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Research Article
Preference for Efficiency or Confusion? A Note on
a Boundedly Rational Equilibrium Approach to Individual
Contributions in a Public Good Game
Luca Corazzini1,2 and Marcelo Tyszler3
1Department of Law Science and History of Institutions, University of Messina, Piazza XX Settembre 4,
98122 Messina, Italy
2ISLA, Bocconi University, Via Roentgen 1, 20136 Milan, Italy
3Center for Research in Experimental Economics and Political Decision Making (CREED), University of Amsterdam,
Roetersstraat 11, 1018 WB Amsterdam, Netherlands
Correspondence should be addressed to Marcelo Tyszler; marcelo@tyszler.com.br
Received December ; Revised March ; Accepted March
Academic Editor: Dimitris Fotakis
Copyright © L. Corazzini and M. Tyszler. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
By using data from a volu ntary contr ibution mechanism exper iment with heterogeneous endowments and asymmetric information,
we estimate a quantal response equilibrium (QRE) model to assess the relative importance of eciency concerns versus
noise in accounting for subjects overcontribution in public good games. In the benchmark specication, homogeneous agents,
overcontribution is mainly explained by error and noise in behavior. Results change when we consider a more general QRE
specication with cross-subject heterogeneity in concerns for (group) eciency. In this case, we nd that the majority of the subjects
make contributions that are compatible with the hypothesis of preference for (group) eciency. A likelihood-ratio test conrms
the superiority of the more general specication of the QRE model over alternative specications.
1. Introduction
Overcontribution in linear public good games represents
one of the best documented and replicated regularities in
experimental economics. e explanation of this apparently
irrational behaviour, however, is still a debate in the literature.
is paper is aimed at investigating the relative importance of
noise versus preference for eciency. In this respect, we build
andestimateaquantalresponseequilibrium(henceforth,
QRE []) extension of the model presented by Corazzini et al.
[]. is boundedly rational model formally incorporates
both preference for eciency and noise. Moreover, in con-
trast to other studies that investigate the relative importance
of error and other-regarding preferences, the QRE approach
explicitly applies an equilibrium analysis.
To reconcile the experimental evidence with the standard
economic framework, social scientists developed explana-
tions based on renements of the hypothesis of “other-
regarding preferences”: reciprocity [–], altruism and spite-
fulness [–], commitment and Kantianism [,], norm
compliance [], and team-thinking [–].
Recently, an additional psychological argument to explain
agents’ attitude to freely engage in prosocial behavior is
gaining increasing interest: the hypothesis of preference for
(group) eciency. ere is evidence showing that exper-
imental subjects oen make choices that increase group
eciency, even at the cost of sacricing their own payo
[,]. Corazzini et al. [] use this behavioral hypothesis
to explain evidence from linear public good experiments
basedonprizes(alottery,arstpriceallpayauction,and
Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2015, Article ID 961930, 8 pages
http://dx.doi.org/10.1155/2015/961930
Journal of Applied Mathematics
a voluntary contribution mechanism used as a benchmark),
characterized by endowment heterogeneity and incomplete
information on the distribution of incomes. In particular,
they present a simple model in which subjects bear psy-
chological costs from contributing less than what is ecient
for the group. e main theoretical prediction of their
model when applied to linear public good experiments is
that the equilibrium contribution of a subject is increasing
in both her endowment and the weight attached to the
psychological costs of (group-) inecient contributions in
the utility function. e authors show that this model is
capable of accounting for overcontribution as observed in
their experiment, as well as evidence reported by related
studies.
However, as argued by several scholars, rather than being
related to subjects’ kindness, overcontribution may reect
their natural propensity to make errors. ere are several
experimental studies [–]thatseektodisentangleother-
regarding preferences from pure noise in behavior by running
ad hoc variants of the linear public good game. A general
nding of these papers is that “warm-glow eects and random
error played both important and signicant roles” [,p.]
in explaining overcontribution.
ere are several alternative theoretical frameworks that
canbeusedtomodelnoiseinbehavior(boundedrationality)
and explain experimental evidence in strategic games. Two
examples are the “level-” model (e.g., [–]) and (rein-
forcement) learning models (e.g., []). In the “level-”model
of iterated dominance, “level-0” subjects choose an action
randomly and with equal probability over the set of possible
pure strategies while “level-” subjects choose the action that
represents the best response against level-(−1)subjects.
Level-models have been used to account for experimental
results in games in which other-regarding preferences do not
play any role, such as -Beauty contests and other constant
sum games. Since in public good games there is a strictly
dominant strategy of no contribution, unless other-regarding
preferences are explicitly assumed, “level-”modelsdonot
apply. Similar arguments apply to learning models. In the
basic setting, each subject takes her initial choice randomly
and with equal probability over the set of possible strategies.
As repetition takes place, strategies that turn out to be more
protable are chosen with higher probability. us, unless
other-regarding preferences are explicitly incorporated into
the utility function, repetition leads to the Nash equilibrium
of no contribution.
e QRE approach has the advantage that even in the
absence of other-regarding preferences it can account for
overcontribution in equilibrium. Moreover, we can use the
model to assess the relative importance of noise versus
eciency concerns.
We start from a benchmark model in which the popula-
tion is homogeneous in both concerns for (group) eciency
and the noise parameter. We then allow for heterogeneity
across subjects by assuming the population to be partitioned
into subgroups with dierent degrees of ecieny concerns
but with the same value for the noise parameter.
In the QRE model with a homogeneous population, we
nd that subjects’ overcontribution is entirely explained by
noise in behavior, with the estimated parameter of concerns
for (group) eciency being zero. A likelihood-ratio test
strongly rejects the specication not allowing for randomness
in contributions in favor of the more general QRE model. A
dierent picture emerges when heterogeneity is introduced
in the QRE model. In the model with two subgroups, the
probability of a subject being associated with a strictly
positive degree of preference for (group) eciency is approx-
imately one-third. is probability increases to 59%when
we add a third subgroup characterized by an even higher
eciency concern. A formal likelihood-ratio test conrms
the superiority of the QRE model with three subgroups
over the other specications. ese results are robust to
learning processes over repetitions. Indeed, estimates remain
qualitatively unchanged when we replicate our analysis on
the last % of the experimental rounds. e rest of this
paperisstructuredasfollows.InSection , we describe
the experimental setting. In Section ,wepresenttheQRE
extension of the model based on the preference for (group)
eciency hypothesis. Section reports results from our
statistical analysis. Section concludes the paper.
2. The Experiment
We use data from three sessions of a voluntary contribution
mechanism reported by Corazzini et al. []. Each session
consisted of 20 rounds and involved 16 subjects. At the
beginning of each session, each subject was randomly and
anonymously assigned, with equal chance, an endowment
of either 120,160,200,or240 tokens. e endowment was
assigned at the beginning of the experiment and was kept
constant throughout the 20 rounds. e experiment was run
in a strangers condition [] such that, at the beginning
of each round, subjects were randomly and anonymously
rematched in groups of four players. is procedure was com-
mon knowledge. us, in each round, subjects made their
choices under incomplete information on the distribution
of the endowments in their group. In each round, every
subject had to allocate her endowment between an individual
and a group account. e individual account implied a
private benet such that, for each token a subject allocated
to the individual account, she received two tokens. On the
otherhand,tokensinthegroupaccountgeneratedmonetary
returns to each of the group members. In particular, each
subject received one token for each token allocated by her
or by any other member of her group to the group account.
us, the marginal per capita return used in the experiment
was 0.5. At the beginning of each round, the experimenter
exogenously allocated 120 tokens to the group account,
independently of subjects’ choices, thus implying 120 extra
tokens for each group member. At the end of each round,
subjects received information about their payos. Tokens
wereconvertedtoeurosusinganexchangerateof1000
points per euro. Subjects, mainly undergraduate students of
economics, earned 12.25 euros on average for sessions lasting
about 50 minutes. e experiment took place in May
in the Experimental Economics Laboratory of the University
Journal of Applied Mathematics
of Milan Bicocca and was computerized using the z-Tree
soware [].
e features of anonymity and random rematching nar-
row the relevance of some “traditional” behavioral hypothe-
ses used to explain subjects’ overcontribution. For instance,
they preclude subjects’ possibility to reciprocate (un)kind
contributions of group members []. Moreover, under these
conditions, subjects with preferences for equality cannot
make compensating contributions to reduce (dis)advan-
tageous inequality [,]. Rather, the hypothesis of prefer-
ence for (group) eciency as a particular form of warm-glow
[,] appears as a more plausible justication.
3. Theoretical Predictions and
Estimation Procedure
Consider a nite set of subjects ={1,2,...,}.Inageneric
round, subject ∈, with endowment 𝑖∈
+,contributes
𝑖to the group account, with 𝑖∈
+and 0≤
𝑖≤
𝑖.e
monetary payo of subject who contributes 𝑖in a round is
given by
𝑖𝑖,𝑖=2𝑖−
𝑖+ 120 + 𝑖+
−𝑖,()
where −𝑖 is the sum of the contributions of group members
other than in that round. Given (),ifsubjects’utilityonly
depends on the monetary payo, zero contributions are the
unique Nash equilibrium of each round. In order to explain
the positive contributions observed in their experiment,
Corazzini et al. [] assume that subjects suer psychological
costsiftheycontributelessthanwhatisoptimalforthegroup.
In particular, psychological costs are introduced as a convex
quadratic function of the dierence between a subject’s
endowment (i.e., the social optimum) and her contribution.
In the VCM, player ’s (psychological) utility function is given
by
𝑖𝑖,𝑖,𝑖=
𝑖𝑖,𝑖−
𝑖𝑖−
𝑖2
𝑖,()
where 𝑖is a nonnegative and nite parameter measuring
the weight attached to the psychological costs, (𝑖−
𝑖)2/𝑖,
in the utility function. Notice that psychological costs are
increasing in the dierence between a subject’s endowment
and her contribution. Under these assumptions, in each
round, there is a unique Nash equilibrium in which individual
contributes:
NE
𝑖=2𝑖−1
2𝑖𝑖.()
e higher the value of 𝑖,thehighertheequilibrium
contribution of subject . e average relative contribution,
𝑖/𝑖, observed in the VCM sessions is 22%, which implies
= 0.64.
Following McKelvey and Palfrey [], we introduce noisy
decision-making and consider a Logit Quantal Response
extension of (). In particular, we assume subjects choose
their contributions randomly according to a logistic quantal
response function. Namely, for a given endowment, 𝑖,
and contributions of the other group members, −𝑖,the
probability that subject contributes 𝑖is given by
𝑖𝑖,𝑖,𝑖,= exp 𝑖𝑖,𝑖,𝑖/
∑𝑤𝑖
𝑔𝑗=0 exp 𝑖𝑖,𝑗,𝑖/,()
where ∈R+isanoiseparameterreectingasubject’s
capacity of noticing dierences in expected payos.
erefore each subject is associated with a 𝑖-
dimensional vector q𝑖(𝑖,g𝑖,𝑖,) containing a value of
𝑖(𝑖,𝑖,𝑖,)for each possible contribution level 𝑖∈g𝑖≡
{0,...,𝑖}.Let{q𝑖(𝑖,g𝑖,𝑖,)}𝑖∈𝑃 be the system including
q𝑖(𝑖,g𝑖,𝑖,), ∀ ∈ . Notice that since others’ contribu-
tion, −𝑖, enters the r.h.s of the system, others’ 𝑖will also
enter the r.h.s. A xed point of {q𝑖(𝑖,g𝑖,𝑖,)}𝑖∈𝑃 is, hence, a
quantal response equilibrium (QRE), {qQRE
𝑖(𝑖,g𝑖,𝑖,)}𝑖∈𝑃.
In equilibrium, the noise parameter reects the disper-
sion of subjects’ contributions around the Nash prediction
expressed by (). e higher the ,thehigherthedispersion
of contributions. As tends to innity, contributions are
randomly drawn from a uniform distribution dened over
[0,𝑖]. On the other hand, if is equal to 0,theequilibrium
contribution collapses to the Nash equilibrium.(more specif-
ically, for each subject equilibrium contributions converge to
𝑖(𝑖,NE
𝑖,𝑖,0)=1and 𝑖(𝑖,𝑖,𝑖,0) = 0, ∀𝑖=
NE
𝑖.)
In this framework, we use data from Corazzini et al. []
to estimate and , jointly. We proceed as follows. Our
initial analysis is conducted by using all rounds ( = 20)
and assuming the population to be homogeneous in both
and .isgivesusabenchmarkthatcanbedirectly
compared with the results reported by Corazzini et al. [].
In our estimation procedure, we use a likelihood function
that assumes each subject’s contributions to be drawn from
a multinomial distribution. at is,
𝑖𝑖,g𝑖,,=!
∏𝑤𝑖
𝑔𝑗=0𝑗!
𝑤𝑖
𝑔𝑘=0QRE
𝑖𝑖,𝑘,,𝑛(𝑔𝑘),
()
where (𝑗)is the number of times that subject contributed
𝑗over the rounds of the experiment, and similarly for
(𝑘). e contribution of each person to the log-likelihood
is the log of expression (). e Maximum Likelihood
procedure consists of nding the nonnegative values of and
(and corresponding QRE) that maximize the summation
of the log-likelihood function evaluated at the experimental
data. In other words, we calculate the multinomial probability
of the observed data by restricting the theoretical probabili-
ties to QRE probabilities only.
We then extend our analysis to allow for cross-subject
heterogeneity. In particular, we generalize the QRE model
above by assuming the population to be partitioned into
Journal of Applied Mathematics
T : Homogeneous population (all rounds).
Data CFS () ,() , () ,
—— 1. [19.69;24.34] . [39.11; 44.34]
—0.64 0.64 0.64 [0;0.01]
(Predicted) avg. contributions
Overall endowments . . . . .
𝑖= 120 . . . . .
𝑖= 160 . . . . .
𝑖= 200 . . . . .
𝑖= 240 . . . . .
log −. −. −.
Obs.
is table reports average contributions as well as estimates and predictions from various specications of the model based on the eciency concerns
assumption using all rounds of the experiment. CFS refers to the specication not accounting for noise in subjects’ contributions while (), (), and ()
are Logit Quantal Response extensions of the model. In () 𝛼and 𝜇are constrained to . and , respectively. In (), the value on 𝛼is set to ., while
𝜇is estimated through (). Finally, () refers to the unconstrained model in which both 𝛼and 𝜇are estimated through (). e table also reports, for each
specication, the corresponding log-likelihood. Condence intervals are computed using an inversion of the likelihood-ratio statistic, at the . level, subject
to parameter constraints.
subgroups that are characterized by the same but dierent
. In this case, the likelihood function becomes
𝑖𝑖,g𝑖,1,2,...,𝑆,
1,
2,...,
𝑆,
=𝑆
𝑠=1𝑠!
∏𝑤𝑖
𝑔𝑗=0𝑗!
𝑤𝑖
𝑔𝑘=0QRE
𝑖𝑖,𝑘,𝑠,𝑛(𝑔𝑘),()
where 1,
2,...,
𝑆,with∑𝑆
𝑠=1 𝑠=1, are the probabili-
ties for agent belonging to the subgroup associated with
1,2,...,𝑆,respectively.isallowsustoestimatethevalue
of forthewholepopulation,thevalueof1,2,...,𝑆
for the subgroups, and the corresponding probabilities,
1,
2,...,
𝑆. For identication purposes we impose that 𝑠≤
𝑠+1. e introduction of one group at a time accompanied by
a corresponding likelihood-ratio test allows us to determine
the number of -groups that can be statistically identied
from the original data. In the following statistical analysis,
estimates account for potential dependency of subject’s con-
tributions across rounds. Condence intervals at the 0.01
level are provided using the inversion of the likelihood-ratio
statistic, subject to parameter constraints, in line with Cook
and Weisberg [], Cox and Hinkley [], and Murphy [].
4. Results
Using data from the rounds of the experiment, Table
reports (i) average contributions (by both endowment type
and overall) observed in the experiment, (ii) average con-
tributions as predicted by the model not accounting for
noise in subjects’ contributions, and (iii) estimates as well as
average contributions from dierent parameterizations of the
Logit Quantal Response extension of the model. In particular,
specication (1) refers to a version of the model in which
both and are constrained to be equal to benchmark values
basedonCorazzinietal.[]. Under this parameterization, is
xed to the value computed by calibrating () on the original
experimental data, 0.64,whileis constrained to 1.(Ta b l e
shows the Maximum Likelihood estimation value of when
we vary .Itispossibletoseethatforalargerangeofvaluesof
this value is close to 0.64.Wechoose=1as a suciently
low value in which the estimated is close to 0.64 and thus
provideanoisyversionofthebasemodelwhichcanbeused
for statistical tests.)
As shown by the table, specication (1) closely replicates
predictions of the original model presented by Corazzini
et al. [] not accounting for noise in subjects’ contributions.
In specication (2),is xed to 0.64,whileis estimated
by using ().evalueofincreases substantially with
respect to the benchmark value used in specication (1).
A likelihood-ratio test strongly rejects specication (1) that
imposes restrictions on the values of both and in favor
of specication (2) in which canfreelyvaryonR+(LR =
10460.33; Pr{2(1) > LR} < 0.01). However, if we compare
the predicted average contributions of the two specications,
we nd that specication (1)better approximates the original
experimental data. is is because a higher value of the noise
parameterspreadthedistributionsofcontributionsaround
the mean. erefore even with mean contributions further
fromthedata(inducedbythexedvalueof)thespread
induced by the noise parameter in specication (2) produces
a better t. is highlights the importance of taking into
account not only the average (point) predictions but also the
spread around it. It also suggests that allowing to vary can
improve t.
In specication (3),and arejointlyestimatedusing
(),subjectto≥0. If both parameters can freely vary
over R+,reducestozeroandreaches a value that
is higher than what was obtained in specication (2).As
conrmed by a likelihood-ratio test, specication (3) ts
the experimental data better than both specication (1)
Journal of Applied Mathematics
T : Homogeneous population (last 5rounds).
Data CFS ()
,
() ,
() ,
—1. [9.80;13.95] . [24.14; 30.17]
. . . [0;0.03]
(Predicted) avg. contributions
Overall endowments . . . . .
𝑖= 120 . . . . .
𝑖= 160 . . . . .
𝑖= 200 . . . . .
𝑖= 240 . . . . .
log −. −. −.
Obs.
is table reports average contributions as well as estimates and predictions from various specications of the model based on the eciency concerns
assumption using the last rounds of the experiment only. e same remarks as in Ta ble apply.
T : Heterogeneous subjects (all and last 5rounds).
,1,and 2(=20), 1,and 2(=5),1,2,and 3(=20),1,2,and 3(=5)
. [25.88;31.26] . [12.90;17.64] . [20.56; 23.95] . [12.04;16.85]
1[0;0.01] [0; 0.02] [0;0.01] [0;0.02]
2. [0.46;0.60] . [0.47; 0.61] . [0.39;0.46] . [0.40;0.56]
3. [0.92;1.16] . [0.53;1.01]
1. [0.53;0.78] . [0.50;0.75] . [0.33; 0.46] . [0.49; 0.64]
2. [0.43;0.55] . [0.23;0.40]
(Predicted) avg. contributions
Overall endowments . . . .
𝑖= 120 . . . .
𝑖= 160 . . . .
𝑖= 200 . . . .
𝑖= 240 . . . .
log −. −. −. −.
Obs.
is table reports estimates and predictions from two specications of the model with eciency concerns accounting for cross subject heterogeneity inthe
value of 𝛼. e analysis is conducted both by including all experimental rounds and by focusing on the last ve repetitions only. Parameters are estimated
through (). Given the linear restriction ∑𝑆
𝑠=1 𝛾𝑠=1, we only report estimates of 𝛾1,𝛾
2,...,𝛾
𝑆−1. Condence intervals are computed using an inversion of the
likelihood-ratio statistic, at the . level, subject to parameter constraints.
(LR = 11086.54; Pr{2(2) > LR} < 0.01)andspecication
(2) (LR = 626.21; Pr{2(1) > LR} < 0.01). us, under the
maintained assumption of homogeneity, our estimates sug-
gest that contributions are better explained by randomness in
subjects’ behavior rather than by concerns for eciency.
In order to control for learning eects, we replicate our
analysis using the last ve rounds only.
Consistent with a learning argument, in both speci-
cations (2) and (3),thevaluesofare substantially lower
than the corresponding estimates in Table . us, repetition
reduces randomness in subjects’ contributions. e main
results presented above are conrmed by our analysis on
the last ve periods. Looking at specication (3),inthe
model with no constraints on the parameters, the estimated
value of again drops to 0. Also, according to a likelihood-
ratio test, specication (3) explains the data better than both
specications (1) (LR = 1578.83; Pr{2(2) > LR} < 0.01)
and (2) (LR = 203.85; Pr{2(1) > LR} < 0.01).
ese results seem to reject the preference for (group)
eciency hypothesis in favor of pure randomness in subjects’
contributions. However, a dierent picture emerges when
we allow for cross-subject heterogeneity. In Tab l e we drop
the assumed homogeneity. We consider two models with
heterogeneous subjects: the rst assumes the population to
be partitioned into two subgroups (=2)andthesecond
into three subgroups (=3). (We have also estimated a
model with =4. However, adding a fourth subgroup does
not signicantly improve the goodness of t of the model
compared to the specication with =3.Inparticular,with
=4, the point estimates for the model with all periods
are = 21.81,1=0,2=0.38,3=0.61,4= 1.04,
1= 0.39,2= 0.42,and3=0.09.) As before, we conduct
Journal of Applied Mathematics
T
Log-likelihood
1000.00 0 −.
500.00 0 −.
333.33 0 −.
250.00 0 −.
200.00 0 −.
166.67 0 −.
142.86 0 −.
125.00 0 −.
111.11 0 −.
100.00 0 −.
90.91 0 −.
83.33 0 −.
76.92 0 −.
71.43 0 −.
66.67 0 −.
62.50 0 −.
58.82 0 −.
55.56 0 −.
52.63 0 −.
50.00 0 −.
40.00 0 −.
30.30 0.14 −.
20.00 0.33 −.
10.00 0.50 −.
9.09 0.52 −.
8.00 0.53 −.
7.04 0.55 −.
5.99 0.56 −.
5.00 0.57 −.
4.00 0.58 −.
3.00 0.59 −.
2.00 0.60 −.
1.00 0.61 −.
0.90 0.61 −.
0.80 0.61 −.
0.70 0.61 −.
0.60 0.61 −.
0.50 0.61 −.
0.40 0.61 −.
is table reports Maximum Likelihood estimates of 𝛼for selected values of
𝜇(see ()). e last column reports the corresponding log-likelihood value.
our analysis both by including all rounds of the experiment
andbyfocusingonthelastverepetitionsonly.
We nd strong evidence in favor of subjects’ heterogene-
ity. Focusing on the analysis over all rounds, according to
the model with two subgroups, a subject is associated with
1=0with probability 0.66 and with 2= 0.53 with
probability 0.34. Results are even sharper in the model with
three subgroups: in this case 1=0and the two other -
parameters are strictly positive: 2= 0.43 and 3= 1.04.
Subjectsareassociatedwiththesevalueswithprobabilities
0.41,0.50,and0.09, respectively. us, in the more parsi-
monious model, the majority of subjects contribute in a way
that is compatible with the preference for (group) eciency
hypothesis.eseproportionsareinlinewithndingsof
previous studies [,,] in which, aside from confusion,
social preferences explain the behavior of about half of the
experimental population.
Allowing for heterogeneity across subjects reduces the
estimated randomness in contributions: the value of
reduces from 41.59 in specication (3) of the model with
homogeneous population to 28.50 and 22.14 in the model
with two and three subgroups, respectively. According to a
likelihood-ratio test, both the models with =2and =3t
the data better than the (unconstrained) specication of the
model with homogeneous subjects (for the model with =2,
LR = 117.25; Pr{2(2) > LR} < 0.01, whereas for the model
with =3,LR= 174.66; Pr{2(4) > LR} < 0.01). Moreover,
adding an additional subgroup to the model, with =2,
signicantly increases the goodness of t of the specication
(LR = 57.42; Pr{2(2) > LR} < 0.01). As before, all these
results remain qualitatively unchanged when we control for
learning processes and we focus on the last 5experimental
rounds.
In order to check for the robustness of our results in
Table ,wehavealsoestimatedadditionalspecications
accounting for heterogeneity in both concerns for (group)
eciency and noise in subjects’ behavior. Although the log-
likelihood of the model with both sources of heterogeneity
signicantly improves in statistical terms, the estimated
values of the -parameters remain qualitatively the same as
those reported in the third column of Tab l e .
5. Conclusions
Is overcontribution in linear public good experiments
explained by subjects’ preference for (group) eciency or,
rather, does it simply reect their natural attitude to make
errors? In order to answer this fundamental question, we
estimate a quantal response equilibrium model in which, in
choosing their contributions, subjects are inuenced by both
a genuine concern for (group) eciency and a random noise
in their behavior.
In line with other studies, we nd that both concerns
for (group) eciency and noise in behavior play an impor-
tant role in determining subjects’ contributions. However,
assessing which of these two behavioral hypotheses is more
relevant in explaining contributions strongly depends on
the degree of cross-subject heterogeneity admitted by the
model. Indeed, by estimating a model with homogeneous
subjects, the parameter capturing concerns for (group) e-
ciency vanishes while noise in behavior entirely accounts
for overcontribution. A dierent picture emerges when we
allow the subjects to be heterogeneous in their concerns for
eciency. By estimating a model in which the population
is partitioned into three subgroups that dier in the degree
of concerns for eciency, we nd that most of the subjects
contribute in a way that is compatible with the preference
for (group) eciency hypothesis. A formal likelihood-ratio
Journal of Applied Mathematics
test conrms the supremacy of the QRE model with three
subgroups over the other specications.
Previous studies [–] tried to disentangle the eects of
noise from other-regarding preferences by mainly manipulat-
ing the experimental design. Our approach adds a theoretical
foundation in the form of an equilibrium analysis. In contrast
to studies which focus mostly on (direct) altruism, we follow
Corazzini et al. [] and allow for preference for eciency.
Our results are in line with the literature in the sense that we
also conclude that a combination of noise and social concerns
plays a role. Our results, however, are directly supported by a
sound theoretical framework proven valid in similar settings
(e.g., []).
Recent studies [,] have emphasized the importance
of admitting heterogeneity in social preferences in order to
better explain experimental evidence. In this paper we show
that neglecting heterogeneity in subjects’ social preferences
may lead to erroneous conclusions on the relative importance
ofthelovefor(group)eciencyhypothesiswithrespectto
theconfusionargument.Indeed,asrevealedbyouranalysis,
the coupling of cross-subject heterogeneity in concerns for
(group) eciency with noise in the decision process seems
to be the relevant connection to better explain subjects’
contributions.
Appendix
Table shows the Maximum Likelihood value of and the
log-likelihood according to () as decreases from 1000 to
0.4.Asshownbythetable,forhighvaluesof,theestimated
value of is 0.Whenis equal to , the estimated value of
alpha is 0.50.Moreover,forlower than 2.00,theestimated
value of is 0.61. For the specication tests presented in
Section ,weset=1. is is a suciently low value of
in order to generate a noisy version of the base model. Two
arguments indicate why this choice is valid. First, for a range
of values including =1,theestimatedis stable. Moreover,
since the log-likelihood of a model with = 0.61 and =1
is higher than that corresponding to a model with = 0.4
(and similarly for = 0.64),thechoiceofanylower than
1forthebenchmarkvaluewouldonlyreinforcetheresults
of Section . More specically, both likelihood-ratio statistics
comparing specications (1)with specications (2)and (3) of
Tables and would increase.
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
Acknowledgments
e authors thank Arthur Schram, Jens Grosser, and par-
ticipants to the Credexea Meeting at the University
of Amsterdam, IMEBE in Barcelona, and Annual
Meeting of the Royal Economic Society in London for useful
comments and suggestions.
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