Leader as Role in Public Goods Game:
An Agent Based Simulation
Chen Yu you, University of Chinese Academy of Science
April 27, 2015
Abstract
We simulate leader as role in public game. This simulation is very interesting because
simulated result suggest that leader either improve or decrease welfare in a group and the result
depend on leader choice. Our simulation show that leader’s decision effect followers in his group
strongly, in some case his lose can be compensated by followers, but not always. Heterogeneity of
leader and followers impact the ability to sustain voluntary public goods contribution. Free rider
as leader do his best in pubic goods game can improve public benefit than any other type of person.
Key words: Leadership, cost, public goods, Heterogeneity.
Introduction
Leaders – politicians, government officials, and managers – may serve as role models for what
is considered appropriate and may thus shape their followers’ beliefs about the behavior of others.
For instance, leaders who behave too selfishly, evade taxes, consume unwarranted privileges,
accept bribes, etc. may induce people to do the same (as suggested by our opening quotes) and
may nurture people’s beliefs that other people will do the same. This may exacerbate the problem
to the extent that people’s behavior is not only shaped by the leader’s example but also by their
beliefs about other people’s actions. Of course, if the leader behaves as a positive role model, the
opposite conclusions may hold.
Voluntary leadership is generally studied as a costless. Nevertheless, in real situations, actions
directed to achieve the role of first mover could be individually costly. It is not clear whether the
cost of a person as leader is covered by follower’s benefit from leader as great example.
In addition, how the social preference heterogeneity affect the leader and followers’ action is
not depicted.
Our main contribution is simulating this situation. More generally, we aim to contribute to a
better understanding how leaders effect people’s contribution in public goods game.
Previous Research
Monetary compensation to leaders can be used to increase public goods provision, but
also that it may create a social crowding-out effect of moral motivation(Alexander et al. 2014).
A purely team-based compensation scheme induces agents to voluntarily cooperate while
introducing an additional relative reward increases effort and efficiency only when the bonus is
substantial(Irlenbusch & Ruchala 2008).
Strong Reciprocators make higher Leader contributions than Selfish subjects is in line with
studies from trust games where subjects play both roles. Strong Reciprocators believe that they
will be paired with another Strong Reciprocator, whereas Selfish subjects believe they will be paired
with another Selfish subject. Beliefs are strongly correlated with distributional preferences. groups
will perform better when led by individuals who are willing to sacrifice personal benefit for the
greater good. Self-interested Leaders could, in principle, do anything that an optimistic
reciprocator does, their cooperation preferences and expectations about others make them less
likely to provide effective leadership(Gächter et al.2008).
The behavioral relevance of conditional cooperation has also been shown in field experiments.
For instance, in a field study on voluntary donations to social funds at the University of Zurich, Frey
and Meier (2004) show that students who are informed that 64 percent of the other students
contributed to the social funds are more likely to contribute than students who are told that only
46 percent donated to the social funds. Meier (2006) replicates this finding in a follow-up study.
Rustagi et al. (2010) study forest management groups in Ethiopia and find that groups comprising
more conditional cooperators are also more successful in managing their forest commons.
As shown, naturally occurring field evidence as well as lab and field experiments suggest that
people contribute more to the public good the more they believe others contribute. It follows from
this observation that any factor that shifts beliefs will shift behavior. Leaders, whose behavior is
visible to followers, are in a particularly powerful position to influence their followers’ beliefs.
Therefore, we chose a leader-follower public goods game, which we detail in the next section, as
one framework for our analysis. Since belief effects also matter in the absence of leaders, which
characterizes many of the situations discussed above, we also look at belief effects in a public goods
game without a leader.
Due to the absence of data, previous studies have not investigated how the leader’s social
preference influence the follower’s behavior and social welfare. Whether there is a situation that
leader improve social welfare including his self? We find that leader’s preference play key role in
change social welfare. And in some condition, compensating to leader and enhance all participants
welfare can be reached.
Agent-Based Simulation
An agent-based model has been developed to simulate the contributions of individuals to a
public good. We use the structure of the public goods game, which has been studied widely
experimentally (see Chaudhuri 2011; Ledyard 1995 for reviews). In the typical public goods game,
agents receive an endowment which they can allocate between an individual account and a group
account. Tokens deposited in the individual account are kept by the agent while contributions sent
to the group account are multiplied by some factor and divided evenly between all agents. So long
as the factor is strictly greater than one and strictly less than the group size this game results in a
distinct Nash equilibrium (i.e. participants decide to keep everything) and a distinct social optimal
(i.e. participants decide to contribute all tokens to the group account). Our simulation base on
Gächter and Renner experiment(2014, we use their parameter below), so the factor is set to 0.4.
4
 i  20  gi  0.4 g j
(1)
j 1
The literature is unclear as to whether social preference types arise from corresponding utility
functions or whether they are only strategies. The process through which utility functions and
interact to produce actions is an important avenue for future behavioral research. For purposes of
our simulation, agents are programmed as adhering to a strategy, with a different strategy for each
social preference type. For purposes of the simulation, we will refer to these as the agents'
'contribution preferences' or 'social preference types.'
With multiple social preference types, behavior may be determined by their contribution
preferences and their beliefs, rather than only payoff-maximization. We are able to separate
payoff-maximization in this manner because the payoff-maximizing strategy is always the same in
the game with finite rounds: to contribute zero.
For the purpose of the simulation, the action chosen in a particular period is determined by a
combination of type and beliefs(Gächter & Renner,2014). The agent's programmed strategy is then
executed over ten rounds with the same group members. Specifically, agent i calculates her
contribution to the public good Ci,t at time t using the following equation:
Ci ,t   Pi ,t   Bi ,t
(2)
where Pi,t is agent i's underlying preference at period t and Bi,t is agent i's subjective belief at period
t. Agent i's preference Pi,t may further be a function of her beliefs Bi,t (see eq. 5 below), depending
on preference type. Hence, agent i's contribution is a weighted average of her preferences and
beliefs, where the weights α, β are set by the model.
For all types of players, agent beliefs are determined in the first period through a random
draw between zero and the maximum possible contribution. After the initial round, beliefs are
computed using the following formula (leader use the 3 and other use the 4):
Bi ,t   xt 1   Bi ,t 1
(3)
Thus, Bi,t is simply the weighted average of others' contributions at period t-1, xt–1 and agent i's
own beliefs in period t-1, Bi,t–1 in the case without leader. where the weights ω and ν are set
by the model. (Gächter & Renner 2014)
BL ,t   xt 1   BL ,t 1   CL ,t
(4)
Bi,t and Bi,t-1 are the leader agent belief at period t and t-1 respectively. CL,t is the leader contribution
at period t. where the weights ω, ν, τ are set by the model. (Gächter & Renner 2014)
The general formulation of agent i's underlying preference is given by the following(Lucas,
Oliveira& Banuri, 2014):
Pi ,t   i   i Bi ,t
(5)
where γi and δi are agent i's preference parameters, which are stable over periods and constant for
all individuals of the same preference type. Preference types can be broadly classified as either
unconditional or conditional. Within each of these are further distinctions leading to a total of 3
different preference types:
Unconditional: These agents have underlying preferences that are independent of their beliefs
about the amount contributed by other group members, that is δi = 0 (see e.g. Burlando & Guala
2005; de Oliveira et al. 2013; Fischbacher & Gächter 2010; Fischbacher et al. 2001; Herrmann &
Thöni 2009; Kurzban & Houser 2005). Within this broad categorization, we have five distinct types
of contribution preferences:
 Free Riders (γi =0, δi = 0): The contribution preference is set to zero for all periods, which
corresponds to the Nash equilibrium level of contributions in the finitely-repeated linear VCM
regardless of the other types in the group (e.g. Burlando & Guala 2005; de Oliveira et al. 2013;
Fischbacher & Gächter 2010; Fischbacher et al. 2001; Herrmann & Thöni 2009).
Conditional: These social preference types have underlying preferences that are conditional on
their beliefs about the amount that group members will contribute (i.e. δi = 0):Burlando & Guala
2005; de Oliveira et al. 2013; Fischbacher & Gächter 2010;Fischbacher et al. 2001; Herrmann &
Thöni 2009; Croson 2007). Within this broad categorization, we have four distinct types of
contribution preferences:
 Conditional Cooperators (γi = 0, δi = 1): This type of agent has a preference for contributing
the amount according to the belief regarding how other group members will contribute in a
given period t (e.g. Burlando & Guala 2005; de Oliveira et al. 2013; Fischbacher and Gächter
2010; Fischbacher et al. 2001; Herrmann & Thöni 2009).

Hump (Triangle) Players (γi = 0, δi = 1 if Bi,t < 10) and (γi = 20, δi = -1 if Bi,t ≥ 10):
Individuals with this contribution preference prefer to behave like conditional cooperators for
low levels of beliefs about others giving, and like economic altruists for high levels of beliefs
about others giving (Fischbacher & Gächter 2010; Fischbacher et al. 2001; Herrmann & Thöni
2009). Similar to threshold players, the threshold level has been set to half the maximum
possible contribution for all agents.
The overall group composition is defined by how many agents with each contribution
preference are in the group. Groups can be created with an arbitrary number of participants and
each participant will have a stable social preference type associated throughout the simulation
runs. The simulation model is initialized by setting the group size to four, the maximum possible
contribution to 20.
Every individual agent is created with a contribution preference type and, if set, also a belief
model. Each simulation run is asynchronous (i.e. agent types are processed without a fixed order
to avoid computational path dependency).
We simulate the public goods game with and without leader at same time. Each simulated
period will compute the following:
1. Random to initiate belief value, Bi1, (ranging from zero to the maximum possible contribution)
for every type of agent in the simulated public goods group.
2. Unconditional Low and High cooperators randomly generate a contribution preference value,
3.
4.
5.
6.
Pi, within the lower and upper half of the contribution range, respectively.
If there is a leader in the group, leader’s contribution is exogenous given.
The follower agent computes Cit based on equation 3 if group has leader, otherwise based on
4.
Follower individual agent updates beliefs based on equation 3. And individual agent (without
leader) update beliefs based o equation 4.
If the final period has been reached, halt the simulation, otherwise repeat from step 2.
Each group is simulated independently, so that the simplified pseudo-algorithm described
above is executed in every run, including Bi,t and Pit.
The pseudo-random values included in the simulation model were implemented using the
Mersenne Twister method (Matsumoto & Nishimura 1998) to compute uniform distributions.
These include the aforementioned initial random setting of values for certain agent types. Due to
these non-deterministic features, the model is not fully driven by the initial conditions. Further,
notice that, with the exception of the noisy agent, Pit, Bit and Cit are not calculated with errors.
Instead of completely eliminating noise from the simulation, the noisy player has been included
due to prior observation (Burlando & Guala 2005; Fischbacher et al. 2001). Specifically, we include
the noisy player as one of the possible contribution preference type, not to add noise to the data.
We thus test their role along with the other possible social preference types found in the public
goods game setting.
The simulation model allows to systematically analyze every type of leader and followers, with
regards to contribution preference type. One advantage of using the agent-based approach to this
research problem is to fully account for the heterogeneous interaction occurring at the individual
level, which ultimately leads to insights at the group level. The other advantage is simulating
contribution in a group with or without leader under almost same condition at the same time. The
model has been implemented in Python3.4. Should the reader wish to read the simulation source,
please contact the first author.
Description of Data
Data are generated using the agent-based simulation described in the preceding section. The
dependent variable in the analysis is the average number of tokens contributed to the group
account in period t, ranging from 0 (the Nash equilibrium in the traditional game) to 20 (the social
optimum).
As previously described, we use groups of four agents, which parallels both the experimental
and simulation results in Oliveira& Banuri (2014). Groups are allowed to vary between completely
homogeneous and completely heterogeneous, with all possible combinations in between, making
use of all nine contribution preference types.
In accordance with Fischbacher and Gächter (2010), the data for Belief Model are restricted
to three agent types (Free Riders, Conditional Cooperators, and Hump Players), yielding 15 distinct
group combinations. With 10 rounds executed a 1000 independent times, this process generates
a sample of 15,0000 data points. Fischbacher and Gächter (2010) note that every other type is
excluded in their models because these (approximately 10% of their sample) are denoted as
"confused subjects" meaning that they were unable to classify these agent types. Since we utilize
their parameters, we restrict our sample as well.
Results
We now turn to our analysis of the social welfare achieved in each of these simulated groups.
Recall that the unit of observation is the average number of tokens contributed to the group
account. Since the social optimal in the linear VCM is full (20-token) contributions, higher numbers
are indicative of higher levels of welfare attained.
With three type of participant – free rider, condition cooperator and hump player, there are
no more than 10 kinds of combinatorial number including combination with replacement.
We find that leader’s heterogeneity influence the public goods game in three aspect.
Firstly, leader either can increase welfare or decrease the welfare of all participants, which
depend on leader’s action. If leader’s contribution is more than other three average contributions,
social welfare will be improved. Otherwise, it is worse than without leader. The leader’s
contribution over 13 token usually take good example to followers can increase social welfare, but
leader’s contribution below 10 token may be deemed to a bad example inducing to decrease social
welfare. Leader contributing 10 tokens can be considered as neutral behavior in common sense.
However, in this simulation, contribution less than 10 token will induce to decrease social welfare.
So it is easier for leader inducing to decrease welfare than to improve it when he belief he do a
neutral action. All the data support this conclusion are in appendix 1.
Secondly, the leader and followers’ social preference play key role in public goods game. At
the condition that all participants social preference unchanged, a free rider become leader is better
than any other in improving social welfare. A free rider become leader means minus a free rider
who reduce social welfare at most. All the data support this conclusion are in appendix 1.
The third, comparing without leader, follower’s revenue from improved situation owing to
leader’s good example, can compensate leader costs in some case, but not always. In our
simulation, follower’s compensation is over to leader’s cost in 30% situation. And it is only relevant
to follower’s composition in an group. The result suggest that internal compensation mechanism
do not always work, some extremal stimulation should be included, such as reputation incentive.
The composition of follower
Free Rider, Free Rider, Free Rider
Free Rider, Free Rider, Conditional Cooperator
Free Rider, Free Rider, Hump Player
Free Rider, Conditional Cooperator, Conditional Cooperator
The leader
Over or Not
Free Rider
Yes
Conditional Cooperator
Yes
Hump Player
Yes
Free Rider
Yes
Conditional Cooperator
Yes
Hump Player
Yes
Free Rider
Yes
Conditional Cooperator
Yes
Hump Player
Yes
Free Rider
No
Conditional Cooperator
No
Hump Player
No
Free Rider
No
Conditional Cooperator
No
Free Rider, Conditional Cooperator, Hump Player
Free Rider, Hump Player, Hump Player
Conditional Cooperator, Conditional Cooperator, Conditional Cooperator
Conditional Cooperator, Conditional Cooperator, Hump Player
Conditional Cooperator, Hump Player, Hump Player
Hump Player, Hump Player, Hump Player
Hump Player
No
Free Rider
No
Conditional Cooperator
No
Hump Player
No
Free Rider
No
Conditional Cooperator
No
Hump Player
No
Free Rider
No
Conditional Cooperator
No
Hump Player
No
Free Rider
No
Conditional Cooperator
No
Hump Player
No
Free Rider
No
Conditional Cooperator
No
Hump Player
No
Discussion
We conduct an agent-based simulation of individual contributions introducing leader as role
to public goods. We identified three distinct preference types in the literature, and constructed
agents to be simulated based on these types. We then allowed agents to be configured with beliefs,
and utilized those respectively updated values to calculate contributions in each simulated period
based on the findings of Gächter and Renner (2014). We then systematically varied the leader’s
preference for 4-agent groups playing a public goods game. We find leader’s action may influence
strongly follower’s behavior and then change the social welfare.
A large stream of literature demonstrates leader as role in public game(Centorrino & Concina
2013; Levati 2005; Gächter et al 2008). We complement this literature by investigating leader with
different type of heterogeneity: social preference heterogeneity. We find that heterogeneity of
leader will take difference effect on follower. The composition of followers also impact the welfare
of group.
Leader can improve social welfare, but who pay for leader? This is an important question in
public goods. Additional future work in this field should be focus on the compensation mechanism.
The designed agent-based experiment allowed the systematic analysis of how different leader
and followers social preference heterogeneity affect contributions in a public goods game setup.
We have discussed a computational model that has been developed to provide a flexible tool to
test, via simulations, experimentally observed behaviors under different circumstances.
Reference
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Economics Discussion Paper, 2013 (10).
Centorrino S, Concina L. A Competitive Approach to Leadership in Public Good Games[J]. General
Information, 2013.
CHAUDHURI, A. (2011). 'Sustaining Cooperation in Laboratory Public Goods Experiments: A
Selective Survey of the Literature.' Experimental Economics. 14(1): 47–83.
CROSON, R.T.A. (2007). 'Theories of Commitment, Altruism and Reciprocity: Evidence from
Linear Public Goods.' Economic Inquiry. 45(2): 199–216.
DE OLIVEIRA, A.C.M., Eckel, C. & Croson, R.T.A. (2013). 'One Bad Apple? Heterogeneity and
Information in Public Goods Provision.'
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leading-by-example[J]. 2008.
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Appendix 1
The table below include all composition of this simulation. Different social preference leader
cooperate with different social preference followers.
The composition of follower
Free Rider, Free Rider, Free Rider
Free Rider, Free Rider, Conditional Cooperator
Free Rider, Free Rider, Hump Player
Free Rider, Conditional Cooperator, Conditional Cooperator
The leader
Title of picture
Free Rider
FreeRider0
Conditional Cooperator
ConditionalCooperator0
Hump Player
HumpPlayer0
Free Rider
FreeRider1
Conditional Cooperator
ConditionalCooperator1
Hump Player
HumpPlayer1
Free Rider
FreeRider2
Conditional Cooperator
ConditionalCooperator2
Hump Player
HumpPlayer2
Free Rider
FreeRider3
Free Rider, Conditional Cooperator, Hump Player
Free Rider, Hump Player, Hump Player
Conditional Cooperator
ConditionalCooperator3
Hump Player
HumpPlayer3
Free Rider
FreeRider4
Conditional Cooperator
ConditionalCooperator4
Hump Player
HumpPlayer4
Free Rider
FreeRider5
Conditional Cooperator
ConditionalCooperator5
Hump Player
HumpPlayer5
Free Rider
FreeRider6
Conditional Cooperator
ConditionalCooperator6
Hump Player
HumpPlayer6
Free Rider
FreeRider7
Conditional Cooperator
ConditionalCooperator7
Hump Player
HumpPlayer7
Free Rider
FreeRider8
Conditional Cooperator
ConditionalCooperator8
Hump Player
HumpPlayer8
Conditional Cooperator, Conditional Cooperator, Conditional
Cooperator
Conditional Cooperator, Conditional Cooperator, Hump Player
Conditional Cooperator, Hump Player, Hump Player
Hump Player, Hump Player, Hump Player
Free Rider
FreeRider9
Conditional Cooperator
ConditionalCooperator9
Hump Player
HumpPlayer9
All picture below, their titles can be find in the table above. In those picture, the horizontal
axis is the leader’s contributions, which from 0 to 20. The vertical axis is the average improved
welfare. the green line is the follower’s benefit from the leader’s action. The red line depict that
cost of leader. The blue one is leader’s cost plus follower’s benefit. On other hands, the blue line
show the all participants’ welfare in an group.