting data beyond that
period, we decided not to include the group in the data analysis. In the
first five periods, punishment in this group was six times higher than
the treatment average, which would cause a severe bias if we included
this group in our statistical analysis. Moreover, a post-experimental
questionnaire indicated that the behavior of the punished person was due
to a misunderstanding.
(9) The observations for the symmetric treatments 2_2 and 3_3 are
taken from Nikiforakis and Normann (2008), who recruited individuals
from the same subject pool and used the same experimental procedures.
(10) We ran alternative regressions in which the Strong dummy is
equal to 1 for strong players in the asymmetric treatments and for all
players in the symmetric treatments. Results from these regressions did
not differ qualitatively from the ones reported here.
(11) We also ran regressions (not reported in the article)
interacting Strong with Period to see whether strong players behave
differently over time relative to the other players. The interaction
term was always far from being significant.
(12) The estimation was made with the use of Generalized Linear
Latent and Mixed Models (GLLAMM) (Rabe-Hesketh and Skrondal 2005). The
downside of using GLLAMM is that we cannot use a Tobit specification to
account for the considerable concentration of observations with
contributions of 20 ECU. If we use Tobit with random effects at either
the individual or group level and compare the results with the
respective linear regressions, results are qualitatively similar, and
most importantly, none of our main results are affected.
(13) An oddity in the literature on punishment in social dilemmas
is the tendency to punish cooperators. Despite the presence of strong
players in asymmetric institutions, the percentage of punishments aimed
toward cooperators is equally high in symmetric and asymmetric
institutions (18.1% and 17.6%, respectively).
(14) We also ran a regression in which the omitted category was
Weak (rather than Symmetric). The coefficient of Strong remained
insignificant at all conventional levels, showing that the difference in
the propensity to punish between strong and weak players is not
significant once we control for individual and group random effects.
Nikos Nikiforakis, * Hans-Theo Normann, ([dagger]) and Brian
Wallace ([double dagger])
* Department of Economics, The University of Melbourne, 3010,
Victoria, Australia; E-mail n.nikiforakis@ unimelb.edu.au; corresponding
author.
([dagger]) Department of Management and Applied Microeconomics,
Goethe University Frankfurt am Main, Grueneburgplatz 1, 60629 Frankfurt,
Germany; E-mail normann@econ.uni-frankfurt.de.
([double dagger]) ELSE & Department of Economics, University
College London, Gower Street, London, WC1E 6BT, UK; E-mail
brian.wallace@ucl.ac.uk.
The authors gratefully acknowledge funding from the Economic and
Social Research Council (project RES-000-22-0948). We also thank the
editor, Laura Razzolini, two anonymous referees, and Christoph Engel,
Dirk Engelmann, Simon Gachter, Steffen Huck, and Andreas Nicklisch for
their helpful comments, as well as seminar participants at La Trobe
University, Max Planck Institute (Bonn), Royal Holloway (University of
London), University College London, and University of Melbourne.
Finally, we would like to thank participants at the American Meetings of
the Economic Science Association in Tucson in 2005 and the European
Meetings of the Economic Science Association in Alessandria in 2005 and
in Nottingham in 2006.
Received February 2008; accepted February 2009.
Table 1. Experimental Design
Average Effectiveness Effectiveness
Effectiveness of Strong of Weak
Treatment ([bar.e]) (a) ([e.sub.s]) (b) ([e.sub.w]) (b)
2_2 2 2 2
4_1.3 2 4 1.3
5_1 2 5 1
3_3 3 3 3
4_2.6 3 4 2.6
6_2 3 6 2
Asymmetry No. of
Treatment Level (l) (c) participants
2_2 1 24
4_1.3 3 20
5_1 5 24
3_3 1 24
4_2.6 1.5 20
6_2 3 20
(a) Average Effectiveness is the average punishment
effectiveness of the group members; that is,
([e.sub.s] + [3e.sub.w])/4.
(b) Effectiveness refers to the income reduction in ECU
caused to the recipient by a single punishment point.
(c) Asymmetry Level is defined as [e.sub.s]/[e.sub.w].
Table 2. Summary Statistics (All Entries Are Average ECU)
Contribution Contribution Contribution
by All by Strong by Weak
Treatment (1) Players (2) Players (3) Players (4)
2_2 11.83 -- --
4_1.3 12.36 13.42 12.00
5_1 12.49 10.30 13.22
Asymmetric
([bar.e] = 2)
(b) 12.43 11.72 12.67
3_3 15.87 -- --
4_2.6 15.92 15.64 16.01
6_2 15.17 14.86 15.27
Asymmetric
([bar.e] = 3)
(b) 15.55 15.25 15.65
Punishment (a) Punishment (a) Punishment (a)
by All by Strong by Weak
Treatment (1) Players (5) Players (6) Players (7)
2_2 0.71 -- --
4_1.3 0.69 1.49 0.42
5_1 0.34 0.92 0.15
Asymmetric
([bar.e] = 2)
(b) 0.50 1.18 0.27
3_3 0.84 -- --
4_2.6 0.65 0.80 0.61
6_2 1.07 2.80 0.49
Asymmetric
([bar.e] = 3)
(b) 0.86 1.80 0.55
Earnings of Earnings Earnings
All Players of Strong of Weak
Treatment (1) (8) Players (9) Players (10)
2_2 23.90 -- --
4_1.3 24.34 23.24 24.69
5_1 26.00 28.40 25.20
Asymmetric
([bar.e] = 2)
(b) 25.24 26.05 24.97
3_3 26.15 -- --
4_2.6 26.91 27.85 26.56
6_2 24.99 26.17 24.59
Asymmetric
([bar.e] = 3)
(b) 25.93 27.01 25.57
(a) "Punishment" refers to the earnings reduction caused
by assigning points.
(b) "Asymmetric" pools data from the asymmetric treatments
ignoring the level of asymmetry.
Table 3. Determinants of Contributions
Dependent Variable: Contribution of Player i,
([c.sub.i])
(1) (2) (3)
Asymmetric 0.06 (1.69) 0.15 (1.57) 0.32 (1.58)
Three 3.45 ** (1.51) 3.45 ** (1.51)
Strong -0.68 (0.61)
Period
Period*Three
Asymmetry
level: 1.5
Asymmetry
level: 3
Asymmetry
level: 5
Constant 13.85 *** (1.35) 12.13*** (1.46) 12.13*** (1.46)
Observations 1320 1320 1320
Subject variance 4.31 [0.86] 4.31 [0.86] 4.24 [0.85]
Group variance 20.22 [5.35] 17.25 [4.62] 17.27 [4.62]
Log likelihood -3870.96 -3868.53 -3867.91
Dependent Variable: Contribution of Player i,
([c.sub.i])
(4) (5)
Asymmetric 0.32 (1.58)
Three 0.96 (1.57) 0.99 (1.89)
Strong -0.68 (0.61) -0.68 (0.61)
Period 0.03 (0.05) 0.03 (0.05)
Period*Three 0.45 *** (0.08) 0.45 *** (0.08)
Asymmetry 0.50 (2.49)
level: 1.5
Asymmetry 0.09 (1.86)
level: 3
Asymmetry 0.55 (2.36)
level: 5
Constant 11.97 *** (1.49) 11.96 *** (1.58)
Observations 1320 1320
Subject variance 4.34 4.34 [0.85]
Group variance 17.27 17.24 [4.62]
Log likelihood -3830.77 -3830.75
Linear regression: Estimation was done in STATA 10.1 using GLLAMM with
random effects at the individual and group level, and adaptive
quadrature. Standard errors are in parentheses. Square brackets
include the covariance of the random effects.
** Significant at the 5% level.
*** Significant at the 1% level.
Table 4. Determinants of Punishment
Dependent Variable: Punishment Inflicted on Player j
(1) (2) (3)
Asymmetric -0.10 (0.17) -0.10 (0.16) -0.37 ** (0.17)
Three 0.28 * (0.15) 0.28 * (0.15)
Strong 1.07 *** (0.19)
Period -0.08 *** (0.01)
[Group_Neg_Diff.
sub.j,t]
[Group_Pos_Diff.
sub.j,t]
[Own_Pos_Diff.
sub.j,t]
[Own_Neg_Diff.
sub.j,t]
Asymmetry
level: 1.5
Asymmetry
level: 3
Asymmetry
level: 5
Effectiveness
Constant 0.78 *** (0.13) 0.64 *** (0.15) 1.09 *** (0.16)
Observations 3960 3960 3960
Subject variance 0.55 [0.10] 0.55 [0.10] 0.37 [0.08]
Group variance 0.03 [0.06] 0.02 [0.05] 0.06 [0.05]
Log likelihood -8931.54 -8929.98 -8893.3
Dependent Variable: Punishment
Inflicted on Player j
(4) (5)
Asymmetric -0.38 ** (0.16)
Three 0.44 *** (0.14) 0.45 *** (0.16)
Strong 1.15 *** (0.19) 1.19 *** (0.18)
Period -0.05 *** (0.01) -0.05 *** (0.01)
[Group_Neg_Diff. 0.20 *** (0.02) 0.20 *** (0.02)
sub.j,t]
[Group_Pos_Diff. 0.02 (0.02) 0.02 (0.02)
sub.j,t]
[Own_Pos_Diff. -0.00 (0.01) -0.00 (0.01)
sub.j,t]
[Own_Neg_Diff. 0.14 *** (0.02) 0.14 *** (0.02)
sub.j,t]
Asymmetry -0.66 *** (0.22)
level: 1.5
Asymmetry -0.17 (0.17)
level: 3
Asymmetry -0.54 ** (0.21)
level: 5
Effectiveness
Constant 0.20 (0.16) 0.19 (0.16)
Observations 3960 3960
Subject variance 0.42 [0.07] 0.43 [0.07]
Group variance 0.02 [0.04] 0.00 [0.00]
Log likelihood -8406.73 -8403.57
Dependent Variable: Punishment
Inflicted on Player j
(6) (7)
Asymmetric 0.06 (0.20)
Three -0.14 (0.22) -0.12 (0.23)
Strong -0.61 (0.55) -0.63 (0.54)
Period -0.05 *** (0.01) -0.05 *** (0.01)
[Group_Neg_Diff. 0.20 *** (0.02) 0.20 *** (0.02)
sub.j,t]
[Group_Pos_Diff. 0.02 (0.02) 0.02 (0.02)
sub.j,t]
[Own_Pos_Diff. -0.01 (0.01) -0.00 (0.01)
sub.j,t]
[Own_Neg_Diff. 0.14 *** (0.02) 0.14 *** (0.02)
sub.j,t]
Asymmetry -0.15 (0.27)
level: 1.5
Asymmetry 0.29 (0.22)
level: 3
Asymmetry -0.07 (0.24)
level: 5
Effectiveness 0.58 *** (0.17) 0.58 *** (0.17)
Constant -0.94 ** (0.37) -0.97 ** (0.38)
Observations 3960 3960
Subject variance 0.39 [0.07] 0.37 [0.06]
Group variance 0.03 [0.04] 0.01 [0.03]
Log likelihood -8400.97 -8398.33
Linear regression: Estimation was done in STATA 10.1 using GLLAMM
with random effects at the individual and group level and adaptive
quadrature. Standard errors are in parentheses.
Square brackets include the covariance of the random effects.
* Significant at the 10% level.
** Significant at the 5% level.
*** Significant at the 1% level.
Table 5. Likelihood of Punishing Player j
Dependent Variable: Likelihood of Punishing Player j
(1) (2) (3)
Asymmetric -0.01 (0.04) -0.01 (0.04) -0.02 (0.04)
Three 0.02 (0.03) 0.02 (0.03)
Strong 0.03 (0.03)
Period -0.02 *** (0.00)
[Group_Neg_Diff.
sub.j,t]
[Group_Pos_Diff.
sub.j,t]
[Own_Pos_Diff.
sub.j,t]
[Own_Neg_Diff.
sub.j,t]
Asymmetry
level: 1.5
Asymmetry
level: 3
Asymmetry
level: 5
Effectiveness
Constant 0.16 *** (0.03) 0.15 *** (0.03) 0.24 *** (0.03)
Observations 3960 3960 3960
Subject variance 0.01 [0.00] 0.01 [0.00] 0.01 [0.00]
Group variance 0.01 [0.00] 0.01 [0.00] 0.01 [0.00]
Log likelihood -1404.77 -1404.67 -1365.01
Dependent Variable: Likelihood of Punishing
Player j
(4) (5)
Asymmetric -0.02 (0.03)
Three 0.03 (0.03) 0.03 (0.03)
Strong 0.04 (0.03) 0.04 (0.03)
Period -0.01 *** (0.00) -0.01 *** (0.00)
[Group_Neg_Diff. 0.03 *** (0.00) 0.03 *** (0.00)
sub.j,t]
[Group_Pos_Diff. 0.00 (0.00) 0.00 (0.00)
sub.j,t]
[Own_Pos_Diff. -0.01 *** (0.00) -0.01 *** (0.00)
sub.j,t]
[Own_Neg_Diff. 0.01 *** (0.00) 0.01 *** (0.00)
sub.j,t]
Asymmetry -0.04 (0.04)
level: 1.5
Asymmetry 0.02 (0.03)
level: 3
Asymmetry -0.06 (0.04)
level: 5
Effectiveness
Constant 0.14 *** (0.03) 0.14 *** (0.03)
Observations 3960 3960
Subject variance 0.01 [0.00] 0.01 [0.00]
Group variance 0.00 [0.00] 0.00 [0.00]
Log likelihood -886.6 -884.3
Dependent Variable: Likelihood of Punishing
Player j
(6) (7)
Asymmetric 0.01 (0.04)
Three -0.01 (0.04) -0.02 (0.04)
Strong -0.09 (0.09) -0.09 (0.09)
Period -0.01 *** (0.00) -0.01 *** (0.00)
[Group_Neg_Diff. 0.03 *** (0.00) 0.03 *** (0.00)
sub.j,t]
[Group_Pos_Diff. 0.00 (0.00) -0.01 *** (0.00)
sub.j,t]
[Own_Pos_Diff. -0.01 *** (0.00)
sub.j,t]
[Own_Neg_Diff. 0.01 *** (0.00) 0.01 *** (0.00)
sub.j,t]
Asymmetry -0.01 (0.05)
level: 1.5
Asymmetry 0.05 (0.04)
level: 3
Asymmetry -0.03 (0.04)
level: 5
Effectiveness 0.04 0.04 (0.03)
Constant 0.05 0.05 (0.06)
Observations 3960 3960
Subject variance 0.01 [0.00] 0.01 [0.00]
Group variance 0.00 [0.00] 0.00 [0.00]
Log likelihood -885.36 -883.07
Probit regression: Estimation was done in STATA 10.1 using GLLAMM
with random effects at the individual and group level and adaptive
quadrature. Standard errors are in parentheses. Square brackets
include the covariance of the random effects.
*** Significant at the 1%a level.
Table 6. Evolution of Contributions
Dependent Variable: Change in Contribution
between Periods t and t + I, ([c.sup.t+1.sub.i]
- [c.sup.t.sub.i])
(1) (2)
Asymmetric 0.34 (0.045) 0.35 (0.41)
Three 0.25 (0.43) 0.22 (0.39)
Strong 0.11 (0.56) 0.23 (0.55)
Punishment inflicted 0.30 *** (0.03) 0.08 (0.08)
to i in t
Average contribution -0.00 (0.04) 0.09 ** (0.04)
of others in t
Period -0.31 *** (0.06) -0.15 ** (0.07)
Asymmetry level: 1.5
Asymmetry level: 3
Asymmetry level: 5
Constant 2.02 *** (0.69) -2.07 *** (0.59)
Observations 434 491
Subject variance 1.40 [0.67] 0.00 [0.00]
Group variance 0.00 [0.00] 0.10 [0.29]
Log likelihood -1141.20 -1361.91
Dependent Variable: Change in Contribution
between Periods t and t + I, ([c.sup.t+1.sub.i]
- [c.sup.t.sub.i])
(3) (4)
Asymmetric
Three 0.08 (0.52) 0.74 (0.45)
Strong 0.1 (0.58) 0.42 (0.56)
Punishment inflicted 0.30 *** (0.03) 0.08 (0.08)
to i in t
Average contribution -0.00 (0.04) 0.09 ** (0.04)
of others in t
Period -0.31 *** (0.06) -0.15 ** (0.07)
Asymmetry level: 1.5 0.75 (0.71) -0.43 (0.65)
Asymmetry level: 3 0.23 (0.51) 0.15 (0.46)
Asymmetry level: 5 0.22 (0.67) 1.08 ** (0.53)
Constant 2.08 *** (0.70) -2.25 *** (0.57)
Observations 434 491
Subject variance 1.41 [0.67] 0.00 [0.00]
Group variance 0.00 [0.00] 0.00 [0.02]
Log likelihood -1140.93 -1359.89
Linear regression: Regressions (1) and (3) are run for subjects who in
period t contributed less than the group average. Regressions (2) and
(4) are run for subjects who in period t contributed more than the
group average. Estimation was done in STATA 10.1 using GLLAMM with
random effects at the individual and group level and adaptive
quadrature. Standard errors are in parentheses. Square brackets
include the covariance of the random effects.
** Significant at the 5% level.
*** Significant at the 1% level.
Table 7. Determinants of Earnings
Dependent Variable: Earnings of Player i,
([[pi].sup.2.sub.i])
(1) (2) (3)
Asymmetric 0.54 (1.28) 0.57 (1.26) 0.26 (1.27)
Three 1.26 (1.22) 1.26 (1.22)
Strong 1.25 ** (0.51)
Period
Period*Three
Asymmetry
level: 1.5
Asymmetry
level: 3
Asymmetry
level: 5
Constant 25.03 *** 24.40 *** 24.40 ***
(1.02) (1.18) (1.18)
Observations 1320 1320 1320
Subject variance 1.69 [0.63] 1.69 [0.63] 1.44 [0.60]
Group variance 11.50 [3.10] 11.10 [3.01] 11.16 [3.01]
Log likelihood -4120.66 -4120.13 -4117.24
Dependent Variable: Earnings of Player i,
([[pi].sup.2.sub.i])
(4) (5)
Asymmetric 0.26 (1.27)
Three 0.01 (1.32) 0.27 (1.55)
Strong 1.25 ** (0.51) 1.25 ** (0.51)
Period 0.37 *** (0.07) 0.37 *** (0.07)
Period*Three 0.23 ** (0.10) 0.23 ** (0.10
Asymmetry 0.78 (1.96)
level: 1.5
Asymmetry -0.68 (1.47)
level: 3
Asymmetry 1.43 (1.86)
level: 5
Constant 22.35 *** 22.22 ***
(1.23) (1.28)
Observations 1320 1320
Subject variance 1.67 [0.60] 1.67 [0.60]
Group variance 11.16 [3.01] 10.62 [2.87]
Log likelihood -4065.55 -4065.8
Linear regression: Estimation was done in STATA 10.1 using GLLAMM
with random effects at the individual and group level and adaptive
quadrature. Standard errors are in parentheses. Square brackets
include the covariance of the random effects.
** Significant at the 5% level.
*** Significant at the 1% level.
RESULT 11. Strong players have significantly higher
earnings than weak players.
