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Analysis of Public Goods Experiments
Using Dynamic Panel Regression Models*
Richard Ashley
Sheryl Ball
Catherine Eckel
Department of Economics, Virginia Tech
Abstract
Laboratory experiments on the provision of public goods follow each subject over time as he interacts
with a small group of others, deciding in each period how much of an initial allocation to contribute to a
group good that yields a return to all group members. These experiments produce data sets that are rich indynamics, as subjects respond not only to the parameters of the experiment, but also to previous
allocation decisions made by themselves and the other individuals in their group. In most early studies,
the data analysis consisted of an informal inspection of the time plots of data for each treatment, averaged
over all participants. Subsequent studies conducted simple statistical tests of average behavior. There hasbeen very little analysis of the individual data. Data sets from these experiments thus provide an
unexploited opportunity to understand how an individuals behavior evolves over time in response to
feedback about the behavior of others. A better understanding of the dynamics of behavior in these
games has the potential to lead to the design of institutions that improve the efficiency of the private
provision of public goods.
Proper analysis of these data presents several challenges. In addition to modeling the (possibly complex)
interactions among agents, the data are almost always substantially censored from both above and below.We analyze the results of two classic public goods experiments (Isaac and Walker (1988) and Andreoni
(1995)) explicitly as double-censored panel data in which both lagged dependent variables and laggedallocations from the remainder of the group play important roles as explanatory variables. We find that
modeling the dynamic interactions and double censoring extract a richer set of results than previously
seen. In particular, failure to take into account the censoring of the data may substantially underestimate
the magnitude of treatment effects. More importantly, there are economically and statistically significant,
and asymmetric, cross-subject dynamics in these data. Subjects respond much more dramatically whentheir contributions are above average than when they are below average. Thus heterogeneity in initial
contribution levels inherently leads to the deterioration of average contributions over time.
JEL Codes: C920, C230, C240, C700
Key Words: Public Good, Voluntary Contributions Mechanism, Panel Data, CensoringCorrespondence: Catherine C. Eckel, Department of Economics (0316), Virginia Tech, 3016 Pamplin
Hall, Blacksburg, VA 24061. ([email protected])
We thank Jim Andreoni and Jimmy Walker for giving us access to their data. John Pepper provided very helpful
comments on an earlier draft. Any remaining shortcomings are, of course, our own. Eckel was supported by a grant
from the National Science Foundation, SES 0094800. This manuscript is VPI Economics Department Working
Paper E2003-8 and is available for download at http://ashleymac.econ.vt.edu/working_papers/E2003_8.pdf.
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1. Introduction
One of the primary social problems studied by economists is the issue of how to fund the
provision of public goods. What makes the public goods problem so interesting to the dismal
science is the especially dismal theoretical prediction that the level of voluntary contributions
should be zero, resulting in the public good never being produced. The theorists approach to
solving this problem has been to design a mechanism that gives agents the incentive to contribute
enough to produce the socially optimal level of the public good (e.g., Laffont, 2000; Chander,
1993; Hurwicz and Walker, 1990; Groves and Ledyard, 1980).
Experimental economists have instead focused on the Voluntary Contribution
Mechanism (VCM), an institution that mimics the incentive structure of a public goods problem
(see Ledyard, 1995 for a survey). Experimental results typically show that contributions within
the VCM framework are not as low as theorists expect, but nevertheless fall much below socially
optimal levels. While a great deal has been discovered about average behavior in this simple
game across a broad range of variations, there is still much to be understood. For example,
overall contributions nearly always deteriorate over time, but as researchers we do not know why
or by what mechanism this occurs. Behavior is highly heterogeneous across subjects, but we
have been largely unable to explain the nature and sources of the heterogeneity. It is clear that
subjects respond to each others actions, but details of the interactions are not well understood.
In this paper we conduct a more in-depth analysis of VCM data from two classic studies in an
attempt to better understand the underlying behavior.
An understanding of what information agents take into account and how they respond to
information can help economists to design mechanisms that encourage the provision of public
goods. Public goods experiments have already shown the critical importance of structural
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factors such as provision points (Marks and Croson, 1999; Rondeau, Schultze and Poe, 1999),
and social factors such as communication (Ostrom, 1990; Ostrom, et al, 1994; Wilson and Sell,
1997) and knowledge of the identity of ones group members (Andreoni and Petri, 2003). Our
work highlights the importance of information about the behavior of other group members, and
the heterogeneity among agents in their initial contributions, as well as responses to changes in
the environment. Fundraisers seem to be well aware of the importance of information about
others contributions, and actively manipulate information in their fundraising activities.
Manipulating information can significantly affect the level of contributions through the selective
announcement of others contributions (Harbaugh, 1998) or the announcement of an initial large
contribution (Vesterlund, 2003; Andreoni, 2002).
In this paper we suggest that data sets from VCM experiments can best be understood by
approaching them explicitly as doubly-censored panels, in which both lagged dependent
variables and lagged allocations from the subjects group play important roles as explanatory
variables. We estimate these models using data from two classic VCM experiments conducted
by Isaac and Walker (1988) and Andreoni (1995). Our models allow us to make a rich set of
inferences about the intensity and asymmetry of individual responses to the recent behavior of
the remainder of the individuals group. Subjects respond much more dramatically when their
contributions are above average than when they are below average. Thus heterogeneity in initial
contribution levels inherently leads to the deterioration of average contributions over time.
In addition, our models reproduce the qualitative results of both studies as to the impacts of the
experimental treatments they considered, but in a more powerful and more statistically credible
framework. Finally, we show that failure to take into account the substantial censoring of the
data may lead to underestimates of the magnitude of treatment effects.
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2. VCM Games and Data Analysis
Each subject in a VCM experiment is followed over time as he interacts with a small
group of other subjects, deciding on how much of an initial allocation to contribute each period
to a group good that yields a return to all members of the group. The individual allocation
decisions in such settings typically exhibit very strong serial correlation. In addition because
subjects respond to the behavior of the rest of the group individual allocations are significantly
related to previous allocation decisions made by others. Moreover, data collected in these
settings are almost always substantially censored: from below by the fact that an individuals
allocation to the group good is constrained to be non-negative, and from above by the fact that an
individuals allocation to the group good cannot exceed his endowment for that period.
The analysis of these VCM data often consists of little more than an informal inspection
of a few time plots or testing whether the mean or median time-averaged allocations of
individuals differ by experimental treatment. The former approach is simple, but forgoes
statistical analysis altogether, thereby disregarding much of the information in the sample data.
The latter approach discards all of the dynamic information in the sample data and also violates
the assumptions underlying the means/medians tests by ignoring the across-individual
correlations in the data induced by their within-group interactions. Both of the studies whose
data we reanalyze present only limited analysis of their data, using techniques such as these.
(Andreoni, 1995; Issac and Walker, 1988).
Several recent papers take a more sophisticated approach. 1 Clark (2002) uses OLS
regression tocompare contribution rates in a VCM experiment when each groups highest
contributor in a given period is announced and when subjects can reward the highest contributor
1Anderson, et al, (1998) present an alternative approach to the analysis of VCM contributions that focuses on
decision errors.
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in their group. He fails to account for dynamics or for the apparent censoring in the data,
however. Solow and Kirkwood (2002) look for relationships between gender, group identity,
and economic behavior in VCM experiments. They account for the fact that their sample data is
censored but, like Clark (2002), do not consider the dynamics. Neither paper accounts for the
possible importance of individual effects due to the panel structure of the data.
Gunnthorsdottir, Houser, McCabe and Ameden (2001) estimate dynamic decision-rule
regressions using VCM data. In a fashion that they note is somewhat arbitrary, they first bisect
their sample into cooperator subjects (who contribute more than 30% of their initial allocation
in period one) and free rider subjects (who contribute less than this in the first period). They
then estimate two separate dynamic regression models to explain the sample variation in the
individual contributions in each period one regression for the cooperator data and one for the
free rider data. They use Tobit regression to account for the substantial double censoring in their
data and attempt to model both the serial and across-subject dynamics in the data using functions
of lagged dependent variables. They then informally compare the sizes and t-ratios of the
estimated coefficients in the two regressions. Gunnthorsdottir, et al, (2001) do not, however,
effectively consider the panel nature of their data in that they ignore the fact that regardless of
whether the ith subject is a cooperator or a free rider this subjects group-good allocation in
period t is substantially determined by a time-invariant person-specific (fixed) effect.
Consequently, the error term in their regression equations is correlated across time for each
person, which violates the assumptions of the Tobit regression framework. Our approach
substantially alleviates this problem.
We chose data sets from two well-known studies for our analysis (Isaac and Walker,
1988; and Andreoni, 1995) because of their importance in the area of VCM studies, and because
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the data sets are readily available. Both experiments were computerized. Subjects were given an
endowment (E) each period, which could be allocated between a private account and a group
account. Contributions to the group account paid to each member of the group the marginal per
capita return (MPCR) times the total of the groups contributions.
Isaac and Walker (IW) test the effect on contributions of changing group size from 4 to
10, and of changing the MPCR from .3 to .75, using a 2x2 experimental design. Within a
session, stable groups of a given size experience ten periods of one value for MPCR, followed by
ten periods of the other value; the order of the MPCR values is blocked. Thus their data consist
of 20 decision periods for each of 84 subjects. Andreoni (1995) tests the effect of positive and
negative framing on contributions, holding group size and MPCR constant. In the positive
frame, the level of the public good is determined by total contributions to the group account. In
the negative frame, subjects withdraw amounts from the group account and transfer it to their
private account. The amount remaining after withdrawals determines the level of the public
good. In Andreonis experiment, subjects are rematched each period into new groups.2
His is a
between-subjects design, so his data consist of ten periods for 80 subjects, 40 in each treatment.
3. A Panel Regression Model for VCM Contributions
We model tiC, , subject is contribution to the group good in period t, expressed as a
fraction of the subjects total period t endowment. The observed contribution is equal to Ci,t*
, a
latent variable that could be characterized as desired contribution, if and only if 10 *, tiC .
Thus,
tiC, = 1 ifCi,t*
> 1
2Some inconsistencies in Andreonis rematching protocol were found in our analysis, but none significantly
affected the statistical results.
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tiC, = Ci,t*
if 10 *, tiC
Ci,t = 0 ifCi,t*
< 0
and Ci,t
*is determined by:
ti
k
i
m
j
i
tiititititi DCCC ,1 1
,2,21,1
*
, +++++= = =
],0[~2
, NIIDti (1)
where the Xt1, Xt2,Xtn include both time invariant variables such as treatment effects, and
dynamic variables such as the deviation of subject is contribution from that of the remainder of
the group in round t-1. itD is a dummy variable specific to subject i, and subsumes all time-
invariant characteristics of the subject. This model allows for both the individual-specific fixed
effects (1, 2,...m) and the strong serial correlation that we expect (and find) in the IW and
Andreoni Ci,t data. This is a standard fixed-effects dynamic panel data regression model, but (so
far as we know) such models have never been considered before with double censoring. Still, in
principle, this model is estimable using the usual maximum likelihood methods for Tobit Type I
models.3
Because of the censoring, some care must be taking in interpreting the estimated
coefficients in this model. To see this consider a simpler model which retains the double
censoring but suppresses all but one explanatory variable.
ci*
= + xi
+ i i~ NIID(0,2) (2)
c i = c i*
for c i*
(0,E)
= 0 for c i* 0
3See Maddala (1983, pp. 160-1 and 186) for the likelihood function, and Amemiya (1973) for a theoretical analysis.
Further complications ensue if lagged values of the latent variable (Ci,t1*
and Ci,t2*
) are used instead of lagged
values of the observed variable. Arellano, et al. (1999) consider a random-effects formulation for this case with one-
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= E for ci* E
This model implies that the ci* are all independently distributed N[+ x
i,2]; thus
(ci* xi) / is a unit normal and ]|*[ ixicE is just + xi .
Clearly, E[ci* | x ] /x i is simply and quantifies the sensitivity of subject is desired
contribution to changes in xi . It is this sensitivity, of course, which is ordinarily the focus of
economic analysis.
Note, however, that the sensitivity of ci, subject is observed contribution to change in
xi, is not equal to the parameter , because observed contributions equal desired contributions
only when ci* is in the interval (0,E). Outside of this interval the observed contribution is a
constant either 0 or E, depending on whether ci* is less than zero or greater than E.
Consequently, the magnitude of the apparent dependence of subject is contribution on the
explanatory variable is (on average) diminished whenever + xi either falls close to (or lies
below) zero or approaches (or exceeds) E. In particular, since (ci* x
i) /~ N(0,1) , this
apparent sensitivity is
]|),0([*/]|[ *i
xEcyprobabiliti
xi
xi
cE i =
=
/)(
/)(
)(
i
i
xE
x
dzzf (3)
where f(z) is the unit normal density function. This probability peaks for xi such that
sided censoring and including only a single lag in the dependent variable; Honore (1993) considers a non-parametric
approach to this special case.
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+ xi = E/ 2 and diminishes to zero as |+ xi E/ 2 | increases. Intuitively, subject is observed
contribution becomes increasingly insensitive to xi as xi becomes sufficiently extreme that
subject i is likely to be either not contributing at all or contributing his entire endowment. This
apparent sensitivity is precisely what is needed so as to analyze the sensitivity of actual average
contributions to changes in explanatory variables - e.g. this sensitivity might be relevant in
predicting the cost of running a particular experiment - but it is itself which quantifies how the
subjects desired behavior depends on x.
Note also that using OLS to estimate the parameters and in the model
iibxa
ic ++= (4)
yields an estimate (bols) which is necessarily an inconsistent estimator ofE[ci
| x i]/x i. The
inconsistency follows directly from the fact that, as noted above, E[ci
| x i]/x i is a function of
xi , whereas plim )( olsb is a constant. In general bols is also an inconsistent estimator of , since it
ignores the distinction between ci
and ci*. The seriousness of these inconsistencies is most
sensibly gauged by comparing bols directly to the maximum likelihood estimator of in the
model given by equation (2); this is the comparison we make in our analysis below.
4. Descriptions of the data sets and variables:
The two data sets we examine are from experiments that were conducted to test different
hypotheses. IW examines the effect of group size (4 or 10) and MPCR (.3 and .75) on an
individuals contributions to the public good. As explained above, each session includes ten
periods with one value of MPCR followed by ten periods using the other value of MPCR. Group
size is varied across sessions. Explanatory variables for this data set include contributions, the
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treatment variables, and measures of how a subjects contribution in the previous round differed
from the average contributions for their group. MPCRhi is a dummy variable equal to 1 for the
high-mpcr sessions. Group size takes on the values 4 or 10 corresponding to the two group sizes.
The dummy variable First 10 periods was set to one in the first set of 10 periods and to zero in
the second set. For subject i, the lagged value of the deviation of his own contribution ( 1, tiC )
from the value of the average contribution of all other members of the group in that round is
denoted Deviation from Group(+) when this value is positive and is otherwise zero. Deviation
from Group(-) is similarly defined for the observations for which this value is negative and is
otherwise zero. These variables thus allow for a possible asymmetry in the responses. The
variable Ci,1 is the subjects initial contribution; this variable is used in Section 7.
Recall that the purpose of the Andreoni study is to test for the effect of positive or
negative framing on contributions to a pubic good. Positive framing is the standard VCM,
whereas subjects withdraw contributions from a common pool in the negative frame treatment.
Variables for this data set include the treatment variable Positive Frame, as well as the variables
defined above. The frame is varied across sessions, which last for ten periods.
Table 1 contains descriptive statistics on these variables for both data sets. Considering
how different these two experiments are, the descriptive statistics are remarkably similar.
Average contributions are 36.0 percent of the endowment in the IW data, and 24.9 percent in the
Andreoni data. The lower average contributions in the Andreoni data are probably are due to the
negative frame treatment, which reduces overall contributions. Some of the difference may also
be due to MCPR, which IW show affects contribution levels: IW use an MCPR of .3 and .75
while Andreoni uses a value of .5 for the MCPR. If we look at the average value the change in
contributions in each period, ci,t , we observe that it is identical in both data sets at -0.03,
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indicating that contributions typically fall by three percentage points per round. The deviations
from group variables also are very similar in the two data sets - 0.145 compared with 0.124 -
indicating a similar range of contributions for both studies. Finally, the initial contribution levels,
Ci,1, are higher, overall, for the IW data, again probably due to the lower contributions in
Andreonis negative frame treatment.
5. Estimation Results: Introducing Censoring and Panel Structure to the Basic Model
Table 2 contains OLS and censored panel estimates for what we term the basic model.
This model, estimated using OLS, is employed in most studies that go beyond graphical
comparisons or comparisons of central tendency (mean or median outcome comparisons) to
analyze individual data. The dependent variable for both data sets is tiC, the level of
contributions in round t expressed as a fraction of endowment. Independent variables include the
round number (a crude attempt capture any time trend in the data) and relevant treatment
variables: MPCRhi and Group size and their interaction in IW, and Positive Frame in Andreoni.
Model 2a is estimated using OLS and adjusts neither for two-sided censoring (at 0 or 100
percent donations) nor for the panel nature of the data. According to these estimates, in both
data sets contributions decrease over time, falling about 2.7 - 2.8 percentage points per round.
Both studies show statistically significant main treatment effects. In IW, a higher MPCR is
estimated to increase contributions by 51 percentage points. Larger group size has a positive
effect when MPCR is low moving from a group size of 4 to a group size of 10 increases
contributions by about 29 percentage points but this effect is offset by the negative coefficient
on the interaction term for the high MPCR case. This result explains why IW can only conclude
there appears to be no support for a pure numbers argument relating increases in group size to
increases in free riding behavior (p. 196). In Andreoni, the positive frame variable carries a
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positive, significant coefficient: the positive frame increases contributions by 17.4 percentage
points.
Model 2b for both data sets uses Tobit estimation to account for the two-sided censoring
in the data, and allows for the panel structure of the data sets by explicitly estimating individual-
specific fixed effects.4
A comparison of 2a and 2b allows us to examine the impact on the
estimates of these two modeling changes. First, note that censoring and fixed effects are clearly
present in both of these data sets: 49 percent of the IW observations are censored, as are 56
percent of the Andreoni observations. In addition, 52 and 67 percent of the fixed-effect dummy
variable coefficients in each of these data sets are significantly different from zero at the 5
percent level, and the joint null hypothesis that all of the fixed effects coefficients are zero is
rejected for each data set with p-value less than .001.5
Individual-specific effects and censoring are clearly present in both data sets. But is
failure to account for these features of the data consequential? We note that the estimates of the
coefficient on the round number variable are nearly twice as large (5-6 percentage points per
round versus 3 percentage points per round in the OLS estimates) once the fixed effects and
censoring are treated appropriately this is a difference of about 4 to 6 estimated standard errors
in each case. In the IW data the coefficient on the MPCR variable is nearly twice as large as in
the OLS estimates; again this is a difference of about 5 estimated standard errors. Furthermore,
the coefficient estimate on the group size variable is no longer significant. Turning to the
Andreoni data, the direction and significance of the treatment effect are unchanged, but as in the
4It should also be noted that fixed effect dummy variable coefficients are not identified for subjects whose
contribution is censored in every round, and are not, in practice, estimable for subjects whose contribution is
censored in every round but one. This was the case for a notable fraction of the subjects in the Andreoni study
(28/80) but only 2/84 for the IW study.5
The individual fixed effect coefficient estimates are suppressed in the table for brevity. The fixed effect dummy
for the last individual in each sample is omitted so that these estimates can be interpreted as the difference in the
intercept relative to this individual.
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IW data, failure to account for the censoring and panel structure substantially reduces the
estimated magnitude of the treatment effect: the coefficient on the positive frame variable is
.174.022 in the OLS model compared with .435.087 in the Tobit model. These results are
consistent with what one might expect from the analysis at the end of Section 3 and, in fact,
primarily stem from this source rather than from the inclusion of the fixed effect dummy
variables in the model.
6. Estimation Results: Introducing Lags and Group Feedback
Adjusting for panel structure and censoring is clearly appropriate for these data sets, but
the estimated models in Table 2 fail to allow for the possibility of correlation in a subjects
decisions over time. In theory the random rematching of the subjects should render the outcomes
independent in each round, since subjects should ignore information about one group in making
a decision with another. A cursory examination of the sample data indicates otherwise. To
allow for these serial correlations, we model the dynamics by including lagged values of the
observed contribution. Results incorporating these dynamics are reported in Tables 3 and 4.
Note that the number of observations is decreased to account for the calculation of the lagged
values. In both tables, results are again reported for both OLS and censored panel regressions.6
In the OLS models, the coefficients on the lagged values are positive and significant,
though their sum is less than one. (If the values sum to one, the model has a unit root.) Clearly
there is substantial correlation in contributions from one round to the next. By comparison with
Model 2a for both data sets in Table 2, we see that adjusting for autocorrelation improves the fit
of the model; the treatment effects now appear smaller in magnitude, however. Turning to the
censored panel estimation results, the biases in the estimates of the treatment effects from
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neglecting the censoring and panel effects is evident: the coefficients on all treatment variables
are larger in the censored panel models. The fixed effects are jointly significant in both data sets
as well.7
Recall that the IW subjects are in stable groups, while the Andreoni subjects are
randomly rematched. The dependence between rounds should be lower in the rematched data
(Table 4). However, that is not the case. The coefficients on the lagged contribution variables
are not substantially different across the two studies. Evidently, random rematching is not as
effective as one might have expected in inducing independence of observations between rounds.
Tables 5 and 6 report models that examine the dynamics in more detail. An additional
factor that is ignored by the basic model is the effect of recent history on subjects decisions.
Each subject experiences a different history, since the identities of the subjects constituting the
remainder of his group in the previous round are unique to him. We test the effect of this intra-
group interaction by incorporating the two additional variables, Deviation from Group (+) and
Deviation from Group (-) defined in Section 4. These estimates are reported for the censored
panel regression model in Tables 5 and 6. (OLS estimates are not reported; the biases are similar
to those reported above.)
Models 5a and 6a allow the subjects response to differ, depending whether his or her
contribution in the previous round was above or below the group average. A positive coefficient
on these variables indicates adjustment away from the group average, while a negative
coefficient means that the subject is adjusting his contribution toward the group average. In both
data sets, if a subjects contribution is above the group average, the estimated coefficients
6 Note that the variable Round Number is not included in the reported estimates. While retaining the Round Number variable inthe models improves the fit slightly, it leaves the coefficients unchanged in magnitude and significance, except for the firstvariable in the IW data, which changes sign.7 Adjusting for fixed effects alone increases the treatment effect slightly (to .078). Adding censoring alone raises the coefficientto .170. These alternative specifications are available on request from the authors.
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indicate that he will reduce his contribution significantly, by about 60 percent of the excess.
However, if his contribution is below the group average, there is no corresponding increase; the
insignificance of this coefficient for both studies indicates no response. The null hypothesis that
these two coefficients are equal is easily rejected: for the IW data, 2
(1) = 17.0, p < .001; and for
the Andreoni data, 2
(1) = 6.8, p = .009.8
As in the previous models, we again observe positive
and significant coefficients on the lagged values, and significant treatment effects.
Models 5b and 6b extend the analysis by allowing the response to group feedback to vary
by treatment. Here we interact the Deviation from Group variables with the treatment variables.
In the IW data, the estimated treatment effects appear relatively stable across the two
specifications. None of the interaction effects is significant on its own, but a LR test rejects the
null hypothesis that the effect of the interactions is zero (2
(4) = 11.2, p = .024).9
In the Andreoni data, for both frames the coefficient on Deviation from Group (+) is
negative, though the magnitude of the response is substantially greater in the negative frame.
Thus subjects in the negative frame are quicker to adjust their contributions downward, toward
the group average. If the subjects contribution last period was below average, the negative
coefficient on Negative Frame X Deviation from Group(-) indicates that the negative frame
subjects are still more responsive. They adjust their contributions toward the group average more
than the positive frame subjects, who do not respond significantly at all in this case. A
likelihood ratio test for the null hypotheses that coefficients are equal across treatments can be
rejected (2
(2) = 12.0, p = .002). Thus we can conclude that the response to feedback differs by
treatment; Andreonis framing treatments affect contributions in part by affecting the subjects
8In both cases the restricted model (not shown) yields a negative coefficient estimate that is smaller in magnitude
but still significantly different from zero at p
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response to what others are doing.
7. Does the initial contribution level signal type?
Each of the fixed effect models above shows that there is heterogeneity among subjects.
The evidence adduced above that subjects react to feedback about their groups contribution
level suggests that a stable group consisting of relatively high initial contributors might lead to
sustainable high contribution levels. GHMA test this hypothesis by sorting subjects into
groups of others with similar contribution levels based on their initial first-period contributions.
The validity of this procedure depends on whether a subjects initial contribution level contains a
unique signal of their type that is, their propensity to contribute in subsequent periods
which is distinct from the effects of the various treatments, such as MPCR, group size, or
positive/negative framing. Tables 7 and 8 contain estimates for models that include Ci,1, the first
period contribution level, as an explanatory variable. Note that the coefficient on Ci,1 is positive
and significant for both the IW and the Andreoni data.10
Using Ci,1 as an indicator of type is problematic, however, since the level of initial
contributions is no doubt affected by the treatments. If so, then Ci,1 andCi,t for t>1 are jointly
determined by the treatment effects, in which case the coefficient estimates on C i,1 become
difficult to interpret as quantifying the effect of subject type on contributions.
Ideally, one would like to have a measure of type that is independent of treatment, as in
Park (2000). Failing that, however, the hypothesis that Ci,1 reflects type (and that type matters)
can be tested by interacting Ci,1 with the treatments and observing whether or not the impact of
Ci,1 on contributions is significant (and in the same direction) regardless of the treatment.
10Ci,t is in part a substitute for individual effects and, as a consequence, we found that that the MLE routine did not
always converge for models including both the fixed effects and explanatory variables including Ci,1; where
necessary, results for models omitting the fixed effect dummy variables are quoted in Tables 7 and 8.
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Models 7b and 7c for the IW data and Model 8b for the Andreoni data address this issue.
The results in Model 7b indicate that the coefficient on Ci,1 is significantly different from zero
and essentially identical regardless of whether the MPCR is .75 (MCPRhi=1) or .30
(MCPRlow=1). Similarly, as shown in Model 7c, since the estimated coefficient on Ci,1 X
GroupSize is not significantly different from zero, the coefficient on Ci,1is not significantly
different for the subjects in groups of size 4 than for subjects in groups of size 10.
Turning to the results on the Andreoni data in Table 8, the estimated coefficient on Ci,1is
clearly positive and significant in Model 8a, but when the coefficient is separately estimated over
the data for the positive and negative framing treatments using Ci,1 X Positive Frame and Ci,1 X
Negative Frame, both coefficients become individually insignificant. They are also not
significantly different from each other using a likelihood ratio test.11
Thus, it appears that the estimated coefficient on Ci,1 is not significantly sensitive to the
treatment for either of these two data sets. Consequently, it seems reasonable to interpret the
positive estimated coefficient on Ci,1 in Model 5 as quantifying the effect of subject type on
contributions.
9. Conclusions
For the most part, our analysis confirms the direction of the treatment effects found by
the original investigators in both data sets. We find (as did IW) that the MPCR is an important
determinant of behavior in the VCM. Group size is a significant determinant of the allocation
level when MPCR is low, but not when it is high. IW could only conclude that the effect is
weak and ambiguous. We also are able to determine that the impact of MPCR on allocations
depends inversely on group size. Similarly, our results support Andreonis conclusion that
11The value of the log likelihood function for Model 8a without the fixed effect dummy variables is 375.53, so the
likelihood ratio test statistic is .32~2(1), p=.572.
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subjects contribute more when the public good is positively framed, but we also are able to show
that the frame affects the speed with which subjects adjust their contributions to match those of
the group.
Treating the data as a fixed-effects panel had limited effect on the qualitative results.
Although the fixed-effect dummy variables in the allocation regression are highly significant as a
group, the key economic results are not sensitive to them. Indeed, even omitting these variables
altogether does not substantially alter the conclusions, at least in these two data sets. Of course,
one cannot know that is the case without (appropriately) including them. However, this
robustness result reduces our concern that, because their number increases as one expand the
number of subjects, such coefficients cannot be consistently estimated.
In contrast, we find that failure to appropriately deal with the censored nature of VCM
data is quite consequential. While the direction of the treatment effect is rarely affected, its
magnitude is sometimes substantially different when censoring is properly specified. It seems
clear from our results that censoring is not an aspect of the data that can be safely ignored.
More importantly, we find economically and statistically significant, asymmetric cross-
subject dynamics in the data. When a subjects allocation is above the average of the other
members of the group, he reduces his contribution toward the group average; when he discovers
his contribution is below average, he does not adjust toward the others. In other words, when a
subject finds that he/she has contributed more than the rest of the group, that subject responds!
This pattern of behavior is consistent with recent work on inequality aversion (Fehr and Schmidt,
1999), which asserts that people care about inequality, but that they care more when their income
is below than when it is above others income. Statistically, is not appropriate to adopt the past
practice of analyzing data using methods that assume that the allocations are independently
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distributed across subjects. Allocations are correlated over time for a given individual, and
across individuals.
Comparing the two data sets, we also conclude that random rematching has virtually no
effect on the extent to which subjects respond to information about others contributions.
Random rematching is commonly used in economics experiments, and the practice is used to
justify the assumption of independence of decisions over time. Our results show that this is
clearly not the case. In our data, random rematching does nothing to alter the dependence of
decisions in a given period on feedback from the decisions in previous rounds. We also find that
random rematching fails to eliminate serial correlation in individual contribution decisions. Thus
dynamic regression analysis is imperative for valid statistical analysis of data generated by VCM
experiments.
Finally, we examine the validity of using initial contributions as an indicator of subject
type. We find that initial contributions do enter our models with a significant positive
coefficient, which does not seem to be sensitive to the treatment regime. We conclude that, for
these data sets at least, initial contribution may be a useful measure of subject type.
Our results suggest that the institutional design for the private provision of public goods
could be improved by careful attention to the types of the participants, and the information that
they receive. Selection of types into groups may result in at least some groups that attain Pareto
efficient outcomes. In addition, information feedback that reassures contributors can keep
contributions high.
This work builds on the points made by Roth (2002) regarding the use of experimental
data in engineering market and nonmarket institutions. Game theory is extremely helpful in
understanding the incentives and examining the equilibria inherent in any given situation.
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However, experiments play a unique role in understanding how an individuals preferences and
attitudes interact with the incentives in a game to produce actual behavior. Our results show that
much more is going on in these games than attention limited soley to the incentive structure of
the monetary payoffs would indicate. Subjects care about their own payoffs, but not so much
that they will allow free riding to keep them from exploiting the gains to cooperation. When
such gains are possible, subjects are willing to cooperate, but only if they observe others
cooperating at least as much as they do. Institutional design that takes these preferences into
consideration are likely to be much more effective at attaining appropriate levels of public goods
production.
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Table 1
Descriptive Statistics: Isaac-Walker Data
Variable N Mean Std.
Dev.
Min Max
Ci,,t Fraction of endowment contributed
to public good
1680 0.360 0.370 0 1
ci,t Change in contribution to public
good: Ci, t-1 - Ci,,t
1512 -0.030 0.319 -1 1
Group size Size of group: (4 or 10) 1680 8.286 2.711 4 10
MPCRhi Dummy variable = 1 for
High MPCR treatment
1680 0.494 0.500 0 1
Deviation
from Group(+)i,,t-1
Ci,,t-1 less (average contribution by
others in group)t-1, if >0
1512 0.145 0.216 0 0.907
Deviation
from Group
(-)i,,t-1
Ci,,t-1 less (average contribution by
others in group)t-1, if 0
800 0.124 0.235 0 100
Deviation from Group (-)i,t-1 Ci,t-1 less (average contribution by
others in group)t-1, if
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Table 2: Basic Models:
Dependent Variable, Ci,,t, Contribution Levels: Fraction of Endowment Contributed to Public Good
(Standard Errors in Parentheses)
Isaac-Walker Data: OLS and Censored Panel Regressions
Model 2a: OLS Model 2b: Censored Panel Regression
Variable Coefficient(standard
error)
p-value2-tailed test Coefficient(standard error) p-value2-tailed test
Round number -0.028
(0.003)
p < .001 -0.054
(0.004)
p < .001
First 10 rounds -0.296
(0.033)
p < .001 -0.536
(0.050)
p < .001
MPCR = HI 0.510
(0.053)
p < .001 0.957
(0.084)
p < .001
Group Size 0.029
(0.004)
p < .001 -0.017
(0.023)
p = .459
MPCRhi X Group Size -0.035
(0.006)
p < .001 -0.067
(0.009)
p < .001
Constant 0.460
(0.059)
p < .001 1.026
(0.203)
p < .001
Fixed Effects:
smallest p-value p < .001
% p-values < .05 52
p-value for Ho: all 0 2(51) = 741.6, p < .001Fraction censored at 0% .344
Fraction censored at 100% .142
N 1680 1680
Adjusted R2
0.158
Pseudo R2
0.296
LLF -1109.6
Andreoni Data: OLS and Censored Panel Regression
Model 2a: OLS Model 2b: Censored Panel Regression
Variable Coefficient
(standard error)
p-value
2-tailed test
Coefficient
(standard error)
p-value
2-tailed test
Round number -0.027
(0.004)
p < .001 -0.056
(0.008)
p < .001
Positive Frame
(=1 for this treatment)
0.174
(0.022)
p < .001 0.435
(0.087)
p < .001
Constant 0.313
(0.026)
p < .001 -0.217
(0.071)
p = .002
Fixed effects:
smallest p-value p < .001
% p-values < .05 67p-value for Ho: all 0 2(79) = 183, p < .001
Fraction censored at 0% .489
Fraction censored at 100% .072
N 800 800
Adjusted R2
0.125
Pseudo R2
0.213
LLF -547.6
*This test is compared to a Tobit model omitting the fixed effect dummy variable.
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Table 3: Isaac-Walker Data: Introducing Lags: OLS and Censored Panel RegressionDependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good
(Standard Errors in Parentheses)
Model 3a: OLS Model 3b: Censored
Panel Regression
Variable Coefficient
(standard
error)
p-value
2-tailed test
Coefficient
(standard error)
p-value
2-tailed test
Ci,t-1 0.441
(0.027)
p < .001 0.511
(0.052)
p < .001
Ci,t-2 0.207
(0.026)
p < .001 0.161
(0.052)
p = .002
First 10 rounds 0.013
(0.015)
p = .372 0.045
(0.028)
p = .100
MPCR = HI 0.190
(0.050)
p < .001 0.657
(0.100)
p < .001
Group Size 0.010
(0.004)
p = .009 -0.048
(0.027)
p = .077
MPCRhi X Group Size -0.012
(0.006)
p = .031 -0.047
(0.011)
p < .001
Constant -0.048
(0.035)
p = .171 0.233
(0.231)
p = .314
Fixed effects:
smallest p-value p < .001
% p-values < .05 39.4
p-value for Ho: all 0
2(78) = 197.2
p < .001
Fraction censored at 0% .362
Fraction censored at 100% .134
N 1344 1312
Adjusted R2
0.273 -
Pseudo R2
- 0.329
LLF - -842.8
*This test is compared to a Tobit model omitting the fixed effect dummy variable.
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Table 4: Andreoni Data: Introducing Lags: OLS and Censored Panel RegressionDependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good
(Standard Errors in Parentheses)
Model 4a: OLS Model 4b: Censored
Panel Regression
Variable Coefficient
(standard error)
p-value
2-tailed test
Coefficient
(standard error)
p-value
2-tailed test
Ci,t-1 0.361
(0.035)
p < .001 0.529
(0.075)
p < .001
Ci,t-2 0.251
(0.004)
p < .001 0.449
(0.074)
p < .001
Positive Frame
(=1 for this treatment)
0.057
(0.021)
p = .005 0.204
(0.087)
p = .028
Constant 0.031
(0.026)
p = .043 -0.588
(0.066)
p < .001
Fixed effects:smallest p-value p < .001
% p-values < .05 39.2
p-value for Ho: all 0
2(51) = 75.2
p = .015
Fraction censored at 0% .512
Fraction censored at 100% .055
N 640 640
Adjusted R2
0.371 -
Pseudo R2
0.34
LLF -348.7
*This test is compared to a Tobit model omitting the fixed effect dummy variable.
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Table 5:Isaac-Waker Data: Introducing Group Feedback
Censored Panel RegressionDependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good
(Standard Errors in Parentheses)
Model 5a: Group Feedback Model 5b: Group Feedback
by TreatmentVariable Coefficient
(standard error)
p-value
2-tailed test
Coefficient
(standard error)
p-value
2-tailed test
Ci,t-1 0.848
(0.110)
p < .001 0.890
(0.112)
p < .001
Ci,t-2 0.150
(0.051)
p = .003 0.137
(0.050)
p = .007
First 10 rounds 0.064
(0.027)
p = .091 0.047
(0.027)
p = .083
MPCR = HI 0.569
(0.112)
p < .001 0.436
(0.122)
p < .001
Group Size -0.011
(0.004)p = .009 -0.030
(0.027)p = .276
MPCRhi X Group Size -0.041(0.011)
p < .001 -0.036(0.012)
p = .004
Deviation from Group (+) -0.649
(0.132)
p < .001 -
Deviation from Group (-) 0.016
(.132)
p = .902
MPCRhi X
Deviation from Group (+)
-0.059
(0.284)
p = .835
MPCRlow X
Deviation from Group (+)
-0.447
(0.317)
p = .158
MPCRhi X
Deviation from Group (-)
-0.208
(0.330)
p = .528
MPCRlow X
Deviation from Group (-)
-0.079
(0.373)
p = .883
Group Size X
Deviation from Group (+)
-0.052
(0.032)
p = .098
Group Size X
Deviation from Group (-)
0.014
(0.038)
p = .693
Constant -0.103
(0.203)
p = .614 -0.119
(0.233)
p = .608
Fixed effects:
smallest p-valuep < .001 p < .001
% p-values < .0534.2 27.4
p-value for Ho: all 0
2
(79) = 199.0p < .001
2
(51) = 79.4p = .007
Fraction censored at 0% .362 .362
Fraction censored at 100% .134 .134
N 1312 1312
Pseudo R2
0.339 0.344
LLF -829.8 -824.2
*This test is compared to a Tobit model omitting the fixed effect dummy variable.
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Table 6: Andreoni Data: Introducing Group Feedback
Censored Panel RegressionDependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good
(Standard Errors in Parentheses)
Model 6a: Group Feedback Model 6b: Group Feedback
by Treatment
Variable Coefficient
(standard error)
p-value
2-tailed test
Coefficient
(standard error)
p-value
2-tailed test
Ci,t-1 0.939
(0.164)
p < .001 0.906
(0.162)
p < .001
Ci,t-2 0.428
(0.074)
p < .001 0.372
(0.074)
p < .001
Positive Frame
(=1 for this treatment)
0.217
(0.094)
p = .022 0.300
(0.126)
p = .017
Deviation from Group (+) -0.607
(0.189)
p = .001
Deviation from Group (-) 0.045
(.178)
p = .798
Positive Frame X
Deviation from Group (+)
-.379
(0.214)
p = .078
Negative Frame X
Deviation from Group (+)
-.721
(0.213)
p = .001
Positive Frame X
Deviation from Group (-)
.203
(0.212)
p = .340
Negative Frame X
Deviation from Group (-)
-.508
(0.298)
p = .089
Constant -0.570(0.072) p < .001 -0.638(0.085) p < .001
Fixed effects:
smallest p-value p < .001 p < .001
% p-values < .05 29.4 27.4
p-value for Ho: all 0
2(51) = 70.0
p = .040
2
(51) = 79.4
p = .007
Fraction censored at 0% .512 .512
Fraction censored at 100% .055 .055
N 640 640Pseudo R
20.347 0.358
LLF -343.5 -337.5
*This test is compared to a Tobit model omitting the fixed effect dummy variable.
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Table 7 Isaac-Walker Data: Do initial contributions indicate type?
Dependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good
(Standard Errors in Parentheses)
Model 7a: Ci,1=type
Censored Regressiona
Model 7b: Ci,1=type
Censored Regressiona
Interactions: Type X
MPCRhi
Model 7c: Ci,1=type
Censored Regressiona
Interactions: Type X
Group Size
Variable Coefficient
(standard
error)
p-value
2-tailed
test
Coefficient
(standard
error)
p-value
2-tailed
test
Coefficient
(standard
error)
p-value
2-tailed
test
Ci,t-1 0.924
(0.102)
p < .001 0.924
(0.102)
p < .001 0.930
(.103)
p < .001
Ci,t-2 0.318
(0.050)
p < .001 0.317
(0.050)
p < .001 .317
(.050)
p < .001
First 10 rounds 0.048
(0.028)
p = .090 0.046
(0.028)
p = .105 .048
(.028)
p = .088
MPCR = HI 0.343
(0.119)
p = .004 0.323
(0.124)
p = .009 .337
(.119)
p = .005
Group Size 0.029
(0.010)
p = .003 0.029
(0.010)
p = .003 .025
(.012)
p = .035
MPCRhi X Group Size -0.031
(0.012)
p = .014 -0.031
(0.012)
p = .014 -.030
(.012)
p = .016
MPCRhi X
Deviation from Group (+)
-0.157
(0.280)
p = .573 -0.147
(0.280)
p = .573 -.152
(.280)
p = .585
MPCRlow X
Deviation from Group (+)
-0.580
(0.317)
p = .067 -0.557
(0.319)
p = .067 -.572
(.317)
p = .072
MPCRhi X
Deviation from Group (-)
0.162
(0.287)
p = .571 0.158
(0.287)
p = .571 .163
(.287)
p = .569
MPCRlow X
Deviation from Group (-)
0.272
(0.342)
p = .428 0.266
(0.342)
p = .428 .273
(.342)
p = .425
Group Size X
Deviation from Group (+)
-0.020
(0.031)
p = .513 -0.021
(0.031)
p = .513 -.022
(.031)
p = .483
Group Size X
Deviation from Group (-)
-0.002
(0.033)
p = .961 -0.002
(0.033)
p = .961 -.002
(.033)
p = .944
Ci,1 0.161
(0.042)
p < .001 .095
(.136)
p = .483
Ci,1 X MPCRhi 0.185
(0.058)
p = .001
Ci,1 X MPCRlow 0.139
(0.057)
p = .016
Ci,1 X Group Size .008
(.015)
p = .610
Constant -0.523
(0.081)
p < .001 -0.513
(0.083)
p < .001 -.494
(.099)
p < .001
Fraction censored at 0% .378 .378 .378Fraction censored at 100% .131 .131 .131
N 1344 1344 1344
Pseudo R2
0.289 0.289 0.289
LLF -916.1 -915.9 -915.9
aNote that the censored panel regression failed to converge when C1 was included.
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Table 8 Andreoni Data: Do initial contributions indicate type?
Dependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good
(Standard Errors in Parentheses)
Model 8a: C1=type
Censored Panel Regression
Model 8b: C1=type
Censored Panel Regressiona
Interactions: Type X Frame X
Group Feedback
Variable Coefficient
(standard
error)
p-value
2-tailed test
Coefficient
(standard
error)
p-value
2-tailed test
Ci,t-1 0.892
(0.160)
p < .001 .912
(.154)
p < .001
Ci,t-2 0.291
(0.074)
p < .001 .460
(.074)
p < .001
Positive Frame
(=1 for this treatment)
0.264
(0.129)
p = .042 .160
(.088)
p = .068
Positive Frame X
Deviation from Group (+)
-.503
(0.215)
p = .019 -.604
(.196)
p = .002
Negative Frame X
Deviation from Group (+)
-.683
(0.211)
p = .001 -.454
(.195)
p = .020
Positive Frame X
Deviation from Group (-)
0.188
(0.214)
p = .382 .397
(.194)
p = .041
Negative Frame X
Deviation from Group (-)
-.601
(0.297)
p = .044 -.048
(.277)
p = .863
C1 0.474
(0.109)
p < .001
Ci,1 X Positive Frame .140
(.078)
p = .072
Ci,1 X Negative Frame .074
(.095)
p = .437
Constant -0.760
(0.093)
p < .001 -.379
(.066)
p < .001
Fixed effects:
smallest p-valuep < .001
% p-values < .0527.4
p-value for Ho: all 0 2(51) = 94.6
p < .001
Fraction censored at 0% .512 .512
Fraction censored at 100% .055 .055
N 640 640
Pseudo R2
0.376 0.422
LLF -328.2 -303.5
*This test is compared to a Tobit model omitting the fixed effect dummy variable.
a
Note that the censored panel regression failed to converge when Ci,1 was replace by Ci,1 X Positive Frame and Ci,1 X Negative Frame.
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