Enforcement of Contribution Norms in
Public Good Games with Heterogeneous
Populations∗
Ernesto Reuben
IZA and Columbia University, e-mail: [email protected]
Arno Riedl
CESifo, IZA, and Maastricht University, e-mail: [email protected]
Abstract
We investigate the emergence and enforcement of contribution norms to public goods in
homogeneous and heterogeneous groups. With survey data we demonstrate that uninvolved
individuals hold well-defined yet conflicting normative ideas of fair contributions related to
efficiency, equality, and equity. In the experiment, all groups converge towards free-riding in
the absence of punishment. With punishment, strong and stable differences in contributions
emerge across groups and individuals in different roles. These differences result from the
enforcement of different relative contribution norms. Lastly, individuals in different roles
largely agree on the enforced norm, even when it entails pronounced differences in earnings.
This version: January 2011
JEL Codes: H41, C92, D63
Keywords: public good, heterogeneous groups, punishment, cooperation, social norms, norm
enforcement
∗
Financial support from the Dutch Science Foundation (NWO) through the “Evolution & Behavior” grant
051-12-012 and from the EU-Marie Curie RTN ENABLE (MRTM-CT-2003-505223) is gratefully acknowledged.
1
Introduction
The need for cooperation among people with heterogeneous characteristics is an undeniable fact
of social and economic life. At the work place, teams are composed of workers who may differ in
their productivity, ability, and motivation (Hamilton et al. 2003). Irrigation systems are often
jointly used and maintained by farmers with different plot sizes and water needs.1 People also
can derive very different benefits from public goods. For example, the elevation of dams along the
Mississippi River gives very different benefits to individuals who live close to the river compared
to those who live further away. In the international political and economic arena, countries
that greatly differ in size and wealth are often confronted with situations that require them to
find joint agreements in order to overcome social dilemmas. Sandler and Hartley (2001) discuss
this problem in the framework of international military alliances. Other prominent examples
of international cooperation include the Kyoto Protocol, which aims to reduce greenhouse gas
emissions, fishing quotas for European Union members to mitigate the over-fishing of open waters,
and the Global Disease Detection Program spearheaded by the United States that seeks early
detection of infectious diseases.
As diverse as the above examples seem, they can all be viewed as special cases of a more
general public goods problem where the enforcement of cooperation by third-parties is infeasible
or very limited (e.g., due to the actions of others being unobservable or as the result of the
lack of a supranational institution with coercive power). In such situations, cooperation has to
be promoted through other mechanisms, such as informally enforced social norms (Elster 1989;
Coleman 1990).2 For instance, Ostrom (1990) describes, among others, the case of fishermen in
Alanya, Turkey, who overcame the commons problem through self-devised rules that are defended
violently if necessary. The private lobster management in Maine provides another prominent case
where it is reported that interlopers (free-riders) of the established self-regulatory system were
“discouraged by surreptitious violence” (p. 92, Berkes et al. 1989). More recently, research on
decentralized forest management has shown that effective community monitoring and sanctioning
1
For instance, in the western states of the United States, family farms dependent on irrigation vary in annual
farm sales from below $100,000 to above $500,000 (U.S.D.A. 2004).
2
Social norms are a widespread empirical phenomenon (Becker 1996; Hechter and Opp 2001; Posner 2002).
Numerous examples of the impact of norms on behavior have been meticulously documented (e.g., Roethlisberger
and Dickson 1947; Whyte 1955; Hywel 1985; Sober and Wilson 1997; Gurven 2004). For examples of the role of
norms in the use of common resources see Ostrom (1990).
2
is an important factor for institutional success (Ostrom and Nagendra 2006; Coleman and Steed
2009).
The importance of social norms for sustaining cooperative behavior in public goods environments has also been suggested in a number of controlled laboratory experiments.3 However,
this evidence is based on homogeneous-group environments and neglects the important fact that
people differ. This is a potentially serious shortcoming because people who differ may also adhere to different norms, which may lead to conflicts and inefficiencies. In this paper, we study
the existence, emergence, and informal enforcement of contribution norms in homogeneous and
heterogeneous groups.
In their seminal paper, Fehr and Gächter (2000) showed that in homogeneous groups high
contributions to the public good are sustained because some people reliably sanction those who
contribute less than they do. Such behavior is intuitively appealing and consistent with the
enforcement of norms grounded in general fairness principles such as efficiency, equality, and,
given that individuals are symmetric, also equity (Konow 2003).4 In heterogeneous groups it is
a lot less obvious what contribution norm may emerge, if one emerges at all. Different fairness
principles will often stipulate different actions in heterogeneous groups, which makes it less clear
what the normatively correct behavior is. For example, even uninvolved individuals might find it
hard to unambiguously answer questions such as: should high-income individuals contribute more
to the public good even when they benefit equally from it? Such ambiguity in fairness principles
is also well known in the public finance literature where it is discussed as the benefit-received
versus the ability-to-pay principle (see, e.g. Musgrave 2008).
Due to self-serving interpretations of fairness principles (Babcock and Loewenstein 1997)
heterogeneity may make it difficult to agree on normatively correct behavior, even if individuals
have clear ideas about it. For instance, if people have different tastes for the public good, equal
contributions may be preferred by those who derive a lot of pleasure from it, whereas those
who enjoy the public good less may prefer a norm with unequal contributions. In contrast
to homogeneous groups, the experimental evidence regarding contributions to public goods in
3
For recent reviews see Fehr and Fischbacher (2004) and Gächter and Herrmann (2009).
4
Efficiency, equality, and equity considerations are commonly called upon in normative research and have been
extensively discussed by philosophers (e.g., Aristotle 1925; Rawls 1971; Corlett 2003). Equality is also commonly
invoked in social choice theory as axioms of symmetry and anonymity (e.g., Moulin 1991; Gaertner 2006).
3
heterogeneous groups is much less conclusive,5 and evidence of the enforcement of contribution
norms in heterogeneous groups is basically absent.6
In this paper we investigate the emergence and informal enforcement of different contribution norms in a linear public good game when people differ in either their endowment or their
preference for the public good. In total, we study three types of group heterogeneity. We look
at groups with unequal endowments, where heterogeneity is introduced by providing one person
(out of three) with an endowment that is twice as high as the endowment of the other group
members. To isolate the effect of extending the contribution possibilities due to a larger endowment, in some groups, we restrict the contribution possibilities to be the same for all group
members, whereas in others, we allow for contributions up to the entire endowment. In the third
type of group heterogeneity all group members have the same endowment but one group member
receives a 50 percent higher marginal benefit from the public good. In addition, we also explore
homogeneous groups.
We analyze data from two sources. First, we use a questionnaire study to examine whether the
ideas of uninvolved individuals regarding normatively desirable behavior are related to the various
fairness principles mentioned above and whether these ideas are affected by group heterogeneity.
Second, we study directly the emergence and enforcement of contribution norms in a laboratory
experiment. We implement eight treatments consisting of the four types of groups described
above, each with and without punishment possibilities. Hence, our design allows us to isolate
the effect of unequal endowments, unequal contribution possibilities, and unequal preferences for
the public good on contribution behavior as well as their interaction with sanctioning possibilities
within one experimental setting. We find that in both homogeneous and heterogeneous groups,
the fairness principles of efficiency, equality, and equity strongly influence what is considered
5
Experiments investigating endowment heterogeneity report mixed results. Ostrom et al. (1994), van Dijk et al.
(2002), and Cherry et al. (2005) find that inequality leads to lower contributions, Chan et al. (1996) and Buckley
and Croson (2006) report a positive effect, and Chan et al. (1999) and Sadrieh and Verbon (2006) no effect. With
respect to heterogeneity in the marginal benefit from the public good, Fisher et al. (1995) find that individuals
with a high marginal benefit contribute more than those with a low marginal benefit.
6
To our knowledge, the only experiment that combines endowment heterogeneity and punishment possibilities
is Visser and Burns (2006). They report that among South African fishermen punishment effectively promotes
cooperation in both homogeneous and heterogeneous groups. Reuben and Riedl (2009) show that in privileged
groups—that is, groups in which one player has a dominant strategy to contribute a positive amount—punishment
does not promote contributions as effectively as in homogeneous groups. Tan (2008) finds a similar result for nonprivileged heterogeneous groups and Noussair and Tan (2009) report that voting on punishment is not effectively
increasing contributions in such groups.
4
normatively desirable behavior. However, unlike in homogeneous groups, uninvolved individuals
in heterogeneous groups agree considerably less on which specific behavior is normatively correct.
Interestingly, the fairness principle guiding the modal normative belief changes with the type of
group heterogeneity. This suggests that normative behavioral rules, that could be the basis of a
contribution norm, are not constant across different forms of heterogeneity.
In the laboratory experiment, we find that without punishment possibilities, heterogeneity
does not matter much. In all groups free-riding is relatively frequent and steadily increases
over time. In other words, we do not find evidence for a contribution norm emerging. With
punishment, the picture changes drastically. In both homogeneous and heterogeneous groups,
contributions are much higher with punishment than without punishment and they do not decrease over time. More importantly, in line with the observations from the questionnaire study,
the contribution patterns differ strongly depending on the type of group heterogeneity. In groups
with unequal endowments and unrestricted contribution possibilities, contributions are proportional to endowments. In contrast, in groups with unequal endowments but constrained contribution possibilities, group members with large endowments do not contribute more than other
group members despite the fact that their endowment is twice as large. Likewise, in groups with
unequal marginal benefits from the public good, contributions are similar irrespective of the
marginal benefits. We show econometrically that contribution patterns do not differ accidently,
but are the result of informal enforcement of different contribution norms in the different treatments. Interestingly, irrespective of differences in endowments, individuals within group types
largely agree on which contribution norm to enforce—even when the norm implies that some
individuals benefit relatively more.
2
Design and Procedures
The game used in both the questionnaire study and the experiment is a linear public good game
with groups of three players. The game in its basic form consists of a contribution stage in which
each player i receives an endowment of yi points. Players simultaneously decide how many points
they want to contribute to the public good, ci ∈ [0, c̄i ] where c̄i is i’s maximum contribution.
Every point contributed to the public good by any group member increases i’s earnings by αi
points and every point not contributed by i increases i’s earnings by one point. If αi < 1 for
P
all i and i αi > 1, then each point contributed increases the sum of earnings in the group but
decreases the earnings of the contributing player, which creates a tension between individual and
5
group interest. At the end of the contribution stage each player is informed of the contributions
of the other group members and their earnings up to that point, which (for player i) are equal
to
πi = yi − ci + αi
X
cj .
j
We implement four types of groups. The first type is the homogeneous case where each group
member i has the same endowment yi = 20 points and receives the same marginal benefit from
the public good αi = 0.50. We refer to this group type as Equal. In the remaining group
types we introduce heterogeneity. Specifically, in each group, one player receives either a higher
endowment or a higher marginal benefit from the public good than the other two players. For
convenience we refer to the former as the high player and to the two latter ones as low players.
In the second group type, the high player receives an endowment of yH = 40 points whereas
low players get yL = 20 points. Importantly, in these groups contributions are restricted to a
maximum of 20 points for all players (i.e., c̄L = c̄H = 20 points). We refer to this group type as
the unequal-restricted-endowment type or URE. In the third group type, the high player again
receives yH = 40 points and low players yL = 20 points. However, in contrast to URE, the
contributions of the high player is only restricted by its endowment (i.e., c̄L = 20 points and
c̄H = 40 points). We refer to this group type as unequal-unrestricted-endowment or UUE. In the
fourth group type, all players receive the same endowment of 20 points but the high player earns
a marginal benefit from the public good equal to αH = 0.75 while low players earn αL = 0.50.
Correspondingly, we refer to it as the unequal-marginal-benefit group type or UMB. The four
types of groups are summarized in Table 1.
Equal and URE differ only in the higher endowment of one group member. Thus, comparing
these groups allows us to isolate the effect of endowment heterogeneity. Moreover, because the
contributions of high players are restricted in URE, we can be sure that any observed differences
in behavior are due to high players possessing a higher endowment and not due to the capacity
to contribute more to the public good. The effect of a higher capacity to contribute can be
examined by comparing URE with UUE. Finally, comparing Equal with UMB allows us to
investigate the effect of differences in the marginal benefits from the public good for given equal
endowments.
6
Table 1: Group types
Group
Player’s
type
role
yi
αi
c̄i
questionnaire
no punishment
punishment
Equal
low
20 points
0.50
20 points
39
21
33
low
20 points
0.50
20 points
high
40 points
0.50
20 points
60
21
33
low
20 points
0.50
20 points
high
40 points
0.50
40 points
64
18
33
low
20 points
0.50
20 points
high
20 points
0.75
20 points
62
21
30
URE
UUE
UMB
Parameters
Respondents to the
Subjects in treatments with
The Questionnaire Study
In order to elicit the normative ideas of uninvolved individuals regarding desirable rules of behavior in the public goods game, we conducted an online questionnaire study. Specifically, we asked
students from the subject pool of the University of Amsterdam’s CREED laboratory to take part
in a 15-minute questionnaire. Importantly, the respondents had experience reading experimental
instructions but had not participated in any of our experimental sessions (described below). To
increase the response rate, students who participated in the questionnaire had a 2% chance of
receiving e100 irrespective of their answers.
Each respondent took part in one of four versions of the questionnaire. Each version corresponds to one of the four group types (i.e., either Equal, URE, UUE, or UMB). At the
beginning of the questionnaire, respondents were told that they will be answering normative
questions concerning an experiment that had been run in the CREED laboratory. To ensure
their understanding, respondents read the instructions seen by the subjects that participated in
the experiment and answered a series of control questions. Respondents who did not answer the
control questions correctly were not allowed to continue. In total 225 students completed the
questionnaire. Table 1 contains the number of respondents in each version of the questionnaire.
The questionnaire contains two sets of questions. The first set consists of one question asking
respondents to specify “what is the fair amount that each of the group members should contribute
to the group project? [the public good]”. Respondents indicated a contribution for each group
member and could choose any contribution between 0 and c̄i . The second set corresponds to
a series of questions of the form: “what is the fair amount that group member i and group
7
member j should contribute if group member k contributes x tokens to the group project? [the
public good]”. Respondents indicated a contribution for group members i and j and could choose
contributions between 0 and respectively c̄i or c̄j . In all questions, respondents are asked to put
themselves in the position of a neutral uninvolved arbitrator. Moreover, in the heterogeneous
treatments, respondents are reminded of the characteristics of each group member (i.e., their
endowment, maximum contribution, and marginal benefit from the public good). The precise
wording of the questions can be found in the online appendix.
The Laboratory Experiment
We conducted the laboratory experiment to study the emergence and enforcement of contribution
norms when individuals actually interact with each other. Each subject took part in one of eight
treatments, which vary along two dimensions: (i) the type of group they are in (either Equal,
URE, UUE, or UMB), and (ii) whether or not they have the option to punish other group
members. In all our treatments, subjects were matched with the same group of three subjects
for ten consecutive periods. Moreover, in heterogeneous groups, subjects were randomly assigned
to roles (high or low) at the beginning of the first period and kept the same role throughout the
experiment.
In treatments where subjects cannot punish each other, they simply play the linear public
good game described above. Therefore, their earnings at the end of a period correspond to their
earnings after the contribution stage. In the remaining treatments, subjects can punish each
other as in Fehr and Gächter (2002). In these treatments, the contribution stage is followed
by a punishment stage in which each subject i simultaneously decides how many punishment
points, pij ∈ [0, 10], to assign to each subject j 6= i in the group. Each punishment point costs
the punisher one point and reduces the earnings of the punished subject by three points.7 After
the punishment stage, subjects are informed of the total number of punishment points assigned
to them. As in Fehr and Gächter (2000, 2002), subjects do not receive specific information
7
Subjects can identify other group members through a randomly assigned ID number that remained constant
throughout the experiment. Moreover, the values of yi , αi , c̄i of all group members is common knowledge. We
impose an upper limit on the amount of punishment i can assign to each j (as Fehr and Gächter (2002) and
others). This restriction prevents subjects with higher earnings from having the capacity to punish more than
subjects with lower earnings (as punishment is funded through their own earnings).
8
concerning who punished whom. In the treatments with punishment, at the end of a period,
earnings of a subject i are given by8
πi = yi − ci + αi
X
cj − 3
j
X
j6=i
pji −
X
pij .
j6=i
The computerized experiment was conducted in the CREED laboratory using the typical
procedures of anonymity, neutrally-worded instructions, and monetary incentives. In total, 210
subjects participated in the one-hour long experiment. About half of the subjects were female.
Also, around half were students of economics (the other half came from other fields such as
biology, engineering, political science, and law). Average earnings equaled e13.83 (≈US$17.50).
After arrival in the lab’s reception room, each subject drew a card to be randomly assigned
to a seat in the laboratory. Once everyone was seated, the instructions for the experiment were
read aloud (a translation of the instructions, which are originally in Dutch, can be found in the
online appendix). Thereafter, subjects answered a few questions to ensure their understanding
of the instructions. When all subjects had correctly answered the questions, the computerized
experiment (programmed in z-Tree, Fischbacher 2007) started. After the ten periods, subjects
answered a short debriefing questionnaire and were confidentially paid their earnings in cash.
3
Focal and Conflicting Contribution Norms
If it is common knowledge that all subjects are rational and maximize solely their monetary
earnings, all individuals in all treatments are predicted not to contribute to the public good.
However, existing experimental evidence from homogeneous groups suggests that: (i) without
punishment there is some initial cooperation that decreases to low levels over time (Ledyard
1995), and (ii) with punishment, sanctions are used to enforce high contribution levels that do not
decline with repetition (Fehr and Gächter 2000; Gächter and Herrmann 2009). On the theoretical
side, models assuming other-regarding preferences have been proposed to explain these behavioral
patterns.9 Specifically, it has been shown that, with the appropriate assumptions on the strength
8
As in Fehr and Gächter (2002), in order to avoid excessive losses due to punishment by others, subject i in
P
P
P
j cj − 3
j6=i pji ] −
j6=i pij . This way, a subject cannot be punished below
fact earns: πi = max[0, yi − ci + αi
zero points but must always pay to punish others.
9
These include, among others, Levine (1998), Fehr and Schmidt (1999), Bolton and Ockenfels (2000), Charness
and Rabin (2002), Dufwenberg and Kirchsteiger (2004), Falk and Fischbacher (2006), and Cox et al. (2007). For
an overview and discussion, see Sobel (2005).
9
of other-regarding preferences, social dilemma games are transformed into coordination games
with multiple equilibria, most of which include positive contribution levels (see, e.g., Rabin 1993
and Propositions 4 and 5 in Fehr and Schmidt 1999).10
Our main interest is the possible emergence of social norms regarding contribution behavior,
which we call contribution norms, in homogeneous groups and in groups with different types of
heterogeneity. For that purpose we discuss here what we understand as a contribution norm in
the public goods game.
In the literature on (social) norms there is, by and large, agreement on the definition of
conventions. For instance, Young (2008) refers to conventions as simple rules of behavior that
are self-enforcing because they are part of an equilibrium of the underlying game. Similarly,
Bicchieri (2006) refers to conventions as a special case of descriptive norms, which function
purely as a coordination device. In other words, given the right expectations, everyone follows a
convention because it is in each individual’s best material interest to do so.
When it comes to the definition of a social norm, views are more divergent. In the tradition
of Sudgen (1986) and Coleman (1990), Bicchieri (2006) makes a clear distinction between conventions and social norms by arguing that the function of the latter is not to solve a coordination
problem but to enforce non-equilibrium behavior in situations where there is a tension between
individual and collective material welfare. Although Young (2008, p. 647) agrees that next to
“simple rules that are self-enforcing at a primary level” there are also “more complex rules that
trigger sanctions against those who deviate”, he takes a different stance by arguing that it is not
useful to make a strict distinction between conventions and norms. In his view, the term “norm”
can only apply to games with multiple equilibria and, hence, cannot serve as an enforcement
mechanism for non-equilibrium behavior. For the public good game, the definition of Bicchieri
(2006) is most suitable when assuming narrow self-interest, implying a unique no-contributions
equilibrium, whereas the definition of Young (2008) can be applied when assuming that subjects
have other-regarding preferences, implying the existence of multiple equilibria.
Although there is some disagreement on the precise definition of a social norm, there is largely
agreement on the necessary conditions for a norm to exist and on the mechanisms by which a
norm is held in place and enforced (Bicchieri 2006; Young 2008). Specifically, for a behavioral
rule to be a social norm, the following must hold for sufficiently many people: first, it is known
10
With the distribution of other-regarding parameters as assumed in Fehr and Schmidt (1999) their model
predicts relatively low contribution levels in all our treatments without punishment. For the treatments with
punishment a large number of equilibria, including some with high contribution levels are predicted.
10
that such a behavioral rule exists; second, the involved people are motivated to follow the rule
under the condition that (a) it is believed that sufficiently many others also conform to the rule
(empirical expectations), and either (b) it is believed that sufficiently many others expect oneself
to follow the rule (normative expectations), or (b’) it is believed that sufficiently many others
are willing to sanction deviations from the rule (normative expectations with sanctions).11
An important implication of the second part of this definition is that it allows a social norm
to exist while not being followed and, hence, not being observed. Only if sufficiently many people
have the appropriate expectations about others’ behavior and, if necessary, are able and ready
to punish transgressions, an existing social norm will also be followed. In addition, it has to
be noted that the existence of a social norm does not imply that all people care with the same
strength about norm compliance.12 This implies that even if all involved people have the ‘right’
empirical or normative expectations, actual sanctioning may have to take place in order to make
people with a weak inclination toward norm obedience follow the norm.
Hypotheses
We are now ready to formulate hypotheses on the basis of the developed framework. We hypothesize that contribution norms have two dimensions related to motivations for which there
is mounting evidence indicating they are normatively and behaviorally important. The first dimension is the maximization of collective welfare (for evidence see, Charness and Rabin 2002;
11
This definition is based on the one proposed by Bicchieri (2006), which is in strong accordance with the
view of Young (2008) on the necessary elements for the existence and enforcement of a norm. Young (2008)
also assumes that people know that a behavioral rule exists because such a rule is an equilibrium strategy of the
underlying game. In addition, he defines three mechanisms that are necessary for a norm to be held in place.
First, some norms are “held in place by shared expectations” (p. 648), which parallels the above assumption of
empirical expectations together with self-enforcement of an equilibrium. Second, norms are enforced “through (...)
internalization” (p. 648), which parallels the above assumption of normative expectations (without sanctions).
Third, to be held in place, some norms need “the threat of social disapproval or punishment for norm violations”
(p. 648), which parallels the above assumption of normative expectations with sanctions. Elster (1989) also defines
a social norm in a similar way when he writes that “[for] norms to be social, they must be (a) shared by other
people and (b) partly sustained by their approval and disapproval” and “norms are social [in] that other people
are important for enforcing them” (p. 99, emphasis in original).
12
Ostrom (2000) observes that “social norms may lead individuals to behave differently in the same objective
situation depending on how strongly they value conformance with (or deviance from) a norm” (p. 144) and
Andreoni and Bernheim (2009) find that subjects in a dictator game experiment adhere with different intensities
to the 50-50 division norm (see also, Krupka and Weber 2010).
11
Engelmann and Strobel 2004), which can be thought of as an absolute norm of efficiency that
prescribes that one ought to contribute as much as possible to the public good.13 The second dimension is reciprocity (see, Sobel 2005), which in public goods games translates into conditional
cooperation (for evidence see, Keser and van Winden 2000; Fischbacher et al. 2001; Fischbacher
and Gächter 2010) or, in our context, a relative contribution norm, meaning that one should
contribute a ‘fair amount’ in relation to what others contribute. What such a fair amount is may
depend on the circumstances, which we discuss in detail below.
Since collective welfare increases with contributions in all group types, we expect to find
support for the absolute efficiency norm in homogeneous as well as heterogeneous groups. However, since social norms are not absolute prescriptions of behavior, such as ‘always contribute
everything to the public good’, but instead are to be followed only if sufficiently many others
do so as well, we do not expect maximal contributions to be the (only) normatively desirable
rule under all circumstances. In particular, in situations where maximum material efficiency is
unattainable or when contributing maximally strongly violates conditional cooperation.
We also expect support in all group types for a relative contribution norm. Given the symmetry of homogeneous groups (Equal), we expect respondents to largely agree that the most
attractive relative contribution norm is for everyone to contribute an equal amount. In heterogeneous groups, it is less obvious what the most appealing relative contribution norm will be.
Since, an important characteristic of social norms is that they are “local and context dependent”
(Bicchieri 2008, p. 229), in heterogeneous groups, normative ideas about the relative contributions of high and low players may depend on particularities that may be interpreted differently
by different people. Yet, the literature on fair allocation rules provides two prominent fairness
concepts that can be used to discuss the appeal of different relative contribution norms: equality
and equity (Konow 2003; Konow et al. 2009). Equality is generally thought of as the equalization
of outcomes with no necessary link to individual characteristics such as capacity. In contrast, equity is mostly interpreted as the dependence of fair outcomes—in a proportional way—on effort
or individual characteristics such as ability. In Table 2, we summarize the relative contribution
norms implied by the various interpretations of these two fairness concepts when applied to the
group types we investigate. In the following paragraphs we discuss them in turn.
If people apply the concept of equality to contributions, then it trivially follows that the
13
Since, ex post, public good games have multiple Pareto efficient allocations, we refer here to efficiency from an
ex ante perspective.
12
Table 2: Focal and conflicting relative contribution norms
Equality of
contributions
Equality of
earnings
Proportional to
endowments yi
Proportional to
capacities c̄
Proportional to
benefits α
ci = cj
ci = cj
ci = cj
ci = cj
ci = cj
URE
cH = cL
cH = 20, cL = 0
cH = 2cL
cH = cL
cH = cL
UUE
cH = cL
cH = 20 + cL
cH = 2cL
cH = 2cL
cH = cL
UMB
cH = cL
cH = 2cL
cH = cL
cH = cL
cH = 1.5cL
Equal
equal-contributions norm will be the one that is viewed as normatively most attractive in both
homogeneous and heterogeneous groups. If instead they apply equality to earnings (as argued
by Dawes et al. 2007), then in both URE and UUE, high players should contribute 20 points
more than low players, which in URE implies that low players do not contribute at all. In UMB
equality in earnings means that high players contribute twice as much as low players.
Alternatively, if the concept of equity is applied, then contributions ought to be proportional
to some individual characteristic.14 In UUE, proportionality—irrespective of whether it is applied
to endowments yi or capacities c̄i —implies a relative contribution norm in which high players
contribute twice as much as low players. In the other heterogeneous treatments the possibility of
conflicting interpretations of proportionality arises. In URE, proportionality can be applied to
endowments, which implies that high players should contribute twice as much as low players, or,
consistent with Major and Deaux (1982), it can be applied to the capacity to contribute, which
implies that all players should contribute the same. Similarly, in UMB, contributions proportional
to marginal benefits αi entail a relative contribution norm in which high players contribute 50
percent more than low players, whereas proportionality with respect to the capacities translates
into equal contributions for both roles.
In summary, the application of these fairness concepts considerably narrows down the set
of relative contribution norms that respondents might find normatively attractive. However,
only in homogeneous groups do all concepts lead to one focal relative contribution norm. In
heterogeneous groups, the discussed fairness concepts still allow for a variety of normatively
appealing rules of behavior making it a priori difficult to predict which one will prevail.
14
We do not discuss norms where earnings are the ones that ought to be proportional to endowments, capacities,
or benefits. Although, doing this would be possible, it is inconsistent with modern theories of equity that invoke
responsibility or accountability as a fairness principle (see Konow 1996).
13
The knowledge of one or more normatively appealing rules of behavior in public good games is
only a necessary condition for a contribution norm to be observed in an experiment. In addition,
a contribution norm has to be followed. As pointed out by Bicchieri (2006) and Young (2008),
this may happen without sanctioning if the empirical and normative expectations are strong
enough. However, given the existing evidence on public good games without punishment, we
do not expect this to be the case. Even with similar expectations regarding the contribution
norm that applies, individual differences regarding the willingness to comply with the norm
together with the nonexistence of an effective tool to sanction norm violators, it is to be expected
that low contribution levels will be observed. Since, we have no a priori reason to assume
that the willingness to comply with norms varies with a groups’ type, we expect relatively low
and decreasing contribution levels in all groups without punishment. In other words, without
punishment we do not expect to observe contribution norms.
If it is possible to punish others we expect to see—in concordance with evidence from homogeneous groups public goods experiments—higher and stable contributions in all group types. Furthermore, we expect that (some) subjects use sanctions to (try to) enforce the absolute efficiency
norm as well as a relative contribution norm. It is impossible to predict the precise contributions
level because subjects are likely to differ in their willingness to comply with norms as well as
their willingness to sanction norm violators. However, we can make some predictions concerning
punishment patterns. Since higher contribution levels are more efficient in all group types, the
existence of the absolute efficiency norm predicts that deviations from the maximum contribution will be punished similarly in all group types. Furthermore, based on the above discussion of
possible relative contribution norms, in homogeneous groups we expect that punishment will be
used to (try to) make all group members contribute the same amount. In heterogeneous groups,
we expect punishment to be in accordance with relative contribution norms based on equality
or equity, but due to the multiplicity of attractive relative contribution norms, it is difficult to
tell theoretically which specific norm will emerge, if a unique norm emerges at all. We can note,
however, that in our experiment heterogeneous groups are composed of two low players and only
one high player. Therefore, if individuals interpret fairness concepts in a self-serving manner (for
evidence see, Babcock and Loewenstein 1997), then the higher joint punishment capacity of low
players might make relative contribution norms that favor low players more likely.
14
4
Empirical results
This section consists of two parts. In the first part (Section 4.1), we analyze our questionnaire
study to explore whether individuals uninvolved in the experiment have ideas of normative rules
of behavior applicable in our public good games. In addition, we seek an answer to the following
two questions. Are the elicited normative rules related to the fairness principles discussed in the
previous section? And, do they coincide across the different group types? In the second part
(Sections 4.2, 4.3, and 4.4), we study the behavior of people involved in the public good experiment. We first analyze the contributions to the public good in all treatments with and without
punishment. We concentrate on the behavior of high and low players and on whether players in
different roles display different contribution patterns in the different treatments. Thereafter, we
investigate econometrically the enforcement of contribution norms through punishment. Lastly,
we analyze the effects of enforcement on earnings.
4.1
Normative rules of behavior
A necessary condition for a contribution norm to exist is that such a behavioral rule is known (see
Section 3). To find out whether this is the case, we use the answers to the questionnaire study
as they reflect the normative ideas of how one ought to behave in the various group types and
roles. In particular, we are interested in knowing (a) whether and to what extent people allude
to the absolute efficiency norm, (b) whether any of the relative contribution norms discussed in
Section 3 play a role in the respondents’ normative judgments, and (c) whether these contribution
norms depend on the type of group heterogeneity.
To investigate whether respondents adhere to the absolute efficiency norm, we analyze the
first question of the questionnaire. In this question, respondents state what the fair contribution
to the public good is for each of the three group members (see Section 2). We find that in all group
types the modal answer is for all group members to contribute as much as they can. Specifically,
in Equal, 59 percent of the respondents say that contributing fully is the normatively fair rule.
Interestingly, in the heterogeneous groups this percentage decreases to 50, 39, and 31 percent in
URE, UUE, and UMB, respectively. A likelihood ratio χ2 -test shows that the differences across
group types are statistically significant (p = 0.022).15
In all group types, almost all deviations from efficiency are consistent with one of the relative
15
Pair-wise comparisons reveal that, compared to Equal, the reported percentages are significantly smaller in
UUE and UMB (p ≤ 0.049). This percentage is also smaller in UMB compared to URE (p = 0.029).
15
contribution norms described in Section 3. In Equal, 38 percent of the choices are not efficient
but are consistent with a norm of equality of contributions, which implies that only 3 percent
of all choices are inconsistent with either the absolute efficiency norm or one of the discussed
relative contribution norms. In heterogeneous groups, the percentage of choices inconsistent
with any of the discussed contribution norms increases slightly but is still quite low: 8 percent
in URE, 9 percent in UUE, and 15 percent in UMB. We also find that the percentage of choices
consistent with each of the relative contribution norms varies across group types. Of all inefficient choices, between 12 percent (UMB) and 36 percent (UUE) are consistent with equality
of contributions, between 23 percent (UMB) and 50 percent (URE) with proportionality with
respect to endowments/benefits, and between 13 percent (UUE) and 44 percent (UMB) with
equality of earnings.
Hence, the data confirm that the absolute efficiency norm is an important normative rule
of behavior. However, a substantial number of respondents perceive other contributions as
normatively more appealing, and an overwhelming majority of these contributions is consistent
with the relative contribution norms discussed in Section 3.
It is important to note that the absolute efficiency norm coincides with different relative
contribution norms in the different group types, which makes it impossible to disentangle the
weights given by respondents to the various relative contribution norms. To deal with this
problem, we use the answers to the second set of questions, which were designed to exclude fullyefficient outcomes. As described in Section 2, we asked respondents to indicate the normatively
fair contributions of i and j given that k contributes ck < c̄k points to the public good. We
varied k’s type (either high or low) and the amount ck so that the discussed relative contribution
norms can be identified as clearly as possible in each group type (see the online appendix for the
specific questions and answers used to identify each norm).
Table 3 presents for each group type the fraction of answers that coincides with the discussed
relative contribution norms.16 For completeness, it also reports the percentage of answers consistent with the absolute efficiency norm (interpreted as choosing ci = c̄i and cj = c̄j irrespective
of ck ) as well as the percentage that is not consistent with any of the discussed norms. Lastly,
in cases where various norms inevitably overlap (as they do in Equal), we assign the answers
to the equality of contributions norm and leave the others blank. As can be seen in the table,
16
Note that we do not differentiate relative contribution norms based on proportionality to capacities, c̄, since
they coincide with equality of contributions in Equal, URE, and UMB and with proportionality to endowments
in UUE (see Table 2).
16
Table 3: Normatively perceived contribution norms
Note: Fraction of answers to the second set of questions in the questionnaire study that coincides
with the contribution norms discussed in Section 3. In group types where various norms inevitably
overlap answers are assigned to the equality of contributions norm.
Group
Equality of
Proportional to
Proportional to
Equality of
type
contributions
endowments
benefits
earnings
Equal
0.74
–
–
URE
0.32
0.24
UUE
0.12
UMB
0.16
Efficiency
Other
–
0.13
0.13
–
0.14
0.03
0.27
0.36
–
0.26
0.02
0.24
–
0.24
0.29
0.00
0.31
when fully efficient outcomes are unattainable, the attractiveness of the absolute efficiency norm
decreases strongly. In Equal only 13 percent of the respondents perceive the absolute efficiency
norm as the most attractive rule of behavior. In heterogeneous groups this percentage drops
to between 0 percent (UMB) and 3 percent (URE). A second noteworthy observation is that a
large majority of respondents indicate contributions that are consistent with at least one of the
discussed relative contribution norms: between 69 percent (UMB) and 74 percent (Equal). This
is clear evidence that the relative contribution norms based on equality and equity are strong
focal rules of behavior.
As hypothesized in Section 3, in homogeneous groups there is widespread agreement on the
relevant relative contribution norm: 74 percent of the respondents think that equal contributions
are normatively fair. In contrast, in heterogeneous groups we observe less agreement on the
normatively correct way to behave: the most-commonly chosen norm accounts for 36 percent of
all answers. As before, we also observe clear differences in the relative attractiveness of the focal
rules of behavior across the different group types. While in URE more than 30 percent think
that everybody should contribute equally, only 12 percent apply this rule in UUE. Conversely,
the modal perception in UUE is that contributions ought to be proportional to the (unequal)
endowments, whereas in URE only 24 percent subscribe to this view. Lastly, in UMB the reported
normative ideas are most diverse. The modal relative contribution norm is for contributions that
equalize earnings, implying that the high player contributes twice as much as the low player.
However, a relatively large fraction of 31 percent reports contributions that do not fit into any
of the discussed relative contribution norms. It seems that unequal benefits from the public
good provide a less salient clue on which to base a normative rule of behavior than unequal
17
endowments do. A likelihood ratio χ2 -test rejects the hypothesis that the distribution of relative
contribution norms is the same across the three group types (p = 0.049).
In summary, relative contribution norms based on equality and equity explain the majority
of the normative rules of behavior favored by our respondents. However, only in homogeneous
groups there is consensus on how one should behave. Moreover, the most favored normative
rule varies with the type of group heterogeneity. Given this plurality of normatively attractive
behavioral rules in heterogeneous groups, at the outset, it is not clear whether behavior consistent
with any of the discussed contribution norms will be observed when people are not uninvolved
but have stakes to gain or lose. This is explored in the next sections where we examine the
contribution rates and punishment behavior of subjects in our experimental sessions.
4.2
Contribution rates
Without punishment, contributions in Equal show the commonly-observed decreasing pattern
after initially positive contributions. The Spearman rank-order correlation between mean group
contributions and periods is significantly negative (ρ = −0.584, p ≤ 0.001). In each of the
heterogeneous group types, we observe a similar decreasing pattern for both low (ρ ≤ −0.373,
p ≤ 0.003) and high players (ρ ≤ −0.315, p ≤ 0.008).
Table 4 reports the average contributions across all periods depending on the treatment and
the subjects’ role. Any emerging contribution norm should be reflected by persistent differences
in behavior. Therefore, we also report the averages for the second half of the game. Given
that high players have twice the endowment of low players in URE and UUE, and a 50 percent
higher marginal benefit from the public good in UMB, it is reasonable to expect that their
contributions will be higher than those of low players. Surprisingly, the descriptive statistics
for treatments without punishment (see first two rows labeled low and high in Table 4) show
only small differences in their average contributions. On average, high players contribute slightly
more than low players, but as can be seen from the contributions in the last five periods, the
difference decreases over time (in UUE it even reverses). This visual impression is corroborated
by nonparametric statistical tests.17 Taking all periods into account, Wilcoxon signed-rank tests
(henceforth WSR tests) do not detect a statistically significant difference in contributions between
high and low players in URE and UUE (p > 0.128), and only a weakly significant difference in
17
Throughout the paper, the units of observation used in nonparametric tests are group means (for the relevant
subject type and periods).
18
Table 4: Descriptive statistics
Note: Average contributions and punishment points given depending on the subjects’ role and treatment. Averages are calculated for all periods and for the last five periods. Standard deviations using
group averages as the unit of observation are in parentheses.
All periods
Equal
URE
Last 5 periods
UUE
UMB
Equal
URE
UUE
UMB
without punishment
low
Contributions
high
4.21
6.81
7.48
8.63
2.23
4.24
5.18
6.71
(2.00)
(2.97)
(5.94)
(4.96)
(2.11)
(2.52)
(5.50)
(5.08)
–
–
9.41
(3.94)
9.05
(7.42)
9.87
(5.18)
–
–
5.63
(4.22)
4.10
(4.29)
7.77
(7.18)
15.58
(4.87)
16.44
(4.78)
12.24
(5.95)
30.22
with punishment
low
Contributions
Punishment
16.22
(2.88)
15.68
(3.26)
15.39
(4.63)
12.21
(5.16)
28.31
16.87
(3.62)
high
–
–
15.32
(5.31)
(10.27)
14.59
(5.61)
–
–
15.96
(5.62)
(10.14)
14.34
(6.47)
low→low
0.51
(0.36)
0.40
(0.50)
0.30
(0.41)
0.45
(0.37)
0.38
(0.39)
0.42
(0.72)
0.19
(0.33)
0.53
(0.47)
low→high
–
–
0.88
0.66
0.69
0.42
0.71
(0.63)
(0.79)
–
–
0.86
(0.96)
(1.15)
(0.63)
(0.87)
high→low
–
–
0.48
0.72
0.71
0.45
0.46
(0.70)
(0.60)
–
–
0.41
(0.55)
(0.64)
(0.74)
(0.39)
UMB (p = 0.051). In the last five periods, the contributions of high and low players are not
only considerably lower but are also statistically indistinguishable from each other (WSR tests,
p > 0.205). Table 4 also reveals that mean contributions across group types are very similar.
Kruskal-Wallis tests comparing the contributions of each player role across all treatments without
punishment do not reject the hypothesis that contributions are drawn from the same distribution
(p = 0.409 for low players and p = 0.797 for high players; in the last five periods the respective
p-values are p = 0.374 and p = 0.711).
Furthermore, in all group types contributions decrease over time towards zero,18 and as a
consequence, by the last five periods, there are no statistically significant differences in contributions. In other words, despite the fact that the absolute efficiency norm is embraced by
18
For example, in the last period, a supermajority of at least two-thirds of the subjects contributed nothing to
the public good in all group types: 76.19% in Equal, 71.43% in URE, 88.89% in UUE, and 66.67% in UMB.
19
uninvolved individuals (Section 4.1), if there are no punishment opportunities we do not observe
behavior consistent with this norm in either homogeneous or heterogeneous groups. Instead,
(almost) full free-riding emerges as the prevalent form of behavior.
When punishment is possible no significant decline in the contributions of either high or low
players in either group type is observed (Spearman rank-order correlation coefficients for low
players ρ ≥ 0.022 and high players ρ ≥ 0.035). Consequently, overall contributions tend to be
higher than in the treatments without punishment. However, the size of this increase varies
considerably across group types. For instance, the contributions of low players go up by 12.01
points in Equal but only by 3.58 points in UMB. Similarly, the contributions of high players
increase by 19.26 points in UUE but only by 4.72 points in UMB. Using Fligner-Policello robust
rank order tests (henceforth RRO tests), we find that the introduction of punishment leads to
a significant increase in the contributions of high and low players in all group types (p ≤ 0.030)
except for low players in UMB (p = 0.074).
Introducing punishment opportunities has a strong differential effect on the contributions
of high and low players, with interesting differences across group types. In URE, high and
low players contribute almost the same amount: 15.32 vs. 15.68 points on average (WSR test,
p = 0.789). This stands in stark contrast to UUE where the mean contribution of high players is
almost twice as high as that of low players: 28.31 points vs. 15.39 points (WSR test, p = 0.003).
In UMB, we observe that high players contribute a bit more than low players: 14.59 vs. 12.21
points (WSR test, p = 0.114). The differential effect of punishment is also reflected in a clear
difference in the behavior of high players across group types. A Kruskal-Wallis test rejects the
null hypothesis that the contributions of high players are drawn from the same distribution
(p = 0.002). Pair-wise comparisons reveal that the contributions of high players in UUE are
significantly different from those of high players in URE and UMB (RRO tests, p ≤ 0.001). In
other words, punishment has the strongest effect on the behavior of high players in UUE. In
this group type, high players contribute about six times more with punishment than without
punishment, whereas in the other two group types there is ‘only’ a two- to threefold increase in
contributions. In contrast, the contributions of low players are very similar across all group types
(including Equal). For low players, a Kruskal-Wallis test does not reject the null hypothesis
that contributions come from the same distribution (p = 0.241).19
19
All the reported significant differences also hold if we concentrate only on the last five periods. The same is
true for differences that are not statistically significant except that high players in UMB contribute significantly
more than low players (WSR test, p = 0.047).
20
These results suggest that subjects are following different contribution norms in the different
group types. Despite the multiplicity of possible contribution norms in heterogeneous groups and
the variety of normative ideas found in the questionnaire there are clear contribution patterns
emerging, when punishment is possible. First, the strongly increased contributions suggest a
role for the absolute efficiency norm in all group types. Second, the distinctive differences in
contributions between group types and player roles point toward the importance of different
relative contribution norms. In URE, contributions are consistent with a norm of equality of
contributions. In UUE, behavior is consistent with a norm of contributions proportional to
endowments. Note that all these norms, if applied to the homogeneous case, imply that everyone
should contribute the same amount. In UMB, the relative contributions of high and low players
are somewhere in between with high players contributing slightly more (around 20%). Given
that in treatments without punishment contributions are very similar across group types and
roles, it is likely that the differences we observe in treatments with punishment are the result
of differences in punishment behavior. In particular, subjects might be using punishment to
enforce different contribution norms in the different group types. In the following section we
explore precisely this conjecture.
4.3
Punishment and the enforcement of heterogeneous norms
In Table 4 (bottom panel), we show the average number of punishment points assigned to each
other subject depending on the role of the punishing and the punished subject. We do not
observe large differences between group types or roles. Kruskal-Wallis tests do not reject the
hypothesis that punishment points are drawn from the same distribution for all group types,
irrespective of whether we look at overall punishment (p = 0.966), punishment from low players
to low players (p = 0.319), low players to high players (p = 0.910), or high players to low players
(p = 0.541). Similarly, we do not find significant differences within each group type in the
amount of punishment given or received by high and low players (WSR tests, p > 0.140). The
lack of significant differences in the amount of punishment across group types and roles indicates
that the differences in contributions are not due to the quantity of punishment applied. It rather
suggests that these differences are the result of different ways in which punishment is used.
In order to analyze the way in which subjects punish, we build on our discussion in Section 3
where we hypothesize that normative ideals in public good games are guided by a concern for
collective material welfare (i.e., an absolute efficiency norm), and conditional cooperation (i.e.,
a relative contribution norm). Testing whether subjects punish deviations from an absolute
21
efficiency norm is simple since any contribution below the maximum contribution constitutes
such a deviation. By contrast, evaluating whether subjects also punish deviations from a relative
contribution norm is not as straightforward. In homogeneous groups, where there is consensus on
the type of behavior that is normatively desirable, it seems reasonable to assume that individuals
will punish deviations from equal contributions. In heterogeneous groups, where a variety of
normative ideas regarding fair contributions prevail, this assumption would not be justified.
In principle, one could assume a specific motivation for punishment (e.g., enforce contributions
proportional to endowments) and then use it to make treatment comparisons. However, given the
plurality of relative contribution norms, we opt for a more flexible empirical approach. Namely,
we elicit the relative contribution norm that is most consistent with the observed punishment
data, without imposing any a priori restriction on the potentially enforced norm.20
In order to elicit relative contribution norms, we assume that, ceteris paribus, subject i punishes j depending on how much their contributions differ according to the following expression:
(1−µ)cj −µci , where the term µ ∈ [0, 1] captures the relative contribution norm used to compare
one’s contribution to those of others. For example, if µ = 0.50 = 1 − µ then the above expression
is negative if (and only if) cj < ci , which from i’s perspective implies a negative deviation from
the norm by j. Alternatively, if µ = 0.75 and therefore 1 − µ = 0.25 then subject i considers
that a negative deviation from the norm occurs when cj < 3ci (i.e., the norm prescribes that
j should contribute three times as much as i). In the extremes, if µ = 0, the norm prescribes
that i contributes everything and j contributes nothing, and vice versa if µ = 1. In Equal, as
subjects are in symmetric positions, we estimate one value for µ. In the heterogeneous treatments, we distinguish between roles and estimate a value for µ in each of the following cases:
high players punishing low players (µH→L ), low players punishing high players (µL→H ), and low
players punishing low players (µL→L ).21 Specifically, we estimate the following model in each
group type:
20
Our approach is similar to that of Carpenter and Matthews (2008), who elicit norms in public good games
with homogeneous groups. They find that the best description of the data is attained with a norm akin to our
absolute efficiency norm. However, they do not consider, as we do, that subjects might be punishing due to both
efficiency and reciprocal motivations.
21
Since there is only one high player, there is no punishment of high players to other high players (µH→H ).
22
H→L
H→L
H→L
pijtk
= βpos
max (1 − µH→L )cjt − µH→L cit , 0 + βneg
max µH→L cit − (1 − µH→L )cjt , 0
pL→H
ijtk
+ λ(c̄j − cj ) + γ H→L + ηk + υi + ijt
L→H
L→H
= βpos
max (1 − µL→H )cjt − µL→H cit , 0 + βneg
max µL→H cit − (1 − µL→H )cjt , 0
pL→L
ijtk
+ λ(c̄j − cj ) + γ L→H + ηk + υi + ijt
L→L
L→L
= βpos
max (1 − µL→L )cjt − µL→L cit , 0 + βneg
max µL→L cit − (1 − µL→L )cjt , 0
+ λ(c̄j − cj ) + γ L→L + ηk + υi + ijt .
H→L is the amount of punishment points assigned (not inflicted) by a subject
The variable pijtk
i in the high role to a subject j in the low role in period t and group k. The variables pL→H
ijtk
and pL→L
ijtk are analogously defined. The first two terms in each equation capture the effect of
deviations from the relative contribution norm µH→L , µL→H , or µL→L . As can be seen, by
estimating two coefficients for each norm, we allow for the possibility that positive and negative
deviations from the norm are punished with different intensities. The third term in the equations,
(c̄j − cj ), is simply the amount of points j did not contribute to the public good, which captures
the effect of deviations from an absolute efficiency norm. Note that we estimate the same λ in
all equations as an absolute efficiency norm ought to be applied equally by high and low players.
Finally, γ H→L , γ L→H , and γ L→L , respectively, corresponds to the constant in each equation, ηk
to group dummy variables, υi to unobserved individual characteristics, and ijt is the error term.
Given that punishment is bounded by zero and ten, we use Tobit estimates. Furthermore,
we treat the unobserved individual characteristics as random effects. In order to be consistent
with our definition of norms, we assume that deviations from the norm have a positive effect on
punishment (i.e., we restrict the values of all β’s as well as that of λ to be greater than or equal
to zero).
To find the combination of µ’s that best explains the data, we repeatedly estimate the model
until—using the simplex method developed by Nelder and Mead (1965)—we find the values of
µH→L , µL→H , and µL→L (rounded to two decimal points) that give the best fit.22 Note that we
do not restrict the values of the different µ’s to be consistent with a mutually-shared contribution
norm. In other words, we allow for cases where low players are punishing high players according
to one norm, whereas high players are punishing low players according to another norm. In all
22
We estimate the model using one regression and interaction variables to separate the coefficients that vary
across the three equations. We use the log likelihood of this regression to measure the fit of a set of µ’s.
23
1.00
Normalized Log Likelihood
Normalized Log Likelihood
1.00
0.75
0.50
0.25
0.75
0.50
0.25
High to low
Low to high
Low to low
Low to low
0.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.00
0.00
1.00
0.10
0.20
0.30
0.40
µ
0.70
0.80
0.90
1.00
(b) URE
1.00
Normalized Log Likelihood
1.00
Normalized Log Likelihood
0.60
µ
(a) Equal
0.75
0.50
0.25
0.00
0.00
0.50
High to low
Low to high
Low to low
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
µ
0.75
0.50
0.25
0.00
0.00
High to low
Low to high
Low to low
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
µ
(c) UUE
(d) UMB
Figure 1: Goodness of fit for different values of µ
Note: Log likelihood obtained using different values of µH→L , µL→H , and µL→L . The log
likelihood is normalized such that 0 equals the log likelihood of the worst-fitting regression and
1 that of the best-fitting regression.
group types, we find a unique combination of µ’s, µ∗H→L , µ∗L→H , and µ∗L→L , that globally
maximizes the log likelihood function, which is visualized in Figure 1. The figure shows the log
likelihood of the estimated model as we vary one of the µ’s keeping the other two µ’s constant at
their optimal value. For convenience and the sake of comparison, we normalize the log likelihood
such that 0 equals the log likelihood of a regression without a relative contribution norm (i.e.,
one where all βneg ’s and βpos ’s are equal to zero), and 1 equals the log likelihood of the regression
with the optimal µ’s. The precise values of µ∗H→L , µ∗L→H , and µ∗L→L are available in Table 5.
In Equal, one can see that the fit of the model has a clear maximum almost exactly at
24
Table 5: Values of µ that best describe the punishment data
URE
Equal
µ
All
L→L
µ
H→L
µ
L→H
UUE
µ
L→L
µ
H→L
µ
L→H
UMB
µ
L→L
µ
H→L
µL→H µL→L
0.51
0.51
0.50
0.50
0.34
0.67
0.50
0.48
0.51
0.52
Uncooperative groups 0.51
0.51
0.50
0.52
0.34
0.67
0.50
0.44
0.65
0.47
Additional controls
0.51
0.51
0.54
0.52
0.33
0.68
0.50
0.48
0.51
0.51
Low punishers
0.52
0.52
0.50
0.52
0.33
0.67
0.50
0.48
0.51
0.52
µL→L = 0.50, and deviations from this value monotonically worsen the model’s performance. In
other words, the best fit is obtained for the µ that implies that subjects should contribute the
same amount. In URE we find values for µ∗H→L , µ∗L→H , and µ∗L→L that are all very close to
0.50. That is, low players enforce that high players contribute as much as they do and vice versa,
which is consistent with an equality of contributions norm. In stark contrast, in UUE, both high
and low players clearly enforce that high players contribute more than low players, and judging
by the values µ∗H→L = 0.34 and µ∗L→H = 0.67, the expectation is for high players to contribute
twice as much as low players (i.e., in proportion to their endowment). In UMB, we find once
again µ∗ ’s around the 0.50 value, in accordance with an equal contributions norm.
This evidence clearly points to the informal enforcement of relative contribution norms as
the explanation for the observed differences in the contributions of high players in treatments
with punishment. Importantly, the estimated optimal values µ∗H→L and µ∗L→H indicate that
both high and low players agree on which relative contribution norm to enforce. Therefore, it
is not simply a matter of players in one role using punishment to coerce others to contribute
more. Of the possible relative contribution norms (see Table 2), an equity-based norm where
contributions are proportional to the capacity to contribute explains the data best. However, a
reasonable concern with this interpretation of the punishment data may be that the observed
values of µ∗H→L , µ∗L→H , and µ∗L→L are not the result of unobservable relative contribution
norms but are due to other factors. Therefore, in the following paragraphs we present a series
of robustness checks to verify the norm-based interpretation.
An alternative explanation for the elicited relative contribution norms is that they are driven
by behavior in highly cooperative groups. By design, if a group successfully enforces an absolute
efficiency norm, it will exhibit equal contributions by high and low players (around 20 points)
in all treatments but UUE, where high players would end up contributing twice as much as low
players (40 vs. 20 points). Our punishment data are not entirely consistent with this explanation
25
as we find that deviations from both the absolute efficiency and the relative contribution norm
have a significant effect on determining punishment.23 However, it might still be the case that
highly cooperative groups are biasing our results. To explore this, we divide groups into cooperative groups (those that contribute more than the median group) and uncooperative groups (the
rest) and examine the contribution rates and the contribution ratio of high to low players in these
subsets of groups. Contributions in cooperative groups are indeed quite close to the maximum
contribution (on average 93 percent in URE, 92 percent in UUE, and 86 percent in UMB). In
contrast, contributions in uncooperative groups are well below the maximum (on average, 65
percent in URE, 59 percent in UUE, and 44 percent in UMB). Hence, in uncooperative groups
high players are not artificially constrained and can in principle contribute substantially more
than low players.
As expected, in cooperative groups, the ratio of contributions of high to low players is approximately equal to two (1.99) in UUE and is higher than in URE (1.06) and UMB (1.13)
where it is close to one. Importantly, we see a similar pattern for uncooperative groups even
though contributions in these groups are far from the maximum contribution. In URE, the ratio
of contributions of high to low players is again close to one (0.87) and significantly lower than
in UUE where it is 1.72 (RRO test, p ≤ 0.011). The average ratio of contributions in UMB is,
with 1.49, in between that of URE and UUE, but it displays a larger variance that makes it
statistically indistinguishable from either one (RRO tests, p > 0.429). Lastly, if we test, within
each group type, whether the ratio of contributions is significantly different between cooperative
and uncooperative groups, we find that this is not the case (RRO tests, p > 0.100 in URE,
p > 0.278 in UUE, and p > 0.796 in UMB).24
The observed average relative contributions of high to low players in cooperative and uncooperative groups do not support the idea that the differences between group types are a result of
highly cooperative groups. This observation is corroborated by the following analysis. If we look
for the values of µ∗H→L , µ∗L→H , and µ∗L→L using only uncooperative groups (see Table 5), we
see that in both URE and UUE the µ∗ ’s remain practically identical. Interestingly, in UMB, we
23
The regression coefficients are found in Table 6. We discuss them in detail in subsequent paragraphs.
24
The differences in the ratio of contributions of high to low players are stable over time. If we concentrate on
the last five periods, the ratio of contributions within each group type is similar in cooperative groups (1.01 in
URE, 2.00, in UUE, and 1.10 in UMB) and uncooperative groups (1.02 in URE, 1.70 in UUE, and 1.35 in UMB).
Moreover, even among uncooperative groups, it is significantly lower in URE than in UUE, and the ratio in UMB
is statistically indistinguishable from those in URE and UUE (RRO tests, p ≤ 0.022 and p > 0.405, respectively).
26
observe a shift in µ∗H→L and µ∗L→H . Among uncooperative groups in UMB, low players punish
as if they expect high players to contribute about twice as much as they do, which is consistent
with equalizing earnings, whereas high players punish as if they expect to contribute only 25
percent more than low players, which is midway of equal contributions and contributions in proportion to the benefit received from the public good. Hence, it appears that in UMB there is less
consensus on what the ratio of contributions of high players to low players ought to be. Overall,
the elicited relative contribution norms are consistent with the differences in contributions across
group types also when only taking uncooperative groups into account. This indicates that the
observed differences are not an artifact of the different contribution capabilities.
In addition to group cooperativeness, we also checked whether the values of µ∗H→L , µ∗L→H ,
and µ∗L→L are robust to controlling for other motivations for punishment. Specifically, we include
as additional variables in the regressions: the period number to capture strategic motivations
such as punishing in early periods to elicit cooperation in latter periods, and a lagged variable for
the punishment received in the previous period to capture punishment that might be motivated
by revenge (Denant-Boemont et al. 2007; Nikiforakis 2008). As can be seen in Table 5, these
additional control variables have little effect on the elicited µ∗ ’s. Lastly, we also check whether
our results are driven by the punishment behavior of only a few subjects. To do so, we elicit
µ∗H→L , µ∗L→H , and µ∗L→L excluding the 4 subjects in each group type that punished the most.
These subjects represent less than 15 percent of all subjects but account for more than a third
of all punishment (39% in Equal, 46% in URE, 36% in UUE, and 36% in UMB). By excluding
them we check whether other less belligerent subjects are enforcing the same behavior. Once
again, we find very similar values for the µ∗ ’s (see Table 5).25
In addition to establishing which contribution norm is enforced, it is interesting to see how
severely subjects punish deviations from the elicited norm. This can be observed in Table 6, which
reports the estimated coefficients of the regressions using the best-fitting µ’s and the simplest
specification (i.e., for the first row in Table 5). The regressions for the other specifications are
available in the online appendix. To indicate the explanatory power of each regression, the
table also shows each regression’s log likelihood and, as a baseline, the log likelihood obtained
if we assume that none of the contribution norms is enforced (β’s = λ = 0). It is easy to
25
By and large, the µ∗ ’s are also robust to using Logit estimates and treating punishment as a binary decision
(either punish or not). This indicates that, for the purpose of identifying the norms, what matters most is not
the amount of punishment but what behavior is or is not punished. The only case in which we see a substantial
variation is in UMB where µ∗H→L = 0.42, which is close to the value elicited in that same group type for
uncooperative groups. These regressions are available in the online appendix.
27
Table 6: Regressions for the best-fitting contribution norms
Note: Coefficients of the best-fitting regressions (i.e., for the first set of µ∗ ’s in Table 5).
Tobit estimates with subject random effects and censoring at pij = 0. In addition to
the shown coefficients, all regression have group dummies, a constant, and dummies
indicating the role of i and j. Standard errors are in parentheses. The symbols ∗ ,∗∗
indicate statistical significance at the 5, and 1 percent level.
Coefficient
UUE
UMB
0.50
(0.17)
0.12
(0.14)
0.73∗∗
(0.24)
H→L
βneg
0.56∗∗
(0.16)
0.64∗∗
(0.16)
0.73∗∗
(0.16)
L→H
βpos
0.09
(0.16)
0.28
(0.20)
0.06
(0.16)
L→H
βneg
0.47∗∗
(0.17)
0.43∗∗
(0.17)
1.01∗∗
(0.24)
H→L
βpos
URE
Equal
∗∗
0.27∗∗
(0.09)
0.15
0.49
0.40
(0.20)
(0.27)
(0.22)
L→L
βneg
0.58∗∗
(0.13)
0.73∗∗
(0.17)
1.18∗∗
(0.24)
0.79∗∗
(0.17)
λ
0.17∗∗
(0.06)
0.27∗∗
(0.06)
0.20∗∗
(0.05)
0.17∗∗
(0.06)
660
660
660
600
L→L
βpos
# obs.
log likelihood
–501.86
–437.82
–445.10
–502.70
log likelihood (β’s = λ = 0)
–578.02
–528.73
–512.66
–561.37
see that the contribution norms significantly improve the fit of all regressions.26 Moreover,
the regressions reveal three important facts. First, the coefficients for negative deviations from
the absolute efficiency norm (λ) and the elicited relative contribution norm (the βneg ’s) are
statistically significant in all regressions. Second, the value of the coefficients is such that in
all regressions, free riding is not a profit-maximizing action.27 Third, negative deviations from
26
If we use OLS estimates in order to obtain the easily-interpretable R2 statistic, we find that introducing the
contribution norms improves the R2 from 0.07 to 0.37 in Equal, from 0.11 to 0.35 in URE, from 0.10 to 0.24 in
UUE, and from 0.08 to 0.20 in UMB.
27
For example, the linear estimates indicate that reducing one’s contribution by 1 point in Equal increases
the punishment received from others by 0.91 points and therefore reduces one’s earnings by 2.72 points. In the
heterogeneous groups, the same calculation yields a reduction that varies between 2.04 points (for high players
in UUE) and 4.22 points (for low players also in UUE). If one separates the effect of each norm, in all cases the
28
the absolute efficiency as well as the estimated relative contribution norm seem to be punished
similarly across treatments. In the case of the absolute efficiency norm, we cannot reject the
hypothesis that the value of λ in Equal is equal to its value in the other group types (Wald
tests, p ≥ 0.212). Similarly for the relative contribution norm, Wald tests do not reject that the
L→L in Equal is equal to all of the other β
L→L
value of βneg
neg ’s (p ≥ 0.121) except for βneg in UUE
(p = 0.028). This suggests that the motivation to punish negative deviations from norms is by
and large independent of the type of group heterogeneity and the individuals’ roles.28
In summary, the absolute efficiency norm is similarly enforced in all group types. Therefore,
the observed differences in the contributions of high players in the different group types can
be attributed to the informal enforcement of different relative contribution norms. Both high
and low players largely agree on the contribution norms that are to be enforced and spend
similar amounts of effort punishing deviations from them. Of the possible relative contribution
norms (see Table 2), the one that is enforced depends on the type of heterogeneity in the group.
In URE, both roles enforce a norm consistent with equal contributions, despite the fact that
high players’ earnings are (almost) twice as high as low players’ earnings. In UUE, subjects
enforce relative contributions proportional to endowments. This holds for highly cooperative as
well as uncooperative groups. Finally, in UMB, the enforcement behavior in highly cooperative
groups is consistent with an equal contributions norm. However, in uncooperative groups, low
players enforce a contribution norm consistent with equal earnings, which is not matched by the
enforcement behavior of high players. Hence, the differences in marginal benefits seem to be a
weak guide as to which contribution norm ought to be followed.
4.4
Earnings
In the last part of our empirical analysis, we test whether the enforcement of contribution norms
in heterogeneous groups increases material welfare. As we have seen, punishment possibilities
lead to significantly higher contributions to the public good in all group types. This leads us to
hypothesize that earnings are higher with rather than without punishment. For this to be true,
absolute efficiency norm accounts for around 35% of the increase in punishment and the relative contribution norm
for the remaining 65%.
28
Unlike negative deviations, significant punishment of positive deviations from the relative contribution norm,
the so-called antisocial punishment (see, Herrmann et al. 2008), is not very common. It occurs only in Equal and
in the punishment of low players by high players in URE and UMB.
29
it must be the case that the increase in contributions is enough to compensate for the material
loses caused by the costs of giving and receiving punishment.
If we look at the average earnings in the last five periods of the game,29 we indeed find that
earnings in homogeneous groups are significantly higher with punishment than without: 25.5 vs.
21.1 points (one-sided RRO test, p < 0.001). In heterogeneous groups, the effect of punishment is
less clear. In all group types, average earnings are higher with punishment than without: 30.1 vs.
29.0 points in URE, 34.4 vs. 29.1 points in UUE, and 25.4 vs. 25.3 points in UMB. However, if we
test for statistically significant differences we find that earnings are statistically higher in UUE
(one-sided RRO test, p < 0.050), but not in URE and UMB (one-sided RRO tests, p > 0.380).
Therefore, it looks like some heterogeneous groups perform less well in terms of delivering higher
earnings. On closer inspection, one can see that this is due to a smaller decrease in the amount of
punishment over time in URE and UMB compared to UUE (Spearman’s ρ between punishment
points and periods are −0.130 and −0.148 for the former and −0.392 for the latter). This
result could be due to subjects taking more periods to settle on one of the (conflicting) relative
contribution norms. In this sense, it is telling that punishment increases earnings more in UUE,
where there is only a conflict between equality and equity, compared to URE and UMB, where
there is an additional conflict between different interpretations of equity.
5
Conclusions
We investigate the existence and informal enforcement of different contribution norms in public
good games with homogeneous and heterogeneous groups. We define absolute efficiency and
relative contribution norms, where the former prescribes to contribute as much as possible and
the latter is defined relative to the contributions of others. According to scholars in social
norm research (Bicchieri 2006; Young 2008) a norm is a rule of behavior that exists and can be
observed if (a) people have knowledge of such a behavioral rule, and (b) sufficiently many people
are following the rule, where the latter can come about through internalized beliefs or through
(expected) sanctions in case of a norm violation. To investigate the knowledge of contribution
norms we use data from a questionnaire study that elicits individuals’ normative ideas of how
much one ought to contribute. In order to explore the actual existence and enforcement of
29
As Fehr and Gächter (2000), we concentrate on the last five periods because they are more representative of
long-run earnings. The initial periods are characterized by lots of punishment as the norm is being established.
30
contribution norms we use data from a laboratory public good experiment with and without
punishment opportunities.
The questionnaire study reveals that uninvolved individuals indeed have clear ideas of what
constitutes normatively desirable behavior. For homogeneous as well as heterogeneous groups
we find support for both types of contribution norms. The absolute efficiency norm is favored for
all group types, especially when full efficiency is feasible. In addition, we find that the prevailing
ideas of relative contribution norms are those induced by the fairness concepts of equality and
equity. For homogeneous groups, where ideas of equality and equity all lead to the relative
contribution norm of equal contributions, we observe a consensus on this normative rule. For
heterogeneous groups, while there is consensus that normatively desired contributions should be
related to equality or equity, the various possible interpretations of equality and equity lead to
considerably less agreement on the specific behavior that is considered normatively correct.
In the laboratory experiment, we find that, in the absence of punishment possibilities, contributions steadily decline in all group types to the point where the prevalent behavior is (almost)
full free-riding. In other words, in spite of there being the strong normative idea of absolute
efficiency, we do not see this contribution norm emerging. The ubiquitous trend towards freeriding also dissipates any potential differences between the various forms of group heterogeneity
or between individuals with different induced characteristics.
In stark contrast, when punishment is possible, contributions are higher in all group types
and also exhibit considerable differences in contributions across group types and between players
with different roles. We show that the absolute efficiency norm is a guiding principle independent of group heterogeneity as deviations from full efficiency are similarly punished in all group
types. In addition, we provide evidence that the observed differences in contributions of players
in different roles and their sanctioning behavior are consistent with the enforcement of different relative contribution norms. Specifically, in groups with unequal endowments, we find that
punishment behavior is consistent with the enforcement of a relative contribution norm that
prescribes contributions that are proportional to the maximum feasible contribution. This punishment behavior can be readily reconciled with the normative ideas elicited in the questionnaire.
Indeed, despite the fact that there exists a plurality of normative ideas about the fair rule of
behavior, the norm that is enforced coincides with the norm that is favored by the (relative)
majority of questionnaire respondents.
In groups with unequal marginal benefits from the public good, there is a less clear pattern
regarding the enforced relative contribution norm. When looking at all groups we find support for
31
an emerging equal contributions norm. However, when looking only at uncooperative groups we
find disagreement between high and low players on the relative contributions that are enforced.
The latter points to an interesting avenue for future research on contribution norms. Namely,
due to insufficient punishment data, we have to assume that players that share the same role
within a type of group heterogeneity enforce the same relative contribution norm. However, it
is conceivable that different players with the same characteristics will (try to) enforce a different
norm. Such an investigation of norm enforcement at the individual group level could build on our
work, but it calls for a different design where, for instance, there is considerably more punishment
data at the group level.
The main messages of our study are, first, that with heterogeneity there exists a plurality of
normatively fair rules of behavior that are potential candidates for emerging contribution norms.
In this respect, our paper contributes to the recent empirical literature on fairness ideals (e.g.
Konow 2003; Konow et al. 2009). However, while the existing studies mainly examine two-person
allocation tasks (Cappelen et al. 2007) or bargaining situations (Gächter and Riedl 2006), we
extend this research to the more complex decision problem of contributions to a public good.
Second, in spite of the observed plurality of normative rules, those who are involved in the public
good game are mostly able to coordinate on a unique relative contribution norm, provided that
transgression of the norm can be sanctioned. Third, the enforced relative contribution norm is
based on ideas of equality and equity and clearly related to salient features of the environment
players are immersed in. In this sense behavior in public good games with heterogeneous groups
is no less predictable than the behavior of homogeneous groups. Although, there is one caveat to
be made that is suggested by both the larger variety of answers in our questionnaire study and
the more ambiguous behavior of experimental groups with unequal benefits from the public good:
when the features of the environment allow for too many different normatively-fair contribution
rules then the emerging contribution norms of specific groups may be more difficult to predict.
Recent theoretical and empirical studies underscore the importance of norms in diverse areas
of the economy and society. On the empirical side, Kim et al. (2009) find that norms of departmental productivity strongly influences the individual productivity of academics, and Goette
et al. (2006) report that group membership increases cooperation and the willingness to enforce
cooperative norms in platoons of the Swiss Army. Norms have been found to influence behavior even after individuals have moved across societies. For example, Fisman and Miguel (2007)
show that norms strongly influence illegal parking behavior of U.N. diplomats in New York, and
Guiso et al. (2006) demonstrate how the level of trust exhibited by decedents of immigrants to
32
the United States correlates with the level of trust of the country from which their ancestors
emigrated. On the theoretical side, Fischer and Huddart (2008) show that norms can put restrictions on optimal organizational design. Our study confirms that important differences in
behavior between groups can be attributed to the informal enforcement of different norms, and
are not necessarily due to collectively different preferences of group members. Moreover, we add
to this literature the insight that heterogeneity and salient variations in the environment can
shift attention from one focal norm to another, resulting in considerably different outcomes.
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