Heterogeneous cooperation preferences and the
dynamics of free riding in public goods†
Urs Fischbacher, University of Zurich*
Simon Gächter, University of St. Gallen, CESifo and IZA**
25 August 2004, PRELIMINARY VERSION
Abstract
We study the role of heterogeneous cooperation preferences in the voluntary provision of
public goods. The novel feature of our design is that we devise an experiment to elicit
people’s cooperation preferences which we then use to make a point prediction about each
individual’s contribution to a linear public good. In our preference-elicitation game we
observe 23 percent free riders and 55 percent conditional cooperators who contribute to the
public good if others do so as well. We find a high degree of cross-game consistency. We
also show that the interaction of heterogeneously motivated agents explains 82.5 percent of
contributions, and together with reduced errors, the dynamics of free riding.
Keywords: Public goods; Voluntary contributions; Conditional cooperation; Free riding;
Errors.
JEL classification: H41, C91, D23, C72.
†
This paper is part of the MacArthur Foundation Network on Economic Environments and the Evolution of
Individual Preferences and Social Norms. Support from the EU-TMR Research Network ENDEAR (FMRXCT98-0238) is gratefully acknowledged. Sybille Gübeli, Eva Poen and Beatrice Zanella provided very able
research assistance. We also thank Nick Bardsley, Rachel Croson, Armin Falk, Ernst Fehr, Björn Frank, Sybille
Gübeli, Radosveta Ivanova-Stenzel, Ekkehart Schlicht, Martin Sefton, and seminar participants at CES Munich,
the Max Planck Institute in Jena, the University of Hannover, the Wissenschaftszentrum Berlin, the University of
Konstanz, and the ESA conference in Amsterdam for their helpful comments. Simon Gächter thanks CES
Munich for the hospitality he enjoyed while working on this paper.
*
Institute for Empirical Research in Economics, Blümlisalpstrasse 10, CH-8006 Zürich. E-mail:
[email protected].
**
FEW-HSG, Varnbüelstrasse 14, CH-9000 St. Gallen. E-mail: [email protected].
1
1. Introduction
An important result in economic theory is that there will be an inefficient undersupply of a
voluntarily provided public good because agents have an incentive to take a free ride. A
similar prediction extends to various social dilemma situations, like collective actions or
common pool resources, whose tragedy lies in its overuse by selfishly motivated individuals.
While the logic of self-interest is straightforward, the facts seem to be at odds with theoretical
predictions derived under the joint assumptions of rationality and selfishness. The fact that
people vote, take part in collective actions, often manage not to overuse common resources,
care for the environment, mostly don’t evade taxes on a large scale, donate to public radios, as
well as to charities, etc. suggests that the strict self-interest hypothesis is inconsistent with the
facts. The clearest evidence in favor of this conclusion probably comes from controlled
laboratory public goods experiments (e.g., Ledyard 1995). The stylized facts show that
people contribute to the public good even if they have a dominant strategy to free ride. This
holds both in one-shot and in repeated public goods environments. In the latter, however,
contributions dwindle over time to rather low levels, leading to almost complete free riding.
What explains these observations? In this paper we concentrate on voluntary contributions
to public goods and investigate the hypothesis that people are heterogeneous with respect to
their willingness to make voluntary contributions. Our results show that a few people are
unconditional altruists who are willing to sacrifice resources to benefit others. Some people
are free riders, who, if at all, only make contributions for strategic reasons. Others are
conditional cooperators who are prepared to contribute to the public good if they think that
others are contributing as well. We believe that this heterogeneity explains both why some
people contribute to public goods and why contributions often decline in repeated setups.
That people differ in their social preferences is suggested by existing experimental
evidence and recent theories of social preferences (see, e.g., Camerer 2003, Chap. 2, for an
overview). In this paper we go beyond the observation that there is heterogeneity in behavior
and address a theoretically important question that follows from it. The question is whether
there are ‘types’ of players. To be theoretically useful, a ‘type’ of player (as identified by his
or her social preferences) should behave similarly in comparable environments. Thus, if
someone is a free rider, he or she should take a free ride in different social dilemma situations
where free loading is in an agent’s material self-interest. Likewise, conditional cooperators
should cooperate conditionally on others’ contributions in comparable public goods situations.
To our knowledge, not much is known about the cross-game stability of social preferences.1
To assess the cross-game stability of behavior of different types of agents in the context of the
voluntary provision of public goods is the first contribution of our paper.
Our second contribution is to explain the dynamics of free riding as a result of the
interaction of different types of players. The intuition is that if conditional cooperators realize
1
There are a few exceptions. Ashraf et al. (2003) observe people in trust and dictator games. In the context of
public goods provision, Offerman et al. (1996) and van Dijk et al. (2002) elicit social value orientations and
compare them to behavior in public good environments. They find that the social value orientation is positively
correlated with contribution to public goods. Brandts and Schram (2001) use questionnaires to classify people as
free riders and cooperators. However, their designs do not permit to use the value orientations or the
questionnaire-based classifications to make point predictions about contributions, which is what we aspire to do.
2
that some people take a free ride they will reduce their own contribution because they don’t
want to be suckered. In most previous accounts – which we will discuss in Section 5 –
‘errors’ play an important role in explaining (the decline of) cooperation. Thus, a third
contribution is assessing the role of errors relative to the power of explaining (the decline of)
cooperation as resulting from the interaction of heterogeneously motivated agents.
To accomplish our goals, we use the tools of experimental economics that have proved
useful for purposes like ours (see Ledyard 1995 for an extensive discussion of public goods
experiments). Our ambition is to measure people’s preferences toward voluntary contributions
to public goods and then to use these preferences for prediction purposes and for assessing the
role of errors in contributions. To live up to our ambition, our design, which we detail in the
next section, consists of three elements. The first element is a standard linear public good,
where people can contribute between 0 and 20 tokens to a linear public good. The second
element is the elicitation of beliefs about other group members’ contributions. This allows us
to assess the relationship between one’s own contribution and the expected contributions of
others. Beliefs will also be crucial in combination with the third element of our design, which
is an incentive-compatible measure of people’s preferences toward voluntary contributions.
Our instrument is a variant of the strategy method that uses the same strategy set as the
standard public good game. This latter feature allows us to measure preferences such that we
can use the measured preferences together with the elicited beliefs to make a point prediction
about a person’s contribution to the public good. Thus, we can assess the cross-game stability
of behavior, because we measure people’s preferences and observe the same person in another
comparable environment. We can also observe deviations from predicted behavior
(‘prediction errors’) and analyze them separately.
Our results in Section 4 concern the heterogeneity of motivations and cross-game stability
of types. They are as follows. First, we find a high degree of heterogeneity of elicited
cooperation preferences among our subjects. Roughly 55 percent are Conditional Cooperators
who increase their contributions in other group members’ contributions. Twenty-three percent
are strict Free Riders, who contribute nothing irrespective of other people’s contribution. The
rest shows more complicated patterns. Second, by and large, we find a high degree of
consistency of behavior across the two games – the preference elicitation game and the
standard public good game. Thus, there really seem to exist ‘types’ of players. People who
we classified as Conditional Cooperators in the preference elicitation game also behaved as
Conditional Cooperators in the standard contribution game. People classified as Free Riders
contributed significantly less than all others, but, quite surprisingly, they did contribute to the
public good in the contribution game.
In Section 5 we assess the relative importance of errors and heterogeneous motivations in
explaining the often observed decay in contributions. Many previous accounts stress the
importance of reduced errors in explaining the decay. By contrast, we find that the interaction
of free riders and conditional cooperators explains 82.5 percent of contributions and the decay
in cooperation. Reduced errors account as well for the decline of contributions, but to a much
smaller degree than in previous studies.
We believe that our results have potentially important implications for theory and policy.
We discuss them briefly in Section 6. The theoretical significance is that there really seem to
3
be different ‘types’ of agents, who differ in their social preferences but who behave largely
consistently across different but comparable situations. The policy relevance of our results
comes from the observation that many people are conditional cooperators, which may have
consequences for welfare policies, behavior at the workplace, tax morale, corruption and
charitable donations.
2. Design and procedures
The main novel feature of our experimental design is that we combine two different
experiments. The first type of experiment (labeled ‘P-experiment’) applies a variant of the
strategy method to elicit people’s contribution preferences in a public goods game. By
contrast, in the second type of experiments, subjects actually make contribution choices in a
standard linear public goods environment (labeled ‘C-experiment’) for ten rounds in the
random matching mode (“Strangers” – see Andreoni 1988). All subjects play both types of
experiments. For example, in the P-C experiments, subjects first go through the preference
elicitation experiment, before they make their contribution choices in an ordinary linear public
goods game. Our C-P experiments counterbalance the order of experiments to control for
possible sequence effects. Table 1 gives an overview of our design.
The central idea of our within-subjects design is to use the P-experiments to measure
people’s contribution preferences (i.e., determine their ‘type’) and then to see how these types
behave in the C-experiments and how their interaction can explain the dynamics of
cooperation and free riding over time.
Table 1
Design overview
P-C sequence
C-P sequence
1st experiment
2nd experiment
Preference elicitation (P)
(strategy method)
Ten rounds of linear public good with
random matching (C)
(Actual contribution choices)
Ten rounds of linear public good with
random matching (C)
(Actual contribution choices)
Preference elicitation (P)
(strategy method)
Note: P denotes the preference elicitation experiment and C denotes the contribution choice experiment.
We now describe the two experiments in detail. We start with the P-experiments. Our
preference elicitation study has the purpose of directly eliciting subjects’ willingness for
conditional cooperation. To what degree are subjects willing to cooperate given other
subjects’ degree of cooperation? Being able to observe contribution preferences without using
deception requires observing contributions that can be contingent on others’ contribution.
Fischbacher, Gächter and Fehr (2001, henceforth FGF) introduced an experimental design
4
that accomplishes this task.2 The central idea of FGF is to apply a variant of the so-called
“strategy method” (Selten 1967). The subjects’ main task in the experiment is to indicate for
each average contribution level of other group members how much they want to contribute to
the public good. Before we describe how exactly this is done, we first outline the basic
decision situation.
The basic decision situation is a standard linear public goods game (Ledyard 1995). The
subjects are randomly assigned to groups of four people. Each subject is endowed with 20
tokens, which she can either keep for herself or contribute to a ‘project’, the public good. The
payoff function is given as
4
π i = 20 − g i + 0.4∑ g j ,
(1)
j =1
where the public good is equal to the sum of the contributions of all group members.
Contributing a token to the public good yields a private marginal return of 0.4 and the social
benefit is 1.6. Under standard assumptions the prediction is therefore complete free riding by
all subjects, i.e., gj = 0 for all j.
The above public good problem was explained to the subjects in the instructions (see
Appendix). We also took great care to ensure that subjects really understood the game and
the incentives, since we want to measure their preferences as accurately as possible.
Therefore, after subjects had read the instructions, they had to answer ten control questions
that tested their understanding of the basic decision situation. We did not proceed until all
subjects had answered all questions correctly. Thus, we can safely assume that people
understood the game and the incentives.
In the P-experiment subjects were asked to make two types of contribution decisions, a socalled ‘unconditional contribution’, and filling in a ‘contribution table’. In the ‘unconditional
contribution’ subjects just had to make a single decision on a particular contribution to the
public good (i.e., they had to choose a number between 0 and 20). For the contribution table
subjects had to indicate their contribution dependent on the average contribution of the other
three group members (rounded to integers). Specifically, subjects were shown a table of the
21 possible values of the average contribution of the other group members (from 0 to 21) and
were asked to state their corresponding contribution for each of the 21 possibilities. Whereas
the unconditional contribution decision just asked for the ‘usual’ type of decision, the
contribution table elicits a contribution schedule (i.e., a vector of 21 contributions). Subjects
were under no time pressure when they made their decision.
2
The FGF paper is a methodological contribution that was evaluated with a rather small number of subjects
(n=44). Here we adopt their method and use it with n=140 subjects. Other researchers follow similar goals but
have devised other instruments. Kurzban and Houser (2001) have developed a ‘circular public goods game’.
Bardsley (2000) and Bardsley and Moffatt (2003) use a ‘conditional information lottery’. Independently of FGF,
Ockenfels (1999) has developed a similar design. A further method is eliciting beliefs about others’ contribution
(see Croson 2000, 2002; Neugebauer et al. 2004), a method that we will also adopt in our C-experiments. The
drawback of this method is that beliefs are endogenous and beyond the control of the experimenter. They also
typically don’t cover the whole range of possible contributions.
5
After the subjects had made their unconditional and conditional contribution decisions, we
asked them on a new screen to estimate the average of the unconditional contributions of their
three other group members. We gave the subjects financial incentives for correct estimates.3
After all subjects had made all entries, including the estimation, a random device (throw
of a die) selected one subject of the group with each subject having the same probability of
being chosen. This subject contributed according to his or her contribution table. The other
three contributed their unconditional contribution. This random device made both decisions
potentially outcome relevant. Hence, subjects had an incentive to take both decisions
seriously.4
In the C-P experiments, the sequence of experiments was simply reversed. Subjects first
played ten rounds of the C-experiment in the random matching mode and were then
introduced to the P-experiments.
What are the predictions for the P-experiment? It helps to think of our experiment in terms
of the following extensive form game played with the strategy-method: nature chooses three
players who make their contribution decisions simultaneously. The fourth player learns the
(rounded) average contribution of the other players and then decides how much to contribute.
Since we play this game in the strategy-method, all subjects have to make contribution
decisions in the role of the fourth player, which is what they do when they fill in the
contribution table. All players learn whether they actually are the fourth player or not. If they
are not chosen to be the fourth player, they do not learn who is chosen. For rational and selfish
players, we get the following prediction: for the fourth player it is optimal to contribute zero –
independent of the contributions of the other players. Hence, with the strategy method rational
and selfish players should have only ‘0’ entries in their contribution tables. Assuming
common knowledge of rationality and selfishness, also the players who have to make
simultaneous contribution decisions will contribute zero to the public good. If we lift the
assumption of common knowledge of rationality, the latter prediction does not necessarily
hold anymore. If players assume that a ‘fourth player’ is a ‘Conditional Cooperator’ who
displays a pattern of increasing contributions in her schedule then it may be optimal to make a
‘non-zero’ unconditional contribution. However, for the prediction of the conditional
contribution, only rationality and selfishness is assumed. In this paper, we are only interested
in the contribution schedule and not in the unconditional contribution.
3
If their estimation was exactly right, subjects received 3 experimental money units in addition to their other
experimental earnings. They received 2 (1) additional money units if their estimation deviated by 1 (2) point(s)
from the actual average contribution of the other group members, and no additional money if their estimation
was off the actual contribution by more than three points.
4
An example illustrates. Assume that the four group members make an unconditional contribution of 4, 6, 8 and
10 tokens, respectively. Assume that the random device determines that for the fourth subject, whose
unconditional contribution is 10 tokens, the contribution table becomes the payoff-relevant decision, while for
the other three group members their unconditional contributions are relevant. Hence, the average of their
unconditional contribution is six tokens. Assume that the contribution table of the fourth subject says that she
will contribute 5 tokens in case the others contribute 6 tokens, then her contribution to the public good was taken
to be five tokens. Thus, the sum of contributions in this example is 23 tokens. Individual payoffs can now be
calculated according to payoff function (1).
6
The experiment was only played once, i.e., there were no repetitions and the subjects
knew this. The rationale is that we can elicit subjects’ preferences, without intermingling
preferences with strategic considerations. For example, if a subject chooses a contribution
table that is increasing in the average contribution of others, this cannot be due to reputation
formation or any kind of repeated game consideration. Instead, we can take it as an
unambiguous measure of the subject’s willingness to be conditionally cooperative. Since we
identify the ‘type’ of a subject with his or her preference, the P-experiments will allow us to
classify our subjects into types and see how consistently they behave across the two games
(see next two sections).
In the P-C experiments, after subjects had finished the P-experiment, they were informed
that they would play another experiment. Subjects were then told that the second experiment
(which we call the ‘C-experiment’) is just a ten times repeated play of the basic decision
situation, where each group member simultaneously makes his or her contribution choice.
We emphasized that the groups of four would be randomly reshuffled in each period. In
addition to their contribution decisions subjects also had to indicate their beliefs about the
average contribution of the other three group members in the current period. We paid
subjects according to the accuracy of their estimates. Incentives were the same in each period
as in the P-experiment.
We elicited beliefs for two reasons. First, we can assess the correlation between beliefs
and contributions, which we expect to differ between types of players. For instance, free
riders are expected to have a zero correlation between their beliefs about what others are
going to contribute and their own contribution (which is predicted to be zero). By contrast,
Conditional Cooperators are expected to have a positive correlation of beliefs and
contributions. Second, as we will explain in great detail below, we can use the beliefs and the
elicited schedules from the P-experiment to make point predictions about an individual’s
contribution in the C-experiment.
All experiments were computerized, using the software z-Tree (Fischbacher 1999). The
experiments were conducted in the computer lab of the University of Zurich. Our participants
were undergraduates from various disciplines (except economics) from the University of
Zurich and the Polytechnic University (ETH). We conducted six sessions (three in P-C and
three in C-P), with 24 subjects in each session (except one with 20 subjects). A postexperimental questionnaire confirmed that participants were largely unknown to each other.
Our 140 subjects were in each session randomly allocated to the cubicles, where they took
their decisions in complete anonymity from the other subjects. On average subjects earned 35
Swiss Francs (roughly $35 including a show-up fee of 10 Swiss Francs) in experiments which
lasted roughly 90 minutes.
7
3. Types of preferences
The first step in our analysis is the definition of types and the classification of our
subjects. We define four different types: ‘Conditional Cooperators’, ‘Free Riders’, ‘Triangle
Contributors’, and ‘Others’.
The precise rules for the determination of types are as follows: All subjects who show
either a monotonic pattern with at least one increase or have a positive Spearman rank
correlation that is significant at the 1%-level are classified as ‘Conditional Cooperators’. All
subjects who choose to contribute 0 in any case are classified as ‘Free Riders’. ‘Triangle
Contributors’ are subjects who have a significantly increasing scheme up to some maximum
and a significantly decreasing scheme thereafter, again using the Spearman rank test at the
1%-level as the criterion (FGF call this pattern ‘hump-shaped contributions’). All subjects
who cannot be classified this way fall into the category ‘Others’.5
Fig. 1 and Table 2 contain the results of this classification. Fig. 1 shows the pooled
distribution of types. In Table 2 we will report the disaggregated distribution of types in our
P-C and C-P sequences, respectively.
20
20
Conditional Cooperators: 55.0%
Own contribution according to the
'Contribution table'
18
Free Riders: 22.9%
Triangle Contributors: 12.1%
16
16
Others: 10.0%
14
18
14
12
12
10
10
8
8
6
6
4
4
2
2
0
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Average contribution level of other group member
Fig. 1. Average own contribution level per type for each average contribution level of other group members
(diagonal = perfect Conditional Cooperator).
5
The category ‘Others’ contains two ‘Unconditional Cooperators’ who always contribute 20 and one
‘Negatively Conditional Cooperator’. For not complicating the analysis we have subsumed them under the
category ‘Others’.
8
The graphs in Fig. 1 show the average contribution of all subjects of a particular type. For
instance, the graph ‘Conditional Cooperators’ depicts the average contribution across the
schedules of all subjects who we classify as Conditional Cooperators. This also holds for the
other types.
The average contribution vector is not characterized by complete free riding. Although it
was common knowledge that this game will be conducted only once, the average contribution
taken over all 140 subjects is monotonically increasing in the contributions of the other group
members. On average, therefore, our subjects clearly displayed conditional cooperation.
Conditional Cooperators are the most prevalent type. Fifty-five percent of our subjects
display a monotonically increasing contribution schedule (at least in the statistical sense).
Our Conditional Cooperators contribute nothing if the others contribute nothing and increase
their contribution up to roughly 15 if the others contribute 20. Thus, on average, the
Conditional Cooperators do not match the others’ contribution perfectly; contributions are
self-servingly biased. Seventeen percent of the Conditional Cooperators are perfect
Conditional Cooperators, whose schedule lies exactly on the diagonal. The second most
frequent type is the Free Rider, who never contributes anything to the public good,
irrespective of the others’ contribution. Slightly less than a quarter of our subjects fall into
this category. A somewhat peculiar type is the Triangle Contributor who increases his or her
contribution up to a certain level and from then on decreases his or her contribution. Twelve
percent of our subjects belong into this category. Ten percent of our subjects (‘Others’)
cannot be classified and most show a rather random pattern.
In Table 2 we report the percentage distribution of types. We separate the classification
according to the P-C and the C-P sequences and we also report the pooled distribution. As a
benchmark, we list the results from FGF. The last two columns contain the p-values of pairwise Fisher exact tests to test whether the distribution of types is different between (i) the P-C
and the C-P sequence (i.e., to see whether there is a sequence effect) and (ii) FGF.
Table 2
Distribution of types (percentages)
Our experiments
FGF
p-values of pair-wise
Fisher exact tests
P-C
(n=72)
C-P
(n = 68)
Pooled
(n=140)
(n=44)
56.9
52.9
55.0
50.0
Free Rider
19.4
26.5
22.9
29.6
0.421
0.419
Triangle Contributor
15.3
8.8
12.1
13.6
0.305
0.512
6.8
0.579
0.686
Conditional Cooperator
Other
8.3
11.8
10.0
Note: For the exact definition of types see the main text.
P-C vs. C-P
Pooled vs. FGF
0.734
0.772
Two interesting observations can be drawn from Table 2. First, there is no sequence effect
in our elicitation of types. Put differently, the measured frequency of types is not affected by
9
the order in which the P-experiment was conducted, i.e., whether it came first (as in the P-C
sequence) or second (as in the C-P sequence). This is also confirmed by a χ2-test that
compares the distribution of types under P-C and C-P, respectively (p=0.481). Pair-wise
Fisher exact tests, performed separately for each type, all return p-values > 0.30, i.e., the null
hypothesis of an equal distribution of types under C-P and P-C cannot be rejected (see
penultimate column). A second robustness test is a comparison with FGF. As a comparison
of columns ‘Pooled’ and FGF shows, we replicate their results quite closely. Again, a χ2-test
cannot reject the null hypothesis of an equal distribution of types (p=0.729). This also holds
for all separate pair-wise Fisher exact tests (see last column, where all p-values > 0.4).
Together, we conclude from these results that our measurement of types is quite robust and
therefore provides a sound basis for looking at behavior in the C-experiments.
4. Types and their contribution behavior
With 5.2 in the C-P sequence and 4.5 in the P-C sequence, the mean contributions are very
similar and so are the temporal patterns. Contributions start between 7 and 9 and decline
smoothly and almost identically to an average contribution of two tokens.6 The c.d.f.’s are
also nearly identical. Thus, there is no sequence effect in our data and we will therefore pool
the sequences for the rest of our analysis.
The main purpose of this section is to see whether the subjects who we classified into four
different types actually behave differently from one another, i.e., we want to assess the crosscame stability of behavior. The following analysis consists of five parts. First, we will look at
the contribution levels and the beliefs our types hold about their group members’ average
contribution. Second, we investigate the correlation between beliefs and actual contributions
of the types at the aggregate level. Third, we study individual behavior. Fourth, we investigate
how predictable types are, given their observed behavior in the C-experiment. Last, we check
the consistency of actual and predicted contributions according to their elicited contribution
schedules.
4.1 Contribution and belief levels
What are our predictions how different types will behave in the C-experiments? We start
with the contribution levels. If there would be a perfect match between our classification of
types in the P-experiment and their behavior in the C-experiments, we should see the
following patterns in the C-experiments: Free Riders should contribute nothing in all rounds,
since they have a dominant strategy to take a free ride in each round. Conditional Cooperators
and Triangle Contributors should have higher contributions than Free Riders and show a
declining pattern, since they will reduce their contribution when they realize that they have
been suckered. ‘Others’ should show a random pattern.
6
Our results are comparable to previous research (e.g., Andreoni 1988; Weimann 1994; Croson 1996, 2002;
Burlando and Hey 1997; Keser and van Winden 2000; Fehr and Gächter 2000, 2002).
10
Fig. 2 depicts the temporal development of contributions. Free Riders start out with much
lower contributions (less than 5 tokens) than all other types, whose starting contributions
amount to roughly 9 tokens and are statistically indistinguishable (Kruskal-Wallis test, p =
0.8911). The difference in initial contributions between Free Riders and all other types is
highly significant according to a Mann-Whitney test (p = 0.0012, two-tailed). In all periods,
Free Riders contribute less than all others. Their average contribution is 2.49 tokens. Seventy
percent of all contributions of Free Riders are exactly zero. By contrast, Conditional
Cooperators on average contribute 5.64 tokens; Triangle Contributors spend 4.88 tokens on
the public good and ‘Others’ invest 5.66 tokens.
14
Conditional Cooperators - Mean: 5.64
Free Riders - Mean: 2.49
Triangle Contributors - Mean: 4.88
Others - Mean: 5.66
Mean contribution
12
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10
Period
Fig. 2. Mean contributions over time per type.
Remember that in each period we have also elicited the beliefs of our subjects about their
group members’ actual contribution. Since our subjects had an incentive to correctly estimate
their group members’ contribution, we predict that there should not be any difference in
beliefs between types. A competing hypothesis comes from the ‘false consensus effect’ and
the observation, made by psychologists (e.g., Kelley and Stahelski 1970, Shafir and Tversky
1992) that people have a tendency to believe that others behave similarly to themselves. If this
were the case we should observe that Free Riders hold consistently lower beliefs than, e.g.,
Conditional Cooperators.
Fig. 3 reports the temporal pattern of the beliefs our subjects hold about their current
group members’ contribution. Two facts are noteworthy. First, all types overestimate the
contribution of others, for the mean actual contribution is consistently below the beliefs of all
types. This fact is consistent with the observation from Fig. 1 that on average for all types,
except ‘Others’, contributions are below the diagonal (i.e., people contribute less than others).
Second, the average belief about the others’ contribution does not strongly differ between
types. If we only look at the first period, where subjects have not yet made any experience
11
with other’s contributions in the C-experiment, we find that beliefs don’t differ between types
(Kruskal-Wallis test, p = 0.4531).
14
Conditional Cooperators - Mean: 6.72
Free Riders - Mean: 5.82
Triangle Contributors - Mean: 6.78
Others - Mean: 6.62
Mean actual contribution - Mean: 4.83
12
Mean belief
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10
Period
Fig. 3. Mean belief about others’ average contributions over time per type.
Table 3
Contributions and beliefs of different types.
Dependent variables:
Contributions
Constant
Beliefs
(i) All periods
(ii) Last five periods
(iii) All periods
(iv) Last five periods
6.00***
4.79***
9.96***
9.08***
(0.771)
(0.223)
(0.682)
(0.988)
Conditional Cooperators
3.15***
(0.455)
3.05***
(0.332)
0.90**
(0.277)
0.68
(0.370)
Triangle Contributors
2.39**
(0.650)
1.13*
(0.539)
0.96
(0.517)
0.36
(0.335)
‘Others’
3.17*
(1.19)
3.94**
(1.013)
0.80*
(0.371)
0.83
(0.563)
-0.64***
(0.071)
-0.47***
(0.039)
-0.75***
(0.061)
-0.62***
(0.136)
1400
700
1400
700
162.85***
58.57***
76.39***
38.22***
0.145
0.095
0.288
0.093
Period
F(4,5)
R squared
Note: OLS regressions with robust standard errors clustering on sessions. Free Riders are the reference group. Numbers in
parentheses are robust standard errors. ** significant at 5%; *** significant at 1%. Tobit and ordered probit estimates yield
similar results.
Table 3 provides a statistical account of the above findings. It reports the results of OLS
regressions with robust standard errors of (i) subjects’ contributions over all periods and (ii)
in the last five rounds; (iii) subjects’ beliefs about the average contributions of the other group
12
members over all periods and (iv) in the last five periods. The explanatory variables are
simply dummies for the respective types – Conditional Cooperators, Triangle Contributors,
and ‘Others’. The Free Riders provide the reference case. To control for the time trend
visible in the data we add ‘Period’ as a further variable.
The statistical results confirm the above graphical analysis. All types contribute
significantly and substantially more than Free Riders, who form the reference group. This
holds over all periods, as well as in the last five periods. Models (iii) and (iv) perform the
same analysis for the beliefs subjects hold about the average contribution of the other group
members. Here we find that over all ten periods Conditional Cooperators and Others hold
slightly higher beliefs than Free Riders. Yet, this difference vanishes in the last five periods.
Thus, the false consensus effect does not seem to be very important in our data. According to
all models, contributions strongly decline over time.
4.2 The correlation between beliefs and contributions
In the next step of our analysis we turn our attention to the relation between one’s belief
about others’ contribution and one’s actual contribution. A first indication can be found from
comparing Fig. 2 and Fig. 3. All types contribute less than what they believe the others are
going to contribute. This finding is consistent with the observation from Fig. 1 that all
schedules are on average below the diagonal. Yet, the difference between beliefs and actual
contributions is largest for the Free Riders, who actually contribute 2.49 tokens and believe
that the others contribute 5.82 tokens (i.e., the difference is 3.33 tokens). For the Conditional
Cooperators the difference between beliefs and actual contributions amounts to 1.08 tokens;
for the Triangle Contributors it is 1.90 tokens and for ‘Others’ it is 0.96 tokens.
We are now in a position to investigate in detail the relationship between actual
contributions and the belief about others’ contribution. Remember that in the P-experiment
we asked subjects how much they would like to contribute if the other group members
contribute a certain amount x. Thus, if the elicited schedules from the P-experiment would
perfectly describe people’s preferences in the C-experiment, then a subject who believes that
the others will contribute x should contribute the same amount as the one he or she has
indicated in his or her schedule for others’ contribution of x. Hence, on the basis of Fig. 1 we
predict, apart from some random deviations, a significantly positive correlation of beliefs and
contributions for the Conditional Cooperators, and no correlation for the Free Riders. The
Triangle Contributors should have a hump-shaped relation between their stated beliefs and
their actual contributions. Beliefs and contributions should be unrelated for ‘Others’.
Fig. 4 provides first graphical evidence. It shows – separately for each type – the
distribution of beliefs (i.e., the relative frequencies of a particular belief), the mean
contribution for a given belief, and the predicted contribution from the P-experiment (i.e., the
schedules of Fig. 1 are reproduced for the sake of comparison). If the P-experiment would
perfectly predict behavior, then the graphs ‘actual contribution’ and ‘P-experiment’ would
coincide. We will investigate the deviations in subsequent sections.
13
50%
16
45%
14
50%
Free Riders
10
belief distribution
40%
actual contribution
35%
predicted contribution
30%
8
25%
20%
6
15%
4
12
0
0
1
2
3
4
5
6
7
8
9
10
0%
0
12
20%
15%
5%
0%
1
2
3
4
5
6
7
8
9
10 11 12 13+
Belief about the average contribution of others
50%
16
45%
14
actual contribution
40%
predicted contribution
35%
10
30%
8
25%
6
20%
15%
4
25%
0
Mean contribution
belief distribution
30%
10%
2
Frequency of beliefs
Mean contribution
14
35%
predicted contribution
6
5%
10 11 12 13+
Triangle Contributors
40%
actual contribution
8
Belief about the average contribution of others
16
belief distribution
4
10%
2
45%
50%
Others
45%
belief distribution
12
actual contribution
40%
predicted contribution
35%
10
30%
8
25%
20%
6
15%
4
10%
10%
2
5%
0
0%
0
1
2
3
4
5
6
7
8
9 10 11 12 13+
Belief about the average contribution of others
Frequency of beliefs
Mean contribution
12
Mean contribution
14
Frequency of beliefs
Conditional Cooperators
Frequency of beliefs
16
2
5%
0%
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13+
Belief about the average contribution of others
Fig. 4. Distribution of beliefs (secondary axis), mean actual contribution for a given belief,
and predicted contribution (from the P-experiment) per type (primary axis).
Consistent with the evidence from Fig. 3, (i) the distribution of beliefs is rather similar for
all four types and (ii) the mass of the beliefs is below 14 (95 percent of all beliefs are lower
than 14). Thus, to smooth the graphical exposition we put all beliefs ≥ 13 into one category
‘13+’ and calculate the respective averages for all observations in this category.7
In the empirically relevant range of beliefs below 14, all four types have a contribution
pattern that is increasing in their stated beliefs. Ironically, in a sense, all four types behave as
conditional cooperators. The relationship between beliefs and contributions is the strongest
for the Conditional Cooperators, whose actual contribution is very close to their predicted
contribution. Even Free Riders seem to have a positive relation between beliefs and
contributions. Yet, it is markedly weaker than the one of Conditional Cooperators and close
to zero in the range of 0 to 5 (which contains 55 percent of the beliefs of Free Riders).
Apparently, Free Riders ‘change their mind’ in particular, if they believe that others will
contribute a high amount. Apart from high beliefs, the relationship between beliefs and
contribution for the Triangle Contributors seems to follow the predicted pattern. For ‘Others’
we find no relationship between the ‘prediction’ and their actual relationship (which, in fact,
7
For beliefs above 13 we have in many cases only very few (1 or 2) and often no observation. In the statistical
analysis we include all observations, however.
14
is similar to the Conditional Cooperators and the Triangle Contributors in the empirically
relevant range).
A regression analysis confirms the visual impression from Fig. 4. Table 4 reports the
results of a Tobit regression with robust standard errors. The dependent variable is ‘Own
contribution in period t’ and the explanatory variables are ‘Belief’ (which measures the belief
in t about other group members’ mean contribution), and the interaction variables
‘Belief×Conditional Cooperator’, ‘Belief×Triangle Contributor’ and ‘Belief×Others’, which
measure the slope differentials relative to the slope of the benchmark group, the Free Riders.
We estimate three models. The first model comprises all periods. The second and third
models look at the first five and last five periods, respectively.
Table 4
The correlation between one’s own belief and actual contribution
All periods
Periods 1-5
Periods 6-10
Constant
-4.533
-3.957
-4.297
(0.439)***
(0.810)***
(0.557)**
Belief
0.59
0.656
0.124
(0.115)***
(0.150)***
(0.213)
Belief×Conditional Cooperator
0.722
0.572
1.253
(0.102)***
(0.121)***
(0.196)**
Belief×Triangle Contributor
0.536
0.505
0.695
(0.121)***
(0.141)***
(0.230)**
Belief×Others
0.639
0.397
1.389
(0.149)***
(0.165)**
(0.310)**
Observations
1400
700
700
χ2(4)
437.1***
180.1***
189.4***
Note: Tobit regression with robust standard errors. Numbers in parentheses are standard errors. ** significant at 5%;
*** significant at 1%. Free Riders are the reference group.
We start with the first model, which comprises all periods. For all types the higher the
belief about the other group members’ contribution, the higher is, ceteris paribus, the own
contribution.8 Yet, Free Riders have the smallest slope of all types. All other types have a
significantly higher slope than free riders. Thus, the statistical analysis confirms that even
Free Riders behave conditionally cooperatively.9 Yet, if we split up the data in the first and
second five periods, we find that Free Riders have a positive slope only in the first five
periods; in periods 6-10 the coefficient loses significance. We will come back to this
observation in the next section. By contrast, the coefficients on the slope differentials of all
other types remain highly significant.
8
Croson (2000, 2002) and Neugebauer et al. (2002) who, however, don’t distinguish between types, also report
positive correlations between beliefs and contributions. See also Croson (2000) for further references. Moreover,
Croson (2000) finds that eliciting beliefs leads to more selfish behavior than not eliciting them. This result
suggests that our elicitation of beliefs makes it harder to observe non-selfish behavior.
9
Triangle contributors are predicted to have a ‘hump-shaped’ relation between contribution and beliefs. We
therefore ran a regression for Triangle Contributors where we also include a quadratic term (belief2) to capture a
possible non-linearity. The quadratic term is highly insignificant. This is also true for the other types.
15
Fig. 3 and Table 4 have looked at the correlation between beliefs and contributions at the
aggregate level. In the following section we investigate the individual-level correlations and
see whether we get differences between types.
4.3 Individual behavior
To check behavior at the individual level, we calculate for each subject the slope
coefficient (denoted β in the following) of a regression of own contribution on the beliefs of
this subject (i.e., Contributioni = αi + βiBeliefi + εi). We force the regression to go through
the origin (i.e., αi = 0 for all i), for two reasons. First, almost all subjects contribute zero
tokens to the public good if they believe the others contribute zero (see Fig. 4). Second, we
can easily compare the slopes as a measure of a subject’s degree of conditional cooperation.
Fig. 5 shows – separately for each type – the c.d.f. of all individual slope coefficients.
100%
90%
Cumulative frequencies
80%
70%
60%
50%
Free Riders
40%
Triangle Contributors
30%
Conditional Cooperators
20%
Others
10%
2.8
1.51
1.21
1.15
1.11
1.06
1.03
1.01
0.97
0.89
0.85
0.75
0.7
0.61
0.57
0.49
0.44
0.37
0.24
0
0.1
0%
Individual OLS slope regression coefficients of beliefs of others' contribution
on own contribution
Fig. 5. Cumulative distribution for each type of OLS slope regression coefficients between actual contribution
and belief about the other group members’ average contribution in the C-experiments
(with constant forced to zero).
Before we interpret the results notice that a ‘perfect’ Free Rider has a β = 0 and for a
‘perfect’ Conditional Cooperator β = 1. From Fig. 1 we predict that most β’s should be
smaller than 1, because on average schedules are below the diagonal. This is indeed the case
for 69 percent of the subjects; 90 percent have a β ≤ 1.16. Consistent with the previous
results, we find that the c.d.f. of Free Riders lies above the c.d.f.’s of all other types, except
for very high βi’s. Slightly more than 40 percent of Free Riders have a βi of exactly zero,
whereas this is the case for less than 15 percent of the other types. Seventy percent of the
Conditional Cooperators have a βi > 0.5, i.e., their contribution increases by at least 0.5 tokens
16
for each token that the other group members are believed to contribute on average. Thus, Fig.
5 reinforces the messages from Fig. 4 and Table 4.10
4.4. Predicting types
In the previous analysis we have investigated how a particular type, as determined in the
P-experiment has behaved in the C-experiment. We have seen, for instance, that Free Riders
mostly have smaller βi’s than the other types. One can also ask the reverse question: given
that we observe a particular βi of a given subject i, what is the likelihood that subject i is of a
particular type? This question is interesting, for two reasons. First, the above analysis has
shown that, even though we find a high degree of consistency, the matching of type and
behavior is not perfect. Second, in most experiments one does not have direct information
about the type of the particular player as we have it from the P-experiments.
For the purpose of determining the predicted probability of a particular type, we estimate a
multinomial logit model with our four type categories as dependent variable and βi as the sole
explanatory variable (χ2(3) = 13.9; p = 0.0031; thus, the model predicts better than chance).
We calculate the predicted probabilities and plot them in Fig. 6 for each type as a function of
β.11
100%
Predicted probability of type
90%
80%
70%
60%
Conditional Cooperators
Free Riders
Triangle Contributors
Others
50%
40%
30%
20%
10%
0%
0
0.25
0.5
90 % of the subjects
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
10 % of the subjects
Slopes of individual OLS regressions of contribution on belief
Fig. 6. The estimated probability of being a particular type given the observed slope (βi) of the OLS regression of
contribution on belief about the average contribution of other group members (constant suppressed).
First, the likelihood that a particular β identifies a Triangle Contributor or an ‘Others’
type, is almost constant and amounts to roughly ten percent for both types and all β’s.
Second, the probability that a particular β identifies a Conditional Cooperator is 37 percent at
10
Using individual Spearman rank correlation coefficients instead of linear slope coefficients leads to the same
qualitative result. Likewise, if we estimate individual slopes by not repressing the constant and plot them, we
qualitatively get the same picture.
11
Again, using individual Spearman rank correlation coefficients instead of linear slope coefficients or slope
coefficients of unrestricted regressions in the multinomial logit regression leads to the same qualitative results.
17
β = 0, increases in a concave way in β and approaches 80 percent for very high β’s. The odds
of identifying a Free Rider are almost the mirror image of the probability for identifying
Conditional Cooperators. At β = 0 the probability for a Free Rider is 45 percent, which
decreases in a convex way in β and approaches zero for high β’s. The probability that a Free
Rider has a β = 1 (i.e., behaves perfectly conditionally cooperatively) is only 13 percent and
with a probability of 63 percent such a person is a Conditional Cooperator.
4.5 Consistency of predicted and actual choices
In this section we will investigate the consistency of behavior in the P- and the Cexperiments more systematically. Since we have the schedule of each subject from the Pexperiment and since we also have the beliefs in the C-experiment, we can calculate a
predicted contribution that follows from the schedule of a particular subject and compare it to
this subjects’ actual contribution given the stated belief. For the time being, we define a
deviation from the predicted and actual contribution as an ‘error’. There are several sources
for such errors. First, psychological research suggests that in many decisions there exists a
so-called ‘hot-cold empathy gap’ (see, e.g., Loewenstein 2000). Decisions may depend on
whether they are made in a ‘cold’ state, as in our P-experiments, where subjects have to think
about what they do if the others would contribute a certain amount, or whether they are, as in
the C-experiments, actually confronted with contributions that others have made in the
previous period (the ‘hot’ situation).12 Second, subjects may tremble in the implementation of
their preferences. Third, subjects may be confused. Fourth, some subjects may simply
imitate others. Fifth, despite the fact that groups are randomly reshuffled in each period
subjects may believe that they can influence others’ contributions.
In Fig. 7 we plot – separately for each type – the average predicted contribution and the
average actual contribution of each participant in the experiment. Thus, each dot in Fig. 6
represents one individual. On the diagonal the average actual contribution equals the average
predicted contribution. The bold line in Fig. 7 is a trend line, based on OLS. In case of perfect
prediction all observations would be on the diagonal.
12
The evidence on the hot-cold empathy gap in strategy-method experiments and actual game-playing
experiments is mixed. For instance, Brandts and Charness (2000) find no difference, whereas Brosig et al.
(2003) do report differences between methods.
18
Conditional Cooperators
Free Riders
20
20
MAD: 3.15
MSD: 25.7
Std.dev: 3.97
16
MAD: 2.49
MSD: 30.8
Std.dev: 4.97
18
Mean ACTUAL contribution
in the C-experiment
Mean ACTUAL contribution
in the C-experiment
18
14
12
10
8
6
4
2
16
14
12
10
8
6
4
2
0
0
2
4
6
8
10
12
14
16
18
0
20
0
Mean PREDICTED contribution according to the schedule
in the P-experiment
2
4
6
Triangle Contributors
10
12
14
16
18
20
Others
20
20
MAD: 3.14
MSD: 28.8
Std.dev: 4.37
16
MAD: 8.16
MSD: 124.3
Std.dev: 7.62
18
Mean ACTUAL contribution
in the C-experiment
18
Mean ACTUAL contribution
in the C-experiment
8
Mean PREDICTED contribution according to the schedule
in the P-experiment
14
12
10
8
6
4
2
16
14
12
10
8
6
4
2
0
0
0
2
4
6
8
10
12
14
16
18
20
Mean PREDICTED contribution according to the schedule
in the P-experiment
0
2
4
6
8
10
12
14
16
18
20
Mean PREDICTED contribution according to the schedule
in the P-experiment
Fig. 7. Correlation between predicted (according to the schedule in the P-experiment) and actual
contribution (in the C-experiment). Bold lines are OLS trend lines. MSD is the mean squared deviation of
predicted and actual contribution.
A couple of interesting results can be taken away from Fig. 7. First, 68 percent of
Conditional Cooperators deviate on average by at most ± 2 tokens (i.e., ten percent of their
endowment). For Free Riders this holds for 56 percent; for Triangle Contributors for 59
percent and for ‘Others’ for 43 percent. Interestingly, the average contribution of Free Riders
varies quite a bit (between 0 and 8 tokens). Forty percent of the Free Riders (i.e., 13 percent
of all subjects) can be classified as die-hard free riders, because they consistently free rode in
both experiments. In Fig. 7 they are symbolized by the large dot at (0,0). Apart from the
thirteen dedicated Free Riders, the other 19 Free Riders contribute on average between 0.3
and 8.6 tokens.
Second, Fig. 7 also shows that there is a positive correlation between predicted and actual
contributions in the cases of Conditional Cooperators and Triangle Contributors. Regression
analyses (using all individual observations) show that the slopes of both trend lines are less
than unity but clearly significantly positive. Thus, in the relevant range of beliefs, Triangle
19
Contributors behave very similarly to Conditional Cooperators, an observation that is
consistent with Fig. 1. For ‘Others’ the correlation is even negative (though the slope of the
trend line is insignificantly different from zero). By construction, there is no correlation in the
case of Free Riders.
Third, on the basis of all individual data we also calculate three measures of dispersion of
the difference between predicted and actual contribution: the mean absolute deviation (MAD),
the mean squared deviation (MSD) and the standard deviation. Taking the three measures of
dispersion as indicators for prediction error or tremble, we find that Conditional Cooperators,
Free Riders and Triangle contributors are in the same ballpark. The behavior of ‘Others’ is
clearly more dispersed according to all our three indicators than the behavior of the other
three types. In the next section we take a closer look at errors and their role in explaining the
decay in contributions.
5. Heterogeneous preferences and the role of errors in the dynamics of free riding
It is a well-known phenomenon in experimental public good games that contributions to
the public good decline over time. Most explanations for the decay rely on the reduction of
errors that occurs over rounds of play.13 One of the first papers that addressed this question
was Andreoni (1995). He developed a design that allowed him to distinguish ‘kindness’ from
‘confusion’. He concluded that reduced confusion explains the decay in cooperation. In a
recent paper, Houser and Kurzban (2002) extended Andreoni’s design and confirmed his
conclusion. Palfrey and Prisbrey (1996, 1997) developed a design that allowed them to
separate altruism, ‘warm glow’ and errors. They find that altruism (i.e., maximizing group
payoff) does not explain behavior, whereas ‘warm glow’ (a willingness to contribute a certain
constant amount independently of others’ contribution) is significant but differs between
people. The decay in contributions is again explained as a reduction in errors. All designs
have in common that they don’t allow easily for the existence of conditional cooperators.14
Our observation that the contributions of more than 50 percent of our subjects depend on
others’ contributions suggests that conditional cooperation should not be neglected.
Our approach allows us to take a new look at this issue. Our claim will be that the
interaction of heterogeneously motivated people, as well as a reduction in errors, explains the
decay in cooperation. We will also assess the relative importance of errors in explaining the
decay of cooperation.
Fig. 1 already suggests that contributions will decline over time, because the schedules of
all types, with the exception of ‘Others’, are consistently below the diagonal. Based on this
observation, FGF and Fehr and Fischbacher (2004), for instance, have speculated that this can
in principle explain the decay in cooperation. The implicit assumption is that preferences are
stable across games. Our results above support this assumption. Moreover, even if
13
For an analysis of the role of errors in public goods experiments, see, e.g., Andreoni (1995); Palfrey and
Prisbrey (1996, 1997); Keser (1996); Sefton and Steinberg (1996); Anderson et al. (1998); Brandts and Schram
(2001); Willinger and Ziegelmeyer (2001); Houser and Kurzban (2002), Bardsley and Moffatt (2003).
Neugebauer et al. (2004) also argue with conditional cooperation to explain the decay in cooperation.
14
Andreoni (1995), though, suggests an explanation in terms of conditional cooperation.
20
preferences are stable across games, this does not say anything about the actual path of
contributions and the speed and asymptote of the decline. Thus, our goal is to provide
rigorous evidence on the role of heterogeneous social preferences in explaining the decay in
cooperation.
Since we have the schedules of all subjects we can assess type-interaction induced decay
in cooperation by simulating the contribution path that would occur if all participants would
strictly adhere to their schedules. In other words, in our simulation we take the unconditional
contributions as the starting values and let contributions evolve as determined by the
schedules of our participants, given the exact matching structure that was in place in the Cexperiment. Fig. 8 shows the resulting average contribution of this simulation, along with the
average actual contribution. Two observations can be made from this simulation. First, the
interaction of heterogeneous motivations can explain the decline of cooperation. The
asymptote of this process is not zero, but a contribution level slightly below 3. The second
observation is that the simulated contributions drop off too quickly, relative to the decline in
actual contributions.15
14
Actual contribution
Mean contribution
12
Predicted contribution derived from beliefs and schedules
Simulated contribution resulting from schedules only
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10
Period
Fig. 8. Mean actual, mean predicted contribution (derived from beliefs and schedules) and mean simulated
contribution (resulting from interaction of schedules) across all types over time.
It is tempting to attribute the difference between the simulated and the actual contribution
solely to errors. After all, the identifying assumption of the simulation is that by construction
people do not make any errors. The rationale of this argument is that – given the extensive
training and the absence of time pressure in the P-experiment (see Section 2) – the schedules
measure people’s error-free contribution preferences, as a function of other group members’
contributions. Therefore, if we keep the expressed preferences fixed and let them interact we
15
Since we have the schedules of 140 subjects we can form 1404 groups of four and simulate their contributions
by using the unconditional contributions as the starting value. We find that contributions drop off very quickly
and converge to a contribution level of about two. The resulting limit distribution closely resembles the actual
distribution of contribution in the tenth round. This finding suggests that cooperation under conditions of
random matching is very likely to collapse eventually, despite the fact that many people are conditional
cooperators who initially are willing to invest sizeable amounts into the public good.
21
get the ‘error-free’ contribution path. Consequently, the difference between the actual
contribution path and the simulated path (which amounts to 77 percent of actual
contributions) measures the extent of contributions that are due to errors. If we accept this
reasoning then 23 percent of the actual contributions would be due to errors.
However, this is a premature conclusion, because it might overestimate the extent of
errors. The reason is as follows. If, for instance, a Free Rider contributes more than zero,
then this contribution will influence the belief, e.g., the Conditional Cooperator holds about
the average contribution level of the other group members. Since the contribution schedule of
a Conditional Cooperator is increasing in the belief about what others contribute, the
predicted contribution of a Conditional Cooperator increases in the Free Rider’s erroneous
contribution. Thus, in general, the predicted contribution of all types, except the Free Riders,
is endogenous and, via the beliefs, influenced by both the contributions due to the other types’
preferences and their errors.
The bold line in Fig. 8 depicts the mean predicted contribution over time that results if we
evaluate the schedules given the expressed beliefs. As we see, the predicted contribution that
is endogenous to the errors of other players falls as well but is much closer to the actual
contribution level than the simulated contribution that assumes errors away.
Errors can occur in both directions (except for Free Riders who can only over-contribute
relative to their preferences) and the average might hide some over- and under-contributions.
Therefore, we calculate the absolute error, i.e., |actual contribution – predicted contribution|.
On average, errors over all periods account for 17.5 percent of the endowment.
Before we look at the temporal development of errors and turn to an econometric analysis
it is worth looking at the relative ‘responsibilities’ of the different types for the errors we
observe. Intuitively, if all types where equally error-prone then a type’s relative share of
errors should reflect the relative share of this type in the population. We find that Conditional
Cooperators, who represent 55 percent of the participants, are responsible for 49.5 percent of
all errors. Free Riders, who have a share of 22.9 percent in our subject pool, make up for 16.3
percent of the errors. Triangle Contributors who account for 12.1 percent of our participants,
cause 10.9 percent of the errors. ‘Others’, who only characterize 10 percent of the subjects,
are responsible for almost a quarter (23.3 percent) of all errors. Thus, ‘Others’ are clearly
more error-prone than the other types.
Fig. 9 plots the resulting time series for each type and the total MAD. Except for ‘Others’
errors clearly shrink over time. ‘Others’ start at an error level that is above all others and get
even worse over time. By contrast, the consistency of Conditional Cooperators, Triangle
Contributors and Free Riders is very similar over time. All become gradually more consistent
over time. Already after period 4 the errors of the Free Riders and the Triangle Contributors
are persistently below those of Conditional Cooperators.
22
MAD of actual and predicted contribution
60%
50%
40%
Conditional Cooperators
Free Riders
Triangle Contributors
Others
Total
30%
20%
10%
0%
1
2
3
4
5
6
7
8
9
10
Period
Fig. 9. MAD of actual and predicted contribution in percent of endowment per type and period.
We now turn to an econometric analysis of the development of errors over time. We are
in particular interested in estimating the starting and asymptotic values of the deviations of
actual and predicted contributions. The approach is inspired by Noussair et al. (1995, p. 473).
We estimate the following model:
AEijt = β1C DC (1 / t ) + β1F DF (1 / t ) + β1T DT (1 / t ) + β1O DO (1 / t ) +
+ β 2C DC (t − 1) / t + β 2 F DF (t − 1) / t + β 2T DT (t − 1) / t + β 2O DO (t − 1) / t + u
where AEijt is the absolute error (i.e., the deviation of predicted and actual contribution) in
period t of individual i who is of type j. Dj is a dummy for type j ∈ {Conditional Cooperator,
Free Rider, Triangle Contributor, Others}. The coefficient β1 j measures the starting value of
the absolute deviation of type j while the coefficient β 2 j measures the asymptote. As t
increases the weight of β1 j gets small because 1/t goes to zero, while the weight of β 2 j gets
large because (t – 1)/t approaches 1. The estimation approach is OLS.
As a benchmark we also estimate a model where we pool the types:
AEit = β1 (1 / t ) + β 2 (t − 1) / t + ν .
The results are documented in Table 5. The econometric results confirm the impression
from Fig. 9. We start with the pooled model. The estimated initial value of absolute error is
25 percent of the endowment, which is highly significantly different from zero. Over time
absolute errors are reduced almost by half and approach 14.4 percent.
When we distinguish between types we find that both starting values and asymptotic
values of the absolute error differ between types. 16 All types have significantly positive
starting values. The absolute error of Free Riders and Triangle Contributors asymptotically
16
F-tests confirm this conclusions for both the starting values [F(4,5)=34.93; p=0.0008] and the asymptotic
values [F(4,5)=146.64; p=0.0000]. The goodness of fit, as measured by the R2 also increases from 0.34 to 0.42.
23
clearly declines, as β 2 for these types is an order of magnitude smaller than the starting
values. The asymptotic absolute error rates are 5.3 and 7.7 percent of the endowment,
respectively. Erroneous contributions of Conditional Cooperators are only slightly (but
significantly) reduced: The initial error rate of Conditional Cooperators is 18.6 percent, which
declines modestly to an asymptotic error rate of 14.6 percent of the endowment. Only for
‘Others’ error rates even increase from an initial rate of 36.9 percent to an asymptotic value of
42.5 percent.
Table 5
Development of absolute errors over time
Dependent variable:
|actual contribution – predicted contribution|
in percent of endowment
Pooled data
(no distinction between types)
Starting value
Asymptote
β1
β2
0.250***
(0.0236)
0.144***
(0.0107)
Observations
R-squared
Distinguish between types:
Conditional Cooperators
Free Riders
Triangle Contributors
Others
1400
0.34
Starting value
Asymptote
β1j
β2j
0.186***
(0.0337)
0.298**
(0.0772)
0.349**
(0.0926)
0.369**
(0.091)
0.146***
(0.0170)
0.053***
(0.0109)
0.077**
(0.023)
0.425***
(0.0860)
Observations
1400
R-squared
0.42
Note: Robust standard errors (clustering on sessions) in parentheses. * significant at 10%;
** significant at 5%; *** significant at 1%
Taken together our results suggest that both the interaction of heterogeneously motivated
subjects and the reduction of errors explain the decay of cooperation. If people would not
make any errors and stick to the planned contributions as made in the ‘cold’ situation of the Pexperiment, contributions would decay very quickly. Yet, under the ‘hot’ situation of the Cexperiment, even Free Riders contribute to the public good. Likewise, Triangle Contributors
and Conditional Cooperators on average as well contribute more than they have planned to
according to their preferences stated in the ‘cold’ decision situation of the P-experiment.
These errors slow down the decay in contributions because they increase the beliefs and hence
the contributions of the Conditional Cooperators and the Triangle Contributors. It turns out
that Free Riders and Triangle Contributors strongly reduce their erroneous contributions over
time. Conditional Cooperators only slightly reduce their errors, whereas ‘Others’ even
slightly increase their errors over time. Thus, it is in particular the strong reduction in errors
of the Free Riders and the Triangle Contributors and to some small degree also of the
24
Conditional Cooperators that, together with the conditional cooperation exhibited by the
Conditional Cooperators and the Triangle Contributors, explains the decay in cooperation.
So far, our definition of ‘errors’ has been very conservative. To tie our hands, we have
defined any deviation of actual from predicted contribution as an ‘error’. Somewhat
speculatively, apart from the mentioned ‘hot-cold’ empathy gap and the trembles in the
implementation of one’s preferences, we believe that the different types may commit different
sorts of ‘errors’.
Take the Free Riders first. We think it is very unlikely that they were confused in the Cexperiment, given that they were able in the P-experiment to identify the payoff-maximizing
schedule. Rather, they may have experienced the ‘hot-cold’ empathy gap or they may have
reasoned strategically and tried to trigger a kind of contagion effect, despite the fact that the
C-experiment was run in the random matching mode. If they believe (correctly as it turns out)
that many others are conditional cooperators, Free Riders might have tried to build up ‘trust’
to induce the Conditional Cooperators to contribute more in the early periods of the Cexperiment. In the final periods, there is no strategic reason to build trust. Our observation
from Table 4 that the correlation between beliefs and contributions is significantly positive
only in periods 1-5 but not in periods 6-10 supports the interpretation of strategic behavior.
The Conditional Cooperators revealed conditionally cooperative preferences in the ‘cold’
P-experiment and behaved largely consistently in the ‘hot’ C-experiment. Thus, the errors of
Conditional Cooperators are probably not due to a ‘hot-cold’ gap, but simply to trembles. In
our opinion, a similar reasoning holds for the Triangle Contributors, who exhibit similar
preferences in the P-experiment in the behaviorally relevant range of actual contributions
(which are below 13). Given the rather random pattern of revealed preferences of ‘Others’ in
the P-experiment, it is likely that they suffer from trembles and mere confusion. The fact that
‘Others’ also exhibit a positive correlation between beliefs and contributions might be due to
a ‘hot-cold’ gap, conformism, or a (boundedly rational) imitation behavior.
6. Concluding remarks on theory and policy implications
Understanding people’s motivations to make voluntary contributions to public goods is an
important issue in public economics. A proper understanding is important for theoretical
reasons and a variety of policy issues. Take the theoretical reasons first. If people would be
largely motivated by ‘warm-glow’ preferences and if the decay in contributions would be due
to reduced errors, then the modeling approach might be another one than if people were free
riders or conditional cooperators whose interaction explains the decay in contributions. In the
former case, a modeling approach where errors figure prominently might be the preferable
one (see, e.g., Anderson et al. 1998). In the latter case, a theory of social preferences (see,
e.g., Fehr and Schmidt 1999; Falk and Fischbacher 2000; Bolton and Ockenfels 2000) might
be chosen. These theories predict free riding by all agents in the public good game if people
are inequality averse. Our design has also allowed us to assess the free riding dynamics that
leads to the predicted equilibrium outcome. Our results reinforce the theoretical prediction
that in the absence of punishment opportunities (see, e.g., Fehr and Gächter 2000; 2002), the
25
co-existence of heterogeneously motivated agents will lead to almost universal free riding
behavior. Our results show that this holds despite the fact that a majority of people is not
motivated by selfishness. This provides empirical evidence for the theoretical speculations put
forward in, e.g., FGF and Fehr and Fischbacher (2004).
We see our results as evidence for the existence of different ‘types’ of economic agents,
who are characterized by different social preferences. For instance, our results would support
modeling assumptions that assume different types of players in an incomplete information
game. Our results suggest that, for practical modeling purposes, it suffices to concentrate on
two types – the Free Riders and the Conditional Cooperators, because in the relevant range of
actual contributions (and therefore beliefs) Triangle Contributors are very similar to
Conditional Cooperators. Even ‘Others’ behave conditionally cooperatively in the Cexperiments.
Our finding of a high degree of cross-game consistency is also relevant for the
interpretation of recent theories of social preferences mentioned above. These theories also
allow for heterogeneous preferences but assume that for a given individual the relevant
preference parameters are fixed across games. Our results support this crucial assumption.
Our observations may also shed light on interesting social issues. For instance, voting
behavior can be shaped by expectations on how others vote (Tyran 2004). Behavior at the
work place might be influenced by the work morale exhibited by co-workers (Ichino and
Maggi 2000; Falk and Ichino 2004). The prevalence of corruption also seems to be influenced
by motivations similar to conditional cooperation (Abbink et al. 2002). Following a leader’s
example in team work may also be strongly shaped by conditional cooperation (Gächter and
Renner 2004; Potters et al. 2004; Güth et al. 2004).
Policy consequences might be different if most people are conditional cooperators or free
riders than if people contribute for reasons of a ‘warm glow’ or errors. The latter are motives
that are independent of other people’s contribution behavior, whereas conditional cooperation
is contingent behavior. For instance, observers of welfare state policies (e.g., Wax 2000;
Fong 2001; Fong et al. 2002) point out that many people hold reciprocity norms that are akin
to the conditional cooperation observed in our experiments. Fong et al. (2002) even argue
that “people support the welfare state because it conforms to deeply held norms of reciprocity
and conditional obligations to others”. Norms of reciprocity and conditional cooperation
might also influence tax morale. Too many cheaters can spoil tax morale. People might be
more likely not to cheat on their taxes if others honestly pay theirs (e.g., Besley and Coate
1992; Andreoni et al. 1998; Rothstein 2000; Feld and Tyran 2002). Contributions to charities
might also be influenced by what others do (List and Lucking-Reiley 2002; Falk 2004; Frey
and Meier, forthcoming). In our view, these kinds of reactions are hard to square with ‘warmglow’ preferences alone but can be reconciled with the kinds of observations made in our
study.
26
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29
Appendix: Instructions for the experiment
This is a translation of the original German version. We present the instructions of the P-C experiments. The
instructions of the C-P experiments were adapted accordingly. They are available upon request.
Instructions for the P-Experiment
You are now taking part in an economics experiment that has been financed by the Swiss Science Foundation. If
you read the following instructions carefully, you can – depending on your decisions – earn some more money in
addition to the 10 Francs, which you can keep in any case. At the end of the experiment the whole amount of
money, which you have earned with your decisions, will be added up and paid to you in cash. These instructions
are solely for your private information. During the experiment you are not allowed to communicate. If you
have any question, please ask us. Violation of this rule will lead to the exclusion from the experiment and all
payments. If you have questions, please raise your hand. A member of the experimenter team will come to you
and answer your question in private.
During the experiment we will not speak of Francs but rather of points. So at first your whole income will be
calculated in points. At the end of the experiment, the total amount of points you have earned will be converted
to Francs at the following rate:
1 point = 35 Rappen.
All participants will be divided in groups of four members. Except us, the experimenters, nobody knows who
is in which group.
On the following pages the exact process of the experiment will be described.
The decision situation
You will learn later how the experiment will be conducted. We first introduce you to the basic decision situation.
At the end of the description of the decision situation you will find control questions that help you to understand
the decision situation.
You will be a member of a group consisting of 4 people. Each member of this group has to decide on the
allocation of 20 points. You can put these 20 points into your private account or you can invest them fully or
partially into a project. Each point you do not invest into the project, will automatically remain in your private
account.
Your income from the private account:
For each point you put on your private account you will earn one point. For example, if you put 20 points
into your private account (and therefore do not invest into the project) you get exactly 20 points out of your
private account. If you put 6 points into your private account, your income will be 6 points out of your private
account. Nobody except you earns something from your private account.
Your income from the project
From the amount you invested into the project, each group member will earn similarly. On the other hand,
you will also get a payoff out of the investments of the other group members. The income for each group
member will be determined as follows:
Income from the project = sum of all contributions × 0.4
If, for example, the sum of all contributions to the project is 60 points, then you and the other members of your
group get 60 × 0.4 = 24 points out of the project. If four members of the group contribute totally 10 points to the
project, you and the other members of your group get 10 × 0.4 = 4 points.
Total income:
Your total income is the sum of your income from your private account and the income from the project:
Income from your private account (= 20 – contribution to the project)
+ Income from the project (= 0.4 × sum of all contributions to the project)
Total income
30
Control questions:
Please answer the following control questions. They will help you to gain an understanding of the calculation of
your income, which varies with your decision about how you distribute your 20 points. Please answer all the
questions and write down your calculations.
1. Each group member has 20 points. Assume that none of the four group members (including you) contributes
anything to the project.
What will your total income be?
___________
What will the total income of the other group members be?
___________
2. Each group member has 20 points. You invest 20 points to the project. From the other three members of the
group each one contributes 20 points to the project.
What will your total income be?
___________
What will the total income of the other group members be?
___________
3. Each group member has 20 points. The other 3 members contribute in total 30 points to the project.
a)
b)
c)
What will your total income be, if you have – in addition to the 30 points – invested 0 points into the
project?
Your Income
___________
What will your total income be, if you have – in addition to the 30 points – invested 8 points into the
project?
Your Income
___________
What will your total income be, if you have – in addition to the 30 points – invested 15 points into the
project?
Your Income
___________
4. Each group member has 20 points at his or her disposal. Assume that you invest 8 points to the project.
a)
b)
c)
What is your total income if the other group members – in addition to your 8 points – together
contribute 7 points to the project?
Your Income
___________
What is your total income if the other group members – in addition to your 8 points – together
contribute 12 points to the project?
Your Income
___________
What is your income if the other group members – in addition to your 8 points – contribute 22 points to
the project?
Your Income
___________
The Experiment
The experiment contains the decision situation that we have just described to you. At the end of the experiment
you will get paid according to the decisions you make in this experiment. The experiment will only be conducted
once. As you know you will have 20 points at your disposal. You can put them into a private account or you can
invest them into a project. In this experiment each subject has to make two types of decisions. In the following
we will call them "unconditional contribution" and "contribution table".
• With the unconditional contribution to the project you have to decide how many of the 20 points you want
to invest into the project. Please indicate your contribution in the following computer screen:
31
After you have determined your unconditional contribution, please click "OK".
•
Your second task is to fill in a "contribution table". In the contribution table you have to indicate for each
possible average contribution of the other group members (rounded to the next integer) how many
tokens you want to contribute to the project. You can condition your contribution on the contribution of
the other group members. This will be immediately clear to you if you take a look at the following table. In
the experiment, this table will be presented to you:
The numbers are the possible (rounded) average contributions of the other group members to the project. You
simply have to insert into each input box how many tokens you will contribute to the project – conditional on the
indicated average contribution. You have to make an entry into each input box. For example, you will have to
indicate how much you contribute to the project if the others contribute 0 tokens to the project, how much you
contribute if the others contribute 1, 2, or 3 tokens etc. In each input box you can insert all integer numbers
from 0 to 20. If you have made an entry in each input box, click "OK".
After all participants of the experiment have made an unconditional contribution and have filled in their
contribution table, in each group a random mechanism will select a group member. For the randomly
determined subject only the contribution table will be the payoff-relevant decision. For the other three
group members that are not selected by the random mechanism, only the unconditional contribution will be
the payoff-relevant decision. When you make your unconditional contribution and when you fill in the
contribution table you of course do not know whether you will be selected by the random mechanism. You will
therefore have to think carefully about both types of decisions because both can become relevant for you. Two
examples should make this clear.
EXAMPLE 1: Assume that you have been selected by the random mechanism. This implies that your
relevant decision will be your contribution table. For the other three group members the unconditional
contribution is the relevant decision. Assume they have made unconditional contributions of 0, 2, and 4 tokens.
The average contribution of these three group members, therefore, is 2 tokens. If you have indicated in your
contribution table that you will contribute 1 token if the others contribute 2 tokens on average, then the total
contribution to the project is given by 0+2+4+1=7 tokens. All group members, therefore, earn 0.4×7=2.8 points
from the project plus their respective income from the private account. If you have instead indicated in your
contribution table that you will contribute 19 tokens if the others contribute two tokens on average, then the total
contribution of the group to the project is given by 0+2+4+19=25. All group members therefore earn 0.4×25=10
points from the project plus their respective income from the private account.
EXAMPLE 2: Assume that you have not been selected by the random mechanism which implies that for you
and two other group members the unconditional contribution is taken as the payoff-relevant decision.
Assume your unconditional contribution is 16 tokens and those of the other two group members are 18 and 20
tokens. The average unconditional contribution of you and the two other group members, therefore, is 18 tokens.
If the group member who has been selected by the random mechanism indicates in her contribution table that she
will contribute 1 token if the other three group members contribute on average 18 tokens, then the total
32
contribution of the group to the project is given by 16+18+20+1=55 tokens. All group members will therefore
earn 0.4×55=22 points from the project plus their respective income from the private account. If instead the
randomly selected group member indicates in her contribution table that she contributes 19 if the others
contribute on average 18 tokens, then the total contribution of that group to the project is 16+18+20+19=73
tokens. All group members will therefore earn 0.4×73=29.2 points from the project plus their respective income
from the private account.
The random selection of the participants will be implemented as follows. Each group member is assigned a
number between 1 and 4. As you remember, at the very beginning a participant, namely the one with the number
11, was randomly selected. This participant will, after all participants have made their unconditional
contribution and have filled out their contribution table, throw a 4-sided die. The number that shows up will be
entered into the computer. If participant 11 throws the membership number that has been assigned to you, then
for you your contribution table will be relevant and for the other group members the unconditional contribution
will be the payoff-relevant decision. Otherwise, your unconditional contribution is the relevant decision.
----------------------------------------------------------------------------------------------------------------------------------------
Instructions for the C-Experiment
We now conduct another experiment. This experiment has 10 periods, in which you and the other group
members have to make decisions. Like in the other experiment every group consists of 4 people. The formation
of the group changes at random after every period. So your group consists of different people in all 10
periods. After these 10 periods, the whole experiment is finished.
The decision situation is the same as we described on page 2 of the instructions of the previous experiment. Each
member of the group has to decide about the usage of the 20 points. You can put these 20 points into your
private account or you can invest them fully or partially into a project. Each point you do not invest into the
project, you automatically put into your private account. Your income will be determined in the same way as
before. Reminder:
Income from your private account (= 20 – contribution to the project)
+ Income from the project (= 0.4 × sum of all contributions to the project)
Total income
1 point = 7 Rappen!
The decision screen, which you will see in every period, looks like this:
As you can see, you have to make two inputs:
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1.
First you have to make a decision about your contribution to the project, that is, you have to decide how
many of the 20 points you want to contribute to the project, and how many points you want to put into your
private account. This decision is the same as the unconditional contribution of the previous experiment. In
this experiment you only make unconditional decisions. There is no contribution table.
2.
Afterwards you have to make an estimate about the average contribution to the project (rounded to an
integer number) of the other three group members of this period. You are getting paid for the accuracy of
your estimate:
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If your estimate is exactly right (that is, if your estimate is exactly the same as the actual average
contribution of the other group members), you will get 3 points in addition to your other income from
the experiment;
If your estimate deviates by one point from the correct result, you will get 2 additional points;
A deviation by 2 points still earns you 1 additional point.
If your estimate deviates by 3 or more points from the correct result, you will not get any additional
points.
After these 10 periods are over, the whole experiment is finished and you will get:
+ your income from the first experiment
+ your income from the second experiment (including your income from your right estimates)
= total income from both experiments
+ 10 Francs show up fee !