PRELIMINARY VERSION, JUNE 2008 Playing with the Good Guys:
A Public Good Game with Endogenous Group Formation
Kjell Arne Brekke, Karen E. Hauge, Jo Thori Lind, Karine Nyborg
Abstract
In social dilemmas, conditional cooperators may be able to sustain cooperation if they are matched
with other conditional cooperators. We report results from a public good game experiment where
subjects can choose between two group types: Red and Blue. In Red groups, a fixed amount of each
individual’s payoff is donated to the Red Cross. Choosing Red can be interpreted as a costly signal
that one is of a cooperative type. Slightly less than half of the subjects chose Red. While contributions
in Blue groups show the usual declining pattern, contributions in Red groups stay high, leading to
substantially higher average overall contributions in Red groups.
1 Introduction Although subjects in public good game experiments typically contribute on average about half their
endowment in one-shot games (Ledyard, 1995), contributions tend to decline dramatically as the game
is repeated. Previous literature indicates that this may, to a large extent, be due to conditional
cooperators who may start off contributing relatively high amounts, but who condition their further
cooperation on others’ behavior. In Fischbacher and Gächter (2006), for example, about 55 percent of
subjects are classified as conditional cooperators, that is, subjects who contribute more the more others
contribute; egoists, who typically do not contribute at all to the public good, make up only around 20
percent in this study. Fischbacher et al. (2001) find that about 50 percent of their subjects are
conditional cooperators, while about 30 are free-riders. Hauge (2007) asked her subjects about what
they belived to be the morally ideal contribution. She found that the majority of subjects reported a
conditional moral ideal; that is, if others contributed less, these subjects found it morally right to
reduce their own contributions.
While the presence of free-riders may thus undermine cooperation, conditional contributors may be
able to sustain cooperation among themselves. This is confirmed by experimental studies where
groups, without subjects’ knowledge, are formed exogenously based on previous contribution
behavior (Gächter and Thöni 2005, Gunnthorsdottir et al. 2007, Ones and Putterman, 2007). Another
way to sustain cooperation is to introduce institutions allowing individuals to punish others’ noncooperative behavior (Fehr and Gächter 2000, 2002).
1
PRELIMINARY VERSION, JUNE 2008 While the above studies use exogenous group formation, the most common case in everyday life is
that people self-select into groups. To our knowledge, however, few scholars have studied endogenous
group formation within the context of a public good game. If some groups succeed in sustaining
cooperation, free-riders will have an incentive to invade those groups. This is, indeed, reflected in the
endogenous group formation game investigated by Ehrhart and Keser (1999), who observe that “the
more cooperative subjects are continually on the run from the less cooperative ones. However, the less
cooperative ones keep following them around” (op.cit, p.1).
One route to sustain cooperation is to limit access to (or exit from) endogeneously formed groups
(Ahn et al. 2004, Page et al. 2005). In the present paper, we investigate another possibility, namely
introducing a costly signalling device. If more cooperative individuals can signal their type, and the
signal is sufficiently costly that free-riders prefer not to use it, cooperators may be able to match
without being invaded by free-riders.
In our experiment there are two group types, Red and Blue. Each subject can choose what type of
group he wishes to be part of. Individuals choosing the same group type are then matched randomly
into groups of equal size. The difference between Red and Blue groups is that in Red groups, a fixed
amount is subtracted from individual payoff and donated to the Red Cross.1 We find that contributions
in Red groups are, on average, substantially higher than in Blue groups, although not sufficiently to
leave Red group membership equally profitable in monetary terms.
We believe our result to be relevant for several real life phenomena. One example is corporate social
responsibility. If cooperative behavior is derived from an underlying ethical principle, and the weight
attached to such principles varies between individuals, one would expect to find a correlation between
individuals’ cooperativeness in different contexts. It may be the case, for example, that those least
likely to shirk in teamwork are also more willing to accept a lower wage in order to have a socially
responsible employer. Based on this argument, Brekke and Nyborg (2008) show that in market
equilibrium, socially responsible firms may be able to survive, and possibly even to drive nonresponsible firms out of business altogether, due to their ability to attract more responsible employees.
2
Experimental design
This experiment studies endogenous group formation in a public good game setting. The experiment
consists of 3 parts. All subjects participate in all three parts in the same order. The experiment is
programmed in z-tree (Fischbacher, 2007).
1
For experimental and field studies involving contributions to charity, see Eckel and Grossman 1996, Croson
(199x), Martin, Alpizar et al, Shang and Croson, Frey and Mejer, Heldt.
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PRELIMINARY VERSION, JUNE 2008 The first part of the experiment is a one-shot public good game experiment with 3 subjects in each
group. In the following we will call this the “one-shot game”. Groups are decided exogenously and
formed randomly. Subjects do not receive any feedback about what happens in this first part of the
experiment. The payoffs from part 1 are described below. Subjects are informed that there will be 2
new experiments after the first part (and that their actions in part 1 will not affect their payoffs or
available choices in parts 2 and 3), but they are not informed what the two other parts will be about.
Before part 1 starts, subjects are tested in their understanding of the instructions for part 1.
In part 2 of the experiment, subjects choose, before playing the public good game, which type of group
they prefer to be in; Red or Blue, where Red and Blue groups differ only in their payoffs. To the
largest extent possible, subjects are allocated to groups of their preferred type. If the number of
subjects preferring a given firm type is not divisible by 3, there will be one group consisting of
individuals with mixed preferences. The type of this mixed group, is decided by the type which the
majority of the group wants. In part 2 there are 10 periods and the group composition remains the
same for all 10 periods. This means that if a subjects chooses to in a Red group, and she ends up in a
Red group, she will be in a Red group together with the same two other subjects who also chose to be
Red for the entire 10 periods of part 2. In the following this part therefore will be called the “partner
game”. After every period of the partner game, every subject receives feedback on how many units she
contributed to the public good, how many units were contributed to the public good in her group on
average, and her calculated payoff from that period. The actual payoff from the partner game is the
average payoff from the ten periods. The payoffs are described in detail below. Subjects are tested in
their understanding of the instructions before part 2 starts.
Part 3 of the experiment is quite similar to part 2. The new feature in this part is that the number of
periods are increased to 20, and that subjects can choose which type of group they prefer to be a
member of between each period. In the following this part therefore will be called the “stranger
game”. Among those that have chosen the same group type, groups are formed randomly in every
period, and as in part 2, if the number of subjects preferring a given group type is not divisible by 3,
there will be one mixed group. In this part of the experiment, new groups are formed in every new
period. Between each period, each subject receives information about how many units she contributed
to the public good in that period and how many units were contributed on average in her group. In
addition, each subjects is informed of the average contribution of one Red group and one Blue group.
To vary the information received by different subjects, each subject is either shown the average
contribution of the Red group with the highest average contribution or that of the Red group with the
lowest average contribution, each with 50% probability. Likewise, each subjects is either shown the
average contribution of the Blue group with either the highest or the lowest average contribution, each
with 50% probability. In the instructions, subjects are simply informed that they will be shown the
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PRELIMINARY VERSION, JUNE 2008 average of one Red and one Blue group, but not anything more. The payoff from the stranger game, is
the individuals average payoff over the 20 periods, described in more detail below.
In the experiment the group types are called Z groups and X groups instead of Red and Blue groups to
avoid framing effects. To avoid confusion, on every screen where the participants choose between X
and Z there is a reminder about the difference.
The payoff structure is kept as similar as possible across the three parts of the experiment. In every
period of the partner game and the stranger game, in addition to in the one-shot game in part 1, each
subject is endowed with 60 Norwegian kroner2 (NOK) which each subjects can allocate between
herself and the public good. The endowment is kept constant in order to make the decisions as similar
as possible across the different parts, at the same time as creating a real choice dilemma in the first
one-shot game.
The payoff in the one-shot game is as in a standard public good game, where each participant has the
choice of how much xi to contribute to the public good. The sum contributed to the public good is
doubled, and then divided equally between the three group members. The payoff function in the oneshot game thus is
π = 60 − xi +
2 3
∑ xj
3 j =1
There are two main differences in the payoff functions between the one shot game on the one side and
the partner and stranger game on the other. The first difference is the implementation of endogenous
group formation through the choice of group type and the following difference in payoff between the
two group types. The second difference is that the public good game is repeated instead of being oneshot, and that subjects receive the average payoff over the 10 and 20 periods of the partner and
stranger game respectively. In Red groups, the individual payoff in each repetition is as π above, but
in addition 50 NOKare given to the Red Cross. In the Blue group, each individual in each repetition
earns 50 NOK in addition to his share of the public good:
π B = 60 − xi +
2 3
∑ x j + 50
3 j =1
Subjects in Red groups therefore only get payoff from the public good, while being member of a Blue
group dominates being member of a Red group in monetary terms as long as the average contribution
in Red groups is less than 50 NOK higher than the average contribution in Blue groups.
2
1 NOK is about 0.20 US$.
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PRELIMINARY VERSION, JUNE 2008 3
Hypotheses
If cooperative behavior is derived from an underlying ethical principle, and the weight attached to
such principles varies between individuals, one would expect to find a correlation between
individuals’ cooperativeness in different contexts. In the current setting this would correspond to a
claim that there is a correlation between contributions to public goods and contributions to charity.
According to this line of thought, free-riders are less likely to care about contributions to the Red
Cross, than cooperative types. This leads us to the following hypothesis regarding contribution
behavior:
Hypothesis 1: Contributions in Red groups are higher than in Blue groups.
If we regard the contribution in the one-shot game as an indication about the type of the individual, the
claim regarding correlation in contribution behavior leads us to the following hypothesis regarding
choice of group type:
Hypothesis 2: High contributors in the one-shot game are more likely to choose Red in the partner
game.
In line with the discussion on sustaining cooperation in the introductory chapter, we may formulate a
third hypothesis:
Hypothesis 3: In the partner game, average contributions decline less in Red groups than in Blue
groups.
While the partner game allows us to analyze the development in contribution behavior over time
within groups of different types, it does not allow analysis of the effect of learning on the choice of
group type or the survival of group types over time. The stranger game allows us to look at the
development of group choice over time, and how information about behavior in Red and Blue groups
respectively influences this choice. Unless the difference in contribution between Red and Blue is at
least 50 NOK, out of an endowment of 60, the payoff will be higher in the Blue groups. Such a
difference in contributions is unlikely, and thus we expect subjects to learn that payoffs is higher in the
Blue groups. Note also that in the partner game the groups are fixed over 10 periods of contributions.
This introduces strategic motives in the sense that subjects may make high contributions since they
might expect the other subjects to be reciprocal. High contributions will thus induce higher
contribution from the other group members in later periods. In the stranger gamehowever, each group
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PRELIMINARY VERSION, JUNE 2008 only plays one period together as a group. We believe however that these strategic motives are not the
only reason why subjects in the Red groups contributes more. Hypothesis 1 thus also applies to part 3,
but as the design rules out strategic motives in this part, the higher contributions are more directly
linked to the role of group type as a selection device.
If subject only maximize monetary payoff, the number of individuals choosing Red should decline
towards zero. We believe however that the motive for choosing Red is not purely strategic, thus our
hypothesis is
Hypothesis 4: The share of subject choosing Red in the partner game will stabilize at a positive level.
We expect payoffs to be higher in Blue groups, but contributions to be higher in the Red groups. In
this case a subject looses payoff by choosing the Red group, but less than the 50 kroner given to the
Red Cross. Subject assessment of the difference in payoffs between the two groups is thus an
assessment of the cost of giving 50 kroner to the Red Cross. The less costly it is to give 50 kroner to
the Red Cross, the more attractive we expect this option to be. We compute subjects assessment of the
difference in payoff based on Bayesian updating. Our Hypothesis is then
Hypothesis 5: Subjects will be more likely to choose Red groups the higher the estimated difference in
contributions between Red and Blue groups is.
4
Results
The experiment was conducted at the Oeconlab at the University of Oslo in February 2008. 87
subjects recruited among students from several departments at the University of Oslo participated in
the experiment.
4.1 Contribution behavior The average contributions by Red and Blue groups for each period in the entire experiment is shown
in figure 2. The x-axis of both the upper and the lower graph shows the period number, where the oneshot game is called period 0, the partner game period 1-10 and the stranger game period 11-30.The top
graph shows contributions, while the bottom graph shows the t-statistic from student’s t-test (solid
line) and the z-statistic from a Mann-Whitney test (dashed line); the horizontal dotted lines show the
0.05 and 0.01 significance levels.
Note that in the one-shot game (period 0), subjects did not choose group type; in the figure, group type
in period 0 is assigned according to the subject’s (later) choice of group type in the partner game
(periods 1-10).
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PRELIMINARY VERSION, JUNE 2008 10
20
Contribution
30
40
50
60
10
1
10
20
30
20
30
t− and z−value
2
5
8
1
Period
Fig 2: Contributions and test statistics
The average contribution in Red groups is higher than the average contribution in Blue groups in all
periods, and from period 1 to 10 it looks as though the difference between the average contributions in
Red and Blue groups increases. The contributions in Blue groups follow the well known pattern of
deteriorating cooperation (Ledyard, 1995), where contributions start off high and then gradually
decline; while there is no similar trend in the Red groups, except for an end-game effect.
Turning to the test statistics, we note that the average contributions of Red and Blue groups become
significantly different once the contributions in Blue groups star to deteriorate. According the t-test, a
5% level of significance is reached from period 2 in the partner game and onwards. According to the
Mann-Whitney test the difference is significant on the same level from period 4 onwards. Note also
that the difference in contribution is no less in the partner game, where the strategic motive related to
reciprocity is absent.
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PRELIMINARY VERSION, JUNE 2008 0
Contribution
20
40
60
0
5
10
15
Period
Blue groups
Average contribution
25th and 75th percentiles
20
25
30
Red groups
Average contribution
25th and 75th percentiles
Fig 3: Contributions by period including 25-75 percentile
Figure 3 shows that the variance of contributions is much higher in blue than in red groups. In addition
to showing the average contribution by type and period, the vertical lines and bars in this figure show
the 25 to 75 percentile contributions in each group type. For red groups, the 25 to 75 percentile is
illustrated with the red vertical line, while this is shown with the blue bars for blue groups. It is
obvious that the variance in blue groups is much higher. In fact, the median in the red groups is 60 in
every period.
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PRELIMINARY VERSION, JUNE 2008 100
120
Profit
140
160
0
10
20
30
Period
Blue
Red
Red and Red Cross
Fig 4: Average profit by group type
It is interesting also to look at the profit earned in the two group types. As illustrated by figure 4, the
average profit in Blue groups was higher than in Red groups in all periods. Because contributions are
higher in Red than in Blue groups, the difference in profit between the two group types is less than the
amount given to the Red Cross from Red groups (NOK 50). This means that the sum of what is earned
in Red groups and what is given to the Red Cross from the Red groups, is higher than the amount
earned in Blue groups.
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PRELIMINARY VERSION, JUNE 2008 10
20
Contribution
30
40
50
60
0
10
20
30
Period
Red females
Blue females
Red males
Blue males
Fig 5: Average contribution by group type and gender
The behavior of males and females appear rather similar. Interestingly, however, notice the difference
in the behavior of men and women between the one-shot game (period 0) and the first period of the
partner game (period 1): While females show a quite similar behavior in the one-shot game
irrespective of whether they later choose Red or Blue, there is a quite pronounced difference in what
the males that later choose Red and those who later choose Blue contribute in the one-shot game.
Table 1 gives a panel data analysis of the contribution to the public good in the partner game of the
experiment, that is, period 1 to 10. The dependent variable is here individuals i’s contribution to the
public good in period t. In the first specification, a dummy with the value one for the group type Red,
is the only dependent variable. The coefficient for group type is positive and significant, with a value
of about 15, meaning that individuals in red groups on average contribute 15 out of 60 units more to
the public good than in blue groups. Nothing much happens to this coefficient when background
variables are included (Column (2)), or when period dummies are included (Column (3)).
Specification (3) and (4) show that period 9 and 10 have negative and significant coefficients, which
illustrates the end game effect, both with and without the presence of background variables. However,
as revealed by Column (5), this effect is much weaker in red groups than in blue ones.
In specifications (6) and (7), we include lagged own contribution and lagged average of the
contribution of the others on the individuals group. Now the coefficient of group type is markedly
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PRELIMINARY VERSION, JUNE 2008 reduced. The reason is that high contribution in later periods is partly explained by the subject’s own
high contribution and the other group members’ high contribution the period before. The positive
coefficient of the lag of the individual’s contribution shows some individual consistency in behavior
over time, while the positive coefficient of the lag of the average contribution of the group suggests
conditional cooperation. Conditional cooperation may explain the sustained contribution we observed
in Figure 2: When high contributors are in the same (Red) groups, their contributions are not
discouraged by the low contribution from opportunists.
To further explore the long run dynamics of contributions, Column (8) includes a two-period lag of
own contribution and the average contribution of others. Twice lagged own contribution has a positive
effect, indicating that a subject’s contributions depend on long run past behavior, not only last period.
There is no significant effect of twice lagged period average contributions of the others in the group,
so reaction to others occur immediately. Column (9) includes the average of the contribution of the
group in all previous periods. This does not seem to have any significant effect on contributions when
controlling for average contribution in the last period.
Since group type no longer was a significant variable when the lagged variables were included, it is
interesting to look at whether the two group types react differently to information from the previous
periods. In Column (10), an interaction variable of group type and the lag of the group’s contribution
is included. This coefficient is negative, suggesting that members of red groups are conditional
cooperators who contribute more the more others contribute, and less the less others contribute, but to
a less extent than members of blue groups. In other words, they do not react so strongly to the
behavior of others, but are more willing to forgive low contributors, and this is mainly what makes the
Red groups able to sustain the high levels of contributions over time. When a subject in a Blue group
experiences that the average in his group falls, he will react by reducing his own contribution in the
following period, and thus make the average in his group even lower in the following period. In Red
groups on the other hand, a person experiencing a fall in the average contribution in his group, will not
react so strongly as a Blue type, and thereby contribute to sustaining the level of cooperation in Red
groups.
Table 1 about here
4.2 Choice of group type As seen in Figure 1.
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PRELIMINARY VERSION, JUNE 2008 Number of groups
10
20
29
Blue groups
0
Red groups
10
20
30
20
30
Fraction of agents
.25
.5
.75
1
1
Blue
0
Red
1
10
Period
Fig 1: Number of red groups by period
45% of the subjects are in a red group, while 55% are in a blue group in the partner game. In the
figure, contributions are shown in chronological order, such that the one-shot game is called period 0,
the partner game period 1-10 and the stranger game period 11-30. It is also interesting to note that in
the stranger game, when subjects have the possibility to choose group type between every period, the
proportion who choose red groups varies throughout the 20 periods, revealing that some subjects are
switching between the two group types. During the 20 periods the number of Red groups varies
between 8 and 13, that is: between 28 and 45%. Even after 20 choices between Red and Blue, over
40% of the groups are Red in the last period. This figure does not show any clear downward trend in
the number of subjects choosing Red, thus this does not indicate that learning drives out the existence
of Red groups. This result is consistent with Hypothesis 4.
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PRELIMINARY VERSION, JUNE 2008 Table 2 about here
Table 2 shows several specifications of a logit on the choice of group type in the partner game, where
the independent variable is the choice of group type and choosing Red takes the value 1. The first
specification has only one dependent variable, namely the contribution in the one-shot game, which is
not significant. This result does not confirm our hypothesis 2. High contributors are slightly more
likely to choose Red, but the difference is not significant. On the other hand we cannot reject the
hypothesis, which is the same conclusion the test statistics in Figure 2 gives.
In Column (2), some background variables are included. Here being female has a positive and
significant coefficient, while being an economist is positive but not significant. 32 % of the subjects in
our sample were economists, mostly undergraduates. This does not seem to influence the results.
We are interested in whether group choice is determined by expected payoffs in the two group types,
or rather the difference in expected payoffs between the two group types. As a proxy for expected
payoffs in the two group types, we use expected contributions to the public good reported in a
questionnaire after the experiment. Plugging the reported expected contributions into the payoff
function gives calculated expected payoffs in each of the groups. The difference between the expected
payoffs in the two group types can be regarded as the expected cost of being member of a Red group,
or the expected cost of giving 50NOK to the Red Cross. One should expect that the higher the
expected cost of being in a Red group, the lower is the probability of wanting to join such a group. The
results confirm this, but the result should be interpreted with care as the post experiment survey may
be a post-rationalization of the experiment behavior. In Columns (3)-(5), the expected cost of being in
a Red group has a negative and significant effect on the choice of being in a red group, also when
controlling for background variables and the contribution to the public good in the one-shot game.
Group type choice might be driven by differences in types. Subjects are classified as either altruistic,
self interested or strategic according to according to their answer to a question in the questionniare
after the experiment on why they chose the group type they did. The answers to this question has been
classified into three categories by a research assistant which did not know the purpose of the study.
The answers are classified as either altruistic (a typical answer in this category is “I wanted to help the
Red Cross”), egoistic (a typical answer in this category is “I am a student and want as much money as
possible”), strategic (a typical answer in this category is “I expect contributions to be higher in red
groups”) or two of the mentioned categories. In specification 6 and 7 dummy variables for these three
categories are included. Again, keep in mind that these answers are given after the experiment, and
therefore might include attempts at justifying behavior. Summing up, females tend to choose red more
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PRELIMINARY VERSION, JUNE 2008 than males, and expectations about payoff also drive the choice of group type. Otherwise we are not
able to say much about which factors drive the choice of red group type.
To explore further how perceptions about contributions in Red and Blue groups affect the choice of
group, we use the differentiated dissemination of information in the stranger game. As explained
above, at the end of each period subjects were informed about the contributions in one Red and one
Blue group in addition to his own group. If expected contributions in different groups has an impact
on group choice, we should expect that subjects who observe high contributions in one group type and
low contributions in the other should have a tendency to go towards the former, assuming that their
expectations are at least partially formed by the information they are given. The rational way to
aggregate this information is though Bayesian updating. Appendix A gives an explanation of how we
model this in the current context. Essentially, we assume that each subject holds a prior distribution of
the mean contribution among Rred and Blue group members. After each period, we assume subjects
use observations from his own group as well as the provided information to update his beliefs
according to Bayes’ rule. The calculated beliefs are then used as explanatory variables in the
estimations shown in Table 3.
Table 3 about here
Column (1) of Table 3 is a simple regression to show the persistence in group choice. Having chosen a
Red group in one period significantly increases the probability of choosing a Red group in the next
period. Column (2) is a regression where we only include the estimated belief of the difference of the
mean contributions between Red and Blue groups. Here μ denote the Bayesian updated best estimate
of contributions in each group type, and Δμ is the expected difference in contributions. We see that a
belief in more generosity among red group members and less generosity among blue group members
tend to increase the likelihood of choosing a red group next period. Given that the potential range of
Δμ is -60 to 60, the quantitative effect is also substantial. Column (3) shows that the same relationship
holds when we control for current group choice, although the strength of the relationship is reduced. In
Column (4) we also include the difference between the two pieces of information provided in the
current round. It seems that agents are quite rational as the rational belief has all the explanatory
power. In Column (5) we disaggregate Δμ into its two components. Although both parts have the
expected sign, it seems that most of the effects come from differences in the beliefs about the behavior
of players in red groups.3
3
As the correlation between the two beliefs is almost zero, this result is not driven by multicollinearity.
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PRELIMINARY VERSION, JUNE 2008 The next question is whether agents who choose Red react different to updated information about μ
than players who choose Blue. Column (6) interacts the belief-variable with an indicator variable for
having chosen a Red group in the former treatment. We see that agents from Red groups in fact do
react more strongly to information, although this difference is only significant at the 10 % level. In
Column (7), we instead use the average number of times the agent has been in a Red group so far in
this treatment. Now it seems that only players who have been in Red groups a substantial fraction of
the time react to updated information as only the interaction variable retains significance.
To test whether these results are driven by unobserved heterogeneity between agents or the timing of
the experiment, Table 4 repeats some of the estimations above using the panel structure to control for
individual level random effects and/or period fixed effects. The results are strikingly similar to those
found in Table 3.
Table 4 about here
5
Conclusions
In this paper we have studied a public good game with endogenous group formation. Before making
their contribution choice, subjects could choose which type of group they preferred to be a member of:
Red or Blue. Groups were then formed randomly, but to the largest extent possible respecting
subjects’ group type preferences. The difference between group types was that a fixed extra amount of
money was donated either to the Red Cross (in Red groups) or to individual subjects (Blue groups).
Average contributions were significantly higher in Red groups, although not sufficiently so to leave
Red group membership as profitable, in monetary terms, as Blue group membership. Through the
rounds, the share of red groups was always between 30 and 45 percent, with no visible trend in either
direction.
This indicates that willingness to contribute to one public good (the Red Cross) may be taken as a
signal of relatively high willingness to contribute in other contexts or to other public goods as well
(here, the group’s material benefit). Thus, it may, for example, be possible for firms to use corporate
social responsibility, combined with slightly lower wages, as a screening device to attract workers who
shirk less when not monitored (Brekke and Nyborg, 2008).
Nevertheless, note that our results seem to follow from a slightly more complex relationship than one
might expect. Our first hypothesis was that high contributors in the one-shot game would be more
likely to choose Red in the partner game. Although this did hold in our data, this result was not
statistically significant. What seems to be driving the main result, that contributions are higher in Red
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PRELIMINARY VERSION, JUNE 2008 groups, seems to be the fact that although both group types start off with quite high contributions, the
decline in contributions over periods is substantially higher for the Blue groups.
One reason may be that since Red contributions were initially lower, members of these groups were
less disappointed with each other and thus reacted less negatively in subsequent periods. However, it
seems unlikely that this would be the whole explanation, since the initial difference between groups
was very small. Our analysis shows, in fact, that the contribution of Red group members were less
sensitive to others’ previous contributions than was the case for Blue group members. Thus, one
interpretation is that willingness to contribute to the Red Cross is not simply a signal of being a
cooperative type, but of being a more persistently cooperative type.
We have also seen that subjects made efficient use of available information. While this information
clearly indicates that the payoff is higher in Blue than in the Red groups, the share of subject choosing
red is quite stable.
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16
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17
Appendix A: Details of the Bayesian updating
In Treatment 3, agents are allowed to choose their group at the beginning of each period.
Furthermore, at the end of each period, they are informed about the average contribution
and profit of one blue and one red group. Formally, assume that the effort of a randomly
chosen individual j is distributed
ej (blue) ∼ N (θ0 , σ 2 )
ej (red) ∼ N (θ1 , σ 2 )
where the variance σ 2 is assumed known but the means θblue and θred are only imperfectly
known. Specifically, it is assumed that initially, they have prior distributions
2
θblue |I0 ∼ N (µblue|0 , ω0|0
)
2
θred |I0 ∼ N (µ1|0 , ωred|0
)
where It is the information set at time t.
Each individual observes his own group i, some groups Γblue and some Γred . He uses
Bayesian updating to construct posterior distributions
2
θblue |I1 ∼ N (µblue|1 , ωblue|1
)
2
θred |I1 ∼ N (µred|1 , ωred|1
)
Let ēi denote the average contribution in i’s own group excluding his own contributions, and
ēblue and ēred denote the averages in Γblue and Γred . Finally, there are N members in both
i’s group in the groups Γblue and Γred . If we let τi take the value 0 if i was in a blue group
and 1 if he was in a red group, then the updated means are given by
µblue|t+1 =
µred|t+1 =
1
µblue|t
2
ωblue|t
+
N
ē
σ2 0
1
2
ωblue|t+1
+
N
σ2
1
µred|t
2
ωred|t
1
2
ωred|t
+
+
1
+ (1 − τi ) Nσ−1
2
N
ē
σ2 1
N
σ2
+ (1 − τi ) Nσ−1
2 ēi
+ τi Nσ−1
2 ēi
+ τi Nσ−1
2
(1)
(2)
and the variances
2
ωblue|t+1
=
2
ωred|t+1
=
N0
N −1
+ 2 + (1 − τi )
2
ωblue|t
σ
σ2
!−1
N0
1
N −1
+ 2 + τi
2
ωred|t
σ
σ2
1
Updating to I2 follows an analogous procedure.
2
!−1
(3)
(4)
7
No
No
No
0.104
870
(1)
15.35***
(4.06)
Yes
No
No
0.131
690
(2)
15.37***
(3.35)
No
Yes
No
0.146
870
(3)
15.35***
(4.06)
-6.184***
(-2.89)
-9.966***
(-4.65)
-17.24***
(-8.05)
Yes
Yes
No
0.166
690
(4)
15.37***
(3.35)
-3.986*
(-1.74)
-8.464***
(-3.69)
-14.52***
(-6.33)
Yes
Yes
Yes
0.174
690
(5)
8.283
(1.50)
-7.308**
(-2.41)
-14.18***
(-4.67)
-20.69***
(-6.82)
13.15***
(2.86)
14.19***
(3.08)
Yes
Yes
No
0.589
621
0.582***
(17.12)
0.296***
(7.33)
(6)
3.217**
(2.22)
-1.490
(-0.58)
-5.280**
(-2.05)
-7.403***
(-2.86)
Yes
Yes
Yes
0.591
621
(7)
3.022
(0.81)
-1.537
(-0.45)
-6.773**
(-1.97)
-8.048**
(-2.32)
3.425
(0.66)
1.464
(0.28)
0.583***
(17.02)
0.295***
(7.23)
Yes
Yes
Yes
0.606
552
(8)
3.334
(0.89)
1.027
(0.30)
-4.578
(-1.33)
-6.812**
(-1.96)
3.436
(0.66)
1.479
(0.28)
0.429***
(9.56)
0.180***
(2.80)
0.260***
(5.60)
0.0538
(0.80)
Yes
Yes
Yes
0.591
621
0.0107
(0.10)
(9)
2.979
(0.79)
-1.555
(-0.45)
-6.798**
(-1.97)
-8.097**
(-2.31)
3.469
(0.66)
1.503
(0.29)
0.582***
(16.19)
0.287***
(3.37)
-0.180**
(-2.05)
Yes
Yes
Yes
0.593
621
(10)
11.71**
(2.08)
-1.273
(-0.37)
-6.438*
(-1.87)
-7.597**
(-2.19)
3.400
(0.65)
0.667
(0.13)
0.571***
(16.45)
0.343***
(7.31)
Notes: All estimation includes individual level random effects. R2 is the overall R2 . Background variables are age, years of education
at university level, and dummies for gender and being an economics student.
Red*lagged average contrib
others
Background variables
Period dummies
Period*Red
R2
N
2 period lagged average contr
of others in group
Average of cum contr others
Lagged average contribution of
others in group
2 period lagged contribution
Lagged contribution
Red * Period 10
Red * Period 9
Period 10
Period 9
Period 8
Group type
Table 1: Contributions under treatment 2
8
Notes: Estimates are logit coefficients.
Pseudo R2
N
Redcross and altruist
Strategy
Expected difference in payoff
(b-r)
Altruist
Pro redcross
Age
Economist
Female
Contribution one shot game
0.0224
87
(1)
0.0183
(1.59)
0.222
69
(2)
0.0151
(0.98)
1.678***
(2.76)
0.348
(0.57)
-0.118*
(-1.72)
1.959**
(2.55)
(4)
(5)
(6)
(7)
0.00288
-0.0348
0.730
(0.17)
(-0.95)
(0.00)
1.847**
1.841**
2.887
51.95
(2.55)
(2.54)
(1.50)
(0.01)
1.027
1.055
5.156*
71.23
(1.36)
(1.36)
(1.70)
(0.01)
-0.126
-0.124
-0.605** -10.24
(-1.60)
(-1.58)
(-2.31) (-0.01)
2.285**
2.293**
5.144**
63.07
(2.54)
(2.54)
(2.05)
(0.01)
-0.0426*** -0.0558*** -0.0556*** -0.0856* -0.0162
(-3.55)
(-3.35)
(-3.31)
(-1.91) (-0.00)
6.687**
137.9
(1.98)
(0.01)
-1.594
52.58
(-0.77)
.
11.64
(0.01)
0.137
0.375
0.376
0.619
1.000
87
69
69
44
33
(3)
Table 2: Choice of red type in partner game
9
1653
1473
0.139***
(17.18)
(2)
1473
(3)
1.916***
(13.24)
0.0904***
(10.37)
1473
(4)
1.921***
(12.68)
0.0900***
(9.44)
0.000342
(0.09)
1473
-0.0127
(-0.71)
0.112***
(11.47)
(5)
1.738***
(11.53)
1473
0.00404
(0.21)
-1.644***
(-3.17)
(6)
1.416***
(8.75)
0.0741***
(2.79)
1473
0.0485**
(2.02)
2.330***
(3.38)
(7)
0.487**
(2.27)
0.0131
(0.79)
Notes: Estimates are logit coefficients with choice of red type next period as the positive outcome.
The µ’s are from the Bayesian updating, i.e. the best guess of the mean contribution of blue and red players.
The variable ∆ Shown is the difference in the contributions actually shown.
Some specifications include individual random effects and period fixed effects.
Robust t-values are reported in parentheses
R2
N
Red group in treatment 3
∆µ× Red in tr 3
Red group in treatment 2
∆µ× Red in tr 2
µ red
µ blue
∆ Shown
∆µ
Group choice
(1)
2.667***
(21.29)
Table 3: Results from Treatment 3
10
No
Yes
0.383
1473
Yes
No
0.372
1473
(2)
0.412***
(14.01)
0.0143***
(11.21)
Yes
Yes
0.383
1473
(3)
0.397***
(13.19)
0.0161***
(11.57)
Yes
Yes
0.383
1473
(4)
0.398***
(12.91)
0.0160***
(10.76)
0.0000489
(0.09)
Yes
Yes
0.405
1473
0.000813
(0.31)
0.0225***
(13.62)
(5)
0.330***
(10.23)
Yes
Yes
0.436
1473
-0.00468*
(-1.95)
-0.150**
(-2.37)
(6)
0.281***
(8.47)
0.0196***
(5.39)
(7)
0.103**
(2.36)
0.00208
(1.00)
0.00784***
(2.97)
0.440***
(4.83)
Yes
Yes
0.458
1473
Notes: Outcome is choice of red type next period.
The µ’s are from the Bayesian updating, i.e. the best guess of the mean contribution of blue and red players.
The variable ∆ Shown is the difference in the contributions actually shown.
Some specifications include individual random effects and period fixed effects.
Robust t-values are reported in parentheses
Individual RE
Period FE
R2
N
Red group in treatment 3
∆µ× Red in tr 3
Red group in treatment 2
∆µ× Red in tr 2
µ red
µ blue
∆ Shown
∆µ
Group choice
(1)
0.397***
(13.19)
0.0161***
(11.57)
Table 4: Results from Treatment 3 - Robustness to individual heterogeneity