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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.

References

[] R. D. McKelvey and T. R. Palfrey, “Quantal response equilibria

for normal form games,” Games and Economic Behavior,vol.,

no.,pp.–,.

[] L. Corazzini, M. Faravelli, and L. Stanca, “A prize to give

for: an experiment on public good funding mechanisms,” e

Economic Journal,vol.,no.,pp.–,.

[] R. Sugden, “Reciprocity: the supply of public goods through

voluntary contributions,” e Economic Journal,vol.,no.,

pp. –, .

[] H. Hollander, “A social exchange approach to voluntary con-

tribution,” e American Economic Review,vol.,no.,pp.

–, .

[] A. Falk and U. Fischbacher, “A theor y of reciprocity,” Games and

Economic Behavior, vol. , no. , pp. –, .

[] U. Fischbacher, S. G¨

achter, and E. Fehr, “Are people condition-

ally cooperative? Evidence from a public goods experiment,”

Economics Letters,vol.,no.,pp.–,.

[] D. K. Levine, “Modeling altruism and spitefulness in experi-

ments,” Review of Economic Dynamics,vol.,no.,pp.–,

.

[] J. Andreoni, “Giving with impure altruism: applications to

charity and ricardian equivalence,” Journal of Political Economy,

vol.,no.,pp.–,.

[] J. Andreoni, “Impure altruism and donations to public goods:

atheoryofwarm-glowgiving,”Economic Journal,vol.,no.

, pp. –, .

[] J. J. Laont, “Macroeconomic constraints, economic eciency

and ethics: an introduction to Kantian economics,” Economica,

vol. , no. , pp. –, .

[] M. Bordignon, “Was Kant right? Voluntary provision of public

goods under the principle of unconditional commitment,”

Economic Notes,vol.,pp.–,.

[] M. Bernasconi, L. Corazzini, and A. Marenzi, “‘Expressive’

obligations in public good games: crowding-in and crowding-

out eects,” Working Papers, University of Venice “Ca’ Foscari”,

Department of Economics, .

[] M. Bacharach, Beyond Individual Choice: Teams and Frames

in Game eory, Edited by N. Gold and R. Sugden, Princeton

University Press, .

[] R. Sugden, “e logic of team reasoning,” Philosophical Explo-

rations,vol.,no.,pp.–,.

[] R. Cookson, “Framing eects in public goods experiments,”

Experimental Economics,vol.,no.,pp.–,.

[] G. Charness and M. Rabin, “Understanding social preferences

with simple tests,” Quarterly Journal of Economics,vol.,no.,

pp. –, .

[] D. Engelmann and M. Strobel, “Inequality aversion, eciency,

and maximin preferences in simple distribution experiments,”

American Economic Review,vol.,no.,pp.–,.

[] J. Andreoni, “Cooperation in public-goods experiments: kind-

ness or confusion?” e American Economic Review,vol.,no.

, pp. –, .

[] T. R. Palfrey and J. E. Prisbrey, “Altruism, reputation and

noise in linear public goods experiments,” Journal of Public

Economics,vol.,no.,pp.–,.

[] T. R. Palfrey and J. E. Prisbrey, “Anomalous behavior in

public goods experiments: how much and why?” e American

Economic Review, vol. , no. , pp. –, .

Journal of Applied Mathematics

[] J. Brandts and A. Schram, “Cooperation and noise in public

goods experiments: applying the contribution function ap-

proach,” Journal of Public Economics,vol.,no.,pp.–,

.

[] D. Houser and R. Kurzban, “Revisiting kindness and confusion

in public goods experiments,” American Economic Review,vol.

,no.,pp.–,.

[]J.K.Goeree,C.A.Holt,andS.K.Laury,“Privatecostsand

public benets: unraveling the eects of altruism and noisy

behavior,” Journal of Public Economics,vol.,no.,pp.–

, .

[] D. O. Stahl and P. W. Wilson, “On players’ models of other play-

ers: theory and experimental evidence,” Games and Economic

Behavior, vol. , no. , pp. –, .

[] T.-H. Ho, C. Cambrer, and K. Weigelt, “Iterated dominance

and iterated best response in experimental ‘p-Beauty Contests’,”

American Economic Review, vol. , no. , pp. –, .

[] D. O. St ahl and E . Haruv y, “Level-nbounded rationality in

two-player two-stage games,” Journal of Economic Behavior &

Organization,vol.,no.,pp.–,.

[] I. Erev and A. E. Roth, “Predicting how people play games:

reinforcement learning in experimental games with unique,

mixed strategy equilibria,” American Economic Review,vol.,

no. , pp. –, .

[] J. Andreoni, “Why free ride? Strategies and learning in public

goods experiments,” Journal of Public Economics,vol.,no.,

pp. –, .

[] U. Fischbacher, “Z-Tree: zurich toolbox for ready-made eco-

nomic experiments,” Experimental Economics,vol.,no.,pp.

–, .

[] M. Rabin, “Incorporating fairness into game theory and eco-

nomics,” e American Economic Rev iew,vol.,no.,pp.–

, .

[] E. Fehr and K. M. Schmidt, “A theory of fairness, competition,

and cooperation,” Quarterly Journal of Economics,vol.,no.

, pp. –, .

[] G. E. Bolton and A. Ockenfels, “ERC: a theory of equity,

reciprocity, and competition,” American Economic Review,vol.

,no.,pp.–,.

[] R. D. Cook and S. Weisberg, “Condence curves in nonlinear

regression,” JournaloftheAmericanStatisticalAssociation,vol.

, no. , pp. –, .

[] D. R. Cox and D. V. Hinkle y, eoretical Statistics, Chapman and

Hall, London, UK, .

[] S. A. Murphy, “Likelihood ratio-based condence intervals in

survival analysis,” JournaloftheAmericanStatisticalAssocia-

tion,vol.,no.,pp.–,.

[] J. K. Goeree and C. A. Holt, “An explanation of anomalous

behavior inmodels of political participation,” American Political

Science Review,vol.,no.,pp.–,.

[] U. Fischbacher and S. Gachter, “Heterogeneous social prefer-

ences and the dynamics of free riding in public goods,” CeDEx

Discussion Paper -, .

[] M. Erlei, “Heterogeneous social preferences,” Journal of Eco-

nomic Behavior and Organization,vol.,no.-,pp.–,

.

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