A Trust Game in Loss Domain
*
Ola Kvaløy, Miguel Luzuriaga and Trond E. Olsen **
January 2017
Abstract: In standard trust games, no trust is the default, and trust generates a potential gain.
We investigate a reframed trust game in which full trust is default and where no trust generates
a loss. We find significantly lower levels of trust and trustworthiness in the loss domain when
full trust is default than in the gain domain when no trust is default. As a consequence, trust is
on average profitable in the gain domain, but not in the loss domain. We also find that subjects
respond more positively to higher trust in the loss domain than in the gain domain.
Keywords: Trust; Reciprocity; Framing; Defaults; Reference points; Experiment
JEL codes: C72; C91
*
We would like to thank the Editor, two anonymous referees, Björn Bartling, Kristoffer W. Eriksen, Martin
Kocher, Petra Nieken, Marie Claire Villeval, Øyvind N. Aas and participants at the Stavanger Workshop in
Behavioral Economics for valuable comments. Financial support from the Norwegian Research Council is greatly
appreciated.
**
Kvaløy: University of Stavanger, Norway, [email protected]. Tel +47 51831582. Luzuriaga: Neu-Ulm
University of Applied Sciences, Germany, [email protected]. Tel +49(0)731/9762-1443. Olsen:
Norwegian School of Economics, Norway, [email protected]. Tel +4755959976
1
1. Introduction
People’s willingness to trust and reciprocate trust is of great importance for economic
prosperity. It enforces incomplete contracts and reduces transaction costs. A large economic
literature has thus evolved investigating the origins of trust, when and why people are willing
to trust, when and why people act trustworthily, and what consequences trust (and the lack
thereof) has for economic welfare. The experimental literature on trust and reciprocity has been
an important contributor in this respect. It is now well established that people have a tendency
to reward kind actions, and are thus willing to trust others since they expect to be reciprocated,
see Fehr et al. (1993), Berg, et al. (1995) and the large body of evidence thereafter. Bohnet and
Croson (2004) and Bohnet (2008) provide comprehensive reviews of the literature.
Except for a paper by Bohnet and Meier (2012), which we will turn to below, a common feature
with experimental papers on trust is that they are variations of investment games where trust
provides a potential gain and where no trust yields no loss. Trust is an active decision, an act of
commission, while no trust is a passive one, an act of omission. Moreover, trustworthiness
typically involves actively returning money (or effort) to the trustor, while no trustworthiness
is a more passive decision, choosing not to return anything. In many situations, however, trust
often involves taking no action, and the return from trust is the absence of loss. We trust others
when we do not lock our door, do not take all necessary precautions, and do not spend resources
in writing detailed contracts that cover every possible contingency. Additionally, we are
trustworthy when we do not steal or do not abuse those who dare make transactions under loose
contracts. In other words, full trust is often the default.
This distinction between trust in the loss domain and the gain domain is potentially important.
From Kahneman and Tversky (1979) we know that people act differently in the loss domain
than in the gain domain, and importantly, that losses loom larger than gains. A consequence is
that defaults or reference points matter, as particularly demonstrated by Thaler (1980) and
Samuleson and Zeckhouser (1988). Reference points and defaults determine whether a given
prospect or value is perceived as being in the loss domain or the gain domain. Related to this,
it has been shown that perceptions of an outcome differ if the outcome resulted from an act of
omission rather than an act of commission (see Kahneman and Tversky, 1982, and Baron and
Ritov, 1994). Given these insights, the absence of trust expectedly feels more severe in the loss
domain when full trust is the default than in the gain domain when no trust is the default.
Furthermore, a given intermediate level of trust may be perceived as less kind in the loss domain
2
than in the gain domain. Hence, if trustees care about the kindness of the trustors’ actions (as
has been demonstrated in a number of experiments, most notably McCabe et al., 2003, and
Charness and Levine, 2007), we will expect less trustworthiness in the loss domain when full
trust is default than in the gain domain when no trust is default. Moreover, since losses loom
larger than gains, we will expect that subjects respond more positively to higher trust in the loss
domain than in the gain domain. However, the effect on trust is more subtle. The pure default
effect calls for higher trust in the loss domain, but since the expected trustworthiness level is
lower, one may consequently also expect lower trust in the loss domain.
In this paper we formalize the hypotheses above, based on Cox et al’s (2007) “Tractable model
of fairness and reciprocity”. We then investigate the hypotheses by running a rephrased version
of Berg et al’s (1995) well known trust game (which they call the investment game). In a
controlled laboratory experiment, subjects were randomly paired as players A and B, which we
here will refer to as trustor and responder, respectively. Both the trustors and the responders
received an endowment of 200 Norwegian kroner (about $34). The responder was then given
the opportunity to take an amount from the trustor’s endowment. But before the responder made
this decision, the trustor could choose to insure an amount that the responder could not take.
The money not insured was then equivalent to money trusted. For each krone that the trustor
insured, the total surplus in the relationship was reduced by two kroner. Hence, full trust was
the default, while the lack of trust generated a loss. In economic terms, this game is identical to
Berg et al, where the trustor could instead send money, and for each krone sent, total surplus
increased by two (i.e. money was tripled). The only difference is that Berg et al. is framed in a
gain domain, while our new trust game is framed in a loss domain.
Indeed, subjects did behave differently in the gain domain than in the loss domain. We find
significantly lower levels of both trust and trustworthiness in the loss domain when full trust
was default, than in the baseline trust game. As a consequence, trust was on average profitable
for the trustor when no trust was default, but not when full trust was default. The treatment
differences are quite large, also in economic terms. The trust and trustworthiness levels were,
respectively, 51% and 32% higher in Baseline than in the loss domain. Moreover, we find that
responders were more responsive to a marginal change of money trusted in the loss domain
than in the gain domain. In Baseline, higher trust did not lead to higher levels of trustworthiness.
In the loss domain, however, the responders were more trustworthy the more they were trusted.
3
Related literature: There is now a large literature on reference-dependent preferences, but the
experimental literature on how reference points and defaults affect pro-social behavior is
limited. The return from pro-social behavior is typically in terms of gains, not the absence of
loss. 1 There are, however, some exceptions: Andreoni (1995) and Willinger and Ziegelmeyer
(1999) study framing in public good games and find that the average contribution to public
goods is higher when the game is framed positively (giving to the public) than negatively
(taking from the public). 2 Lopez and Nelson (2005) study how endowment effects influence
public good provisions, and find that subjects contribute more to a public account when the
money is said to start in their private account than when the money is said to start in their public
account. In contrast, Messer et al. (2007) find that a status quo of giving increases public good
contributions. It has also been shown that framing can effect coordination, see in particular
Ispano and Schwardmann (2013) who find that subjects coordinate better in the loss domain
than in the gain domain.
Framing effects have also been demonstrated in the classic generosity games. Bardsley (2008)
and List (2007) find that reference points affect generosity in dictator games. In particular, it is
shown that subjects are less willing to transfer money when the action set includes taking.
Correspondingly, Abbink et al. (2011) find more antisocial behavior in the loss domain of
money burning tasks. However, Baquero et al. (2013) find a huge generosity effect in the loss
domain of ultimatum games, implying that strategic considerations, which are absent in dictator
games, alter the framing effects.
Closest to our paper is Bohnet and Meier (2012). They introduce a reframed version of Berg et
al (1995), which they call the distrust game. As in our paper, full trust is default and no trust
yields a loss. Yet in contrast to our paper (and Berg et. al), the parties do not start out with the
same endowment. The trustor starts with nothing, while the trustee (responder) has the whole
endowment. Hence, the trustor is never in the loss domain. Moreover, there are two reference
points that are different from the baseline trust game, namely the trust level and the inequality
level, leading to a “double default effect”. In Section 3, we argue that this double default effect
can lead the trustor to make decisions that are closer to the default of full trust/high inequality.
1
There is, though, an extensive literature on positive versus negative reciprocity, but the loss domain should not
be confused with negative reciprocity. See e.g. Abbink et al. (2000) on the relationship between negative and
positive reciprocity.
2
Park (2000) replicates Andreoni's experiment and finds that while there is a significant difference between the
two framing conditions in terms of overall contribution rates, there is no significant difference for subjects who
have strong cooperative value orientation.
4
In other words, the trustor’s default effects are more likely to dominate the strategic response
to an expectedly lower trustworthiness level. Indeed, Bohnet and Meier find lower
trustworthiness in their distrust game than in Baseline, but in contrast to us (and in contrast to
their main trust hypothesis), they find higher trust.
The rest of the paper is organized as follows. In Section 2 we present the experimental design
and procedure, while in Section 3 we present behavioral predictions. In Section 4 we present
the experimental results, while Section 5 concludes.
2. Experimental Design and Procedure
We ran three treatments: One baseline treatment in which we replicated Berg et al. (1995), and
two loss domain treatments, referred to as LD I and LD II. In Baseline, players were randomly
paired as trustors and responders and were endowed with NOK 100 each (about $17). The
trustor was then given the opportunity to send an amount x from her endowment to the
responder. The amount of money sent by the trustor was tripled by the experimenter so that the
responder received 3x. Then the responder had the opportunity to send back to the trustor an
amount y. The trustor’s payoff was then 100 - x + y, while the responder’s payoff was 100 +
3x - y. Total surplus was thus 200+2x.
In the loss domain treatments, players were also randomly paired as trustors and responders,
but were endowed with NOK 200 each. This is the payoff allocation that results in the baseline
treatment if the sender fully trusts, i.e., sends x=100, and the receiver is fully trustworthy, i.e.,
chooses to return y=200. The responder was then given the opportunity to take an amount from
the trustor’s endowment. However, before the responder made this decision, the trustor could
insure an amount between 0 and NOK 100. The responder was not able to take anything from
the amount that was insured. For each krone the proposer insured, total surplus was reduced by
two kroner. This efficiency loss is equivalent to the efficiency gains from sending positive
amounts in the baseline treatment. 3
The surplus reduction was formulated in two different ways. In LD I, then for each krone the
trustor insured, the responder immediately lost two kroner, which is the closest equivalent to
3
In Bohnet and Meier, the responder starts out with the whole endowment (while the trustor starts with nothing),
and hence the money taken by the trustor could simply be divided by three in order to mimic the baseline. Since
we did not want to change the inequality default (i.e. both parties started out with half of the maximum surplus),
the LD treatments required a slightly different formulation of the surplus reduction.
5
Baseline. In LD II, then for each krone the trustor insured, both the trustor and the responder
lost one krone each. In economic terms, the two formulations are equivalent. Yet we did it as a
robustness check to see whether subjects were sensitive to these different formulations. In
particular, we wanted to check whether the responder interpreted an insurance decision from
the trustor more negatively in LD I than in LD II. In reality, he should not, since in both cases
the responder was free to dictate the allocation of the money not insured. In fact, the only
difference between the treatments is the responder’s opportunity to be extra generous towards
the trustor (see discussion below). As it turned out, the behavior in the two LD treatments was
very similar and we will mainly refer to LD, meaning both LD I and LD II.
In all three treatments, the two players played the game once and we see that the payoffs are
equivalent in all three treatments: If money trusted is x, then money insured is 100-x. If money
returned is y in Baseline, then y is equal to money not taken in LD. In LD total surplus is then
given by 400-2(100-x)=200+2x, which is the same as in Baseline. The trustor’s payoff is given
by money insured, plus money not taken, i.e. 100-x+y. The responder’s payoff is total surplus
minus the trustor’s payoff, i.e. 200+2x-(100-x+y)= 100+3x-y. Hence, the players’ payoffs, for
a given trust level, is the same in LD and Baseline.
However, it should be noted that the payoff sets are equivalent only if the responder prefers at
least as high payoff to himself as to the trustor. See Figure A1 in the appendix over the payoff
sets in the different treatments (LD I, LD II, Berg et al., 1995, and Bohnet and Meier, 2012).
The left hand sides of the 45 degree line are the same in all treatments, but the right hand sides
differ. That is, the treatments differ in the opportunities for the responder to be extra generous
towards the trustor, i.e. to let the trustor earn more than himself. We cannot reject the possibility
that this matters. Cox et al. (2008) show that subjects may be affected by the opportunity set
given by their interacting partner, i.e. the responders may differ in their opinion of how nice it
is of the trustors to offer them a given set to choose from. Here we make the implicit assumption
that the responder does not find a given trust level nicer (or less nice) if it differs with respect
to how generous the responder can be towards the trustor. Interestingly, we find no significant
differences in behavior between LD I and LD II, although the designs actually differ in this
respect. 4
4
There were only a few observations at the right hand side of the 45 degree line where trustors earn more than
responders; three in Baseline, seven in LD I and two in Berg et al (1995). Bohnet and Meier does not report on
this. Our main results hold also when excluding these observations.
6
Note that if we were to allow for the exact same payoff set as in Berg et al. (1995) also in the
case where the responder admits more to the trustor than to himself, we would have to allow
the responder to not only take money from the trustor, but also to give money to the trustor (in
addition, we had to restrict the giving amount so that the responder himself were guaranteed at
least 100). By allowing for this, the game would no longer be a perfect transformation from
returning money (as in Berg et al) to taking money, as in our design. Moreover, we could create
a new confound. The fact that the responder could both give and take might in itself matter.
A total of 274 students from the University of Stavanger, Norway participated in the
experiment. We had 90 subjects spread over 3 sessions in Baseline, with 88 subjects in LD I
and 96 subjects in LD II spread over 5 sessions each (i.e. a between-subject design).
5
The
subjects were recruited by e-mail. They were told that by participating in an economic
experiment they would have the possibility to earn a nice sum of money. All instructions were
given both written and verbally. The experiment was conducted and programmed with the
software z-Tree (Fischbacher 2007).
3. Behavioral Predictions
In order to fix ideas and provide some behavioral predictions, we need to investigate
implications from a utility function that captures both social preferences and framing effects.
For our purpose, we take Cox et al’s (2007) “Tractable model of reciprocity and fairness” as
our starting point. Like the intention-based models of reciprocal preferences (Dufwenberg and
Kirschsteiger, 2004; and Falk and Fischbacher, 2006) 6, Cox et al assume that players consider
the kindness of other players’ actions and not only the outcome, before they decide how to
allocate their endowment. But the simple structure of their model is more similar to the
distributional preference models of Fehr and Schmidt (1999) and Bolton and Ockenfels (2000).
Like them, Cox et al only specifies preferences and not strategies and beliefs, and is well suited
for analyzing experimental work on social preferences.
5
The same baseline was also used in Kvaløy and Luzuriaga (2014). The paper studies how people play trust games
with other people’s money. The baseline was then compared with one other treatment, which was identical to
baseline except that the players were playing with a third party’s money.
6
Rabin (1993) was the first to consider the role of intentions, but he restricts his analysis to two player normal
form games. Dufwenberg and Kirschsteiger (2004) and Falk and Fischbacher (2006) develop the theory for n
player sequential games.
7
Let the first mover's (trustor) material payoff be denoted z, while the second mover's (responder)
material payoff is m. The responder’s utility function is given by
𝑢𝑢(𝑚𝑚, 𝑧𝑧) = (𝑚𝑚𝛼𝛼 + 𝜃𝜃𝑧𝑧 𝛼𝛼 )/𝛼𝛼,
𝑢𝑢(𝑚𝑚, 𝑧𝑧) = 𝑚𝑚𝑚𝑚 𝜃𝜃 ,
𝛼𝛼 ∈ (−∞, 0) ∪ (0, 1]
𝛼𝛼 = 0
The factor 𝜃𝜃 is the responder’s emotional state, which determines his/her willingness to pay
own for other’s payoff. More specifically, the emotional state of the responder defines the
marginal rate of substitution (MRS) between own payoff m and trustor’s payoff z. With the
given utility function, MRS equals
𝜕𝜕𝜕𝜕/𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕/𝜕𝜕𝜕𝜕
𝑧𝑧
= 𝜃𝜃 −1 (𝑚𝑚)1−𝛼𝛼 . Hence, the higher 𝜃𝜃, the lower is MRS
and thus the more is the responder willing to give up own payoff in order to increase the trustor’s
payoff.
Cox et al. assume that the emotional state, 𝜃𝜃, is a function of a reciprocity variable r and a status
variable s. The reciprocity variable is specified as 𝑟𝑟(𝑥𝑥) = 𝑚𝑚(𝑥𝑥) − 𝑚𝑚0, where 𝑚𝑚(𝑥𝑥) is the
maximum payoff the responder can guarantee himself given the trustor’s trust level x, and 𝑚𝑚0
is 𝑚𝑚(𝑥𝑥0 ) when the trustor’s choice is considered as neutral, 𝑥𝑥0 . Hence, when the choice is
considered as neutral, then 𝑟𝑟(𝑥𝑥) = 0. The status variable s represents the trustor’s relative
status. This could be related to the trustor’s entitlements, claims or obligations. Cox et al then
hypothesize (and test) that the emotional state 𝜃𝜃(𝑟𝑟, 𝑠𝑠) increases in r and s.
Before we discuss a slight modification of this utility function, we can ask whether 𝜃𝜃(𝑟𝑟, 𝑠𝑠) is
expected to differ between Baseline and LD. Consider 𝑟𝑟(𝑥𝑥) = 𝑚𝑚(𝑥𝑥) − 𝑚𝑚0. Clearly, 𝑚𝑚(𝑥𝑥) is
the same in both treatments, i.e. the maximum payoff the responder can guarantee himself given
the trustor’s choice x is the same. However, 𝑚𝑚0 , which depends on the trustor’s neutral choice
𝑥𝑥0 differs by design. In the gain domain, no trust is the default, while in the loss domain full
trust is the default. We thus assume that 𝑥𝑥0 and thus 𝑚𝑚(𝑥𝑥0 ) is higher in the loss domain than in
the gain domain. It follows that 𝑟𝑟(𝑥𝑥) is higher in the gain domain. Now, if the status variable
s does not vary between treatments, it follows that the emotional state, 𝜃𝜃, for a given trust level
x, is strictly lower in the loss domain. 7
7
There are no differences in the players’ characteristics or objective relative status between the treatments that
call for differences in the status variable. However, there might be a slight difference in entitlements. In the gain
domain, the trustor’s trust level x is an act of commission, while in the loss domain x is a passive action, an act
of omission. For a given x, the trustor’s entitlement might then be higher in the gain domain. We can thus
assume that the status variable is not higher in the loss domain than in gain domain, supporting the assumption
that 𝜃𝜃 is lower in loss domain.
8
There is, however, an aspect with losses that the utility function does not capture. From
experimental evidence leading to the famous prospect theory, it is shown that losses loom larger
than gains (Kahneman and Tversky, 1979). Hence, we might want to multiply the emotional
state 𝜃𝜃 with a variable 𝜆𝜆, which equals one in the gain domain, but which might be larger than
one in situations where the parties are in the loss domain. Like in the experimental design, we
define loss domain as the domain in which total surplus is less than the sum of the initial
endowments. (Recall that the initial endowments sum to the maximal amount of 400 in the loss
domain treatment.) This is a natural assumption also since the responder with θ > 0 cares about
both parties’ monetary payoff. Moreover, a difference between the loss domain treatment and
the gain domain treatment, is that in the former the responder inflict losses on the trustor by
taking money, instead of inducing gains by giving money. 8 The utility function is then
𝑢𝑢(𝑚𝑚, 𝑧𝑧) = (𝑚𝑚𝛼𝛼 + 𝜆𝜆𝜆𝜆𝑧𝑧 𝛼𝛼 )/𝛼𝛼,
𝑢𝑢(𝑚𝑚, 𝑧𝑧) = 𝑚𝑚𝑚𝑚 𝜆𝜆𝜆𝜆 ,
𝛼𝛼 ∈ (−∞, 0) ∪ (0, 1]
𝛼𝛼 = 0
where 𝜆𝜆 = 1 in the gain domain treatment and 𝜆𝜆 ≥ 1 in the loss domain treatment.
To be more specific, in the experiment we have m = 100 + 3x − y, where y ≥ 0 is the money
returned. Moreover, z = 100 − x + y, while total surplus is m + z = 200 + 2x. We can define
the reciprocity variable r(x) = m(x) − m₀ = 100 + 3x − m₀ where m₀ = 100 in the gain
domain and m₀ = 200 in the loss domain. We assume that θ = θ(r(x)) and thus ignore the
Cox et al's status variable s.
The responder maximizes utility, subject to a budget constraint for m + z and the constraint
y ≥ 0, where the latter is equivalent to z ≥ 100 − x:
max 𝑢𝑢(𝑚𝑚, 𝑧𝑧; 𝜃𝜃) 𝑠𝑠𝑠𝑠 𝑚𝑚 + 𝑧𝑧 = 200 + 2𝑥𝑥 ≡ 𝑏𝑏(𝑥𝑥) 𝑎𝑎𝑎𝑎𝑎𝑎 𝑧𝑧 ≥ 100 − 𝑥𝑥 ≡ 𝜁𝜁(𝑥𝑥)
𝑚𝑚,𝑧𝑧
The utility function 𝑢𝑢(𝑚𝑚, 𝑧𝑧) = (𝑚𝑚𝛼𝛼 + 𝜆𝜆𝜆𝜆𝑧𝑧 𝛼𝛼 )/𝛼𝛼 then yields the following explicit solution for
the trustor's monetary payoff:
8
An alternative approach is to say the responder chooses between allocations that may or may not put himself in
loss domain compared to own initial endowment, i.e. that λ is endogenously determined within a treatment.
However, this approach gives rise to implausible behavior in some cases.
9
(1)
(𝜆𝜆𝜆𝜆)𝛽𝛽
z(x) = max �𝜁𝜁(𝑥𝑥), (𝜆𝜆𝜆𝜆)𝛽𝛽 +1 𝑏𝑏(𝑥𝑥)� ,
β = 1/(1 − α) ∈ (0, ∞) for θ = θ(x) ≥ 0
When y > 0 is optimal (and thus z > ζ(x)), the amount z accruing to the trustor is thus a fraction
of the budget 𝑏𝑏(𝑥𝑥), where the fraction is determined by λθ. This will clearly hold only if λθ
exceeds some lower positive bound. For λθ below this bound, and in particular for θ ≤ 0, the
responder will optimally return nothing (y = 0) , and the trustor's monetary payoff will then be
z = ζ(x) = 100 − x.
Now, when positive, the amount returned y is given by
𝑦𝑦 = 𝑧𝑧 + 𝑥𝑥 − 100 =
(𝜆𝜆𝜆𝜆)𝛽𝛽
(200 + 2𝑥𝑥) + 𝑥𝑥 − 100
(𝜆𝜆𝜆𝜆)𝛽𝛽 + 1
We see that the fraction of the budget, and thus the returned amount y is increasing in λθ, all
else equal. Hence, larger λ implies larger y, all else equal. Since θ = θ(r) is increasing in r =
m(x) − m₀, it follows that θ and hence y is decreasing in m₀. Larger m₀ thus implies smaller
y, all else equal. In other words, if we hold λ constant (like in the original Cox et al. model),
and only consider the different reference points in the two treatments, then trustworthiness is
clearly lower in the loss domain treatment.
If the loss domain has a larger λ, however, then there are two opposing effects on the returned
amount y, for given x. If λ is "small" (close to 1) the effect of larger m₀ will dominate, for any
given x. It is only if λ is sufficently large that the effect of larger 𝑚𝑚0 will be dominated, and
hence the amount returned y might be higher in the loss domain treatment. Moreover, this will
then tend to occur for large x, since θ(x) is then largest, and hence the effect of λ on y is then
also largest. In particular, for the case α = 0 and θ = k ⋅ (m(x) − m₀), where k > 0 is a scaling
parameter, it can be seen (see appendix) that the returned amount y (for any x) is lower in the
3
3
loss domain as long as λ < 2. For λ > 2, then y is lower in the loss domain than in the gain
domain, unless x is sufficiently high. We have:
Hypothesis 1: Responders are less trustworthy in the loss domain than in the gain domain,
(possibly) unless the trust level is sufficiently high.
10
We are also interested in the responder's responsiveness to higher trust, i.e. how the treatment
variables m₀ and λ affect
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
. In the appendix we show that there are sets of reasonable
parameters where 𝑑𝑑𝑑𝑑 increases in both m₀ and λ. This is our hypothesis:
Hypothesis 2: Responders respond more positively to higher trust levels in the loss domain
than in the gain domain.
Finally, we are interested in the trustor's choice. There are two countervailing effects, the default
effect and the strategic effect. If also the trustor has an idea of a neutral choice x₀, then this may
in itself affect the trustor's behavior. We may call it a direct default effect, which can affect trust
positively in loss domain, since the neutral choice x₀, presumably is higher when full trust is
default. However, it is natural to assume that the first mover (trustor) maximizes the expectation
of a utility function that has the same form as the second mover's (responder's) utility function.
But, as noted by Cox et al, this is complicated as it raises the question about the first mover's
beliefs about the second mover's behavior. Here we assume that since the trustor must make
her choice before the responder has done anything, the trustor's reciprocity parameter equals
zero. Moreover, we assume that the trustor forms beliefs about the responder's behavior
according to the model above.
From (1) we see that for fixed λθ, the amount z will increase with the budget b(x) when y > 0,
and thus increase with x. As θ increases with x, the amount z will increase further. Thus z is
increasing in x whenever the amount returned is positive (y > 0). The trustor's payoff then
increases in x for two reasons: first, because of a higher budget, and second, because a higher x
increases the emotional state θ. But for sufficiently low x the amount returned will be zero, and
then it is better for the trustor to send nothing, and by that guarantee herself a minimum payoff
(denoted ζ(0) above). Thus it is optimal for her to choose either the maximum amount x = M,
say, or x = 0. Since the returned amount y is expected to be lower in loss domain, then there
are parameters where we will see full trust in gain domain, and no trust in loss domain (see
appendix). However, we expect trustors to also send intermediate levels, which will be
consistent with a model where the trustor is uncertain about the responder's type (see appendix).
We state our final hypothesis as follows:
11
Hypothesis 3: i) If responders are less trustworthy in the loss domain than in the gain domain,
then we will see less trust in the loss domain than in the gain domain. ii) If responders are more
trustworthy in the loss domain than in the gain domain, then we will see more trust in the loss
domain than in the gain domain.
It is instructive to present a specific parameter example that is consistent with the stated
hypotheses. For parameters as in the experiment, and with α = 0 and θ = k ⋅ (m(x) − m₀) as
above, we obtain
(2)
𝑦𝑦(𝑥𝑥) = max �0,
(3𝑥𝑥+100−𝑚𝑚0 )
(3𝑥𝑥+100−𝑚𝑚0 )+
1
λk
(200 + 2𝑥𝑥) + 𝑥𝑥 − 100�
In the gain domain we have λ = 1 and m₀ = 100, while in the loss domain m₀ = 200 and λ ≥
1. Figure 1 depicts, for k = 1/600, the responder’s response (2) in the loss domain for λ =
1 and λ = 2 , as well as the response in the gain domain. 9
Figure 1. Relationship between sent amount x and returned amount y for different levels of λ and 𝑚𝑚0 .
9
The parameter k must be sufficiently small to assure λθ ≤ 1, which is a reasonable assumption, since this is the
relative weight on the other party’s payoff in the utility function.
12
The figure presents the relationship between the sent amount x and the returned amount y. The
black curve is the returned amount in gain domain, the green curve in loss domain with λ =
1 (and 𝑚𝑚0 = 200), and the red curve in loss domain with λ = 2. We see that for λ = 1 the
returned amount is always lower in the loss domain. Moreover, for x = 100 the amount
returned exceeds the amount sent in the gain domain, while it is below the amount sent in the
loss domain. But if λ = 2 the responder is more responsive to higher trust, and so the
returned amount is higher in the loss domain compared to the gain domain for high trust
levels.
Before we proceed to the result section, it can be useful to compare with the Bohnet and Meier
design. They also predict lower trustworthiness (although not formally deduced), and hence
lower trust in their distrust game (DTG). However, in the experiment they find lower
trustworthiness but higher trust. The model presented here can contribute to explain the
difference.
Recall that the predicted treatment differences are driven by the parameters 𝑟𝑟(𝑥𝑥) = 𝑚𝑚(𝑥𝑥) − 𝑚𝑚0
and 𝜆𝜆. By design, both 𝑚𝑚0 and 𝜆𝜆 are assumed to be higher in our LD treatments. The Bohnet
and Meier design, by contrast, does not call for clear assumptions regarding differences in 𝜆𝜆.
Since the trustor starts with nothing, she is never in the loss domain. It is only the responder
that is in the loss domain. This could make treatment differences in 𝜆𝜆 more dubious. Indeed,
Bohnet and Meier do not find higher response to trust in their distrust game. Moreover, in
Bohnet and Meier’s design, there are two changes in defaults; no trust vs full trust and no
inequality vs full inequality. This gives two instead of one reason for change in the neutral
choice 𝑥𝑥0 . This double default effect can lead the trustor to make decisions that are closer to the
default of full trust/high inequality. In other words, the trustor’s default effects are more likely
to dominate the strategic response to an expectedly lower trustworthiness level. Indeed, this is
what seems to be the case, in that they find lower trustworthiness, but higher trust.
4. Experimental Results
In this section we present the main findings. We measure trust level x as the amount sent in
Baseline, and as x =100 – amount insured in LD. Trustworthiness is measured by the return
ratio y/x which is the money returned (in Baseline) and money untaken (in LD) as a proportion
of positive trust levels (x>0).
13
Table 1. Descriptive statistics by treatment.
Table 1 shows the mean and standard deviation of money trusted, return ratio, trustors’, responders’ and pairs’
payoffs (sum of trustor’s and responder’s payoff). The sample size is 90 subjects in Baseline, 88 in LD I, 96 in LD
II, and 184 in LD (pooled data from LD I and LD II). Half of each sample are trustors, and the other half are
responders.
Baseline
Mean
Std
Mean
LD I
Std
LD II
Mean
Std
Mean
Trust
65.04
32.21
41.14
Trustworthiness
1.27
0.83
Trustors' Payoff
113.22
Responders' Payoff
Pairs' Payoffs
LD
Std
43.61
44.58
37.88
42.93
40.53
0.96
0.86
0.96
0.87
0.96
0.86
62.62
106.89
57.87
104.88
50.73
105.84
53.97
216.87
89.25
175.39
113.58
184.29
77.03
180.03
95.82
330.09
64.42
282.27
87.23
289.17
75.76
285.87
81.06
Table 2. Statistical tests between treatments.
Table 2 presents the statistical tests between treatments of: money trusted, return ratio, trustors’, responders’ and
pairs’ payoffs (sum of trustor’s and responder’s payoff). The sample size is 90 subjects in Baseline, 88 in LD I, 96
in LD II, and 184 in LD (pooled data from LD I and LD II). Half of each sample are trustors, and the other half
are responders.
Baseline vs. LD I
z-value
p-value
Baseline vs. LD II
z-value
p-value
LD I vs. LD II
z-value
p-value
Baseline vs. LD
z-value
p-value
Trust
3.00
<0.01
2.84
<0.01
0.51
0.61
3.37
<0.01
Trustworthiness
1.70
0.09
1.60
0.11
0.17
0.87
1.97
0.05
Trustors' Payoff
0.94
0.35
1.26
0.21
-0.55
0.59
1.27
0.20
Responders' Payoff
2.50
0.01
1.80
0.07
1.08
0.28
2.47
0.01
Pairs' Payoffs
3.00
<0.01
2.84
<0.01
0.51
0.61
3.37
<0.01
Table 1 and Table 2 show descriptive statistics by treatment and statistical tests, respectively.
We see that the levels of trust and trustworthiness are lower in both LD I and LD II than in
Baseline. The trust levels are significantly lower (p<0.01), while the differences in
14
trustworthiness levels are marginally significant (p= 0,09 in LD I and p= 0,11 in LD II). We
also see that the subjects’ payoffs are lower in the LD treatments compared with Baseline. 10
The behavior in the two LD treatments is very similar (no significant differences). For the rest
of the paper we will mainly pool the data from LD I and LD II and treat it as one treatment,
called LD, but comment also on the specific treatment effects of the two LD treatments. We see
from Table 1 and 2 that the responders’ behavior in LD pooled is significantly different from
the responders’ behavior in Baseline. The trustworthiness measure (y/x) is 1.27 in Baseline and
0.96 in LD (p<0.05). Hence, as expected (Hypothesis 1), we find:
Result 1: Responders were significantly less trustworthy in the loss domain than in the gain
domain.
Now, from Hypothesis 3, we expect that the trustor will trust less if he or she anticipates lower
trustworthiness, hence lower trustworthiness should give lower trust. This is also what we find.
The difference between trustors’ behavior in LD and Baseline is highly significant, both
statistically (p<0.01) and economically (43 vs. 65). In fact, nearly one third of the sample (30
subjects) in LD did not trust at all (x =0), while in Baseline all subjects exhibited positive trust
levels (x≥10). Hence we have:
Result 2: Subjects trusted significantly less in the loss domain than in the gain domain.
As discussed in Section 3 in relationship with Hypothesis 2, lower trust in the loss domain
indicates that the trustors’ strategic considerations dominate the pure default effect (full trust in
the loss domain). This is in contrast to Bohnet and Meier, who have a double default effect (full
trust and full inequality) and who find higher trust in the loss domain (their “distrust game”).
As a remark, people tend to follow defaults when they use simple heuristics and do not think
thoroughly through their decision. Complexity can thus strengthen default effects and the use
of heuristics (see e.g. Tversky and Shafir, 1992). Despite a slightly more complex formulation
10
The levels of trust and trustworthiness are higher in our baseline than in Berg et al (1995), but are in line with
other trust games run in high-trust countries such as Sweden and Switzerland. In Berg et al. trustors sent on
average a fraction of 0.52 of their initial endowment, compared to 0.65 in our baseline. With respect to
trustworthiness, the returned proportion was 0.47 in Berg et al, compared to 0.78 in our baseline. Johnson and
Mislin (2011) conducted a meta-analysis with data from 162 replications of the trust game from different
geographical regions. Trust and trustworthiness are found to vary significantly across countries. Overall the
average trust level is 0.5 while it is 0.66 in Switzerland and 0.74 in Sweden. In this study the trustworthiness is
measured as money returned/amount available to return. Using this measure, the average trustworthiness is 0.37
the same as in Sweden. In Switzerland it is 0.53 while in our study it is 0.42.
15
in LD than in Baseline, our subjects seem to make strategic considerations that overrule the
pure default effect.
Now, lower trust and trustworthiness in the loss domain give lower payoffs. We see that total
payoff by pairs is significantly lower in LD compared with Baseline (286 vs. 330, p<0.01). The
main part of the loss was borne by the responders (180 vs. 217, p=0.01), while the trustors’
payoff is not significantly different (106 vs. 113, p=0.2). However, note that from the
trustworthiness measures the average rate of return (ROT) for the trustors is significantly
different in the two treatments. With trustworthiness levels of 0.96 and 1.27, respectively, the
average ROT in LD was -4% while it was +27% in Baseline. Hence, on average, trust was
profitable in the gain domain but not in the loss domain.
The smaller differences in trustors’ payoffs compared to differences in ROT are due to lower
treatment differences on high trust levels. To illustrate how trustworthiness varies with different
levels of trust in both domains, we estimate the trustworthiness measure by trust tertiles. From
Figure 2 we see that the trustworthiness level is almost twice as high in Baseline compared to
LD on the lowest trust level (Q1), but that the treatment effect disappears on the highest trust
levels (Q3). This is in line with Hypothesis 2. Interestingly, we also see that trustworthiness
increases steadily with trust in the loss domain. This effect is absent in the baseline trust game.11
11
The amount returned in Baseline increases with amount trusted (sent), while the proportion returned, which is
our measure of trustworthiness, does not increase with trust.
16
2
Trustworthiness
1,55
1,5
1,28
1,14
1,13
0,97
1
0,71
0,5
0
Q1
Q2
Baseline
Q3
LD
Figure 2. Return ratio in Baseline and LD by tertiles. The tertiles are divided over the money trusted, and the
amounts represent the average proportion within each tertile.
Tobit regressions can further illuminate the trustors’ and responders’ behavior.
Table 3. Determinants of trust.
Table 3 presents the Tobit regressions for the money trusted as a function of the treatment (Loss Domain=1),
gender (Female=1), as well as the Two-Way interaction term Loss Domain*Female. Robust standard errors are
reported in parentheses below the estimated coefficients. Individual coefficients are significant at ***p<0.01,
**p<0.05, or *p<0.1 significance level. The R² is pseudo.
Dependent Variable: Trust
Tobit Regressions
Regressor:
Loss Domain
(1)
(2)
(3)
-46.16***
-54.68***
-90.91***
(13.55)
(13.25)
(26.35)
-33.86**
-73.55***
(13.27)
(23.81)
Female
Loss Domain*Female
54.52*
(28.69)
Intercept
84.30***
108.35***
137.93***
(10.44)
(14.57)
(23.14)
R²
0.012
0.02
0.024
F-statistic
N=
0.000
0.000
0.000
137
137
137
17
Regressions (1) and (2) from Table 3 confirm that subjects, controlling for gender, trusted
significantly less money in the loss domain than in the gain domain. The coefficient for Female
in regression (2) also shows that women in general trust significantly less than men. However,
the interaction term Loss Domain*Female in (3) shows that the treatment effect is stronger
among men than among women (p=0.059). Hence, the difference in trust level between men
and women is larger in Baseline than in the loss domain. It is well documented from a number
of experiments that men tend to trust more than women in trust games, which is related to
evidence on higher risk-taking among men than among women (see Croson and Gneezy, 2009,
for a comprehensive overview of gender differences in preferences). However, the interaction
effect we find is a bit surprising. As shown by Croson and Gneezy, women are generally more
sensitive to context and framing than men, while we find the opposite. With respect to trust, the
treatment effect is larger among men. 12
Table 4 examines the responders’ behavior.
12
We ran the same regressions as in Table 3 with LD I and LD II treated separately, see Table A1 and A2 in the
appendix. The main results are the same, with only marginal differences between the treatments. However, we
lose some power, and the interaction effect with gender is not significant in LD I (p=0.14).
18
Table 4. Determinants of trustworthiness.
In this table we present the Tobit regressions for the return ratio as a function of the treatment (Loss Domain=1),
money trusted, and gender (Female=1) as well as the Two-Way interaction terms Loss Domain*Female and Loss
Domain*Trust. Robust standard errors are reported in parentheses below the estimated coefficients. Individual
coefficients are significant at ***p<0.01, **p<0.05, or *p<0.1 significance level. The R² is pseudo.
Dependent Variable: Trustworthiness
Tobit Regressions
Regressor:
Loss Domain
(1)
(2)
(3)
(4)
(5)
-0.490**
-0.488**
-0.462**
-1.23**
-0.506
(0.217)
(0.218)
(0.224)
(0.477)
(0.34)
Trust
0.002
0.003
-0.004
0.003
(0.003)
(0.004)
(0.005)
(0.004)
Female
0.242
0.288
0.189
(0.217)
(0.215)
(0.296)
Loss Domain*Trust
0.012*
(0.007)
Loss Domain*Female
0.097
(0.434)
Intercept
1.22***
1.09***
0.929***
1.35***
0.949***
(0.146)
(0.265)
(0.325)
(0.365)
(0.347)
R²
0.017
0.018
0.022
0.033
0.022
F-statistic
N=
0.026
0.078
0.051
0.026
0.067
107
107
107
107
107
The coefficients for Loss Domain confirm that subjects were less trustworthy in LD compared
with Baseline. This holds after controlling for gender and trust-level. From (3) we see that there
is no gender effect on trustworthiness, and from the interaction term Loss Domain*Female in
(5) we see that the treatment effect is not driven by gender. (Moreover, there is no significant
effect on gender subsample, i.e. both the variable Loss Domain in (5), which measures the effect
on men only, and Female in (5) are insignificant). In (4) we confirm the positive relationship
between trust and trustworthiness in Loss Domain that we illustrated in Figure 2. The interaction
term Loss Domain*Trust is significant at p=0.08. 13 Hence, responders were more responsive to
a marginal change of money trusted in LD than in Baseline, and we have:
13
We ran the same regressions as in Table 4 with LD I and LD II treated separately, see Table A3 and A4 in the
appendix. The main results are again the same, but it turns out that the interaction effect between trust and loss
domain is mainly driven by LD II, with p=0.02. In LD I there is also a positive interaction effect, but it is not
significant (p=0.5).
19
Result 3: In contrast to responders in the gain domain, responders in the loss domain were
more trustworthy the more they were trusted.
This result supports Hypothesis 3 and demonstrates that intentions and emotions play a role in
trust games. If responders only have distributional fairness concerns (as in the theories of Fehr
and Schmidt, 1999, and Bolton and Ockenfels, 2000), one should expect the same
trustworthiness level for all trust levels. However, models that incorporate reciprocal
preferences (like Cox et al. 2007), where the perceived kindness of the players matters, can
contribute in explaining why trustworthiness increases with trust, as it does in the loss domain.
The results also support the presumption leading to Hypothesis 3: If losses loom larger than
gains, it is not surprising that responders are more responsive to trustors’ behavior in the loss
domain than in the gain domain.
5. Concluding remarks
One common feature with experimental papers on trust is that they are variations of investment
games, in which trust provides a potential gain and no trust yields no loss. No trust is default,
hence trust is an active decision, an act of commission, while no trust is a passive decision, an
act of omission. In many situations, however, full trust is often the default and trust involves
taking no action. For instance, parties make transactions without writing detailed contracts on
all possible contingencies that might occur, and people buy credence goods without requiring
documentation or verification of the provided quality.
We thus investigate a reframed trust game in which full trust is default and no trust generates a
loss. We hypothesize that the absence of trust feels more severe in the loss domain when full
trust is the default, than in the gain domain when no trust is the default. Furthermore, a given
level of intermediate trust may be perceived as less kind in the loss domain than in the gain
domain. Hence, if trustees care about the kindness of the trustors, we expect less trustworthiness
in the loss domain than in the gain domain. Cet par, the trustor will thus trust less if he or she
anticipates lower trustworthiness.
The experimental results support our main hypotheses. We find significantly lower levels of
both trust and trustworthiness in the loss domain when full trust was the default, than in the
Baseline trust game where no trust was the default. As a consequence, on average trust was
profitable in the gain domain but not in the loss domain. The treatment differences are quite
20
large. The trust and trustworthiness levels were, respectively, 51% and 32% higher in Baseline
than in the loss domain. Finally, we find that responders were more responsive to higher trust
in the loss domain than in the gain domain. In Baseline higher trust did not lead to higher levels
of trustworthiness, while in the loss domain responders were more trustworthy the more they
were trusted.
Our results have some interesting implications: First, we show that framing is important also in
situations where people trust or act trustworthily. The experiment indicates that if one can
choose framing in trust environments, it is better to frame a situation or transaction as a potential
gain than as a potential loss. One implication is that a trust relationship should “start small”,
trying to build trust, rather than to start ambitiously, risking losing trust. Indeed, starting small
is quite common in trust relationships. For example, in joint ventures and international business
relationships, partners typically start small by first attempting small scale projects before they
let the relationship gradually build (see e.g. Egan and Mody, 1992). Additionally, employment
relationships often start with a probation period before the parties make long term
commitments. 14
Second, the results may contribute in explaining complementarities between trust and legal
institutions. It may seem like a paradox that we find less trust and trustworthiness in situations
where full trust is the default, but its empirical implications may not be that surprising: In
countries with weak contract enforcement, transactions are trust-based, and hence trust is the
default. Any deviation from trust is regarded as a loss. However, in countries with strong
contract enforcement, transactions are less trust-based. Hence, low trust is the default and
deviations yield a potential gain. This can contribute to explain Knack and Keefer’s (1997)
finding that trust and contract enforcement tend to be complements, i.e. lower levels of trust
and trustworthiness in countries with weak legal contract enforcement institutions. 15
This apparent paradox is perhaps the most interesting feature with our results: The loss domain
default of full trust moves behavior in the opposite direction than what the default effect in itself
predicts. However, default effects are mainly demonstrated in simple decision problems, and
not so much in games where strategic considerations play a role. In our experiment, subjects do
not only make isolated decisions, they play a game. It then turns out that the effect of playing
14
See Andreoni and Samuelsson (2006) and Watson (1999, 2002) on complementary arguments for starting small.
Bohnet et al. (2010) provides a different but related argument for why trust and trustworthiness levels are lower
in Gulf countries than in Western Countries. They argue that the reference point for trustworthiness is higher in
the Gulf, making them require a higher minimum trustworthiness level in order to trust.
15
21
in the loss domain overrules the direct default effect. Or to put it differently: deviations from
defaults have strategic implications that seem to drive behavior even further away from the
default. The generality of this pattern should be investigated further, both theoretically and
experimentally.
22
APPENDIX
A 1. Instructions
In Baseline and LD treatments:
Welcome to our experiment. The experiment will last approx. 30 minutes. During the
experiment you will be able to earn money that will be paid out in cash anonymously once the
experiment is over. You will now have plenty of time to read through the instructions for the
experiment. If you have any questions on the instructions, please raise your hand and we will
come over to you. It is not allowed to talk or communicate with the other participants during
the experiment.
Follow the messages that pop up on the screen. There will be some waiting during the
experiment. Please do not press any other buttons than those you are asked to press. When you
are told on the screen that the experiment is over, it is important that you note down your pc
number and the amount earned on the enclosed receipt sheet. When we tell you that you may
leave the room, you can take along the receipt sheet to EAL, office no. H-161, to have the
amount paid out.
Baseline:
All the participants are split into pairs which consist of a sender and a responder. Half of you
are thus given the role as senders and half the role as responders. You do not get to know who
your partner is. Your partner is in the room, but you will not get to know who this person is
during the experiment or after the experiment.
At the start of the experiment all participants receive NOK 100. Sender (S) then gets the
opportunity to send all, some or none of his or her money to the responder (R). The amount that
is not sent is kept by the sender. The amount that is sent to R is tripled. If S chooses to send e.g.
NOK 20 to R, then R receives NOK 60. If S sends NOK 90, then R receives NOK 270. R then
decides how much of this amount he/she wants to keep and how much he/she will send back.
The amount that is sent back is not tripled.
In summary: If S sends an amount x to R and R returns y, the profit will be as follows:
•
S receives NOK 100-x+y.
23
•
R receives NOK 100+3x-y.
Loss Domain I and II:
All the participants are split into pairs which consist of a player A and a player B. Half of you
are thus given the role as player A and half the role as player B. You do not get to know who
your partner is. Your partner is in the room, but you will not get to know who this person is
during the experiment or after the experiment.
At the start of the experiment all participants receive NOK 200. Player B then gets the
opportunity to take money from player A. But before player B makes this decision, player A
can insure an amount between 0 and NOK 100. Thus player B is not able to take any money
from the amount that is insured.
Loss Domain I:
For each krone that player A insures, player B loses 2 kroner. This means that if player A insures
100 kroner, then player B has nothing left, but can instead take up to 100 kroner from player A.
On the other hand, if player A insures nothing, both players keep their corresponding 200
kroner, but player B can then take an amount between 0 and 200 kroner from player A.
Some examples:
If player A insures 80 kroner, then he/she has an uninsured amount of 200-80=120, in addition
to the 80 that he/she has insured. Player B will have 200-2*80=40. In addition, player B can
decide how much from player A’s uninsured amount of 120 that he/she wants to take. Hence,
player A ends up with an amount between 80 and 200, while player B ends up with an amount
between 40 and 160 kroner.
If player A insures 10 kroner, then he/she has an uninsured amount of 200-10=190, in addition
to the 10 that he/she has insured. Player B will have 200-2*10= 180. In addition, player B can
decide how much from player A’s uninsured amount of 190 that he/she wants to take. Hence,
player A ends up with an amount between 10 and 200, while player B ends up with an amount
between 180 and 370 kroner.
Summarizing: If player A insures an amount x, and player B takes an amount y from player A,
then the payoffs are as follows:
24
Player A earns kr. 200-y
Player B earns kr. 200-2x+y
Loss Domain II:
For each krone that player A insures, both players A and B lose 1 krone each. This means that
if player A insures 100 kroner, both players lose 100 kroner and end up with 100 kroner each.
On the other hand, if player A insures nothing, both players keep their corresponding 200
kroner, but player B can then take an amount between 0 and 200 kroner from player A.
Some examples:
If player A insures 80 kroner, then he/she has an uninsured amount of 200-80-80=40 in addition
to the 80 that he/she has insured. Player B will have 200-80=120. In addition, player B can
decide how much from player A’s uninsured amount of 40 that he/she wants to take. Hence,
player A ends up with an amount between 80 and 120, while player B ends up with an amount
between 120 and 160 kroner.
If player A insures 10 kroner, then he/she has an uninsured amount of 200-10-10=180 in
addition to the 10 that he/she has insured. Player B will have 200-10=190. In addition, player
B can decide how much from player A’s uninsured amount of 180 that he/she wants to take.
Hence, player A ends up with an amount between 10 and 190, while player B ends up with an
amount between 190 and 370 kroner.
Summarizing: If player A insures an amount x, and player B takes an amount y from player A,
then the payoffs are as follows:
Player A earns kr. 200-x-y
Player B earns kr. 200-2x+y
25
A 2. Tables and Figures
R
R
400
400
D
D
T
T
400
400
LD I
Berg et al. (1995)
R
R
400
400
D
D
T
400
T
400
Bohnet & Meier (2012)
LD II
Figure A1. The figure shows payoff sets for different trust levels in Berg et al. (1995), Bohnet & Meier (2012),
LD I and LD II. The trustor (T) determines the total surplus, i.e. the isoprofit curve, while the responder (R)
determines the allocation at a given isoprofit cuve. The default (D) is the initial allocation before any actions are
taken.
26
Table A1. Determinants of trust (restricting to LD I).
Table A1 presents the Tobit regressions for the money trusted as a function of the treatment (Loss Domain I=1),
gender (Female=1), as well as the Two-Way interaction term Loss Domain I*Female. Robust standard errors are
reported in parentheses below the estimated coefficients. Individual coefficients are significant at ***p<0.01,
**p<0.05, or *p<0.1 significance level. The R² is pseudo.
Dependent Variable: Trust
Tobit Regressions
Regressor:
Loss Domain I
(1)
(2)
(3)
-50.00***
-62.46***
-95.09***
(18.35)
(18.25)
(31.73)
-45.32**
-75.20***
(18.23)
(24.70)
Female
Loss Domain I*Female
54.39
(36.28)
Intercept
85.64***
117.83***
140.05***
(11.57)
(18.67)
(24.69)
R²
0.014
0.027
0.031
F-statistic
N=
0.008
0.001
0.003
89
89
89
Table A2. Determinants of trust (restricting to LD II).
Table A2 shows the Tobit regressions for the money trusted as a function of the treatment (Loss Domain II=1),
gender (Female=1), as well as the Two-Way interaction term Loss Domain II*Female. Robust standard errors are
reported in parentheses below the estimated coefficients. Individual coefficients are significant at ***p<0.01,
**p<0.05, or *p<0.1 significance level. The R² is pseudo.
Dependent Variable: Trust
Tobit Regressions
Regressor:
Loss Domain II
(1)
(2)
(3)
-38.18***
-46.30***
-75.57***
(13.09)
Female
(12.74)
(23.51)
-35.94***
-63.45***
(12.58)
(19.66)
Loss Domain II*Female
46.53*
(26.26)
Intercept
79.36***
104.64***
125.02***
(9.18)
(13.14)
(18.97)
R²
0.014
0.027
0.032
F-statistic
N=
0.004
0.000
0.001
93
93
93
27
Table A3. Determinants of trustworthiness (restricting to LD I).
Table A3 presents the Tobit regressions for the return ratio as a function of the treatment (Loss Domain I=1),
money trusted, and gender (Female=1) as well as the Two-Way interaction terms Loss Domain I*Female and Loss
Domain I*Trust. Robust standard errors are reported in parentheses below the estimated coefficients. Individual
coefficients are significant at ***p<0.01, **p<0.05, or *p<0.1 significance level. The R² is pseudo.
Dependent Variable: Trustworthiness
Tobit Regressions
Regressor:
Loss Domain I
(1)
(2)
(3)
-0.485*
-0.478*
-0.473*
-0.799
-0.392
(0.280)
(0.280)
(0.282)
(0.599)
(0.436)
Trust
(4)
(5)
-0.003
-0.002
-0.004
-0.003
(0.004)
(0.004)
(0.005)
(0.004)
Female
0.110
0.139
0.169
(0.249)
(0.257)
(0.285)
Loss Domain I*Trust
0.005
(0.009)
Loss Domain I*Female
-0.174
(0.572)
Intercept
1.22***
1.40***
1.32***
1.44***
1.31***
(0.145)
(0.286)
(0.359)
(0.371)
(0.362)
R²
0.018
0.020
0.021
0.023
0.022
F-statistic
N=
0.088
0.182
0.290
0.338
0.334
72
72
72
72
72
28
Table A4. Determinants of trustworthiness (restricting to LD II).
Table A4 shows the Tobit regressions for the return ratio as a function of the treatment (Loss Domain II=1), money
trusted, and gender (Female=1) as well as the Two-Way interaction terms Loss Domain II*Female and Loss
Domain II*Trust. Robust standard errors are reported in parentheses below the estimated coefficients. Individual
coefficients are significant at ***p<0.01, **p<0.05, or *p<0.1 significance level. The R² is pseudo.
Dependent Variable: Trustworthiness
Tobit Regressions
Regressor:
Loss Domain II
(1)
(2)
(3)
(4)
(5)
-0.473*
-0.465*
-0.423
-1.58***
-0.511
(0.253)
(0.250)
(0.261)
(0.575)
(0.366)
Trust
0.003
0.003
-0.004
0.003
(0.004)
(0.004)
(0.005)
(0.004)
Female
0.271
0.308
0.184
(0.239)
(0.231)
(0.295)
Loss Domain II*Trust
0.018**
(0.008)
Loss Domain II*Female
0.218
(0.500)
Intercept
1.22***
1.04***
0.879**
1.35***
0.914**
(0.145)
(0.292)
(0.343)
(0.367)
(0.364)
R²
0.018
0.020
0.025
0.052
0.026
F-statistic
N=
0.065
0.157
0.089
0.022
0.155
80
80
80
80
80
29
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33
A.3. Theory
We …rst verify the formula for z(x). When the constraint z
(x) is not
binding, the …rst order conditions for the optimization problem imply 0 =
m 1 + z a 1 and m = b(x) z. Solving for z then yields the formula.
Consider next the case = 0 (so = 1) and
when the returned amount y is positive, we have
y =z+x
100 =
= k (m(x)
k(m(x) m0 )
b(x) + x
k(m(x) m0 ) + 1
m0 ). Then,
100
In the baseline (BL) treatment we have
= 1; m0 = 100, while in the
loss domain (LD) treatment we have
1; m0 = 200. Consider x such that
yLD > 0. Then we must have m(x) > mLD
= 200 and therefore m(x) > mBL
=
0
0
100. Moreover, yLD
yBL i¤ (m(x) mLD
(m(x) mBL
0 )
0 ), which, since
m(x) = 100 + 3x, is equivalent to
3x=(3x 100). This must hold for x such
> 0, i.e. x > 100=3.
that 100 + 3x mLD
0
3
100, the condition
3x=(3x 100) implies
Since 100
3 <x
2 . Hence
3
we must have yLD < yBL if < 2 .
On the other hand, if
> 32 , this argument shows that yLD > yBL i¤
3x=(3x 100) < , i.e. i¤ x > 3(100 1) . This veri…es the claim in the paragraph
preceding Hypothesis 1 in the text.
Now consider
with z given by
dy
dx .
When y > 0, we have y = z +x 100 and thus
z=
(
(
(r(x)))
b(x)
(r(x))) + 1
dy
dx
=
dz
dx +1,
(x)
b(x)
(x) + 1
Straightforward computations yield
dz
= z( ; x)
dx
0
1
(x) b0 (x)
+
>0
(x) + 1 (x)
b(x)
(A1)
All terms on the RHS are positive, hence z and thus y increases in x. Higher
0
0
(x)
(x)
yields higher and thus larger z, for given x. The terms (x)
and bb(x)
do
not depend on . The …rst term z( ; x) on the RHS increases, while the second
term (in square brackets) decreases when increases. So higher yields two
dy
dz
, and thus also on dx
. The e¤ects of higher m0 on the
opposing e¤ects on dx
derivatives are also not straightforward. We will …rst show that if
1, then
dy
is
increasing
in
.
Note
that
1
is
a
reasonable
condition,
since
is
dx
the relative weight the responder puts on the trustor’s payo¤
Claim 1. If
(r(x))
1, then
To show this, we di¤erentiate
0
(x)
(x)
and
0
b (x)
b(x)
@2y
@ @x
dz
dx
> 0.
with respect to
, noting that he terms
do not depend on . This yields
@2z
@z
=(
@ @x
@
0
0
1
(x) b0 (x)
1
(x) @
+
+ z( ; x)
)
(x) + 1 (x)
b(x)
( (x) + 1)2 (x) @
1
(A2)
Substituting for z( ; x) =
some algebra
(x)
(x)+1 b(x)
@z
@
and
=
1
( (x)+1)2 b(x)
then yields, after
@2z
1
(x) 0 (x) b0 (x) @
b(x)
(
=
+
)
2
@ @x
( (x) + 1)
(x) + 1 (x)
b(x) @
Since (x)
1 when
veri…es the claim, since
(r(x))
@2y
@ @x
1, and
=
@2z
@ @x .
@
@
The next result provides conditions for
Claim 2. For the case
m(x) = 100 + 3x we have
= 0 and
@2y
> 0 if
@m0 @x
> 0, we see that
dy
dx
1=( k)
To verify the claim, …rst note that for (x) =
straightforward computations yield
This implies that
i.e.
@2y
@m0 @x
=
km1 b(x)
k(m1 x + m2 m0 ) + 1
@2z
@m0 @x
@2z
@ @x
> 0. This
to be increasing in m0 .
= k (m(x)
@2z
2 k
=
@m0 @x
( (x) + 1)2
(A3)
m0 ), with k > 0 and
400 + 2m0
(x) = k(m1 x + m2
km1
b(x)
(x) + 1
b0 (x)
2
m0 )
(A4)
is positive i¤ the last parenthesis is positive,
b0 (x)
3(200 + 2x)
=
2
(3x + 100 m0 ) + 1= k
1>0
(A5)
Note that for y(x) > 0 we must have (x) > 0 and therefore here 3x+100 m0 >
0. Also note that for 1= k < 400 + 2m0 , the expression on the RHS of (A5)
exceeds (3x+1003(200+2x)
1. This last expression is increasing in x, and
m0 )+400+2m0
therefore exceeds its value at x = (m0 100)=3, which is 0. This veri…es the
claim.
Remark. As noted above, it is reasonable to assume
(x)
1 for all
x, which holds true here if 400 m0
1= k. A set of parameters where
dy
400 m0
1= k
400 + 2m0 thus implies that (x)
1 and dx
increases
dy
with m0 . By Claim 1 we then have dx increasing in both m0 and in .
We …nally verify the claim that if the sender is uncertain about the responder’s type, then she may optimally choose to send an amount x between the
minimum amount (0) and the maximal amount (x = M = 100).
Suppose there are two responder types; one altruistic as in the text, and
one sel…sh type, who never returns anything (y = 0). For the sel…sh type, the
trustor is left with the monetary payo¤ (x) = M x. Suppose the altruistic
2
type returns more than he receives if the latter is su¢ ciently large, and hence
that z(M ) > M . The sender’s expected utility is then
pU (z(x)) + (1
p)U (M
x)
v(x);
where p is the probability of an altruistic type.
There will be an interior optimal solution 0 < x < M if v(M )
v 0 (M ) = pU 0 (z(M ))z 0 (M )
(1
v(0) and
p)U 0 (0) < 0
These conditions will hold if z(M ) > M , U 0 (0) is su¢ ciently large ( U 0 (0) = 1
is certainly su¢ cient) and p is su¢ ciently close to 1.1
1 It should be noted that the shape of the utility function is not necessarily concave, since
risk loving behavior is more common in loss domain. On the other hand, a common explanation
for di¤erent behavior in trust settings than in pure risk-settings is that in the former, subjects
may be betrayed (see Bohnet and Zeckhauser, 2004). Our empirical results with intermediate
trust levels indicate that the trustors’ utility functions are indeed concave also in the loss
domain.
3