
Research Article


An Experiment on the Level of Trust in an Expanded Investment Game 

M. Grof,
L. Lechova,
V. Gazda
and
M. Kubak



ABSTRACT

The study presented an experimental study of an investment game modification. It introduced a variation based on expanding the traditional twoplayer structure of one sender and one receiver to a structure comprising of one receiver and multiple senders. Using experimental data, it has been shown that the number of senders in the given game structure has an effect on the level of trust and trustworthiness. The analysis also includes other personality traits that influence trust and trustworthiness.





Received: December 15, 2011;
Accepted: April 02, 2012;
Published: June 30, 2012


INTRODUCTION
The question of trust remains a widely discussed one in numerous fields of
study, ranging from institutional and organization trust (Chirico
and Presti, 2011; Rezaiean et al., 2010;
Safakli, 2007; Yilmaz, 2008;
Tokuda and Inoguchi, 2008) to various studies and applications
in information technology (Wang et al., 2011;
Teoh and Cyril, 2008; Jassim et
al., 2011; Rabah, 2004; Dingguo
et al., 2011). This study has presented a game theory approach to
trust based on the widely used investment game.
The investment game (also called trust game) represents an experimental approach
to the trust theory in economy as defined by Berg et
al. (1995). In comparison with the ultimatum or the dictator games,
it deals not only with the concepts of fairness or altruism but also with the
concepts of trust and trustworthiness. The game involves two players, each of
whom is assigned a role of either a sender or a receiver. The role of the sender
is to send a portion of his initial budget to the receiver, simulating an investment
process. This amount is then multiplied by an investment multiplicator before
being received by the receiver. The receiver, having received the multiplied
amount as well as disposing with his initial budget, then returns some money
(if any at all) back to the sender. Provided the existence of rational players,
this situation can be modeled by the game theory. Here, the receiver would maximize
his own utility by returning nothing and on the other hand, the sender, knowing
he would not receive anything back for his investment, will not invest anything
and all players finish the game with their resulting endowment equal to their
initial budget.
According to Berg et al. (1995) the experimental
results are not in concordance with the concept of rationality. They showed
that only 6.25% of senders did not send anything and a total of 15% of senders
sent their whole initial budged. In a game with the initial budget of $10 for
each player, the average amount sent was $5,36 and the average amount returned
was $6,46. Therefore, it can be stated that players in the role of sender do
show a degree of trust (as the amount they sent is a strictly positive amount
of monetary units) and the players in the role of receivers demonstrate a degree
of trustworthiness (as the amount they return is not just strictly positive
but often higher than the amount received). As the Berg
et al. (1995) model of the investment game describes the nature of
most economic interactions, its various modifications became the subject of
a fruitful research.
Cassar and Rigdon (2008), besides other experimental
results, compared the situation of the one senderone receiver game structure
with the situation of two senders and one receiver. Their experiment showed
negligible differences in the demonstrated trust/trustworthiness. This fact
is mention worthy, as one would expect trust and trustworthiness to change with
the increasing number of senders. As most models of the managerial organization
schemes as well as democratic decisionmaking structures are based on the presence
of one central figure (receiver in our case) and a number of subordinates (senders),
we aim to study these relations in extending the number of senders from one
up to four. The question here is, if and how will the trust/trustworthiness
change under these conditions. In the following part, we formulate the mathematical
model of the experiment. The third part includes the experimental design, followed
by the fourth part which presents the experimental data analysis.
MODEL
• 
Players let I = {1,2,…,n} be a set of senders 
• 
Endowment let M≥0 be an initial endowment of each player expressed
in monetary units 
• 
Payoff let π^{S}_{i} be a payoff of sender i and
π^{R} be a payoff of a receiver 
• 
Sender strategy let s_{i}(where 0≤s_{i}≤M) be an
amount sent by sender i to the receiver 
• 
Receiver strategy let: 

Be an amount returned by the receiver to the sender i 
• 
Trust and trustworthiness: The amount s_{i} represents the trust
of the sender i and the ratio r_{i}/s_{i} represents the
trustworthiness of the receiver 
• 
Game stages all players start with their initial endowment M = 10. In
the first stage, sender i sends the amount s_{i} to the receiver,
with the receiver receiving this amount multiplied by 3. In the second stage,
the receiver returns the amount r_{i} back to sender i. After the
second stage, the payoff of sender i is equal to π^{S}_{i}
= Ms_{i}+r_{i} and the payoff of the receiver is equal
to: 
EXPERIMENTAL DESIGN
The experiment had a form of a typical classroom experiment. It involved 62 participants, divided into 5 groups with n = 1, 4 groups with n = 2, 5 groups with n = 3 and 4 groups with n = 4 senders. Each group featured one receiver. All participants were university students of economics and finance in their 4 or 5th year of the study. Before starting the experiment, the participants had been asked to draw a random sealed envelope. This envelope contained the instructions of the experiment, the questionnaire and the playing card marked with the group the participant would be in and the role he would play within the group. This process ensured random matching of the players. The instructions for each sender included the information about the total number of senders in his group, as well as the information that the receiver would respond to each of the senders individually. Since, the participants were instructed to open the envelopes after they had been seated and were not allowed to communicate after this point, the matching was also anonymous. Once all participants were ready, the organizers of the experiment explained the rules of the experiment, the way the experiment would be operated and also the reward the participants would be paid after the experiment. Then, the experiment commenced. The experiment represented a one shot game. Each participant started the game with the initial endowment of M = 10. The experiment itself consisted of 3 stages. In the first stage, the participant that had been assigned the role of sender i had to write down into the playing card the amount s_{i} he wanted to send to the receiver in his group. At the same time, the receiver was asked to write down into the playing card his expected amount s*_{i} (the amount he expects to receive) for each sender i within his group. The first stage ended after the organisers had collected the playing cards of all participants. Before stage 2 commenced, the organisers had marked the amounts s_{i} sent by every sender i within a group into the playing card of the receiver of that group, as well as the total amount the receiver had available at this point to avoid mathematical errors of the participants. During this time, the participants were asked to fill out the provided questionnaires. Once this was done, all participants were returned their playing cards and stage 2 commenced. During this stage, each participant in the role of sender i was asked to write down in to the playing card the expected amount r*_{i}. Each receiver was asked to write down into the playing card the amount returned r_{i} for every sender i within his group. The second stage ended with the organisers collecting the playing cards of all the participants. Before stage 3 commenced, the organisers had marked the amounts r_{i} received by every sender i. Again, during this time the participants were asked to fill in the provided questionnaires. Once all the participants were returned their playing cards, the final stage commenced. During this stage, the participants reviewed their playing cards containing the final outcome of the game and were paid their adjusted total amount achieved during the experiment in real money. ANALYSIS Analyzing the experimental data we first look at the average amount of money sent s_{i}, money returned r_{i} and the level of trustworthiness r_{i}/s_{i} recorded. The results are given in Table 1.
As seen in Table 1, with the rising number of senders n in
a group, the average amount sent s_{i} is getting lower. In case of
the average amount returned r_{i}, this trend is not so clear and is
disrupted by n = 3. The average level of trustworthiness r_{i}/s_{i}
demonstrates no trend with the increasing number of senders. However, we have
to also take into account the personal characteristics of individual players
which require a more complex analysis, regarding data from the personal questionnaire.
Table 1: 
Average amounts sent and returned, the average level of trustworthiness 

Table 2: 
Statistically significant factors influencing sender trust 

Dependent variable s_{i}, Level of significance: ***p>0.01;
**p>0.05; *p>0.1, JB: JarqueBera, BP: BreuschPagan, RESET: Ramsey
specification error test 
The questionnaire was anonymous and contained various questions concerning
the altruism, income, expenditures, risk attitude, gender and family background
of the participants. A copy of the questionnaire is available on demand.
Sender analysis: Since, the individual senders had no way of communicating with each other and had no information about each other, we can use individual senders as the unit of the analysis. Due to incompletely filled questionnaires we excluded 3 senders from the analysis. Aiming to perform a linear regression we firstly excluded multicollinear variables based on the variance inflation factor (VIF>5). Then, proceeding from general to specific, we started with the overparametrized model including all the remaining variables. The statistically insignificant regressors were then excluded taking into account the Akaike information criterion and other standard linear regression tests. The results are given in Table 2. Here, out of all considered variables and questionnaire answers, the ones having statistically significant impact on the amounts sent (s_{i}) are the following:
• 
s*^{•}_{i} (second order expectation)
is the amount of monetary units the sender expects he is expected by the
receiver to send. The higher the expectation of the sender, the higher the
amount the sender sends. We can interpret this as a form of social responsibility 
• 
Gender (0male, 1female). Female senders sent on average 1,44 less than
male senders. This can be accounted for by the higher risk aversion of female
players and the fact that females seem to trust less than males in general
as shown by Chaudhuri and Gangadharan (2003) 
• 
Altruism is a fivedegree scale variable factor identified in the questionnaire.
This question asked the participants, if they would engage in voluntary
activities. As shown, the higher the level of altruism of a given sender,
the more he is willing to send. Here, altruism represents a form of unconditional
kindness as showed by Fehr and Gachter (2000) 
• 
No. senders (n) is the number of senders in the group. Although, the statistical
significance of this factor is quite disputable, removing it from the model
causes an increase of the Akaike information criterion. That is why we decided
not to exclude this variable. The result supports our observation that the
higher the number of competing senders in a group, the lower the average
amount they send. This could be attributed to the fact that if a sender
playing with three other senders in the same group sent the same amount
as a single sender playing in a pair, the resulting profit of the receiver
would be much higher. Therefore, it is reasonable to assume this motivates
senders to send lower amounts in bigger groups as a result of inequity aversion
(Rabin, 1993) 
Table 3: 
Significant factors influencing receiver trustworthinessordinary
least squares model 

Dependent variable r_{i}/s_{i}, ***p>0.01;
**p>0.05; *p>0.1, JB: JarqueBera, BP: BreuschPagan, RESET: Ramsey
specification error test 
Receiver analysis: For the purpose of analyzing the effect of the number of senders on the receiver’s trustworthiness r_{i}/s_{i} we use senderreceiver pair as the unit of the analysis. Here, receivers who returned more than they had received were excluded (7 pairs in total). Senderreceiver pairs with both amount sent and amount returned equal to 0 were also excluded (2 pairs). To avoid multicollinearity, we excluded variables according to their variance inflation factor (VIF>5). Then, using the ordinary least squares regression, we proceeded from general to specific by eliminating variables based on their statistical significance and Akaike Information Criterion (AIC). The results are given in Table 3.
Table 4: 
Significant factors influencing receiver trustworthinesslinear
mixed model 

Dependent variable r_{i}/s_{i}, ***p>0.01;
**p>0.05; *p>0.1 
In the ordinary least squares regression given in Table 2 we neglect to take into consideration the fact that the senderreceiver pairs belonging to a single receiver are influenced by his individual characteristics. To account for this fact, a linear mixed model is used with each receiver representing a group with a specific intercept (random effect in the model). Using the linear mixed model we increase the effectiveness of the estimation of parameters but on the other hand, we have to deal with the small sample bias which pertains in the model. Surprisingly, after using the general to specific methodology, we arrive at the same conclusions regarding the statistically significant factors influencing the receiver’s trustworthiness r_{i}/s_{i}. The results are presented in Table 4. The estimated regression coefficients of the two presented models display only negligible differences. Out of all considered variables and questionnaire answers, the ones with statistically significant impact on the amount (r_{i}/s_{i}) are the following:
• 
r*^{•}_{i}(second order expectation)
is the amount the receiver expects he is expected to return back to the
sender. According to the sign of the estimated regression coefficient, the
higher the receiver second order expectation, the higher his trustworthiness.
This can be interpreted as a form of social responsibility 
• 
Gender (0male, 1female). Female receivers presented a higher level of
trustworthiness than male receivers which can be explained by females being
more generous and demonstrating a higher level of reciprocity than males
in general (Chaudhuri and Gangadharan, 2003) 
• 
No. senders (n) is the number of senders in the group. The higher the
number of senders in the group, the higher the trustworthiness. This can
be interpreted by the fact that in larger groups with more senders, the
receiver is accumulating a higher total amount he will receive at the end
of the game and thus he will return more as a result of his inequity aversion
like in the study of Rabin (1993). On the other
hand, Holm and Danielson (2005) and other studies
showed that trustworthiness is constant with the amount sent in the case
of citizens of developing countries (balanced norm of reciprocity) and is
increasing in case of developed countries (conditional norm of reciprocity).
Then, our results could be the evidence of the conditional norm of reciprocity.
Interpreting this fact we must take into account that the total amount receiver
by the receiver increases with the number of senders in the group, even
though the amounts received from individual senders are decreasing (as showed
by the sender analysis in Table 2) 
• 
Relation represents 1; 0; 1 values based on the sign of the difference
between the amount really received and the amount the receiver expected
to receive from the sender (s_{i}s*_{i}). The expected
positive sign of the estimated regression coefficient could model the punishing/rewarding
effect appropriately (Dufwenberg and Kirchsteiger, 2004,
Falk and Fischbacher, 2006). Surprisingly, the negative
sign expresses the situation where the receiver that received the amount
higher than he had expected has shown a lower level of trustworthiness than
the receiver that has received the amount lower than he had expected to
receive. This fact could be caused by the receivers making their decisions
about the amount they will return back to the senders based partly on the
amount they expected to be sent and not only on the amount they were really
sent 
• 
Profit represents the amount the receiver keeps for himself from the total
amount received from all the senders in his group. The negative sign of
the estimated coefficient represents the material preferences of the receiver 
DISCUSSION
The article introduced an experiment based on a modification of the investment
game. We used a modified structure with a single receiver and a varying number
of senders (14). In the provided analysis we showed that the game structure
does have an effect on the level of trust of senders in a group, with the rising
number of senders lowering the level of trust. The analysis also showed that
the trust is affected by the gender of the senders, with female senders exhibiting
a lower level of trust than male senders. The other statistically significant
factor proved to be the sender altruism. Using a concept of the second order
expectations, the influence of the social responsibility also seems to be statistically
significant.
A similar analysis on the part of receivers proved that the number of senders, the receiver faces in the group, positively affects the receiver trustworthiness. We interpret this as a manifestation of the inequity aversion extensively discussed in literature. Other factors that proved to influence the trustworthiness of receivers include gender, social responsibility and material preferences. The difference between the amount the receiver really received and he had expected to receive proved to be statistically significant, as the receivers also base the amount they return to the senders on the amount they expected to receive and not only on the amount they really received.

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