Three-player trust game with insider communication.
Sheremeta, Roman M. ; Zhang, Jingjing
I. INTRODUCTION
Trust and reciprocity play important roles in economic
interactions. The most frequently used measure of trust and reciprocity
in economics is based on a two-player trust game, proposed by Berg,
Dickhaut, and McCabe (1995). In this game, the first player (trustor)
sends any part of his endowment to the second player (trustee). The
amount sent is tripled and the second player decides how much to return.
Berg et al., as well as many replications, show that most participants
display trust and trustworthiness contrary to self-interested
profit-maximizing behavior (Burks, Carpenter, and Verhoogen 2003;
Glaeser et al. 2000; McCabe, Rassenti, and Smith 1998; McCabe, Rigdon,
and Smith 2003; McCabe and Smith 2000). However, the bilateral relation
in the two-player trust game rules out the multiple levels of trust that
often emerge in real life when more than two agents are involved. For
example, customers trust that the retailer will link them to a reliable
producer. Safari travelers rely on their domestic travel company to
match them with a high-quality foreign travel agent in Africa. Web
businesses connect people with hotels, houses, condominiums, and other
accommodations for rent. In all these relationships, the retailer, the
domestic travel company, and the web businesses serve as a middleman
linking users to goods and services. Whether to purchase via a middleman
depends on the degree to which users are willing to accept vulnerability
based on positive expectations of both the middleman and the provider.
The redistribution of the benefits in these types of transactions is
mainly controlled by the last player in the chain who provides goods or
services to the customer and pays a commission to the middleman for
making the linkage.
Multilevel trust interactions are also common in financial markets.
For example, a person investing in a bond fund must trust the fund
manager to correctly represent the bonds in the fund. The fund manager,
in turn, must trust the bond issuers. The same intuition applies in the
fund of funds (FOF) industry, where the manager of a hedge fund company
invests in other funds instead of individual securities. Thus, multiple
(direct and indirect) levels of trust are required between the
individual investor, the hedge fund manager, and the FOF manager.
Finally, multilevel trust is crucial in workplaces where the workers
must not only trust their managers to report their performance
truthfully to the CEO, but also trust that the CEO will appropriately
reward their performance.
This study provides a framework for understanding multilevel trust
interactions in complex environments involving direct and indirect
interactions among multiple players. We depart from the conventional
two-player trust game of Berg, Dickhaut, and McCabe (1995) by
introducing the third player. In our three-player trust game, the three
players move sequentially. (1) The first player sends any portion of his
endowment to the second player, with the amount being tripled. (2) The
second player then decides how much to send to the third player, with
the amount being tripled again. (3) Finally, the third player decides
the final allocation among three players. The three-player trust game
captures the essential elements of complex multilevel trusting and
reciprocal behavior in a simplified setting.
Moreover, trust in multilevel interactions depends on the thickness
and the pattern of the links between players. One of the indispensable
social lubricants for the network of trust and reciprocity is
communication. The multilevel interactions introduced by adding the
third player provide us a useful platform to explore our second research
question: what are the internal and external effects of communication on
trust and reciprocity? There are many potential channels of
communication that one can investigate in the three-player trust game,
but the considerable complexity that arises with the introduction of
communication is nontrivial. As a first step, we focus on studying
communication between the second and the third player which resembles
insider communication in a group when only a subgroup is allowed to
communicate (as far as we know, this is the first laboratory study of
insider communication). (1) In the FOF example, there is a potential for
privileged insider communication between the FOF manager and the
managers of the hedge funds. Similarly, in the workers-manager-CEO
example, the detailed discussions CEO and managers have in the board
room are often not revealed to workers.
We conducted treatments with and without insider communication. The
results of our experiment indicate that even in the baseline treatment
with no communication, the first and second players send significant
amounts and the third player reciprocates. When we allow communication
between the second and the third player, the amounts sent and returned
between these two increase. The new interesting finding is that there is
an external effect of insider communication: the first player who is
outside communication sends 54% more and receives 289% more than in the
baseline treatment. As a result, insider communication increases
efficiency from 44% to 68%. Content analysis of the communication
reveals that what drives the most efficient outcomes are the proposals
of equal split among three players made by either the second or the
third player. The effect of these types of proposals is strong enough to
overcome tendencies toward collusion between the second and the third
player.
Our three-player trust game is related to a three-player centipede
game of Rapoport et al. (2003) and Murphy, Rapoport, and Parco (2004).
(2) The three-player centipede game is a multistage game which can be
used to address some aspects of indirect trust (Camerer 2003). However,
the strategy space of each player in the three-player centipede game is
restricted to a binary choice, whether to end the game and take some
percentage of the available surplus, or to increase the surplus and
allow other players a chance to end the game. Thus, it allows observing
only whether indirect trust exists but not the magnitude of indirect
trust. The three-player trust game proposed in this study is general
enough to capture both the degrees of direct and indirect trust and
reciprocity by using a continuous strategy space for each player.
Moreover, our game gives us the flexibility to analyze different
communication channels and, in this paper, we focus on the external
effect of insider communication which is new to the communication
literature.
II. THREE-PLAYER TRUST GAME
We introduce a novel three-player trust game, where player I acts
as a trustor, player 2 embodies both the trustor's and
trustee's characteristics, and player 3 always acts as a trustee.
All players 1, 2, and 3 are endowed with [e.sub.1], [e.sub.2], and
[e.sub.3]. Player 1 can send a portion oti2 of his endowment [e.sub.1]
to player 2. The amount sent by player 1 is multiplied by factor k\.
Then player 2 can send a portion [[alpha].sub.23] of his total income to
player 3. The amount sent by player 2 is multiplied by factor [k.sub.2].
Then player 3 can reciprocate to players 1 and 2 by returning portions
of the total money received ([[alpha].sub.31] > 0 and
[[alpha].sub.32] > 0). It is important to emphasize that, in
returning to player 1, player 3 may be motivated by direct reciprocity
and two types of indirect reciprocity, that is, observation-based and
experience-based. (3) Moreover, being reciprocal only requires returning
positive amounts, while being trustworthy requires returning at least as
much as the amount received (McCabe, Rigdon, and Smith 2003).
The unique subgame perfect Nash equilibrium in the three-player
trust game, which assumes that all players maximize their earnings, is
for all players to send nothing. By backwards induction, player 2 knows
that a rational player 3 will not return anything ([[alpha].sub.32] =
[[alpha].sub.31] =0) and therefore player 2 should send nothing
([[alpha].sub.23] = 0). Anticipating this, player 1 should send nothing
to player 2 ([[alpha].sub.12] = 0). In this setting, if player 1 sends
any positive amount ([[alpha].sub.12] > 0), it means he is willing to
take a risky bet that both players 2 and 3 will reciprocate. In other
words, player 1 exhibits direct trust in player 2 and indirect trust in
player 3. It is riskier to trust in this game than in the two-player
game because player 1 is repaid by player 3 and not by player 2.
Therefore, player 1 has to trust that player 2 will pass the money to
player 3 and also trust that player 3 will be trustworthy. The most
efficient outcome is when both players 1 and 2 fully trust player 3 by
sending all of their incomes.
III. EXPERIMENTAL DESIGN AND HYPOTHESES
A. Experimental Design
We conducted an experiment in which each session had two
treatments: a no communication treatment (NC) and a communication
treatment (C). Both treatments lasted for ten periods. We used a random
stranger protocol with fixed roles. In the NC treatment, all subjects
were randomly assigned to a specific role, designated as player 1,
player 2, or player 3. Each subject remained in the same role throughout
the experiment. At the beginning of each period, each player was endowed
with [e.sub.1] = [e.sub.2] = [e.sub.3] = 100 experimental francs and was
randomly regrouped with two other players to form a three-player group,
with each player in a different role. Player 1 made a decision on how
many francs between 0 and 100 to send to player 2 and how many francs to
keep. Each franc sent by player 1 was tripled ([k.sub.1] = 3). After
players 2 and 3 learned the amount of francs sent by player 1, player 2
then made a decision on how many francs to send to player 3. The amount
sent by player 2 was also tripled ([k.sub.2] = 3). Finally, player 3
made a decision on how many francs to return to player 1, how many
francs to return to player 2, and how many francs to keep. All subjects
were told that player 1, player 2, and player 3 can send some, all, or
none of the francs available to them. At the end of each period, the
amounts sent and returned by all players were reported for everyone to
see. Instructions, available in Appendix SI (supporting information),
explain the structure of the game in detail.
To study the effects of insider communication we conducted a
treatment C. The design of the C treatment closely followed the design
of the NC treatment except that, after player 1 made his decision,
players 2 and 3 were able to communicate for 90 seconds in a text-based
"chat room." Communication took place only after players 2 and
3 learned the decision made by player 1. Subjects were told that only
players 2 and 3 would see the messages. In sending messages back and
forth, we requested subjects to be civil to each other and not to reveal
their identities.
A total of 72 undergraduate student subjects from Purdue University
participated in our experiment. The computerized experimental sessions
were run using z-Tree (Fischbacher 2007). We ran two NC-C sessions, in
which a total of 36 subjects were engaged in ten interactions with no
communication and then ten interactions with communication (NC-C
sessions). The other 36 subjects participated in the C-NC sessions,
where we reversed the order of the treatments. (4) After completing all
20 decision periods, four periods were randomly selected for payment
(two periods for each treatment). The earnings were converted into U.S.
dollars at the rate of 100 francs to $1. On average, subjects earned $16
each and the experiment session lasted for about 90 minutes.
B. Hypotheses
Previous studies have shown that subjects care about treating
others fairly (Fehr and Gachter 2000a), they display trust and
trustworthiness contrary to self-interested profit-maximizing behavior
(Berg, Dickhaut, and McCabe 1995; McCabe, Rassenti, and Smith 1998),
they are concerned about efficiency (Engelmann and Strobel 2004), and
they have unconditional other-regarding preferences (Bolton and
Ockenfels 2000; Cox 2004; Fehr and Schmidt 1999). In evolutionary
literature it is found that people exhibit direct and indirect trust in
other people (Buchner et al. 2004; Greiner and Levati 2005). (5) On the
basis of these observations we provide the following hypothesis.
Hypothesis 1: Players 1 and 2 trust player 3 by sending positive
amounts, and player 3 reciprocates.
It is also documented in a two-player trust game that the levels of
direct trust and reciprocity are higher than the levels of indirect
trust and reciprocity (Dufwenberg et al. 2001; Guth et al. 2001; Seinen
and Schram 2006; Wedekind and Milinski 2000). (6) Therefore, we expect
that:
Hypothesis 2: Player 2 trusts more than player 1, and player 3
reciprocates to player 2 more than to player 1.
We base our hypothesis about the effects of insider communication
in the three-player trust game on previous findings in the communication
literature. Several experimental studies of one-shot two-player trust
games show that communication increases cooperation between trustor and
trustee (Ben-Ner and Putterman 2009; BenNer, Putterman, and Ren 2011;
Buchan, Croson, and Johnson 2006; Charness and Dufwenberg 2006; Glaeser
et al. 2000). (7) Communication also improves cooperation in prisoner
dilemma games (Wichman 1972), public good games (Isaac and Walker 1988),
common-pool resource games (Hackett, Schlager, and Walker 1994), voting
experiments (Schram and Sonnemans 1996; Zhang 2012), and contests
(Cason, Sheremeta, and Zhang 2012; Sheremeta and Zhang 2010). Social
psychologists have identified several means by which communication can
increase cooperation: communication creates group identity, thus
improving group welfare, and communication elicits commitments, creating
a promise-keeping norm (Bicchieri 2002; Bornstein 1992; Kerr and
Kaufman-Gilliland 1994). In our three-player trust game, insider
communication occurs between players 2 and 3. Therefore, we expect that:
Hypothesis 3: With insider communication, player 2 trusts player 3
more, and player 3 reciprocates more.
According to the social identity theory (Chen and Li 2009; Tajfel
and Turner 1979), individuals may put themselves and others into
different categories based on perceived similarities and differences
(categorization), identify others as in-group or out-group members
(identification), and discriminate in favor of the in-group and against
the out-group members (comparison). Various methods have been used to
induce saliency of group identity, including communication between group
members (Cason, Sheremeta, and Zhang 2012; Sutter 2009). As in our
experiment insider communication occurs only between players 2 and 3,
these players should identify each other as in-group members, while
categorizing player 1 as an out-group. Such categorization would imply
collusion between players 2 and 3, and thus less trust from player 1. On
the other hand, as discussed previously, communication should enhance
trust and trustworthiness between players 2 and 3, thus increasing their
payoffs (Ben-Ner. Putterman, and Ren 2011). Given that some individuals
have preferences for equal distribution of payoffs (Bolton and Ockenfels
2000; Fehr and Schmidt 1999), it is likely for players 2 and 3 to share
their higher payoffs with player 1, which in turn may increase the trust
level of player 1. In summary, depending on whether the "equal
distribution" effect or the "collusion" effect dominates,
player 1 will either trust more, less, or the same. This is an empirical
question for us to test against the following null hypothesis.
Hypothesis 4: With insider communication, player 1 trusts the same.
IV. RESULTS
Our analysis in Section III.A focuses on the first ten-period data
before switching to a different treatment. We discuss the order effect
in details using all 20-period data in Section III.B. We mainly use
parametric tests and multilevel mixed-effects linear regressions to
analyze individual decisions. (8) The regression models have random
effects at both the individual level and the session level to control
for correlations that may arise between individuals due to the random
regrouping within a session over time. The within-subject residuals are
estimated as being autoregressive of order 2 to account for the repeated
measurement for each individual.
A. Trust and Trustworthiness
Table 1 summarizes the average amount sent and the profit earned by
all players in the C and NC treatments. Among three players, player 1
earns the lowest profit while player 3 earns the highest profit in the
experiment. In line with Hypothesis 1, in the NC treatment, players 1
and 2 trust player 3 by sending significant amounts, and player 3
reciprocates. Moreover, in line with Hypothesis 2, the level of indirect
trust exhibited by player 1, which is represented by 39 francs sent to
player 2 (39% of income), is significantly lower than the level of
direct trust by player 2, which is represented by 96 francs sent to
player 3 (43% of income). The reciprocal behavior of player 3 is also in
agreement with Hypothesis 2, with player 3 returning more to player 2
than to player 1 (57 vs. 35 francs, 10% vs. 7% of income) but the
difference is only marginally significant. (9) On the other hand, on
average, player 3 returns 90% of the amount received from player 1 but
only 59% of the amount received from player 2. Thus, in terms of
trustworthiness, neither player 1's nor player 2's trust pays
off. On average, player 2 passes on 82% of the tripled amount received
from player 1 without risking his own endowment. Without communication,
efficiency is 44%.
When insider communication is allowed between players 2 and 3,
efficiency increases significantly from 44% to 68%. This is because on
average player 2 sends to player 3 the entire tripled amount received
from player 1 plus 50% of his own endowment. As we will show in Section
V, player 1 correctly anticipates the increase in trust player 2 places
on player 3 and sends 60% of his endowment to player 2 (54% more than in
the NC treatment). Player 3 is trustworthy--player 1 receives twice the
amount sent and player 2 receives 107% of the amount sent.
Interestingly, the increased trust and trustworthiness do not change the
distribution of payoffs among three players.
Table 2 reports the estimation results of the mixed-effects linear
regressions, where the dependent variable is the amount sent by each
player in each period and the independent variables are a treatment
dummy-variable and a period trend. (10) As we expected, when
communication is allowed, player 2 exhibits more trust in player 3
(specification 2). Controlling for the amount player 2 receives from
player 1, the share of income sent by player 2 is significantly higher
in the C treatment (specification 6). Anticipating this increase, player
1 sends more to player 2 (specifications 1 and 5). Comparing to the NC
treatment, player 3 returns higher absolute and relative amounts to
players I and 2 in the C treatment (specifications 3, 4, 7, 8). The two
panels in Figure 1 show that the distribution of return ratio is shifted
toward more generous behavior of player 3 in the C treatment as compared
to the NC treatment. These findings are consistent with Hypothesis 3.
Although only players 2 and 3 were allowed to communicate, we find
that the amount player 1 sends to player 2 in the C treatment is
increased by 54%. This finding rejects the null Hypothesis 4. We
conjectured that the trust level of player 1 would fall in the C
treatment if communication would serve as a collusion device between
players 2 and 3. In fact, we do find evidence that insider communication
increases the collusion between players 2 and 3. Table 3 categorizes
player 3's decisions conditional on positive amounts sent by
players 1 and 2. In the C treatment, player 3 returns roughly half of
his income to player 2 and nothing to player 1 in around 11% of the
time. This did not happen once in the NC treatment. Communication also
significantly decreases the percentage of players 3 who are trustworthy
to player 1 but not to player 2 and increases the percentage of players
3 who are trustworthy to player 2 but not to player 1. Then the question
is why would communication increase trust of player 1? The answer turns
out to be very simple. In the NC treatment, player 3 almost never splits
the income equally between three players. In the C treatment, this
happens 28% of the time. Also, there is a significant decrease of the
proportion where player 3 keeps everything to himself from 24% in the NC
treatment to 12% in the C treatment and an increase of the proportion
where player 3 is trustworthy to both players 1 and 2. Therefore, in the
C treatment, player 1 receives 288% more than in the NC treatment. This
means that insider communication has two opposite effects on the amount
player 3 returns to player 1: (1) insider communication enhances
collusion between players 2 and 3, and (2) it also activates fairness
norms and thus increases cooperation between all players. The
cooperation effect dominates the collusion effect leading to significant
efficiency gains. (11) The efficiency in the NC treatment is about 44%
while in the C treatment it is 68% (see Table 1). Moreover, as a result
of communication, all players earn higher payoffs (see Table l). (12)
[FIGURE 1 OMITTED]
To better understand the determinants of trust and trustworthiness,
Table 4 reports estimation results of different regression models, where
the dependent variable is the amount sent by players 1, 2, and 3. To
control for endogeneity we use three-stage estimation for systems of
simultaneous equations with individual subject dummies. Besides a
treatment dummy-variable and a period trend, we also include the
observable decisions in the current period and the average amounts sent
or received by each player across all past periods. (13) Although we
randomly regrouped all players with fixed roles after each period, from
specification 1 we see that the amount player 1 sends to player 2
depends on the average amount player 1 received from all previous
players 3. This finding suggests that player 1 is learning about the
general level of trustworthiness exhibited by player 3. Similarly, the
amount player 2 sends to player 3 depends on the average amount player 3
returned to player 2 in all past periods (specification 2).
Besides the past observable decisions, specifications 2, 3, and 4
show that the current period's observable choices are significant
determinants of the trusting and reciprocal behavior. Specifically, the
more player 1 sends to player 2, the more player 2 passes on to player 3
and the more player 3 returns to player 1. More interestingly, for a
given amount that player 2 sends to player 3, the more player 1 sends to
player 2, the less player 3 returns to player 2 (specification 3) and
the more player 3 returns to player 1 (specification 4). Thus, player 3
reciprocates to player 2 accounting for the decisions made by player 1.
In other words, consistent with Nowak and Sigmund (2005), we find
evidence for both the observation-based indirect reciprocity (the amount
player 3 returns to player 1 increases when player 1 sends more to
player 2) and the experienced-based indirect reciprocity (the amount
player 3 returns to player 2 increases when player 2 sends more to
player 3).
B. Order Effects
We conducted both C-NC sessions and NC-C sessions to examine if
there is a significant order effect. Specifically, one interesting
question is whether cooperation which subjects achieve during the C
treatment could be sustained in the NC treatment when communication is
removed. Figure 2A and B displays the time trend of average amount sent
by all players in different sessions. Figure 2A suggests that
communication in the C treatment indeed influences the behavior of
players in the consecutive NC treatment. The average amount sent by each
player in the NC treatment is higher in the C-NC session (Figure 2A)
than in the NC-C session (Figure 2B).
To further account for order effects, Table 5 reports mixed-effects
regressions of the amount all players sent on treatment and order
variables. Four dummy variables that capture the treatment and order
variations are included. The variable C-treatment x NC-C is equal to 1
if treatment is C and the session is NC-C. The variable C-treatment x
C-NC is equal to 1 if treatment is C and the session is C-NC. We use the
Wald test comparing these two variables to measure the significance of
the order effect for the C treatment (see the second to the last line in
Table 5). Similarly, two variables for the NC treatment in the NC-C
session and C-NC session are included and the corresponding Wald tests
are reported in the last line of Table 5. Clearly, order has a
significant effect on the absolute amount sent by all players in both
treatments. Particularly, communication is more effective in the NC-C
sessions than in the CNC sessions. A possible explanation is that in the
NC-C sessions, after ten periods of the NC treatment, subjects
understand better the efficiency cost of poor cooperation, and thus they
significantly increase cooperation in the following C treatment.
Although there is a decay of cooperation after we disable communication
in the C-NC sessions, the level of cooperation is still significantly
higher than in the first half of the NC-C sessions. (14)
V. BELIEFS AND MESSAGES
A. Beliefs
In both C and NC treatments, after making the decision on how much
to send to player 2, we asked player 1 to make a prediction about the
actions of players 2 and 3 before seeing the outcome screen. (15) Player
1 was asked to guess how much player 2 would send to player 3, how much
player 3 would return to player 2, and how much player 3 would return to
player 1. Subjects were financially motivated to make correct
predictions. They were paid 10 francs for each prediction if the
prediction differed by no more than 5% from the actual decision made.
(16) We chose this belief-elicitation protocol instead of the
quadratic-scoring rule mainly because it is simple and rather easy for
subjects to understand.
[FIGURE 2 OMITTED]
Table 6 reports the average predictions of player 1 on the amounts
sent by player 2 and returned by player 3 and the average percentage
differences from the actual decisions made from the first ten periods.
On average player 1 makes good predictions on the amount player 2 sends
to player 3 and the amount player 3 returns to player 2 in both C and NC
treatments. (17) However, in both treatments, player 1 significantly
overestimates the amount player 3 returns to player 1. (18) This
overestimation may partially explain the high level of trust exhibited
by player 1 in the three-player trust game.
B. Content Analysis of Communication
At this point we know that insider communication enhances
cooperation in the group of three people although only a subgroup of two
people is allowed to communicate. This brings us to the question of what
kinds of messages cause this cooperation. We use content analysis to
answer this question.
The procedure that we used to quantify the recorded messages is as
follows. First, we randomly selected a session to develop a coding
scheme. We classified the messages into 18 categories, shown in Table 7.
Then we employed two undergraduate students to code all messages into
the coding categories independently. The unit of observation for coding
was all messages sent out in a given period before subjects made
decisions. Coders were asked not to start coding until they had finished
reading all messages in a given period. If a unit of observation was
deemed to contain the relevant category of content, it was coded as 1
and 0 otherwise. Each unit was coded under as many or few categories as
the coders deemed appropriate. The coders were not informed about any
hypotheses of the study. (19)
We use Cohen's Kappa A" as a reliability measurement of
the between-coder agreement. This measurement determines to which extent
the coders agree that a certain message belongs to a particular coding
category. Cohen's reliability measurement accounts for the
between-coder agreement by chance (Hayes 2005). (20) Reliability K
greater than zero indicates that the proportion of agreements exceeds
the proportion of agreements expected by chance. According to Landis and
Koch (1977), K between 0.4 and 0.6 corresponds to a moderate agreement
level and K greater than 0.6 corresponds to full agreement. Table 7
displays the coding scheme along with Cohen's reliability indexes
and the frequency of coding for the C treatment. For the vast majority
of categories, K is greater than 0.5. As a result of infrequent coding
there are few categories that have unsatisfactory agreement levels. In
further discussions of categories we use the average of the two
independent codings. Specifically, the value of coding is treated as 1
if two coders agree that a message belongs to a given category, 0 if two
coders agree that a message does not belong to a given category, and 0.5
if the two coders disagree with each other.
Table 8 reports the estimation results of mixed-effects models
which have random effects on both the subjects and session levels and
account for second-order autocorrelation in the within-subject
residuals. The dependent variables are the absolute (specifications 1 to
3) and relative (specifications 4 to 6) amounts sent and returned by
players 2 and 3 and the independent variables are various categories of
messages. In all regressions, we include a trend variable equals the
period number and a constant. The first four independent variables code
the cases when only one proposal was made and differ by who made the
proposal and whether the proposal was to share the profit equally
between players 2 and 3 or among all three players. The next two
variables quantify the cases when both players 2 and 3 proposed the same
strategy. The seventh and eighth variables capture the cases when the
two exchanged different proposals--to collude versus to cooperate. The
last two message variables are the most frequently coded categories
besides making proposals.
There are a number of notable findings. When either player 2 (la)
or player 3 (2a) or both (1a + 2a) propose to collude between
themselves, player 3 returns significantly less absolute and relative
amounts to player 1 (specifications 3 and 6). The collusion proposal
significantly increases the amount player 2 sends to player 3 only when
both of them proposed it (1a + 2a) and has much less effect on the
amount player 3 returns to player 2. When either player 2 (lb) or player
3 (2b) or both (1b + 2b) propose to share equally among all three
players, both the absolute and relative amounts player 2 sends to player
3 and player 3 returns to player 1 significantly increase. The
cooperative proposal significantly increases the absolute amount player
3 sends to player 2 only when player 2 proposes it and has no effect on
the relative amount.
Interestingly, when a collusion proposal is challenged by a
cooperative proposal, the negative effect of collusion proposals on the
amount player 3 returns to player 1 is offset (1a + 2b, specification 3)
or even reversed (1b + 2a, specifications 3 and 6). The positive effect
of cooperative proposals on the amount exchanged between players 2 and 3
also disappears.
Finally, promises made by player 3 and appeals made by player 2 do
not seem to influence the final decisions.
Therefore, content analysis reveals that the proposals of equal
split among three players, especially when such proposals were made by
both players or used to challenge the collusion proposal, significantly
increase cooperation between all players, and thus efficiency.
VI. CONCLUSIONS
This paper presents an experimental study of a novel three-player
trust game. In this game, player 1 acts as a trustor, player 2 embodies
both the trustor's and trustee's characteristics, and player 3
always acts as a trustee. We also investigate the internal and external
effects of insider communication on direct and indirect trust and
reciprocity. Although the three-player trust game requires additional
layers of trust than the standard two-player trust game, we still find a
substantial level of direct and indirect trust even when there is no
communication. Consistent with other studies, we find that the level of
direct trust and reciprocity is higher than the level of indirect trust
and reciprocity.
Regarding insider communication, we find that players 2 and 3 who
are engaged in communication exhibit more trust and trustworthiness. The
most unexpected and positive result of our experiment is the effect
insider communication has on player 1's behavior. Although only
players 2 and 3 are allowed to communicate, we find that player 1's
trust increases by 54%. This is because communication activates stronger
preference for fairness than collusion between players 2 and 3.
Expecting that, player 1 exhibits more trust in players 2 and 3. In
response, player 3 returns higher absolute and relative amounts to
player 1. Belief elicitation reveals that player 1 persistently
overestimates the trustworthiness of player 3, which may also account
for the high level of trust exhibited by player 1. We also find that the
social norms developed during the communication stage carry over to the
no communication stage.
Finally, we use content analysis to study what kinds of messages
enhance cooperation. In the multivariate analysis of communication, we
find that the messages that significantly increase cooperation are the
ones that indicate willingness to split all earnings equally.
Our study provides evidence that economic agents exhibit direct and
indirect trust in multilevel interactions among strangers. One mechanism
that can further promote trust and reciprocity is communication even
when only a subgroup of agents can afford to communicate with each
other. Since communication between insiders may raise the concerns of
forming collusion at the cost of the outsiders, to better use this
mechanism, insiders should deliver the idea that communication activates
more fairness norms toward the outsiders. This suggests that to build
trust with individual investors in FOF. managers have to send clear
signals to investors that their interests of obtaining cooperative,
fair, and efficient outcomes from the investment are perfectly aligned.
As a first attempt to use simplified laboratory experiments to
explore trusting behaviour and effect of communication in multilevel
interactions, caution would be suggested in drawing direct inferences
from our results. Nevertheless, our findings may shed some light on the
important causal factors affecting the emergence of many web-based
auctions and other forms of online businesses which are built on trust
and reciprocity among strangers (Resnick and Zeckhauser 2002). For
example, in the wholesale eBay online auction, as a consumer wholesale
distributor, you can buy products at an unbeatable wholesale price from
suppliers and then set your sale price in online auctions. Advertising
fair trade between you and the wholesale suppliers may help to attract
more buyers.
There are many interesting extensions to our research. Future work
can investigate how trust and reciprocity are affected by different
channels of communication, other interactions between players (e.g.,
player 2 can also directly return to player 1), and factors such as the
size of the endowment and multipliers in the three-player trust game.
ABBREVIATION
FOF: Fund of Funds
SUPPORTING INFORMATION
Additional Supporting Information may be found in the online
version of this article:
doi:10.1111/ecin.12018
Appendix S1. The Instructions for the NC-C Session.
Appendix S2. Instructions for Coders.
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(1.) We chose to study insider communication between players 2 and
3 for several reasons. First, full communication among all players is
less feasible in reality and easier to break down as the size of the
communicating group grows. Second, restricting communication between
insiders allows us to examine the impact of such asymmetric
communication on both the insiders' and outsiders' behavior.
Third, as we discuss in Section III. predictions about the effects of
insider communication are not trivial, and thus conducting a laboratory
experiment is important.
(2.) In a repeated three-player centipede game, Rapoport et al.
(2003) find that neither full cooperation nor full non-cooperation is
supported. In a mixed population of human players and robots, Murphy,
Rapoport, and Parco (2004) find that there is an increase in the
propensity of human players to cooperate over time when a handful of
cooperative robots are added while adding a handful of non-cooperative
robots does not change the cooperation rate.
(3.) In the terminology of Nowak and Sigmund (2005) there is direct
reciprocity and two types of indirect reciprocity, i.e., upstream or
observation-based ("A helps B because B helped C") and
downstream or experience-based ("A helps B because C helped
A"). In our experiment, player 3 may reciprocate to player 1
because player 1 indirectly helped player 3 (direct reciprocity),
because player 1 helped player 2 (observation-based indirect
reciprocity), and because player 2 helped player 3 (experience-based
indirect reciprocity). We report the evidence of the two types of
indirect reciprocity in Section IV.
(4.) Two sessions (one NC-C and one C-NC) had 12 subjects and two
other sessions had 24 subjects.
(5.) Greiner and Levati (2005) use a variant of a trust game in
order to implement a cyclical network of indirect reciprocity where the
first individual may help the second, the second help the third, and so
on until the last, who in turn may help the first. Like in a two-player
trust game, the authors find that pure indirect reciprocity enables
mutual trust in the multi-player environment. Buchner et al. (2004)
compare the trust-reciprocity regimes with the explicit incentive
schemes in the context of a three-person ultimatum game. They find that
mutual trust is as good as incentive contracts in inducing costly
actions by employees.
(6.) Dufwenberg et al. (2001) allow trustees to reciprocate toward
the other trustors, and find that indirect reciprocity induces only
insignificantly smaller donations than direct reciprocity and that
trustees are more rewarding in the case of indirect reciprocity. Guth et
al. (2001) find that indirect reward reduces significantly mutual
cooperation compared to the direct reward. In the same line of research,
Seinen and Schram (2006) and Wedekind and Milinski (2000) provide
experimental evidence on indirect reciprocity in the "repeated
helping game" developed by Nowak and Sigmund (1998). In this game,
donors decide whether or not to provide costly help to the recipients
they are matched with, based on information about the recipients'
behavior in encounters with third parties.
(7.) Glaeser et al. (2000) allow face-to-face communication before
playing the trust game. They find that when individuals are closer
socially, both trust and trustworthiness increase. They conclude that
trusting behavior in the experiments is predicted by past trusting
behavior outside of the experiments. Buchan, Croson, and Johnson (2006)
allow subjects to engage in personal but not task-relevant communication
before playing the trust game and find significant increase of trust and
trustworthiness. Charness and Dufwenberg (2006) allow either trustor or
trustee, but not both, to send free-form messages in a binary trust
game. They find that the messages sent by trustees increase both trust
and trustworthiness. However, no such effect is found when only trustors
can send messages. Ben-Ner and Putterman (2009) allow two-way
communication and find that verbal communication helps subjects to reach
agreement even without visual or auditory contact. Similarly, Ben-Ner,
Putterman, and Ren (2011) allow two-way communication and find that
trust and trustworthiness increase when verbal communication is allowed.
(8.) For a robustness check, we also estimated panel models with
individual subjects representing the random effects (to control for
individual effects), standard errors clustered at the session level (to
control for possible correlation within a session), and a period trend
(to control for learning and experience). The estimation results confirm
our main conclusions and are available from the authors upon request.
All p values reported in the paper are two-sided unless otherwise
stated.
(9.) To formally test Hypothesis 2, we estimated the two-level
mixed-effects model where the dependent variable is the amount sent per
period by players 1 and 2 and the independent variables are a constant,
a period variable, and a player-type dummy. Based on the estimation, the
amount sent by player 1 to player 2 is significantly lower than the
amount sent by player 2 to player 3 (p value < .01). The significance
disappears when we regress on the share of income sent which accounts
for the amount received by player 2 before sending to player 3. A
similar model regressing the amount player 3 returns to players 2 and 1
on the same set of independent variables reports that player 3 sent back
marginally more to player 2 than to player 1 (one-sided p value = .10).
No significant difference is found based on the relative amount sent by
player 3.
(10.) The use of non-parametric tests is not feasible in our
analyses, as the observations are not independent. Instead we reserve to
regressions which control individual effects (since the same subject
makes multiple choices), session effects (since all subjects interact in
the same session), and a time trend (since the trust game is repeated).
(11.) A two-sided proportion test indicates that the number of
cases where player 3 splits equally between all three players is
significantly higher than the number of cases where player 3 splits only
between players 2 and 3 in C treatment (p value < .01).
(12.) Based on the estimation of mixed-effect models where the
dependent variables are the period profits for each player and the
independent variables are a treatment dummy and a period trend, we find
that profits are significantly higher in the communication treatment for
all player types (p value < .01 for players 1 and 2, p value = .01
for player 3). A similar model regressing total earnings of three
players per period on a treatment dummy and a period trend reports
significantly positive communication effect (p value < .01).
(13.) Each subject can see the decisions of all three participants
in the group made at each stage from the outcome screen and we asked
subjects to write down all the decisions for each period in the persona]
record sheet.
(14.) The increase of cooperation when communication is introduced
in NC-C sessions and the decay of cooperation when communication is
removed in C-NC sessions are similar to the findings with respect to the
effect of costly punishment in repeated public goods game with stranger
protocol (Fehr and Gachter 2000b). More interestingly, we find the
communication is less effective in C-NC sessions than in NC-C sessions.
Such order effect is not observed with the punishment mechanism. Thanks
to an anonymous referee for pointing out this analogy.
(15.) We chose to elicit the beliefs of only player 1 for several
reasons. First, the most interesting questions of the current paper are
about player 1's behavior, so eliciting player 1's belief was
a natural choice. Second, player 1 had the most "free" time in
the experiment. After making the decision, player 1 would have to wait
for about 5 minutes before players 2 and 3 communicated and made their
decisions. The fact that players 2 and 3 were more occupied in our
experiment also motivated us not to elicit players 2's and 3's
beliefs. Finally, we felt that subjects assigned as player 1 had the
least interesting roles, in a sense that they had to make the same
unconditional decisions over and over again. So, we decided to provide
player 1 with an additional "productive" task.
(16.) It is also important to emphasize that eliciting beliefs may
cause risk-averse subjects to hedge between choices made in the
experiment and incentivized belief statements. However, Blanco et al.
(2010) find no evidence of such hedging.
(17.) Based on the estimation of random effect models, where the
dependent variable is the amount predicted minus the actual amount sent,
we find that the difference between predicted and actual behavior of
player 2 is not significantly different from zero neither in the NC
treatment (p value = .14) nor in the C treatment (p value = .92).
Similarly, the difference between predicted and actual behavior of
player 3 toward player 2 is not significantly different from zero
neither in the NC treatment (p value = .34) nor in the C treatment (p
value = .83).
(18.) Based on the estimation of random effect models, where the
dependent variable is the amount predicted minus the actual amount sent,
we find that the difference between predicted and actual behavior of
player 3 toward player 1 is significantly different from zero both in
the NC treatment (p value < .01) and in the C treatment (p value =
.02).
(19.) The instructions for coders are available in Appendix S2.
(20.) For binary 0 or 1 coding, agreement by chance is 50%.
ROMAN M. SHEREMETA and JINGJING ZHANG *
* The earlier version of this paper circulated under the title
"Multi-Level Trust Game with 'Insider'
Communication." We thank the associate editors and two anonymous
referees for helpful comments in revising the manuscript. We also thank
Tim Cason, John Dickhaut, Jerry Hurley, Luke Lindsay Stuart Mestelman,
Mohamed Shehata, Anya Savikhin, seminar participants at Purdue
University, and participants at the Economic Science Association meeting
for helpful discussions and comments. This research has been supported
by the National Science Foundation (SES-0721019). J.Z. gratefully
acknowledges financial support from the European Research Council (ERC
Advanced Investigator Grant, ESEI-249433) and Swiss National Science
Foundation (SNSF-135135). We alone are responsible for any errors.
Sheremeta: Assistant Professor, Argyros School of Business and
Economics, Chapman University, One University Drive, Orange, CA 92866.
Phone +1-714-744-7604, Fax + 1-714-532-6081, E-mail sheremet@chapman.edu
Zhang: Assistant Professor, Chair for Organizational Design,
Department of Economics, University of Zurich, CH8006 Zurich,
Switzerland. Phone +41-44-634-3743, Fax +41-44-634-4907, E-mail
jingjing.zhang@uzh.ch
TABLE 1
Summary of Average Amount Sent and Profit
Amount Sent Share of Income Sent
Decision NC C NC C
P1 to P2 39 (39) 60 (40) 0.39 (0.39) 0.60 (0.40)
P2 to P3 96 (107) 231 (143) 0.43 (0.40) 0.82 (0.32)
P3 to P1 35 (61) 136 (172) 0.07 (0.11) 0.14 (0.15)
P3 to P2 57 (111) 247 (189) 0.10 (0.15) 0.30 (0.18)
Share of
Profit total Profit
Decision Player NC C NC C
P1 to P2 P1 96 176 0.20 0.20
P2 to P3 P2 178 297 0.33 0.35
P3 to P1 P3 296 410 0.47 0.45
P3 to P2 Efficiency (%) 43.8 67.9
Note: Standard deviations are in parentheses.
TABLE 2
Treatment Effects
Regression (1) (2) (3) (4)
Amount Sent
Dependent
variable P1 to P2 P2 to P3 P3 to P1 P3 to P2
C-treatment 18.29 * 131.57 *** 100.11 *** 189.22 ***
[1 if C (10.32) (30.95) (27.38) (36.28)
treatment]
Period 0.11 5.71 * -0.72 5.04
[period trend] (1.08) (3.33) (2.88) (3.91)
Constant 41.18 *** 66.50 ** 39.55 29.66
(9.58) (28.53) (25.02) (33.46)
Observations 240 240 240 240
Regression (5) (6) (7) (8)
Amount Sent Relative to Income
Dependent
variable P1 to P2 P2 to P3 P3 to P1 P3 to P2
C-treatment 0.18 * 0.39 *** 0.07 ** 0.20 ***
[1 if C (0.10) (0.10) (0.03) (0.04)
treatment]
Period 0.00 0.02 ** -0.01 ** 0.00
[period trend] (0.01) (0.01) (0.00) (0.00)
Constant 0.41 *** 0.33 *** 0.10 *** 0.08 **
(0.10) (0.09) (0.03) (0.04)
Observations 240 240 240 240
Notes: All regressions are estimated using mixed-effects. The
models have random effects at both the individual level and
the session level and account for second-order autocorrelation
in the within-individual residuals. Standard errors are in
parentheses.
* Significant at 10%; ** significant at 5%;
*** significant at 1%.
TABLE 3
Player 3's Reciprocal Behavior
Classification of player 3's NC C
behavior treatment treatment Z-stat
P3 sent nothing to PI and 0.0% 11.3%
split (almost) equally
between P2 and P3
P3 split (almost) equally 0.0% 27.8%
between P1, P2, and P3
P3 kept everything 23.5% 12.4% -1.88 *
P3 was trustworthy both to 16.2% 26.8% 1.61 *
P1 and P2
P3 was trustworthy to PI but 45.6% 7.2% -5.76 ***
not to P2
P3 was trustworthy to P2 but 2.9% 9.3% 1.61 *
not to P1
P3 was trustworthy neither 11.8% 5.2% -1.55
to P1 nor to P2
Observations 68 97 4.04 ***
Notes: We only included cases where both players 1 and 2 sent
a positive amount. The amount differs less than 10% is counted
as almost equal. The Z-stat reflects the two sample test of
proportions.
* Significant at 10%; *** significant at 1%.
TABLE 4
Determinants of Trust and Trustworthiness
Regression (1) (2)
Dependent variable P1 to P2 P2 to P3
P1 to P2 1.77 ***
[P1 to P2 in the current period] (0.11)
P2 to P3
[P2 to P3 in the current period]
P1 to P2 lag -0.09 -0.75 ***
[P1 to P2 average over all past periods] (0.11) (0.28)
P2 to P3 lag 0.00 0.23 **
[P2 to P3 average over all past periods] (0.03) (0.11)
P3 to P1 lag 0.06 *** -0.02
[P3 to P1 average over all past periods] (0.02) (0.06)
P3 to P2 lag 0.00 0.14 **
[P3 to P2 average over all past periods] (0.02) (0.06)
C-treatment 30.69 *** 52.92 ***
[1 if C treatment] (5.14) (15.39)
Period -1.97 *** -2.23
[period trend] (0.57) (1.60)
Constant 50.71 *** 49.08 ***
(6.01) (16.40)
Observations 648 648
R-squared 0.30 0.59
Regression (3) (4)
Dependent variable P3 to P2 P3 to P1
P1 to P2 -0.75 *** 0.41 ***
[P1 to P2 in the current period] (0.14) (0.13)
P2 to P3 0.89 *** 0.51 ***
[P2 to P3 in the current period] (0.04) (0.04)
P1 to P2 lag -0.24 0.09
[P1 to P2 average over all past periods] (0.32) (0.29)
P2 to P3 lag -0.09 -0.08
[P2 to P3 average over all past periods] (0.11) (0.10)
P3 to P1 lag -0.04 0.18 ***
[P3 to P1 average over all past periods] (0.07) (0.07)
P3 to P2 lag 0.24 *** 0.06
[P3 to P2 average over all past periods] (0.07) (0.06)
C-treatment 77.93 *** -9.97
[1 if C treatment] (15.99) (14.73)
Period -4.29 ** -5.79 ***
[period trend] (1.81) (1.67)
Constant 32.42 * -4.62
(18.63) (17.16)
Observations 648 648
R-squared 0.68 0.53
Notes: All regressions are estimated using a system of simultaneous
equations (SE). In each regression we also control for period,
subject, and session effects. Standard errors are in parentheses.
* Significant at 10%; ** significant at 5%; *** significant at 1%.
TABLE 5
Treatment and Order Effects
Regression (1) (2)
Amount Sent
Dependent variable P1 to P2 P2 to P3
C-treatment x NC-C 38.71 *** 171.62 ***
[1 if C treatment and (9.73) (32.20)
NC-C session]
C-treatment x C-NC -2.57 37.16
[1 if C treatment and (8.35) (29.11)
C-NC session]
NC-treatment x NC-C -13.74 -56.45
[1 if NC treatment and (11.12) (37.53)
NC-C session]
NC-treatment x C-NC 68.00 *** 207.39 ***
[1 if NC treatment and (11.60) (39.79)
C-NC session]
Period -1.68 *** -5.10 **
[period trend] (0.64) (2.21)
Observations 480 480
Wald test for order effect 0.000 0.000
on C-treatment
Wald test for order effect 0.000 0.000
on NC-treatment
Regression (3) (4)
Amount Sent
Dependent variable P3 to P1 P3 to P2
C-treatment x NC-C 130.86 *** 175.25 ***
[1 if C treatment and (32.37) (37.57)
NC-C session]
C-treatment x C-NC -54.31 62.15
[1 if C treatment and (33.82) (37.85)
C-NC session]
NC-treatment x NC-C -99.17 ** -125.72 ***
[1 if NC treatment and (41.38) (46.74)
NC-C session]
NC-treatment x C-NC 222.81 *** 215.88 ***
[1 if NC treatment and (46.40) (51.66)
C-NC session]
Period -9.49 *** -6.08 **
[period trend] (2.61) (2.90)
Observations 480 480
Wald test for order effect 0.000 0.016
on C-treatment
Wald test for order effect 0.000 0.000
on NC-treatment
(5) (6)
Regression
Amount Sent
Relative to Income
Dependent variable P1 to P2 P2 to P3
C-treatment x NC-C 0.39 *** 0.25 **
[1 if C treatment and (0.10) (0.11)
NC-C session]
C-treatment x C-NC -0.03 0.13
[1 if C treatment and (0.08) (0.08)
C-NC session]
NC-treatment x NC-C -0.14 -0.19 *
[1 if NC treatment and (0.11) (0.11)
NC-C session]
NC-treatment x C-NC 0.68 *** 0.67 ***
[1 if NC treatment and (0.12) (0.11)
C-NC session]
Period -0.02 *** -0.00
[period trend] (0.01) (0.01)
Observations 480 480
Wald test for order effect 0.000 0.278
on C-treatment
Wald test for order effect 0.000 0.000
on NC-treatment
(7) (8)
Regression
Amount Sent
Relative to Income
Dependent variable P3 to P1 P3 to P2
C-treatment x NC-C 0.01 0.12 ***
[1 if C treatment and (0.04) (0.04)
NC-C session]
C-treatment x C-NC -0.10 *** 0.10 ***
[1 if C treatment and (0.03) (0.04)
C-NC session]
NC-treatment x NC-C -0.17 *** -0.10 **
[1 if NC treatment and (0.05) (0.05)
NC-C session]
NC-treatment x C-NC 0.30 *** 0.21 ***
[1 if NC treatment and (0.05) (0.05)
C-NC session]
Period -0.01 *** -0.00
[period trend] (0.00) (0.00)
Observations 480 480
Wald test for order effect 0.016 0.711
on C-treatment
Wald test for order effect 0.000 0.001
on NC-treatment
Notes: All regressions are estimated using a random effects error
structure with the individual subject effects. In each regression
we also include dummy variables (not shown in the table) to control
for session effects. Standard errors are in parentheses.
* Significant at 10%; ** significant at 5%; *** significant at 1%.
TABLE 6
Summary of Average Expected Amount Sent and
Percentage Difference
Expected Actual Percentage Difference
Decision Amount Sent Amount Sent from Actual Decisions
NC C NC C NC (%) C (%)
P2 to P3 80 233 96 231 16.8 0.9
P3 to P2 71 251 57 247 24.6 1.6
P3 to P1 60 191 35 136 71.4 40.4
TABLE 7
Coding Table, Reliability Indexes, and Frequency of Coding
Code Description
Messages sent by player 2
1a P2 proposed to send nothing to P1 and (almost) equal split
between P2 and P3
1b P2 proposed (almost) equal split between P1, P2, and P3
1c P2 proposed to send some to P1 and (almost) equal split
between P2 and P3
1d P2 made a positive comment or showed concern for well-being
of P1
1e P2 made a negative comment about P1
1f P2 made any promises or showed trust in P3
1g P2 used threat
1h P2 pleaded or appealed to P3
Messages sent by player 3
2a P3 proposed to send nothing to P1 and (almost) equal split
between P2 and P3
2b P3 proposed (almost) equal split between P1, P2, and P3
2c P3 proposed to send some to P1 and (almost) equal split
between P2 and P3
2d P3 made a positive comment or showed concern for well-being
of P1
2e P3 made a negative comment about P1
2f P3 made any promises or showed trustworthiness
2g P3 mentioned about his or her good qualities
Messages indicating agreement or disagreement between players
2 and 3
3a Agreement was reached on the first proposal
3b Agreement was reached on a different proposal than the
first proposal
3c Agreement was not reached
Cohen's Frequency
Code Kappa K of Coding
Messages sent by player 2
1a 0.53 21.7%
1b 0.75 20.4%
1c 0.81 0.4%
1d 0.76 3.8%
1e 0.50 7.1%
1f 0.39 4.6%
1g 0.39 1.3%
1h 0.53 10.8%
Messages sent by player 3
2a 0.74 32.5%
2b 0.77 24.6%
2c 0.77 1.7%
2d 0.50 5.8%
2e 0.49 6.3%
2f 0.72 9.6%
2g 0.32 0.8%
Messages indicating agreement or disagreement between players 2 and 3
3a 0.70 69.2%
3b 0.67 22.9%
3c N/A 0.0%
Note: The amount differs less than 10% is counted as almost equal.
TABLE 8
Multilevel Mixed-effects Regression on Categories of Messages
Regression (1) (2)
Amount Sent
Dependent variable P2 to P3 P3 to P2
Only one player made a proposal
1a P2 proposed to send nothing -2.37 42.48
to P1 and (almost) equal (37.13) (55.22)
split between P2 and P3
1b P2 proposed (almost) equal 109.46 ** 141.29 **
split between P1, P2, (44.19) (65.76)
and P3
2a P3 proposed to send nothing -16.89 -42.50
to P1 and (almost) equal (29.01) (43.57)
split between P2 and P3
2b P3 proposed (almost) equal 133.98 *** 65.21
split between P1, P2, (29.87) (45.17)
and P3
The same proposal made by players 2 and 3
1a + 2a Both P2 and P3 proposed to 66.34 * -0.78
send nothing to P1 and (35.07) (56.56)
(almost) equal split
between P2 and P3
1b + 2b Both P2 and P3 proposed 173.87 *** 149.93 *
(almost) equal split (57.67) (81.90)
between P1, P2, and P3
Two different proposals made by players 2
and 3
1a + 2b P2 proposed to send nothing 61.12 35.90
to P1 and (almost) equal (51.56) (79.20)
split between P2 and P3
while P3 proposed (almost)
equal split between P1,
P2, and P3
1b + 2a P3 proposed to send nothing 85.11 117.72
to P1 and (almost) equal (67.07) (97.58)
split between P2 and P3
while P2 proposed (almost)
equal split between
P1. P2, and P3
The most frequently used messages
1h P2 pleaded or appealed to P3 67.17 56.99
(50.44) (76.70)
2f P3 made any promises or 25.51 71.86
showed trustworthiness (35.21) (51.97)
Period 6.98 8.47
[period trend] (4.49) (7.53)
Constant 129.78 *** 153.63 ***
(36.25) (54.68)
Observations 120 120
(3) (4)
Regression
Amount sent
Amount Relative
Sent to Income
Dependent variable P3 to P1 P2 to P3
Only one player made a proposal
1a P2 proposed to send nothing -124.17 *** 0.07
to P1 and (almost) equal (39.64) (0.08)
split between P2 and P3
1b P2 proposed (almost) equal 150.30 *** 0.14
split between P1, P2, (48.03) (0.10)
and P3
2a P3 proposed to send nothing -142.32 *** 0.12 *
to P1 and (almost) equal (31.67) (0.06)
split between P2 and P3
2b P3 proposed (almost) equal 139.74 *** 0.16 **
split between P1, P2, (33.06) (0.06)
and P3
The same proposal made by players 2 and 3
1a + 2a Both P2 and P3 proposed to -126.98 *** 0.27 ***
send nothing to P1 and (39.87) (0.08)
(almost) equal split
between P2 and P3
1b + 2b Both P2 and P3 proposed 246.40 *** 0.23 *
(almost) equal split (59.79) (0.12)
between P1, P2, and P3
Two different proposals made by players 2
and 3
1a + 2b P2 proposed to send nothing 31.41 0.09
to P1 and (almost) equal (58.36) (0.11)
split between P2 and P3
while P3 proposed (almost)
equal split between P1,
P2, and P3
1b + 2a P3 proposed to send nothing 211.53 *** -0.06
to P1 and (almost) equal (72.88) (0.15)
split between P2 and P3
while P2 proposed (almost)
equal split between
P1. P2, and P3
The most frequently used messages
1h P2 pleaded or appealed to P3 4.68 0.21 *
(55.92) (0.11)
2f P3 made any promises or 33.61 0.07
showed trustworthiness (36.53) (0.07)
Period 2.63 0.01
[period trend] (4.51) (0.01)
Constant 134.59 *** 0.59 ***
(32.57) (0.09)
Observations 120 120
Regression (5) (6)
Amount sent
Relative to Income
Dependent variable P3 to P2 P3 to P1
Only one player made a proposal
1a P2 proposed to send nothing 0.11 ** -0.13 ***
to P1 and (almost) equal (0.05) (0.03)
split between P2 and P3
1b P2 proposed (almost) equal 0.05 0.10 **
split between P1, P2, (0.06) (0.04)
and P3
2a P3 proposed to send nothing 0.05 -0.15 ***
to P1 and (almost) equal (0.04) (0.03)
split between P2 and P3
2b P3 proposed (almost) equal -0.06 0.10 ***
split between P1, P2, (0.04) (0.03)
and P3
The same proposal made by players 2 and 3
1a + 2a Both P2 and P3 proposed to 0.06 -0.15 ***
send nothing to P1 and (0.05) (0.03)
(almost) equal split
between P2 and P3
1b + 2b Both P2 and P3 proposed 0.00 0.11 **
(almost) equal split (0.07) (0.05)
between P1, P2, and P3
Two different proposals made by players 2
and 3
1a + 2b P2 proposed to send nothing -0.05 0.11 **
to P1 and (almost) equal (0.07) (0.05)
split between P2 and P3
while P3 proposed (almost)
equal split between P1,
P2, and P3
1b + 2a P3 proposed to send nothing -0.03 0.13 **
to P1 and (almost) equal (0.08) (0.06)
split between P2 and P3
while P2 proposed (almost)
equal split between
P1. P2, and P3
The most frequently used messages
1h P2 pleaded or appealed to P3 0.04 0.01
(0.07) (0.05)
2f P3 made any promises or 0.00 0.00
showed trustworthiness (0.05) (0.03)
Period 0.00 -0.00
[period trend] (0.01) (0.00)
Constant 0.25 *** 0.18 ***
(0.06) (0.03)
Observations 120 120
Note: Standard errors are in parentheses. * Significant at 10%;
** significant at 5%; *** significant at 1%.
