Group size effects and critical mass in public goods games
Group size effects and critical mass in public goods games
Mar?a pereda 0 1
Valerio Capraro 2
0 Universidad Polite?cnica de Madrid, Departamento ingenieri?a de Organizacio?n, Administracio?n de empresas y estadi?stica , Madrid , Spain
1 Unidad Mixta interdisciplinar de comportamiento y complejidad Social (UM icc S) , UC3M-UV-UZ, Legane?s, Madrid , Spain
2 Economics Department, Middlesex University London, Business School , The Burroughs, London, NW4 4BT , United Kingdom
3 Grupo Interdisciplinar de Sistemas Complejos, Departamento de Matema?ticas, Universidad Carlos III de Madrid , 28911, Legane?s, Madrid , Spain
4 Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza , 50018, Zaragoza , Spain
5 Institute UC3M-BS for Financial
Understanding whether the size of the interacting group has an effect on cooperative behavior has been a major topic of debate since the seminal works on cooperation in the 1960s. Half a century later, scholars have yet to reach a consensus, with some arguing that cooperation is harder in larger groups, while others that cooperation is easier in larger groups, and yet others that cooperation attains its maximum in intermediate size groups. Here we add to this field of work by reporting a two-treatment empirical study where subjects play a Public Goods Game with a Critical Mass, such that the return for full cooperation increases linearly for early contributions and then stabilizes after a critical mass is reached (the two treatments differ only on the critical mass). We choose this game for two reasons: it has been argued that it approximates real-life social dilemmas; previous work suggests that, in this case, group size might have an inverted-U effect on cooperation, where the pick of cooperation is reached around the critical mass. Our main innovation with respect to previous experiments is that we implement a within-subject design, such that the same subject plays in groups of different size (from 5 to 40 subjects). Groups are formed at random at every round and there is no feedback. This allows us to explore if and how subjects change their choice as a function of the size of the group. We report three main results, which partially contrast what has been suggested by previous work: in our setting (i) the critical mass has no effect on cooperation; (ii) group size has a positive effect on cooperation; (iii) the most chosen option (played by about 50% of the subjects) is All Defection, followed by All Cooperation (about 10% of the subjects), whereas the rest have a slight trend to switch preferentially from defection to cooperation as the group size increases.
Published; xx xx xxxx
In the last decades, a variety of disciplines including anthropology, biology, economics, physics and psychology
have addressed the issue of the inordinate scale of human cooperation1?8. Even in the simplest societies,
individuals cooperate in groups of genealogically distant individuals9,10. In the perpetually connected society11 we live in
today, instances requiring cooperation arise among groups of people of largely different sizes, and it is therefore
of crucial importance to understand if and how group size affects cooperation. This question has been extensively
debated in the literature since the seminal works on cooperation in the 1960s, leading to two different viewpoints.
On the one hand, some scholars suggest that, in so far as the benefits of cooperation decrease with an increasing
number of participants, cooperation should decrease with group size12,13. On the other hand, other researchers
have argued that benefits do not necessarily decrease as the group size increases and, in that case, they reason
that cooperation might even increase with the size of the group14,15. These two viewpoints have been mirrored
by empirical, theoretical, and numerical works, with some finding a negative effect16?22, others finding a positive
effect22?31, and yet others finding an ambiguous or non-significant effect32?34.
A commonly used, stylized setup to probe into the issue of cooperation is the Public Good Game35 (PGG). A
PGG is a cartoon of the problem of provision of resources with three main characteristics: They are jointly
provided, they are non-excludable (it is not possible to prevent those who have not paid for them from having access
to them), and they are also non-rivalrous (resource consumption by one consumer does not prevent
simultaneous consumption by others). Economic theory presents this as a game in which agents have to decide how much
of their resources to contribute to the creation of a public good and how much to spend on private goods. The
agents? contribution is multiplied by some factor (linear PGG) and then distributed equally among everybody,
irrespective of their own contribution. Even in such a simplified situation, the jury is still out on the issue of size
dependence, with some researchers reporting an increase of contribution with the number of players22?24, and
others finding no effects or negative effects21,36; a meta-analysis of 27 experiments37 concluded that there is mild
positive effect of group size.
One way to make sense of these apparently contradictory results is by assuming that the effect of the group size
on cooperation is not domain-general but depends on specific properties of the interaction. As suggested above,
one potential discriminator is the way the benefit of full cooperation increases as a function of the group size. The
intuition is simple: Since it is known that cooperation is positively related to the benefit of full cooperation38,39,
if the benefit for full cooperation increases as the size of the group increases, then larger groups might get more
tempted to cooperate; if, instead, the benefit for full cooperation decreases as the function of the group size, then
larger groups might be less tempted to cooperate. Once again, from the theoretical viewpoint the issue is far from
resolved. Thus, Pe?a considered evolutionary models, finding that the outcome of general nonlinear public goods
games depends not only on the average group size but also on the variance of the group-size distribution in case
groups are heterogeneous40; also, he showed that larger group sizes can have negative effects (by reducing the
amount of cooperation in some cases) and positive effects (by enlarging the basin of attraction of more
cooperative outcomes) on the evolution of cooperation41. Other works addressing multiplayer games from a more general
but still evolutionary viewpoint point to the complexity of this problem42?44.
In order to gain insight into this problem, Capraro & Barcelo45 considered a class of N-person general public
goods game parameterized by a function ?(?, N ), representing the marginal return for cooperation when ?
people out of N cooperate. In the presence of ? cooperators, the payoff of a cooperator is defined as ?(?, N ) ? c
(c > 0 represents the cost of cooperation), while the payoff of a defector is defined as ?(?, N ). In order to have a
social dilemma, two properties are required:
Full cooperation gives a larger payoff than full defection: ?(N, N ) ? c > ?(0, 0);
Defecting is individually optimal, regardless of the number of cooperators: for all ? ? N , one has
?(? ? 1, N ) > ?(?, N ) ? c.
Barcelo & Capraro22,45 argue that, when ?(?, N ) is constant or even decreasing in ?, then group size should
have a negative effect on cooperation, whereas, when ?(?, N ) is increasing in ?, then group size should have a
positive effect on cooperation. They show that these two predictions are satisfied in two experiments and can be
formalized by assuming that agents play according to the cooperative equilibrium model introduced by Capraro46.
Putting these results together, it follows that, if we consider a piecewise linear-then-constant function ?(?, N ),
then group size should have a curvilinear, inverted-U effect on cooperation.
Note that public goods games with piecewise linear-then-constant return for cooperation are interesting
because they approximate those real-life social dilemmas for which the public good has natural limits such that,
from some point on, any new cooperator brings no additional benefit. For example, suppose that a hospital
needs 50 liters of blood for the 50 victims of a terror attack. Of course, every litre of blood is initially important
because it can save one life. However, after reaching the 50 litres, collecting more blood brings no additional
benefit. Another example regards academic collaborations. Suppose that a team of researchers wants to start a
multidisciplinary project that lies between three disciplines, for instance, psychology, anthropology and
physics; and suppose that the team is looking for psychologists, anthropologists and physicists to take part to the
project. Clearly, once each of these three disciplines is sufficiently covered, getting on board one more specialist
would bring no additional benefit. The practical importance of this type of public goods games raises a crucial
question: What is the group size effect on public goods games with a piecewise linear-then constant return for
cooperation? Previous work using field experiments on real-life public goods games have repeatedly shown that
medium-size groups tend to cooperate more47?51. In particular, connecting to one of our examples above,
numerous studies have confirmed that, in academic research, the research quality of a research group is maximized by
intermediate-size groups52?54. These findings are, thus, broadly consistent with the view that public goods game
with a piecewise linear-then-constant return for cooperation should give rise to a inverted-U effect of group size
Following this theoretical prediction, Capraro & Barcelo45 carried out one experiment to probe into the
existence of this curvilinear dependence of the group size. As predicted, they found evidence of an inverted-U
dependence of the number of cooperators with the group size. However, while their prediction was that the rate of
cooperation should start decreasing at N = 10, it turned out that it reached its maximum in groups of size
N = 15. This immediately poses the question as to how the optimal group size for cooperation depends on the
size at which ?(N, N) becomes constant. This is an important question in light of possible applications of the
curvilinear effect: knowing at which group size cooperation attains it maximum has obvious applications to team
formation. In addition, a second question that was not answered by previous work regards the behavior of people
at the individual level and its dependence on the group size. All the empirical studies on group size effect on
cooperation that we are aware of uses a between-subjects design, such that different subjects interact in groups of
different sizes. A major innovation of our work is that we implement a within-subject design, such that the same
subject interacts in groups of different sizes, which will allow us to explore heterogeneity in people?s behavior as a
function of the group size.
Experimental setup. We conducted an online experiment with 200 subjects, that took place in eight
sessions, one every two weeks between October 2017 and February 2018. The participants were Spanish volunteers
from the IBSEN pool of subjects55. In each session, participants played one round of a N-person general public
goods game with Critical Mass (i.e., ?(?, N ) piecewise linear-then-constant). Sessions differed in the size of the
interacting group, from N = 40 in the first session to N = 5 in the last one. Groups were randomly formed at the
end of every session to compute the payoff. However, to avoid iterated game effects, participants received no
feedback about the choices made by others or their own payoff until the last session was finished. To explore the
influence on cooperation of the value where ?(N, N) stabilizes (hereafter, NC), we carried out two different
treatments, namely NC = 10 and NC = 20, each treatment with 100 participants initially assigned to it. Of these 200
volunteers we recruited, only 163 showed up the first day, having as a result a first day dropout of 18.5%. The
experiment ended up with 107 volunteers that made all the decisions (average drop out rate of 5% per session).
The experiment was implemented in IBSEN-oTree55. Participants played online through a web browser in a
computer, tablet or mobile phone; and could made their decisions during eight hours each session day. Email
reminders were sent every two weeks (i.e., before every session) to remind people to participate. Further details can be
found in the Methods section. Only participants that completed the experiment were paid, as it was stated in the
instructions. Participants were paid using PayPal. The average earning was 8 Euros. The 25% participants who
scored highest in each treatment got a chance to enter a lottery of 50 Euros, and correspondingly two participants
earned an additional price of 50 Euros.
Data analysis: Cooperation as a function of the group size. As already indicated above, a total of 93
subjects (47.3% females) dropped out over the course of the experiment. Moreover, because of an internet
connection problem, the condition N = 30 was interrupted before data completion and, as a consequence, only 98
subjects participated in this condition. In view of these circumstances, the analysis is structured as follows: We
first analyze subjects for whom we have all decisions and excluding the condition N = 30. Subsequently, as
robustness checks, we include also subjects who dropped out during the experiment and the condition N = 30.
As we will see below, the overall pattern of results is robust across these exclusions.
To carry out the analysis, we use the following variables: C is equal to 1 if a player cooperates, and 0 if a player
defects; NC is equal to 10 if a player participates in the condition where ?(N, N) stabilizes at N = 10, and equal to
20 if a player participates in the condition where ?(N, N) stabilizes at N = 20; N is equal to the size of the group.
Since C is a binary variable, the analysis will be conducted using a logistic model estimating the probability of
cooperation p(C) as a function of the dependent variables x1, ?, xn (which will be specified later):
1 + e?(?0+?1x1+?+?nxn)
We begin by considering subjects for whom we have all the observations and excluding the condition N = 30.
In Fig.?1 we plot the percentage of people who cooperated. As a first step of the analysis, we observe that logistic
regression predicting C as a function of NC finds no statistically significant effect (p = 0.818). Thus we can
collapse data across the NC treatments. Subsequently, logistic regression predicting C as a function of N finds a
statistically significant positive effect, such that larger groups tend to cooperate more (coeff= 0.017, z = 2.45,
p = 0.014). The aforementioned fact that NC does not affect C is reflected in the fact that the positive effect of N
on C is similar in the two treatments (NC = 10: coeff = 0.016, z = 1.70, p = 0.089; NC = 20: coeff = 0.017,
z = 1.86, p = 0.078).
We now turn to the robustness checks. As a first check, we repeat the analysis by including N = 30 in the data
(19.44% cooperators in NC = 10 and 21.62% cooperators in NC = 20). As before, logistic regression predicting C as a
function of NC finds no statistically significant effect (p = 0.926), and hence we collapse again data across the NC
treatments. In line with the previous results, we find a statistically significant positive effect of N on C (coeff = 0.014,
z = 2.18, p = 0.029), which is similar in the two treatments (NC = 10: coeff= 0.015, z = 1.54, p = 0.123; NC = 20:
coeff = 0.015, z = 1.54, p = 0.123). As a second robustness check, we include subjects who dropped out during the
experiment, while excluding subjects who participated in N = 30. Once again, we find that NC has no statistically
significant effect on C (p = 0.424) and, after collapsing data across the NC treatments, we find a statistically
significant positive effect of N on C (coeff= 0.016, z = 2.66, p = 0.008). Splitting the analysis by NC reveals that the group
size effect is significant only for NC = 20 (NC = 10: coeff = 0.013, z = 1.53, p = 0.125; NC = 20: coeff = 0.019,
z = 2.21, p = 0.027). This suggests that the effect of group size might be stronger forNC = 20. However, the
difference between the two effects is not statistically significant, as revealed by a logistic regression including the NC ? N
interaction term (p = 0.897). Therefore, also in this case we find that the group size effect does not depend on the
critical mass. Finally, in our third and last robustness check, we analyze all data, that is, we include both subjects who
dropped out during the experiment and those who participated in N = 30. The results are qualilatively the same: NC
has no statistically significant effect on C (p = 0.398), and when we collapse data across treatments N has a
statistically significant positive effect on C (coeff = 0.015, z = 2.45, p = 0.014), which is similar in the two treatments
(NC = 10: coeff = 0.012, z = 1.38, p = 0.169; NC = 20: coeff = 0.018, z = 2.07, p = 0.039), with the NC ? N
interaction term turning out to be not statistically significant (p= 0.742). Therefore, our analysis combined with the three
robustness checks we have considered, allows us to draw two main conclusions: First, the size of the group has a
positive effect on cooperation, and second, the positive effect of group size on cooperation does not depend on the
critical mass. We have also conducted the same analyses by adding a quadratic term for the group size effect, in order
to test for the inverted-U effect. Results remain qualitatively the same, and the quadratic term is never significant.
One additional step we have taken to assess the reliability of the above findings is to study separately the
average cooperation among subjects who dropped out during the experiments (44 in the treatment NC = 10 and 49 in
the treatment NC = 20), by comparing it with the average cooperation among subjects who did not drop out. The
idea behind this check is that, as the experiment proceeded, it could be the case that individuals more prone to
cooperation were overrepresented among the dropouts, thus leading to a decrease in average cooperation as the
experiment proceeded from the largest to the smallest size. To test for this possibility, we built a dummy variable
named Incomplete, which takes value 1 if a subject dropped out during the experiment, and 0 if a subject
completed the experiment. We then conducted a set of logistic regressions predicting C as a function of Incomplete, for
each value of N. In doing so, we find all the p-values to be larger than 0.2. This suggests that the average
cooperation among those who dropped out during the experiment is not statistically different than the average
cooperation among those who completed the experiment, which in turn supports the conclusion that our results are
unlikely to be driven by the relatively high rate of dropping out.
Within-subject analysis. In the previous subsection, we have looked at our experiment from the viewpoint of
the global results on cooperation and its dependence on the group size. Now we move on to studying heterogeneity
in people?s behavior. We begin by observing that, since each participant has to take eight independent decisions
about whether to cooperate or not, there are 28 = 256 possible combinations of cooperation and defection. However,
as summarized in Table?1, for each NC treatment, only 25 of them, less than 10%, are actually being played.
Not surprisingly, in both treatments NC = 10 and NC = 20, the most played strategy is AllD (All Defection),
making a total of about 48% of subjects in the treatment NC = 10 (counting also subjects who did not participate
in condition N = 30), and 49% of subjects in the treatment NC = 20. Perhaps more surprisingly, in both
treatments, the second most played strategy, albeit at a large distance from the first, is AllC (All Cooperation), which
makes about 9% of the observations in the treatment NC = 10 and 8% of the observations in the treatment NC = 20
(again counting also subjects who did not participated in the session with N = 30). All other strategies have been
played by less than 4% of the participants. For comparison, Fig.?2 shows the distribution of sequences we would
expect if the decisions were taken at random. Each ?strategy? groups the set of sequences with the same rate of
cooperation; for example, cooperating once and defecting 7 times is ?strategy 0.125?. The plot makes it evident that
participants? choices are largely intentional: People play AllD far more than expected, and the most expected
strategies arising from random choices are barely played. Only highly cooperative strategies (0.75, 0.875) appear to a
similar extent to what would be expected by chance, except for AllC, which appears to be more frequent.
Therefore, about half of the participants played the Nash equilibrium of the game, choosing to defect in all
sessions, irrespective of the value of the critical mass NC. On the other hand, as a consequence of the fact that
the majority of the subjects do not change strategy, the average number of times a player changes strategy over
the course of the eight sessions is relatively low (1.08 out of maximum of 7). In fact, only 43% of the participants
changed their decision at least once: As shown in Fig.?3, there are subjects who change strategy even five times,
but no one changes strategy more than that.
We now explore in more detail the sequence of actions taken by subjects who changed their decision at least
once in the eight sessions. Across the two NC treatments and among the 12 subjects who change strategy only
once, 8 of them (67%), move from C to D, while only 33% move from D to C, i.e., half of those who changed in the
opposite direction. Fourteen subjects change their choice twice, accounting for 28 option changes. 43% of them
go from C to D and back, and 57% do the opposite. As for the rest of the changes, a total of 75, 34 of them (45%)
are switches from D to C, and the remaining 41 go from C to D. Thus, the only net effect of the changes seems to
be in the subjects who changed only one, a behavior that could be a possible reason for the observed increment in
cooperation with size. However, because of the small sample size, this should be read as a hint to be confirmed in
future work, rather than a well-established conclusion.
Finally, we also checked for potential gender effects, especially because previous research suggests that gender
may be related to prosocial preferences in a non-trivial way. In particular, Croson & Gneezy56 argue that women
are more altruistic than men in the Dictator game, but that there are no stable gender differences in cooperation
games such as the Prisoner?s dilemma and the Public Goods game. On the contrary, Molina et al.57 find some
evidence supporting larger cooperativeness among female high school students. Two recent meta-analyses58,59 found
that women are more altruistic than men, especially when acting under intuition59, and that gender differences
in cooperation, if existing, are likely to be very small60. In line with this general picture, also in our cooperation
problem, we found no gender differences in cooperation. Specifically, we conduct a set of logistic regressions
Critical Mass = 10
predicting C as a function of Female, for each value of N and also collapsing all group sizes together, finding all
p-values to be larger than 0.1.
Here we have contributed to the ongoing debate regarding the effect of group size on cooperative behavior. We
have experimentally investigated the possibility of observing nonlinear dependencies of cooperation with group
size in public goods game with a critical mass. Our specific research questions related to the possibility, predicted
theoretically, of observing a maximum in cooperation as a function of the group size around the critical mass at
which additional cooperators do not lead to increase public good production.
In contrast to this prediction, we observe a monotonous increase of the rate of cooperation as a function of
the group size. Furthermore, our results suggest that the critical mass has no influence on the rate of
cooperation. In addition, we have also looked at heterogeneity in individual behavior, finding approximately half of the
subjects never contributing, another 10% that were unconditional cooperators, whereas the rest show a tendency
to change preferentially from defection to cooperation than the converse, as the size of the group increases.
There are several reasons why we may have not observed an inverted-U effect of group size on cooperation in
our experiment. One might have been selection bias: it is possible that subjects who dropped out in the middle of
the experiment were high cooperators with respect to those who completed the experiment, and this might have
killed the pick of cooperation at intermediate-size groups. However, as we have seen in the Results section, our
analysis shows that subjects who dropped out cooperated at the same rate as those who completed the
experiment, thus ruling out this possible explanation. Another possible reason for the discrepancy comes from a closer,
data-driven, inspection of the results reported by Capraro and Barcelo45. Here, the maximum rate of cooperation
was observed for N = 15. However Fig.?1 in Capraro & Barcelo45 highlights that the case N = 15 gave rise to an
unusually high rate of cooperation, suggesting that it might have been an outlier. Treating N = 15 in Capraro &
Barcelo45 as an outlier results in a broad maximum around N = 25, that could only be noticed by going to groups
as large as about 100 people. If the maximum rate of cooperation in public goods game with a critical mass is
attained this far from the critical mass for the case NC = 10, the situation could be further complicated in the case
NC = 20, in which the rate of cooperation could reach its maximum for group sizes not considered in our
experiment ? contributing to mask the effect. It is also possible that the payoff structure of the game affects the results,
as has been observed, e.g., by Nosenzo et al.21, who observed no group size effect for low marginal per capita
returns and negative effect of group size for high marginal per capita returns. Another potential explanation is
that within-subject experiments give rise to fundamentally different dynamics than between-subjects ones,
perhaps for the following reason. In order to have an inverted-U effect of the group size on cooperation in a within
subject design, one would need a significant proportion of subjects that change strategy twice: from defection, to
cooperation, and then back to defection. This somehow conflicts with the literature suggesting that subjects have
preferences for being consistent61, especially when it comes to costly prosocial behavior62. In line with this view,
our analysis of the heterogeneity in people?s behavior shows that about 60% of the subjects never change strategy,
and about 70% of the subjects change strategy only once. This might have partly contributed to the lack of an
inverted-U effect of group size on cooperation. The discrepancy could perhaps be understood in terms of
evolutionary game theory, because from that view point the direction and size of the effect could depend on external
control parameters such as the mutation rate or the intensity of the selection5. Finally, reminding that the
existence of an inverted-U effect was deduced by assuming that subjects play according to the cooperative equilibrium
model46, it is possible that, in our setting, this model does not represent a good approximation of subjects?
behavior. Although this might be true in general, it would not explain why we found a positive effect of group size on
cooperation, while Capraro & Barcelo45 found an inverted-U effect. Further experimental work, involving much
larger number of subjects and groups, is needed to discriminate among these options and have a more complete
understanding of if, when, and how group size impacts cooperative behavior.
All participants in the experiments reported in the manuscript signed an informed consent to participate when
enrolling in the IBSEN volunteer pool55. In agreement with the Spanish Law for Personal Data Protection, their
anonymity was always preserved. This procedure was approved by the Ethics Committee of Universidad Carlos
III de Madrid, the institution responsible for funding the experiment, and the experiment was subsequently
carried out in accordance with the approved guidelines.
The experimental instructions were prepared following as closely as possible those used in45, albeit translated
into Spanish. A screenshot of the decision page is shown in Fig.?4. Here we include a translated version of the
?You are going to participate in an experiment with other people. You will play in groups of N people.
It will take you about 5 minutes. Each wave, your earnings will be shown in points, and only if you complete the 8
waves of the experiment, these earnings will be converted to money. Additionally, if you complete the eight sessions
of the experiment and your total score (of the eight sessions) is among the 25% best, you will participate in a lottery
for a single prize of 50 Euros.
Each session, you will have to decide to join either Group A or Group B. Your earnings will depend on the group you
decide to join and on the size of the two groups, A and B, as follows:
If you join A, and the number of people that also decided to join A (size A) is 3 people, you will earn 15 points.
If you join B, and the number of people that decided to join A (size A) is 5 people, you will earn 35 points.
You will be paid at the end of the experiment. Your earnings will be converted to money and transferred to your
PayPal account, in a maximum period of 7 days counting from the end of the experiment. Please make sure the
address of the Paypal account you provide us is correct, since it will be the address used for the payment. If you want
to correct it, please write to us by email before the end of the experiment. We will not be able to repeat the
payment if the account address is incorrect. For your convenience, these instructions will be available throughout the
At the beginning of each session, the instructions were shown, and afterwards participants had to complete
four control questions to facilitate the understanding of the experiment. They had five opportunities to give the
right answer, and afterwards, the correct answer was shown. The questions were:
Q1: Imagine you decide to join group A, and the number of people that also decided to join A (size A) is 4
people. How many points will you earn?
Q2: Imagine you decide to join group B, and the number of people that also decided to join A (size A) is 8
people. How many points will you earn?
Q3: Imagine you decide to join group A, and the number of people that also decided to join A (size A) is 15
people. How many points will you earn?
Q4: Imagine you decide to join group B, and the number of people that decided to join A (size of A) is 20
people. How many points will you earn?
During the experiment, the only information available to the participants was the group size they were playing
every time and the beta function (as a function of A and B decisions) presented by means of Table?2. Participants
played online through a web browser in a computer, tablet or mobile phone; and could make their decisions
during eight hours each session day. After each of the sessions, which where all one-shot game sessions, groups were
randomly formed only with participants that made their decision in order to compute the payoffs. Participants
membership to these randomly formed groups (of size N) did not remain constant across sessions, but new
random groupings were created after each session. For participants in incomplete groups, we assigned a payoff by
randomly sampling among participants in the corresponding session that made the same decision (cooperate/not
cooperate). The payoffs of all rounds were only shown at the end of the experiment.
This work was partially supported by MINECO/FEDER (Spain) through grant FIS2015-64349-P VARIANCE
(AS) and by the EU through FET Open Project IBSEN (contract no. 662725, AS).
M.P., V.C. and A.S. conceived the original idea for the experiment; M.P. wrote the software interface for the
experiment and carried out the experiment; M.P. and V.C. analyzed the data; M.P., V.C. and A.S. discussed the
analysis results and wrote the paper.
Competing Interests: The authors declare no competing interests.
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