Social Design Engineering Series

SDES-2017-25

Instability in the Voluntary Contribution Mechanism witha Quasi-linear Payoff Function: An Experimental Analysis

Jun Feng
Graduate School of Engineering, Kochi University of Technology

Tatsuyoshi Saijo
Research Institute for Humanity and Nature
Research Institute for Future Design, Kochi University of Technology

Junyi Shen
Research Institute for Economics and Business Administration, Kobe University

Xiangdong Qin
Antai College of Economics and Management,Shanghai Jiaotong University

7th December, 2017

School of Economics and Management
Research Center for Future Design
Kochi University of Technology
KUT-SDE working papers are preliminary research documents published by the School of Economics and Management jointly with the ResearchCenter for Social Design Engineering at Kochi University of Technology. To facilitate prompt distribution, they have not been formally reviewed
and edited. They are circulated in order to stimulate discussion and critical comment and may be revised. The views and interpretations expressedin these papers are those of the author(s). It is expected that most working papers will be published in some other form.

Instability in the Voluntary Contribution Mechanism with a
Quasi-linear Payoff Function: An Experimental Analysis
Jun Fenga,*, Tatsuyoshi Saijob,c, Junyi Shend, and Xiangdong Qinea Graduate School of Engineering, Kochi University of Technology, 2-22 Eikokuji, Kochi

780-8515, Japanb Research Institute for Humanity and Nature, 457-4 Kamigamomotoyama, Kyoto 603-8047,

Japan
c Research Institute for Future Design, Kochi University of Technology, 2-22 Eikokuji,

Kochi 780-8515, Japand Research Institute for Economics and Business Administration, Kobe University, 2-1

Rokkodai, Kobe, Hyogo 657-8501, Japane Antai College of Economics and Management, Shanghai Jiaotong University, 1954

Huashan Road, Shanghai 200030, China* Corresponding author.
Jun Feng
Graduate School of Engineering,
Kochi University of Technology,
2-22 Eikokuji, Kochi 780-8515, Japan
Tel: +81-888217145
E-mail: cb8226@alumni.sjtu.edu.cn.
Abstract:
We conduct experiments to investigate the convergence of contributions in the voluntary
contribution mechanism (VCM) with two quasi-linear payoff functions. One is linear withrespect to private goods and nonlinear with respect to public goods; we call it “QL1.” Theother is linear with respect to public goods and nonlinear with respect to private goods; wecall it “QL2.” The system with QL1, built on the assumption of self-interested players andmyopic Cournot best response dynamics, is not stable, but the system with QL2 has adominant Nash equilibrium. This theoretical result predicts a “pulsing” of contributions in
the VCM with QL1. Our experimental observations demonstrate that individual
contributions are certainly converging to the dominant Nash equilibrium in the experimentwith QL2. In the experiments with QL1, however, the dispersion of individual
contributions increases progressively with repeated trials, and the contributions are still
volatile in the experiments’ last periods, although we do not find a clearly unstable pulsing
in the group’s total contribution.

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Keywords: instability; public goods game; lab experiment; voluntary contribution
mechanism; conditional cooperator
JEL classification: H41

Acknowledgements: We are grateful to two research assistants, Jun Wu and Luqing Yu,
for helping us conduct the experiment, as well as to the participants of the 19th
Experimental Social Sciences Conference at the University of Tokyo, Yoshiro Kamijo, Hizen
Yoichi, Takao Kusakawa, and two anonymous referees for their helpful comments. This
research was supported by JSPS KAKENHI Grant Numbers 24243028, 16K13354, and
17H00980 and by RIHN Project Number 40410000. The authors are responsible for any
remaining errors.

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1 Introduction
The voluntary contribution mechanism (VCM) has been investigated by experimental
economists for many years in order to understand the public goods provision problem.1
Most researchers in this field use linear payoff functions such as u(xi , y) = xi + by, wherexi is a private good of player i, y is a public good, and b is a positive constant. However,
several scholars argue that this linear payoff setting cannot represent real-world situationsof the VCM environment because the self-interested choice (Nash equilibrium) and the
optimal social choice are located at opposite boundaries of the feasible choice set (see, e.g.,
Sefton and Steinberg, 1996; Laury and Holt, 2008).
One way to address this problem is to adopt nonlinear payoff functions to provide aninterior solution for the self-interested choice and the optimal social choice. Thus, twoquasi-linear payoff functions are introduced in the literature. The economic rationale of thefirst payoff function is that private good xi is money; therefore, its marginal return could
be assumed to be constant. However, the marginal return from specific public good 𝑦 isnonlinear. That is, π(xi , y) = xi + t(y) (see Isaac et al., 1985; Isaac and Walker, 1991; Seftonand Steinberg, 1996; Isaac and Walker, 1998; Laury et al., 1999; Hichri and Kirman, 2007).
We call this “QL1.” Conversely, the second payoff function is linear with respect to y andnonlinear with respect to xi. Thus, the function is π(xi , y) = h(xi ) + y (see Sefton andSteinberg, 1996; Keser, 1996; Falkinger et al., 2000; Willinger and Ziegelmeyer, 2001; van
Dijk et al., 2002; Uler, 2011; Maurice et al., 2013; Cason and Gangadharan, 2014). We callthis “QL2.” The second payoff function is used to model a relatively rare situation in which
the marginal return from the private good decreases, whereas it is constant for the public
good.
These two designs lead to completely different theoretical predictions. The VCM with
QL1 induces multiple static Nash equilibria, which produces a coordination problem. By
contrast, the VCM with QL2 induces a unique dominant equilibrium, which is similar tothe VCM with linear payoff functions. Sefton and Steinberg (1996) compared contribution
levels across QL1 and QL2 environments using a randomly re-matched group setting tosuppress the feedback from the results of previous periods in the experiments. They
predicted that the presence of the coordination problem should partially explain why theaverage of individual contributions is significantly above the Nash prediction in theirdesign of the VCM with QL1, although their experimental results indicated only a slightdifference in contribution levels between the two experiments.
In contrast to Sefton and Steinberg (1996), we are interested in the VCM experimentswith QL1 and QL2 using a fixed group setting. Since the fixed group setting transforms thegame into a super game, subjects might be motivated to play strategically in such anenvironment (for details, see the discussion in Sefton and Steinberg, 1996). Furthermore,because the group members are fixed, the feedback from preceding periods contributes tobelief formation much more directly in the fixed group setting than it does in the randomly
re-matched group setting. Healy (2006) provides experimental evidence that subjects
appear to best respond to recent observations in the VCM experiment with QL1 using a
See Ledyard (1995) and Chaudhuri (2011) for surveys on experiments regarding the VCM.
Bergstrom et al. (1986) discuss the basic theoretical properties of the VCM.
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fixed group setting.
Recently, Saijo (2014) showed that, if subjects follow the assumptions of self-interested
players and myopic best response dynamics, all Nash equilibria are not asymptoticallystable in the system of the VCM with QL1.2 This leads to a pulse of contributions(alternating between contributing nothing and contributing everything). This dynamic
analysis predicts that the feedback from repeated trials will worsen the coordination
problem in the VCM with QL1. On the other hand, Laury et al. (1999) found that thesymmetric Nash equilibrium was a poor predictor of individual contributions and thatmean contributions also varied widely among individuals, even within a single experiment.
This result was confirmed by Hichri and Kirman (2007). These observations and theinstability result suggest a complex interaction among subjects in the VCM with QL1.
Analogous arguments of instability were discussed concerning oligopoly competitionin the field of industrial organization (see Cox and Walker, 1998; Rassenti et al., 2000; Hucket al., 2002). Nevertheless, the instability problem in the VCM with QL1 differs from thatexamined in those discussions. As Andreoni (1995) pointed out, subjects are called upon to
generate positive externalities in the VCM environment, whereas they are asked togenerate negative externalities in the experiment of oligopoly competition.3 The positive
and negative framing will lead to different effects on cooperation (see Andreoni, 1995;
Sonnemans et al., 1998; Cookson, 2000; Bowles and Polania-Reyes, 2012). Cooperative
behavior is widely observed in the VCM experiments (for a survey, see Chaudhuri, 2011).
Therefore, an investigation in the VCM environment might provide a new understandingof the effect of instability in an environment that includes cooperation.
More importantly, most experimental studies in the field of VCM experiment haveused the linear payoff function, which might have failed to capture the real-world
instability of the VCM. Therefore, this study investigates that instability and provides
dynamic analyses on the convergence of individual contributions in the VCM with QL1using a fixed group setting. The results of the VCM experiment with QL2 serve as areference.
In contrast to the observation of a tiny difference in contribution levels between the
QL1 and QL2 environments using a randomly re-matched group setting in Sefton andSteinberg (1996), our experimental results show a significant difference in the convergenceof individual contributions between the QL1 and QL2 environments using a fixed groupsetting. We find clear evidence that the dispersion of individual contributions decreases,
indicating the convergence of individual contributions, and that the absolute changes ofindividual contributions diminish, suggesting that individual contributions become steady,in the experiment with QL2.4 Conversely, although we do not find a clearly unstable
An intuitive explanation of asymptotic stability is that an equilibrium 𝑥̂ is asymptoticallystable if all nearby solutions not only stay nearby but also tend to 𝑥̂ (Hirsch and Smale, 1974, p.
180). We provide the formal definition of asymptotic stability in Section 2.
3 The VCM experiments usually frame the subject’s choice as contributing to the provision of
public goods, which could benefit other players within the group, whereas the oligopoly
experiments usually frame the subject’s choice as providing a product, which will lower themarket price and result in a disbenefit to others within the group.
4 Absolute changes are the absolute values of the first-order differences of individual
24pulsing in the group’s total contribution in the experiments with QL1, our observations
show that the dispersion of individual contributions increases progressively with repeatedtrials and that individual contributions are still volatile in the experiments’ last periods.
Hence, individual contributions diverge in the QL1 environment. These observations
suggest that the coordination problem is not alleviated and that individual contributions
are not converging to any equilibrium in the experiments with QL1. Therefore, our main
result is that the experimental observations provide supporting evidence for the
non-convergence of individual contributions in the QL1 environment using a fixed groupsetting, but there is still a significant distance between our theoretical instability argument
and our experimental observations.
Moreover, consistent with the findings of previous studies, our data show
considerable cooperation across players in all experiments. In each experiment, almost 50
percent of the subjects could be considered as following the decision rule of typicalconditional cooperators, and about 20 percent of the subjects are weak free riders.5 Basedon this observation, we discuss possible explanations for the distance between ourtheoretical predictions and the experimental observations in the conclusion.
The remainder of this paper is organized as follows. Section 2 summarizes several
theories concerning the VCM with QL1 and QL2. Section 3 presents our experimental
design. Section 4 reports the experimental observations. Finally, the last section discusses
the results and concludes the study.

2 Theories of the VCM with QL1 and QL2
2.1 VCM with QL1
Suppose that, in an n-player VCM with QL1, all players have the same payoff function andthe same endowment E. A simple quadratic specification is the following:
𝜋𝑖 = 𝐸 − 𝑠𝑖 + 𝑎𝑆 − 𝑏𝑆 2,(1)

where a and b are positive constants, 𝑠𝑖 denotes player i’s individual contribution, and𝑆 = ∑𝑛𝑖=1 𝑠𝑖 represents the group’s total contribution. For this simple game, a list of
individual contributions 𝑠̂ = (𝑠̂1 , 𝑠̂ 2 , ⋯ , 𝑠̂𝑛 ) is a Nash equilibrium if, for all i, 𝜋𝑖 (𝑠̂ 𝑖 , 𝑠̂−𝑖 ) ≥𝜋𝑖 (𝑠𝑖 , 𝑠̂−𝑖 ) for all 𝑠𝑖 ∈ [0, 𝐸], where 𝑠̂ −𝑖 = ∑𝑗≠𝑖 𝑠̂𝑗 . Therefore, from the first-order condition,
the sum of Nash equilibrium contributions is given as𝑎−1𝑆̂ = 2𝑏 , 𝑆̂ ∈ [0, 𝑛𝐸].

(2)This result indicates that any combination of individual contributions constitutes a static
contributions.
5 Typical conditional cooperators are those players who always try to match the averagecontribution of others in the previous period and whose contribution is insignificantly different
from the average contribution of others. Weak free riders are those whose contribution issignificantly below the average contribution of other players in the group and who are affectedby the difference between their individual contributions and the average.

5Nash equilibrium as long as the total contribution equals 𝑆̂ (Bergstrom et al., 1986).
Anderson et al. (1998) introduce decision errors into this model. They show that,though there is a continuum of Nash equilibria, a unique logit equilibrium exists that issymmetric across players. The equilibrium density is a (truncated at the boundary of the
choice set) normal density for the quadratic public goods game (the VCM with QL1).6
Furthermore, they suggest that the quadratic model can easily be generalized to allow for
individual differences in error parameters. The unique symmetric logit equilibrium thus
becomes a unique asymmetric logit equilibrium. Moreover, because the distribution istruncated by the boundary of the choice set, the expected contribution of the logit
equilibrium is also sandwiched between the symmetric Nash equilibrium level and half ofthe endowment. These findings seem consistent with the observations of Isaac and Walker(1998). The rationale behind this comparative static analysis is that the feedback fromrepeated trials will help subjects achieve the equilibrium consistency condition of the logit
equilibrium and solve the coordination problem.7
However, this comparative static analysis is based on the assumption that the dynamic
system of VCM is stable and converging to the unique logit equilibrium. If the equilibrium
consistency condition of the logit equilibrium cannot be reached with belief updating, this
implies that the system is unstable, and the comparative static analysis might thus not besuitable.
Saijo (2014) explores the equilibrium in the VCM with QL1 based on a dynamic
analysis. It is well-known that the best response function in the VCM with QL1 is asfollows:
𝑠𝑖 = − ∑𝑗≠𝑖 𝑠𝑗 + 𝑆̂, 𝑠𝑖 ∈ [0, 𝐸],(3)where 𝑆̂ is the Nash prediction for the aggregate contribution given by equation (2) (seeBergstrom et al., 1986). If players simply follow the myopic Cournot best response
dynamics, player i’s contribution at period t directly responds to the total contribution ofothers in the group at period t-1. The best response function (3) then becomes
𝑠𝑖𝑡 = − ∑𝑗≠𝑖 𝑠𝑗𝑡−1 + 𝑆̂, 𝑠𝑖 ∈ [0, 𝐸].

(4)Now, let us look at the stability property of this dynamic system. We employ thefollowing definition of asymptotic stability.
Definition 1. An equilibrium 𝑥̂ is locally asymptotically stable, if and only if there exists
some open neighborhood 𝑂 of 𝑥̂ such that, for any 𝑥 𝑡 ∈ 𝑂, 𝑥 𝑡 converges to 𝑥̂ as 𝑡
approaches infinity.
A useful conclusion concerning whether the Nash equilibria in the difference equations
See Proposition 3 in Anderson et al. (1998).
The equilibrium consistency condition is that player i’s expectations of other players’ actions
are equal to the means of the actual equilibrium distributions (Anderson et al., 1998).
6

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system of equation (4) are asymptotically stable is the following property (see Bischi et al.,2009; Saijo, 2014). Let k be the slope of the best response function at the Nash equilibrium.
𝑡−1
Property 1. The system 𝑠𝑖𝑡 = 𝑟(𝑠−𝑖), (i = 1, 2, ..., n) is locally asymptotically stable if andonly if |k(n − 1)| < 1.

Since the slope of equation (4) is -1 and n-1≮1 if n≥2, all equilibria are not locallyasymptotically stable, and contributions will alternate between contributing nothing andcontributing everything after a few rounds (if 𝑆̂ ≥ 𝐸) in a simultaneous difference equation
system of the VCM with QL1 under the assumptions of self-interested subjects and myopic
best response. The rationale behind this theoretical result is that the feedback fromrepeated trials will not alleviate the coordination problem, but worsen it. This insight
implies the possibility that the dynamic system of a VCM experiment with QL1 is unstable.

2.2 VCM with QL2
In an n-player VCM with QL2, a simple quadratic payoff function is given as follows:(5)

𝜋𝑖 = 𝑐(𝐸 − 𝑠𝑖 ) − 𝑑(𝐸 − 𝑠𝑖 )2 + 𝑆,

where c and d are positive constants. Then, from the first-order condition, a dominant
Nash equilibrium solution for every player is given as
1−𝑐

(6)

𝑠̂ = 2𝑑 + 𝐸, 𝑠̂ ∈ [0, 𝐸].

Therefore, in the VCM environment with QL2, due to a unique dominant equilibrium,
subjects will face a decision environment similar to the VCM with linear payoff functions.
The only difference is the location of the equilibrium in the choice set. Anderson et al. (1998)
also introduce decision errors into the quadratic model of the VCM with QL2. Since the
distribution of the logit equilibrium is also truncated by the boundary of the choice set,
they suggest that the decision error should partially explain excessive giving when the
Nash equilibrium is less than half of the endowment. Willinger and Ziegelmeyer (2001)
provide experimental observations for this theoretical result.

3 Experimental Design and Procedure
The experiments were conducted at the Vernon Smith Experimental Economics Lab at
Shanghai Jiaotong University (SJTU) in March 2015 (192 subjects) and March 2017 (96subjects). The subjects were SJTU students excluding those from the Department of
Economics and Management. All subjects participated voluntarily and had no experienceof VCM experiments using nonlinear payoff structures. The experiments consisted of 12
sessions. For each session, we recruited more than 30 subjects. We then used a randommechanism to select the participants. Twenty-four subjects were selected in each session,
and we paid a show-up fee to the rest. We used z-Tree to run the experiments (Fischbacher,2007).
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Table 1 shows the parameters of our experimental design.8 We implement four
different experiments. Three of these (QL1N, QL1P, and QL1M) utilize payoffs based on
QL1, which is linear with respect to the private good and nonlinear with respect to the
public good, while QL2N is based on QL2, which is linear in the public good and nonlinear
in the private good.
Following the design of Sefton and Steinberg (1996), we set the following consistency
conditions for the two experiments with nonlinear designs (QL1N and QL2N):
1. The same (symmetric) equilibrium contribution of two tokens per individual.
2. The same (symmetric) optimal contribution of six tokens per individual.
3. The approximately equal reward from (symmetric) equilibrium play.
4. The approximately equal reward from (symmetric) socially optimal play.9(Table 1 is about here)
However, our experimental design differs from that of Sefton and Steinberg (1996) in
two key ways. First, in our design, eight subjects are randomly allocated to a group at the
beginning of the experiment. Their positions are fixed throughout the experiment. In thedesign of Sefton and Steinberg (1996), four individuals are randomly allocated to a group at
the beginning of each period. We use a relatively large group, following Ostrom et al.
(1992), who use an eight-player group setting to study common pool resource
environments.
Second, since assuming the coefficient of linear parts to be equal to one could make iteasy for subjects to understand the nonlinear return structures in payoff tables, we do notconsider the 5th symmetric condition in Sefton and Steinberg (1996)—the same monetaryloss from a one-token unilateral departure from equilibrium play. The result is that theopportunity cost among choices in the QL1N experiment is significantly lower than that in
the QL2N experiment. As Smith and Walker (1993) have shown, the opportunity costamong choices is directly related to the dispersion of individual choices in experiments.
Therefore, the relatively small opportunity costs might affect the convergence of choices.10
To ensure that our experimental observations do not originate from the relatively small
opportunity costs, we design the other two experiments (QL1P and QL1M) for robustness
checks. The QL1P experiment employs a piecewise linear payoff function as the linear
approximation for the nonlinear returns from the public good (see also, the payoff designin Cason and Gangadharan, 2014). We also increase the opportunity costs among choices.11
Payoff lists and instructions translated from the Chinese version can be found among the
supplementary documents. We also present graphs for the relation between returns and tokens
for each account and clearly display which part indicates diminishing marginal returns. This
makes our design close to the DET experiments in Laury et al. (1999).
9 We set an additional payment to make the rewards from equilibrium play and socially
optimal play approximately equal between the two experiments.
10 We thank an anonymous referee for pointing this out.
11 Different from the QL1N experiment, we remove the fixed payment in each period and boost
the magnitude of experimental payoffs by 10 times, but the exchange ratio from experimental
dollars to real money increases by only five times (from 22:1 to 110:1) in the QL1P and QL1M
8

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The QL1M experiment uses the same payoff function as that used in the QL1P experiment
but with a different framing of the payoff table in the instructions. The new payoff table
uses a matrix to directly connect the choices to the payoffs (see, e.g., the design of payoff
tables in Cason et al., 2004).
(Figure 1 is about here)
To clearly illustrate the stability property of our design, we draw the best response
curves for the two environments in Figure 1. In this figure, the horizontal axis is the total
contribution of others in the group, and the vertical axis represents player i’s own
contribution. For the three experiments with QL1, the myopic Cournot response curve (thebold black line “f-w-j-h”) is
𝑠𝑖𝑡 = min{max{− ∑𝑗≠𝑖 𝑠𝑗𝑡−1 + 16, 0}, 8}.

(7)Consider an example. Suppose that every player’s initial contribution is the same at a/7,
which implies that the total contribution of others is initially “a.” Obviously, the best
response to “a” is point “b.” Then, the total contribution of others goes to “c.” Then, we
find the best response to “b” is point “d,” that to “d” is point “f,” and that to “f” is point “h.”Finally, the dynamic difference system will be pulsing between point “f” and point “h.”This example shows that the contributions of subjects will be pulsing between 0 and 8 aftera few rounds. However, for the QL2N experiment, this curve is derived simply as follows:
𝑠𝑖𝑡 = 2.

(8)

Since the best response curve is flat, the best response to any case is contributing two
tokens. Given these theoretical results, we propose the following hypotheses:
Hypothesis 1. In the experiment with QL2 (QL2N), individual contributions will converge
to the unique Nash equilibrium, which indicates that (i) the dispersion of individual
contributions decreases and (ii) individual contributions become steady with repeatedtrials.
Hypothesis 2. In the experiments with QL1 (QL1N, QL1P, and QL1M), individual
contributions will not converge to the symmetric and asymmetric Nash equilibria, whichindicates that (i) the group’s total contribution will be pulsing round after round (thesample autocorrelation statistic should be negative), (ii) the dispersion of individual
contributions might not decrease because of the intergroup level heterogeneity, and (iii)
individual contributions will be volatile even in the last periods.
For each session in the QL1N and QL2N experiments, we employed a random ending
experiments. For the choices around the Nash equilibrium, the opportunity cost in thepiecewise linear design is significantly greater than is that in the nonlinear design.

9rule. Subjects were certain to participate in the first 15 periods. From the beginning of the
16th period, the experiment would continue with a probability of 0.3. This setting helped tosuppress strategic play (e.g., the endgame effect) in a repeated game with the fixed groupsetting.12 Data from the first 15 rounds were used for analysis. Furthermore, to give more
information regarding the convergence of contributing behavior, the public goods gamerepeated 30 periods for each session in the QL1P and QL1M experiments. Since these two
experiments serve as robustness checks for the observations from the QL1N experiment,
we have the third hypothesis.
Hypothesis 3. The dynamic patterns of contributions (concerning dispersion andcontribution volatility) should not be significantly different among the QL1N, QL1P and
QL1M experiments.
At the beginning of each period, each subject received eight tokens. They were called
upon to allocate these tokens to two accounts: the private account and the public account.
All tokens had to be allocated in each period without communication with others, and thefeasible choice set was {0,1, … ,7,8}. Each token in the private account would produce a
private return to oneself. Each token in the public account would produce a public returnto each member of the group. The framing of instructions was similar to that of Sefton andSteinberg (1996) and consistent across experiments.
At the end of each period, the result was reported to each subject. The report consistedof three parts: each subject’s own decisions, the total tokens in the public account, andhis/her own payoff. No subject could observe the individual contributions of othermembers of the same group. This incomplete information setting is consistent with most of
the literature on VCM experiments.
When all 24 subjects entered the lab, the instructions were distributed to each one. A
native speaking research assistant read the instruction loudly. Then, control questions wererequired to be answered correctly to ensure that every subject understood the experimental
procedure. At the end of the experiment, each subject received his/her payment privately
at a preannounced exchange rate of 22 experimental dollars (E$) to 1 Chinese RMB in the
QL1N and QL2N experiments and 110 experimental dollars (E$) to 1 Chinese RMB in the
QL1P and QL1M experiments. The 192 subjects earned RMB 44.5 (7.5 US dollars) each on
average, with a range of RMB 36 to RMB 47 in the QL1N and QL2N experiments. The 96
subjects earned RMB 94 (15 US dollars) each on average, with a range of RMB 80 to RMB
108 in the QL1P and QL1M experiments. Each session lasted about one hour and a half,
including the instruction and payment distribution time.

4 Results
This section consists of four subsections. The first reviews the experimental data. The
second investigates the dispersion of individual contributions. The third shows the
See Dal B´o (2005). However, other studies find no significant difference between the finite
period setting and the random terminated setting (e.g., Selten and Stoecker, 1986;
Engle-Warnick and Slonim, 2004).
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tendency of changes in individual contributions. The final subsection investigates theconditional cooperation in the four experiments and roughly classifies subjects.

4.1 Overview
First, we present an overview of the contributions. Figure 2 shows the averagecontributions to the group account at each period for the four experiments. A decreasing
tendency of average contributions is shared by the four experiments. The Wilcoxon
signed-rank tests show that individual contributions from periods 11 to 15 are significantly
lower than those from periods 1 to 5 in both the QL1N and QL2N experiments (p-values =0.0000) and that individual contributions from periods 21 to 30 are significantly lower thanthose from periods 1 to 10 in both the QL1P (p-value = 0.0171) and QL1M (p-value = 0.0000)
experiments.13(Figure 2 is about here)
Time series plots of the group’s total contribution are provided in Figure 3. The total
contributions of all groups are significantly above the Nash prediction, indicating thepresence of cooperation. Sample autocorrelation statistics (α) of the groups’ total
contributions, reported in Table 2, are positive for all groups. The Wilcoxon rank-sum testshows a slight difference in autocorrelation statistics between the QL1N and QL2N
experiments (p-value = 0.0781). Figure 4 shows that the group’s total contribution ispulsing more in some groups in the QL1N experiment than in the QL2N experiment.
However, the unstable pulsing seems to have been considerably smoothed compared to theprediction of instability in Saijo (2014), whereby it should generate a negative serial
correlation in the experiments with QL1. These observations reject the first prediction of
hypothesis 2.
(Figure 3 is about here)(Table 2 is about here)

4.2 Dispersion
Next, we show the dynamics of dispersion in the four experiments. A common way to do
this in statistics is using the coefficient of variation to compare dispersion between two
samples with different averages. However, we focus on the dispersion of choices rather
than the dispersion of numbers. In this context, each number of contributions represents
each position of actions in the choice set. Here, two contribution samples of {0,0,1,1,2,2,3,3}
and {5,5,6,6,7,7,8,8} share an identical dispersion although their averages are different.
For all Wilcoxon signed-rank tests in this paper, we first compute two averages acrossperiods 1 to 5 and 11 to 15 for each subject in the QL1N and QL2N experiments and acrossperiods 1 to 10 and 21 to 30 for each subject in the QL1P and QL1M experiments. Then, we
conduct the Wilcoxon signed-rank tests over two samples of averages to eliminate correlation
across periods.
1311

Therefore, we still use the standard deviation as a measure of dispersion.
Result 1 (Dispersion): Although average contributions are declining in all four experiments,
the standard deviation of individual contributions is ascending in the three experimentswith QL1 at the aggregate level, whereas it is descending in the QL2N experiment. The
ascending standard deviation of individual contributions at the aggregate level stems fromthe intragroup level in the three experiments with QL1.
Support: Figure 4 shows the standard deviations of individual contributions at each period
for the four experiments. At the beginning of the experiment, the standard deviations of thefour experiments are close. However, the Spearman’s rank correlation tests reveal an
ascending tendency shared by the three experiments with QL1 (ρ = 0.7857, p-value < 0.001
for QL1N; ρ = 0.7130, p-value < 0.001 for QL1P; ρ = 0.7433, p-value < 0.001 for QL1M),
yet a descending tendency appears in the QL2N experiment (ρ = −0.9464, p-value <0.001).
(Figure 4 is about here)
Time series plots of the standard deviation for each group in the four experiments areprovided in Figure 5. In the three experiments with QL1, the Spearman’s rank correlation
tests show that eight out of 12 groups from the QL1N experiment, three out of six groups
from the QL1P experiment, and five out of six groups from the QL1M experiment share asignificantly increasing pattern (p-values < 0.1 for 16 groups; p-values < 0.05 for 11 groups);
and no group shows a significantly decreasing pattern. By contrast, eight out of 12 groups
share a significantly decreasing pattern (p-values < 0.05), and no group shows asignificantly increasing pattern in the QL2N experiment.
(Figure 5 is about here)
To sum up, the observation that the standard deviation of individual contributions is
ascending at the aggregate level stems from the intragroup level in the three experimentswith QL1. These observations do not support that individual contributions are converging
to a symmetric equilibrium in the experiments with QL1. However, we also notice that theincreasing dispersion at the aggregate level stems mainly from the intragroup level rather
than the intergroup level.14 This observation is inconsistent with the reasoning of ourinstability argument.
Therefore, Result 1 supports the first prediction of hypothesis 1, but rejects thereasoning of the second prediction of hypothesis 2. Furthermore, the observation that all
We also check the dynamical tendency of the standard deviation of the group’s total
contributions across periods in the four experiments. The Spearman’s rank correlation tests
show that ρ = 0.2536 and p-value = 0.3618 for QL1N, ρ = 0.5537 and p-value = 0.0015 for
QL1P, ρ = 0.0007 and p-value = 0.9972 for QL1M, and ρ = −0.6643 and p-value = 0.0069 for
QL2N. These results indicate that, in two of the three experiments with QL1, the dispersion at
the intergroup level does not increase with repeated trials.
1412the three experiments with QL1 share similar dynamics of dispersion supports hypothesis
3.

4.3 Absolute Changes in Individual Contribution
We use the absolute value of the first-order difference of individual contributions(|sit − sit−1 |, t ≥ 2; hereafter “AVFD”) to measure the pulsing of individual contributions. If
the system is approaching an equilibrium, the degree of contribution pulsing on average
will diminish.
Result 2 (Absolute changes): The absolute changes on average are diminishing in the QL1P
and QL2N experiments. In the QL1N and QL1M experiments, however, they do notdiminish relative to the beginning of the experiment.
Support: Figure 6 shows the average of AVFDs at each period for the four experiments. By
comparing sample 1 (the AVFDs from periods 2 to 6) with sample 2 (the AVFDs fromperiods 11 to 15), the Wilcoxon signed-rank test shows a significant decrease in the QL2N
experiment (p-value = 0.0000), but an insignificant result for the QL1N experiment (p-value= 0.1312). Furthermore, for the QL1P and QL1M experiments, by comparing sample 1 (theAVFDs from periods 2 to 11) with sample 2 (the AVFDs from periods 21 to 30), theWilcoxon signed-rank test shows a significant decrease in the QL1P experiment (p-value =0.0012) yet an insignificant result for the QL1M experiment (p-value = 0.4817). Although
there is also a decreasing tendency in the QL1P experiment, the AVFDs in the last 10
periods of the QL1P experiment are still significantly greater than those in the last five
periods of the QL2N experiment (p-value = 0.0124, by the Wilcoxon rank-sum test).
(Figure 6 is about here)
Combined with previous observations of standard deviations, the decreasing AVFDs
in the QL2N experiment indicate that the experimental system is converging to thedominant equilibrium, which is symmetric across players. Conversely, the decreasing
AVFDs in the QL1P experiment might indicate that some groups in the experiments with
QL1 are converging to some asymmetric equilibrium. Therefore, we further check the
AVFDs at the group level. Comparing sample 1 with sample 2 in each group of the threeexperiments with QL1 reveals a significant decrease in four groups (p-value = 0.0138 forgroup 10 in the QL1N experiment; p-value = 0.0117 for group 2 and p-value = 0.0687 forgroup 4 in the QL1P experiment; and p-value = 0.0929 for group 1 in the QL1M
experiment). However, by checking the individual data in these four groups, we find thatthe individual contributions from a part of the group members are still volatile in the lastperiods of the experiment. This is not compatible with the experimental system’s
converging to a static asymmetric equilibrium.
Therefore, Result 2 supports the second prediction of hypothesis 1 and the group level
observations also support the third prediction of hypothesis 2. Furthermore, although theobservation in the QL1P experiment at the aggregate level is different from those in the13other two experiments with QL1, the group level observations show that individual
contributions are volatile in the last periods of all the three experiments with QL1, which isconsistent with the prediction of hypothesis 3.
Overall, our experimental data reveal a clear pattern showing that contributions areconverging to the static equilibrium in the QL2N experiment. By contrast, our observations
do not suggest the existence of a process whereby the dynamic system is approaching asymmetric or asymmetric equilibrium and that the coordination problem is alleviated in
the three experiments with QL1. However, we also notice that there is not a significant
pulsing in the group’s total contributions in the three experiments with QL1 and that theincreasing dispersion of individual contribution comes mainly from the intragroup level.
Our instability theory cannot explain these observations. Therefore, in the following
subsection, we investigate the heterogeneity among individuals in order to generate
insights concerning these observations through a categorization of the subjects.

4.4 Conditional Cooperation
In the VCM experiments with linear payoff functions, players are often divided into several
categories. The three most common categories are free riders, conditional cooperators, andunconditional cooperators. Free riders account for only around 20 percent of the total
population. However, conditional cooperators account for around 50 percent (seeFischbacher et al., 2001; Sonnemans et al., 1999; Keser and van Winden, 2000; for a survey,
see Chaudhuri, 2011). These findings indicate that the experimental environment might bemuch more complex than the assumption in Saijo (2014) implies. For the QL1 environment,
since Laury et al. (1999) found that average contributions varied widely among individuals,
even within a single experiment, there might be a systematic difference in the motivation
for cooperation between the experiments with QL1 and QL2.15 In this subsection, we
attempt to investigate the conditional cooperation from a myopic perspective to see
whether there is a systematic difference in conditional cooperation across the experiments.
The individual decision rule, used to isolate the motivation of conditional cooperation,is assumed to take the following form.
1𝑠𝑖𝑡 − 𝑠𝑖𝑡−1 = 𝛼𝑖 + 𝛽𝑖 (𝑠𝑖𝑡−1 − 7 ∑𝑗≠𝑖 𝑠𝑗𝑡−1 ) + 𝜀𝑖 , 𝑡 ≥ 2,

(9)where εi is the residual term of player i. Equation (9) is estimated using the SeeminglyUnrelated Regressions (SUR) method for each group of eight players in the four𝛼experiments. In this regression, − 𝛽𝑖 approximately denotes the overall distance between𝑖player i’s contribution and the average contribution of other players in the group. Thus,
this regression allows us to check two aspects of the subjects’ contribution behavior: first,
how many players are reacting to the difference between their own contribution and theaverage contribution of others (or how many players try to match the average contributionof others in the previous period); second, the overall distance between player i’s
Here, the term “systematic difference in the motivation for cooperation” is used to indicate
the difference in the distribution among different types of subjects.
15

14

contribution and the average contribution of other players. If αi > 0 and βi < 0, subject i’scontribution is significantly above the average contribution of other players in the groupand is also affected by the difference between his/her contribution and the average. This
result indicate that this subject is a weak unconditional cooperator (WUC).16 In turn, if αi <0 and βi < 0, a weak free rider (WFR) is indicated. A typical conditional cooperator (TCC)should have αi = 0 and βi < 0, which implies a person who always tries to match theaverage contribution of others in the previous period and whose contribution is
insignificantly different from the average contribution of others. Moreover, unconditional
cooperators (UC) are those who persisted in contributing a fixed number of at least six
tokens; conversely, free riders (FR) are those who persisted in contributing a fixed numberof no more than two tokens. Hence, By examining αi and βi, we can roughly divide all
subjects into six categories.17
Result 3 (Conditional cooperation): No systematic difference in conditional cooperation isobserved across the four experiments. The individual estimates from the SUR show thataround 50 percent of the players could be categorized as typical conditional cooperators;
weak free riders and weak unconditional cooperators each account for about 20 percent of
the total population in all experiments.
Support: Table 3 summarizes the results of the individual regressions. Briefly, by comparing
the number of subjects in each type, we find no systematic difference in conditional
cooperation across the four experiments. In all experiments, almost half of the players
could be regarded as typical conditional cooperators, while weak free riders and weakunconditional cooperators each account for about 20 percent of the total population. This
result is consistent with the findings in the linear environment of the VCM experiment. The
presence of conditional cooperators might be a reason for the smoothed pulsing in thegroup’s total contribution in the experiments with QL1.
(Table 3 is about here)

5 Discussion and Conclusion
In this study, we conducted experiments to investigate the dynamic pattern of contributing
behavior in the VCM with two quasi-linear payoff functions. We find clear evidence thatthe system is converging to the dominant equilibrium in the QL2N experiment. The
average contribution decreases with repeated trials and individual contributions converge
and become steady. By contrast, in the experiments with QL1, although contributions on
average are also decreasing with no clearly unstable pulsing in the group’s total
We call them “weak unconditional cooperators” to distinguish them from those unconditional
cooperators who always contribute six tokens throughout the experiment.
17 There is one subject from the QL1N experiment who should be classified as α < 0 and β =ii0. Because p-value = 0.049 for αi < 0 and only one observation is considered, we take thisobservation as an unimportant exception and assign this subject into category αi = 0 and βi =
0.
16

15

contribution, individual contributions diverge and change continuously.
These observations do not support the hypothesis that the system of the VCM with
QL1 is converging to an equilibrium, indicating that a comparative static analysis alone
might not be suitable for the VCM with QL1 using a fixed group setting. On the other hand,
our observation is consistent with the finding of previous studies on the VCM experimentwith linear payoff functions that most players in the lab VCM experiment follow thedecision rule of conditional cooperators. This might be a reason for the growing dispersion
in the three experiments with QL1.
Considering a repeated VCM game with two types of players—free riders andconditional cooperators—if the game has a dominant strategy, such as that of a linear
environment, the decay of the average contribution could be explained by the classicalscenario of the interaction between free riders and conditional cooperators. Once theconditional cooperators become frustrated by free riding, they start reducing theircontributions. Then, the average contribution becomes close to the dominant equilibrium.
Our experimental evidence suggest that this may also be true in the VCM with QL2 inwhich there is a dominant equilibrium.
The observations of the dispersion and the absolute changes in the three experimentswith QL1 indirectly suggest another possible interpretation of the interaction between freeriders and conditional cooperators in the VCM with QL1. When the conditional
cooperators become frustrated by free riding, they will reduce their contributions to acertain level. The free riders would then have to increase their contribution to increase their
payoffs if they expect that the total contribution of others will become less than the sum of
the Nash equilibrium contributions. When conditional cooperators find that the total
contribution is increasing, they will seek to sustain this total contribution level. However,the free riders will then begin to free ride again, and a new round of decreasing total
contribution will begin. We thus conjecture that starting from the dynamic analysis of Saijo(2014) and incorporating the interaction between several different types of players mightoffer insights into the ascending dispersion we observed in this study.
Finally, two empirical implications of our experimental observations are worth
mentioning. First, the experimental observation of the growing dispersion indicates thatthe stability property of the mechanism itself might also be a reason for the diversity of
individual contributions, in addition to the social preference heterogeneity among theplayers. Second, and more importantly, the experimental observation of non-convergenceindicates that the Nash equilibrium might not be a suitable theoretical benchmark to use inempirical analyses of the real-world VCM environment if the system is not converging to it.

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18

Tables

Table 1 Parameters of the experiments
Experiments
Payoff function

QL1N

QL1P

QL1M

QL2N

QL1N: (𝐸 − 𝑠𝑖 ) + 1.4484𝑆 − 0.0137(𝑆)2 + 28
10(𝐸 − 𝑠𝑖 ) + 15𝑆, 𝑆 ≤ 16;
QL1P and QL1M: {10(𝐸 − 𝑠𝑖 ) + 5(𝑆 − 16) + 240, 16 < 𝑆 ≤ 48;
10(𝐸 − 𝑠𝑖 ) + (𝑆 − 48) + 400, 48 < 𝑆 ≤ 64.
QL2N: 11.5(𝐸 − 𝑠𝑖 ) − 0.875(𝐸 − 𝑠𝑖 )2 + 𝑆

Endowment (Tokens)

8

8

8

8

Additional payment (E$)

28

0

0

0

2(53.7)2(300)

2(300)

2(53.5)Socially optimal s ∗ (Payoff)6(68)6(420)

6(420)

6(67.5)

Payment ratio

22:1

110:1

110:1

22:1

15(Random ending)30

30

15(Random ending)

12/96

6/48

6/48

12/96

(symmetric)
Nash choice ŝ (Payoff)(symmetric)

Periods
Groups/Subjects

𝑠𝑖 denotes the individual contribution of player i; E represents the endowments; and S denotes the group’s total
contribution.

1

Table 2 Sample autocorrelation statistics
The QL1N experiment
Group

1

2

3

4

5

6

7

8

9

10

11

12

α

0.43

0.27

0.21

0.39

0.55

0.27

0.18

0.11

0.03

0.33

0.53

0.34

The QL1P experiment
Group

1

2

3

4

5

6

α

0.68

0.62

0.83

0.71

0.41

0.65

Group

1

2

3

4

5

6

α

0.42

0.47

0.29

0.68

0.35

0.53

The QL1M experiment

The QL2N experiment
Group

1

2

3

4

5

6

7

8

9

10

11

12

α

0.45

0.37

0.10

0.65

0.48

0.56

0.53

0.57

0.45

0.38

0.36

0.09

2

Table 3 Conditional cooperation1sit − sit−1 = αi + βi (sit−1 − ∑j≠i sjt−1 ) + εi , t ≥ 2

Form

7

Individual results
Category

QL1N (96 subjects)

QL1P (48 subjects)

3 (3%)𝑏1 (2%)

2 (4%)

1 (1%)WUC (αi > 0 and βi < 0𝑎 )19 (20%)11 (21%)7 (15%)

17 (18%)

TCC (αi = 0 and βi < 0)

40 (42%)21 (44%)27 (56%)48 (50%)WFR (αi < 0 and βi < 0)

21 (22%)11 (23%)

11 (23%)20 (21%)1 (1%)

0 (0%)

0 (0%)

5 (5%)

12 (12%)4 (8%)

1 (2%)

5 (5%)UC

FR
Unclassified (αi = 0 and βi = 0)

QL1M (48 subjects) QL2N (96 subjects)aBoth αi and βi of individual regressions (SUR) are judged by a two-tailed test at the 5% significance level.

bPercentages of the total population are reported in parentheses.

3Graphs

The experiments with QL1

The QL2N experimentFig. 1 Stability property of the design1Fig. 2 Average contributions in the four experiments2Panel A: The three experiments with QL1Panel B: The QL2N experimentFig. 3 Time series plots of groups’ total contributions

3

Fig. 4 Standard deviation of individual contributions

4Panel A: The three experiments with QL1Panel B: The QL2N experimentFig. 5 Time series plots of standard deviations in groups5Fig. 6 Average of AVFDs at each period

6Instructions and payoff tables in our experiments.
There are four sets of instructions and payoff tables in this supplementary document.
The QL1N experiment, the QL1P experiment, the QL1M experiment, and the QL2N experiment.

(The QL1N experiment)
Instructions

This is an experiment concerning economic decision-making. At the end of today's
session, you will be paid in private and in cash. It is essential that you remain silent and do notwatch at other people's decisions. Please shut down your cell phone and don’t talk with others.
If you have any questions or need assistance of any kind, please raise your hand. If you
exclaim out loudly or violate any of the rules explained below, you may be asked to leave andwill not be paid. This is necessary for our experiment.
We thank you very much for your cooperation in this regard.

Overview

There will be at least 15 decision-making rounds in this experiment. You will each makea decision in each of these rounds. When the first 15 rounds are finished, the experiment willcontinue with a probability of 30%. In other words, the experiment will be directly terminatedwith a probability of 70%. At the end of each round, you will be informed your earnings forthat round on the PC screen. The rules are similar in every round.
In the first round, you will be randomly assigned to a group. Each group consists of eight
members. The composition of your group will be fixed throughout the experiment. You will
not know which of the other people in the lab are in your group in any given round. You will
be paid the total of your earnings of all rounds at the end of today's session.

Rules

In each round, you have eight tokens to allocate. You must decide the number of tokens
to place into either or both of two accounts: a private account and a group account. All tokens
must be placed in one account or the other. Each token you placed in the private account1

generates a return to you (and to you alone), and each token you placed in the group accountgenerates a return to every member of your group. Returns from the two accounts are listed in
the Earning Tables. Everybody has the exact identical Earning Tables. When the experiment
begins, you need to enter your decisions in blanks on the screen. Your entries on the blanks
must be whole numbers between 0 and 8 and must be summed to be 8.
After everyone has made a decision, the computer will compute the total number oftokens placed in the group account for your group in this round, and prepare a “Report to
Subject” for each of you. You can record the number that the computer has reported on the line
entitled "Total Number of Tokens Placed By Your Group in Your Group Account was" andcompute your earnings. Your earnings in each period are the sum of your earnings from theprivate account, your earnings from the group account, and an additional fixed payoff 28 E$.
To determine your earnings from the private account, you need to find the number from thecolumn headed "Private Account(E − xi )” and “Private return (E$)” on the Earning Tables,
according to your decisions. To determine your earnings from the group account, you need tofind the correct number in the column headed “Group account (∑ x)” and “Individual returnfrom the group account (E$)” on the Earning Tables. This part reports the amount you will
earn from the group account. Your total payoff will be reported on the PC screen
corresponding to the number of tokens you have placed in your private account and yourgroup has placed in the group account in that round.
Next, you should check to see if your calculation is consistent with the computer’s reporton the screen. It is extremely important that we both make this calculation and the results are
consistent. If your calculation differs from the computer’s or if you are unsure on how tocompute your earnings in any round, please raise your hand. When all things are correct, thenext round will begin.
Finally, at the end of experiment, the earnings you have gotten in today’s session will be
exchanged for Chinese yuan at an exchange rate of 22:1.

Final Remarks(1) All subjects have the same Earning Tables.
(2) This session will comprise of at least 15 rounds. From 16th round, the experiment willcontinue with a probability of 30%.
(3) In each round, you and other members of your group will each have 8 tokens to allocate.
(4) In each round, you should decide the number of tokens to place in your private accountand the number of tokens to place in your group account. You must distribute all 8 tokens ineach round.

2(5) Your earnings from the private account depend only on your decision (the number oftokens that you placed in the account).
(6) Your earnings from the group account depend upon the total number of tokens your groupplaced in this account.
(7) The members in your group will be fixed throughout the experiment.
(8) The exchange rate from experimental dollars to Chinese yuan is 22:1.
(9) Do not discuss your decisions with other subjects.

Are there any questions?

If all things are clear, please click “next” on your screen and finish those questions. Note thatthe purpose of those questions is only to make you understand the instructions and youranswers will not affect your earnings in the experiment.

3

Appendix
Earning Tables:
Total payoff = Private return + Individual return from the group account+28
Privateaccount(E − xi )Private return
(E$)

Group
account (∑ x)

Individual
return fromthe group
account (E$)

Group
account (∑ x)

0
1
2
3
4
5
6
7
8

0
1
2
3
4
5
6
7
8

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

0
1.4
2.8
4.2
5.6
6.9
8.2
9.5
10.7
11.9
13.1
14.3
15.4
16.5
17.6
18.7
19.7
20.7
21.6
22.5
23.4
24.3
25.2
26.1
26.9
27.7
28.4
29.1
29.8
30.5
31.1
31.7
32.3

33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64

4

Individual
return fromthe group
account
(E$)
32.9
33.4
33.9
34.4
34.9
35.3
35.7
36.1
36.4
36.7
37
37.2
37.4
37.6
37.8
38
38.1
38.2
38.3
38.3
38.3
38.3
38.2
38.1
38
37.9
37.8
37.6
37.4
37.2
36.9
36.6

Tables of two kinds of return

Private return (E$)
9
8
7
6

5
Private return (E$)

4
3
2
1
0
0

2

4

6

8

10

Individual return from the group account (E$)
45

40
35
30
25

Individual returnfrom the groupaccount (E$)

20
15
10
5
0
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64

5Explanation for the calculator

This part is acalculator to help
you query the
Earning Tables.

You can enter any
number of thetokens in yourprivate account andgroup account here.

When you press the
button of “Calculate”,
the results will appear
here.

6

(The QL1P experiment)
Instructions

This is an experiment concerning economic decision-making. At the end of today's
session, you will be paid in private and in cash. It is important that you remain silent and do
not watch at other people's decisions. Please shut down your cell phone and don’t talk withothers. If you have any questions or need assistance of any kind, please raise your hand. If you
exclaim out loudly or violate any of the rules explained below, you may be asked to leave andwill not be paid. This is necessary for our experiment.
We thank you very much for your cooperation in this regard.

Overview

There will be 30 decision-making rounds in this experiment. You will each make adecision in each of these rounds. At the end of each round, you will be informed your earnings
for that round on the PC screen. The rules are similar in every round.
In the first round, you will be randomly assigned to a group. Each group consists of eight
members. The composition of your group will be fixed throughout the experiment. You will
not know which of the other people in the lab are in your group in any given round. You will
be paid the total of your earnings of all rounds at the end of today's session.

Rules

In each round, you have eight tokens to allocate. You must decide the number of tokens
to place into either or both of two accounts: a private account and a group account. All tokens
must be placed in one account or the other. Each token you placed in the private accountgenerates a return to you (and to you alone), and each token you placed in the group accountgenerates a return to every member of your group. Returns from the two accounts are listed in
the Earning Tables. Everybody has the exact identical Earning Tables. When the experiment
begins, you need to enter your decisions in the blanks on the screen. Your entries on the blanks
must be whole numbers between 0 and 8 and must be summed to be 8.
After everyone has made a decision, the computer will compute the total number oftokens placed in the group account by your group in this round, and prepare a “Report to
Subject” for each of you. You can record the number that the computer has reported on the line
entitled "Total Number of Tokens Placed By Your Group in Your Group Account was" and7compute your earnings. Your earnings in each period are the sum of your earnings from boththe private account and the group account. To determine your earnings from the private
account, you need to find the number from the column headed "Private Account(E − xi )” and“Private return (E$)” on the Earning Tables, according to your decision. To determine yourearnings from the group account, you need to find the correct number in the column headed“Group account (∑ x)” and “Individual return from the group account (E$)” on the Earning
Tables. This part reports the amount you will earn from the group account. Your total payoff
will be reported on the PC screen corresponding to the number of tokens you have placed inyour private account and your group has placed in the group account in that round.
Next, you should check to see if your calculation is consistent with the computer’s reporton the screen. It is extremely important that we both make this calculation and the results are
consistent. If your calculation differs from the computer’s or if you are unsure on how tocompute your earnings in any round, please raise your hand. When all things are correct, thenext round will begin.
Finally, at the end of experiment, the earnings you have gotten in today’s session will be
exchanged for Chinese yuan at an exchange rate of 110:1.

Final Remarks(1) All subjects have the same Earning Tables.
(2) This session will last 30 rounds.
(3) In each round, you and other members of your group will each have 8 tokens to allocate.
(4) In each round, you need to decide the number of tokens to place in your private accountand the number of tokens to place in your group account. You must distribute all 8 tokens ineach round.
(5) Your earnings from the private account depend only on your decision (the number oftokens that you placed in the account).
(6) Your earnings from the group account depend upon the total number of tokens your groupplaced in this account.
(7) The members in your group will be fixed in each round.
(8) The exchange rate from experimental dollars to Chinese yuan is 110:1(9) Do not discuss your decisions with other subjects.

Are there any questions?

8If all things are clear, please click “next” on your screen and finish those questions. Note thatthe purpose of those questions is only to make you understand the instructions and youranswers will not affect your earnings in the experiment.

9Appendix
Earning Tables:
Total payoff = Private return + Individual return from the group accountPrivateaccount(E − xi )
0
1
2
3
4
5
6
7
8

Private return
(E$)

Group
account (∑ x)

Individual
return fromthe group
account (E$)

0
10
20
30
40
50
60
70
80

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
245
250
255
260
265
270
275
280
285
290
295
300
305
310
315
320

10

Group
account
(∑ x)

33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64

Individual
return fromthe group
account
(E$)
325
330
335
340
345
350
355
360
365
370
375
380
385
390
395
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416

Tables of two kinds of return

Private return (E$)
90
80
70
60
50
40
30
20
10
0
0

1

2

3

4

5

6

7

8

Individual return from the group account (E$)
450
400
350
300
250
200
150
100
50
0
0

10

20

30

40

11

50

60

Explanation for the calculator

This part is acalculator to help
you query the
Earning Tables.

You can enter any
number of thetokens in yourprivate account andgroup account here.

When you press the
button of “Calculate”,
the results will appear
here.

12

(The QL1M experiment)
Instructions

This is an experiment concerning economic decision-making. At the end of today's
session, you will be paid in private and in cash. It is important that you remain silent and do
not watch at other people's decisions. Please shut down your cell phone and don’t talk withothers. If you have any questions or need assistance of any kind, please raise your hand. If you
exclaim out loudly or violate any of the rules explained below, you may be asked to leave andwill not be paid. This is necessary for our experiment.
We thank you very much for your cooperation in this regard.

Overview

There will be 30 decision-making rounds in this experiment. You will each make adecision in each of these rounds. At the end of each round, you will be informed your earnings
for that round on the PC screen. The rules are similar in every round.
In the first round, you will be randomly assigned to a group. Each group consists of eight
members. The composition of your group will be fixed throughout the experiment. You will
not know which of the other people in the lab are in your group in any given round. You will
be paid the total of your earnings for all rounds at the end of today's session.

Rules

In each round, you have eight tokens to allocate. You must decide the number of tokens
to place into either or both of two accounts: a private account and a group account. All tokens
must be placed in one account or the other. Your earnings are listed in the Earning Table.
Everybody has the exact same Earning Tables. When the experiment begins, you need to enter
your decisions in the blanks on the screen. Your entries on the blanks must be whole numbers
between 0 and 8 and must be summed to be 8.
After everyone has made a decision, the computer will compute the total number oftokens placed in the group account by the other members in your group in this round, andprepare a “Report to Subject” for each of you. You can record the number that the computer
has reported on the line entitled "Total Number of Tokens Placed By the Other Members ofYour Group in Your Group Account was" and compute your earnings. In each round, yourearnings depend on the tokens placed by your own and the total tokens placed by the other13

seven group members into the group account. In the Earning Table, you can find the column
corresponding to the number of tokens placed to the group account by you own and the line
corresponding to the total number of tokens placed to the group account by the other sevengroup members. The number at the intersection of the line and the column is your earning inthat round. Your earnings will be reported on the PC screen corresponding to the tokens you
have placed in the group account and the total tokens that the other seven members of yourgroup have placed in the group account in that round.
Next, you should check to see if your calculation is consistent with the computer’s reporton the screen. It is extremely important that we both make this calculation and the results are
consistent. If your calculation differs from the computer’s or if you are unsure on how tocompute your earnings in any round, please raise your hand. When all things are correct, thenext round will begin.
Finally, at the end of experiment, the earnings you have gotten in today’s session will be
exchanged for Chinese yuan at an exchange rate of 110:1.

Final Remarks(1) All subjects have the same Earning Table.
(2) This session will last 30 rounds.
(3) In each round, you and other members of your group will each have 8 tokens to distribute.
(4) In each round, you need to decide the number of tokens to place in your private accountand the number of tokens to place in your group account. You must allocate all 8 tokens ineach round.
(5) Your earnings depend on the tokens that you have placed in the group account and thetotal tokens that the other seven members of your group have placed in the group account.
(6) The members in your group will be fixed in each round.
(7) The exchange rate from experimental dollars to Chinese yuan is 110:1(8) Do not discuss your decisions with other subjects.

Are there any questions?

14

If all things are clear, please click “next” on your screen and finish those questions. Note thatthe purpose of those questions is only to make you understand the instructions and youranswers will not affect your earnings in the experiment.

15

Appendix
Earning Table:
The tokens that you haveplaced in the groupaccount
The total tokens that theother seven members ofyour group has placedin the group account.

0

1

2

3

4

5

6

7

8

0

80

85

90

95

100

105

110

115

120

1

95

100

105

110

115

120

125

130

135

2

110

115

120

125

130

135

140

145

150

3

125

130

135

140

145

150

155

160

165

4

140

145

150

155

160

165

170

175

180

5

155

160

165

170

175

180

185

190

195

6

170

175

180

185

190

195

200

205

210

7

185

190

195

200

205

210

215

220

225

8

200

205

210

215

220

225

230

235

240

9

215

220

225

230

235

240

245

250

245

10

230

235

240

245

250

255

260

255

250

11

245

250

255

260

265

270

265

260

255

12

260

265

270

275

280

275

270

265

260

13

275

280

285

290

285

280

275

270

265

14

290

295

300

295

290

285

280

275

270

15

305

310

305

300

295

290

285

280

275

16

320

315

310

305

300

295

290

285

280

17

325

320

315

310

305

300

295

290

285

16

18

330

325

320

315

310

305

300

295

290

19

335

330

325

320

315

310

305

300

295

20

340

335

330

325

320

315

310

305

300

21

345

340

335

330

325

320

315

310

305

22

350

345

340

335

330

325

320

315

310

23

355

350

345

340

335

330

325

320

315

24

360

355

350

345

340

335

330

325

320

25

365

360

355

350

345

340

335

330

325

26

370

365

360

355

350

345

340

335

330

27

375

370

365

360

355

350

345

340

335

28

380

375

370

365

360

355

350

345

340

29

385

380

375

370

365

360

355

350

345

30

390

385

380

375

370

365

360

355

350

31

395

390

385

380

375

370

365

360

355

32

400

395

390

385

380

375

370

365

360

33

405

400

395

390

385

380

375

370

365

34

410

405

400

395

390

385

380

375

370

35

415

410

405

400

395

390

385

380

375

36

420

415

410

405

400

395

390

385

380

37

425

420

415

410

405

400

395

390

385

38

430

425

420

415

410

405

400

395

390

39

435

430

425

420

415

410

405

400

395

40

440

435

430

425

420

415

410

405

400

41

445

440

435

430

425

420

415

410

401

42

450

445

440

435

430

425

420

411

402

43

455

450

445

440

435

430

421

412

403

17

44

460

455

450

445

440

431

422

413

404

45

465

460

455

450

441

432

423

414

405

46

470

465

460

451

442

433

424

415

406

47

475

470

461

452

443

434

425

416

407

48

480

471

462

453

444

435

426

417

408

49

481

472

463

454

445

436

427

418

409

50

482

473

464

455

446

437

428

419

410

51

483

474

465

456

447

438

429

420

411

52

484

475

466

457

448

439

430

421

412

53

485

476

467

458

449

440

431

422

413

54

486

477

468

459

450

441

432

423

414

55

487

478

469

460

451

442

433

424

415

56

488

479

470

461

452

443

434

425

416

18

Explanation for the calculator

This part is acalculator to help
you query the
Earning Table.

You can enter thenumbers of any
allocation here.

When you press the
button of “Calculate”,
the results will appear
here.
19

(The QL2N experiment)
Instructions

This is an experiment concerning economic decision-making. At the end of today's
session, you will be paid in private and in cash. It is important that you remain silent and do
not watch at other people's decisions. Please shut down your cell phone and don’t talk withothers. If you have any questions or need assistance of any kind, please raise your hand. If you
exclaim out loudly or violate any of the rules explained below, you may be asked to leave andwill not be paid. This is necessary for our experiment.
We thank you very much for your cooperation in this regard.

Overview

There will be at least 15 decision-making rounds in this experiment. You will each makea decision in each of these rounds. When the first 15 rounds are finished, the experiment willcontinue with a probability of 30%. In other words, the experiment will be directly terminatedwith a probability of 70%. At the end of each round, you will be informed your earnings forthat round on the PC screen. The rules are similar in every round.
In the first round, you will be randomly assigned to a group. Each group consists of eight
members. The composition of your group will be fixed throughout the experiment. You will
not know which of the other people in the lab are in your group in any given round. You will
be paid the total of your earnings for all rounds at the end of today's session.

Rules

In each round, you have eight tokens to allocate. You must decide the number of tokens
to place into either or both of two accounts: a private account and a group account. All tokens
must be placed in one account or the other. Each token you placed in the private accountgenerates a return to you (and to you alone), and each token you placed in the group accountgenerates a return to every member of your group. Returns from the two accounts are listed in
the 'Earning Tables'. Everybody has the same Earning Tables. When the experiment begins,
you need to enter your decisions in the blanks on the screen. Your entries on the blanks must
be whole numbers between 0 and 8 and must be summed to be 8.
After everyone has made a decision, the computer will compute the total number oftokens placed in the group account by your group in this round, and prepare a “Report to20

Subject” for each of you. You can record the number that the computer has reported on the line
entitled "Total Number of Tokens Placed By Your Group in Your Group Account was" andcompute your earnings. Your earnings in each period are the sum of your earnings from boththe private account and the group account. To determine your earnings from the private
account, you need to find the number from the column headed "Private Account(E − xi )” and“Private return (E$)” on the Earning Tables according to your decisions. To determine yourearnings from the group account, you need to find the correct number in the column headed“Group account (∑ x)” and “Individual return from the group account (E$)” on the Earning
Tables. This part reports the amount you will earn from the group account. Your total payoff
will be reported on the PC screen corresponding to the number of tokens you have placed inyour private account and your group has placed in the group account in that round.
Next, you should check to see if your calculation is consistent with the computer’s reporton the screen. It is extremely important that we both make this calculation and the results are
consistent. If your calculation differs from the computer’s or if you are unsure on how tocompute your earnings in any round, please raise your hand. When all things are correct, thenext round will begin.
Finally, at the end of experiment, the earnings you have gotten in today’s session will be
exchanged for Chinese yuan at an exchange rate of 22:1.

Final Remarks(1) All subjects have the same Earning Tables.
(2) This session will consist of at least 15 rounds. From 16th round, the experiment will continue
with a probability of 30%.
(3) In each round, you and other members of your group will each have 8 tokens to distribute.
(4) In each round, you need to decide the number of tokens to place in your private accountand the number of tokens to place in your group account. You must allocate all 8 tokens ineach round.
(5) Your earnings from the private account depend only on your decision (the number oftokens that you placed in the account).
(6) Your earnings from the group account depend upon the total number of tokens your groupplaced in this account.
(7) The members in your group will be fixed in each round.
(8) The exchange rate from experimental dollars to Chinese yuan is 22:1
(9) Do not discuss your decisions with other subjects.

21Are there any questions?

If all things are clear, please click “next” on your screen and finish those questions. Note thatthe purpose of those questions is only to make you understand the instructions and youranswers will not affect your earnings in the experiment.

22

Appendix
Earning Tables:

Total payoff = Private return + Individual return from the group accountPrivateaccount(E − xi )Private return
(E$)

Group
account (∑ x)

Individual
return fromthe group
account (E$)

Group
account
(∑ x)

0
1
2
3
4
5
6
7
8

0
10.6
19.5
26.6
32
35.6
37.5
37.6
36

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64

23

Individual
return fromthe group
account
(E$)
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64

Tables of two kinds of return

Private return (E$)
40
35
30
25
20

Private return (E$)

15
10
5
0
0

2

4

6

8

10

Individual return from the group account (E$)
70

60

50

40

Individual returnfrom the groupaccount (E$)

30

20

10

0
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64

24

Explanation for the calculator

This part is acalculator to help
you query the
Earning Tables.

You can enter any
number of thetokens in yourprivate account andgroup account here.

When you press the
button of “Calculate”,
the results will appear
here.

25