Monetary Stakes and Socioeconomic Characteristics in Ultimatum
Games: An Experiment with Nation-Wide Representative Subjects
Tsu-Tan Fu
Center for Survey Research and Institute of Economics, Academia Sinica,
Nankang, Taipei 115, Taiwan
Wei-Hsin Kong
Institute of Industrial Economics, National Central University
C.C. Yang *
Institute of Economics, Academia Sinica, Nankang, Taipei 115, Taiwan
Department of Public Finance, National Chengchi University, Wenshan, Taipei 116, Taiwan
June 2007
[Abstract] We conduct an experiment on ultimatum games with subjects who are
representative of a nation. Our focus is on the size effect of monetary stakes when
experimental subjects are “real” people rather than students as in previous studies. It is
found that: (i) raising stakes substantially reduce the number of “outliners” in both offers
and rejections; (ii) higher stakes exert a significant impact on players’ offer and rejection
behavior as the standard economic theory predicts even for inexperienced or one-shot
play; (iii) socioeconomic characteristics dominate responders’ behavior when stakes are
low, whereas monetary stakes dominates responders’ behavior when stakes are high; (iv)
age has a lifecycle effect on players’ behavior when stakes are low: those subjects who
are young and old offer less and reject less often than those who are in the middle age;
and (v) women reject less often than men, but there is no gender difference in offer
behavior.
JEL classification: C78; C91; C93
Key words: Ultimatum games; Monetary stakes; Socioeconomic characteristics
*Corresponding author, C.C. Yang.
Nankang, Taipei 115, Taiwan.
Mailing address: Institute of Economics, Academia Sinica,
E-mail: [email protected]
1
1.
Introduction
Do stakes matter?
Put differently, will observed behavior be more consistent with
the prediction of economic theory as the size of monetary stakes involved gets larger?
This is perhaps the question that people ask most often about experimental economics. 1
Roth et al. (1991) compare the market and the bargaining environment in
experiments.
Nine proposers make simultaneous offers to a single responder in the
market game, while one proposer makes an offer to a single responder in the bargaining
game.
According to the standard economic theory, one player is predicted to receive all
or almost all of the “pie” in both games. 2
Although the market game does exhibit a
vigorous convergence to its subgame-perfection prediction in the experiments, the
bargaining game does not show any tendency toward its subgame perfection.
The bargaining environment considered by Roth et al. (1991) is the so-called
“ultimatum game.”
This is a simple game in which there are two players: a proposer
and a responder. The proposer offers a share of a fixed sum of money to the responder
and keeps the rest to herself. If the responder accepts the offer, the money is divided
accordingly. If the responder rejects the offer, both players receive nothing. Although
simple, this “divided-the-dollar” game is important because it helps crystallize more
complicated bargaining games.
In fact, it constitutes the last round of the finite-horizon
version of the alternating-offer bilateral bargaining game a la Stahl (1972) and Rubinstein
(1982). 3
In response to the finding that subjects’ behavior in ultimatum games does not show
any tendency toward its subgame-perfection prediction, a number of papers have
1
Smith and Walker (1993) and Camerer and Hogarth (1999) provide surveys on the issue.
2
The player who is predicted to receive almost all of the pie is the responder in the market game, while it is the
proposer in the bargaining game..
3
The seminal work on experimental ultimatum games is Guth et al. (1982). For literature surveys, see Guth
(1995), Roth (1995) and Camerer (2003).
2
conducted experiments to investigate if the deviation between observation and theory is
an artifact and will diminish once stakes are raised. 4
Camerer (2003, Section 2.2.2)
surveys this line of the literature and concludes:
Taken together, these studies show that very large changes in stakes (up to several
months’ wages) have only a modest effect on rejections. Raising stakes also has
little effect on Proposers’ offers, presumably because aversion to costly rejection
leads subjects to offer closer to 50 percent when stakes go up. (p. 61)
It is interesting to note that previous ultimatum game studies on the size effect of
monetary stakes almost all use student subjects in their experiments.
In fact, the studies
surveyed by Camerer (2003) all use student subjects in their experiments.
The
confinement to student subjects naturally provokes a question: Will stakes still matter
little if “real” people rather than students play the ultimatum game?
A possible route to
answer the question, suggested by Harrison and List (2004), is to go ahead with the
employment of “real” people in experiments.
We adopt the route in this paper.
The finding that subjects’ behavior in ultimatum games does not show any tendency
toward its theoretic prediction suggests the possibility that players’ preferences value not
only their own monetary wealth but also something else. 5
This possibility leads several
authors, notably Rabin (1993), Fehr and Schmidt (1999) and Bolton and Ockenfels
(2000), to extend the standard wealth maximization or pure self-interest model to
incorporate non-wealth factors such as a concern for fairness into players’ preferences.
These authors in effect explore players’ other-regarding preferences or the so-called
4
These papers include Roth et al. (1991), Straub and Murninghan (1995), Tompkinson and Bethwaite (1995),
Hoffman et al. (1996), Slonim and Roth (1998), Cameron (1999), List and Cherry (2000), Munier and Zaharia
(2003), and Carpenter et al. (2005).
5
As pointed out by Camerer (2003, p. 48), whether to accept an ultimatum offer requires no strategic thinking
since it is simply a choice. Hence, players’ self-interest preferences rather than game-theoretic reasoning seem to
be the culprit responsible for the failure of the prediction.
3
“social preferences.”
This paper considers a different way of tackling the finding.
The
Rabin-Fehr-Schmidt-Bolton-Ockenfels approach attempts to account for experimental
data by theoretically building an extended model in which both wealth and non-wealth
factors in players’ preferences are counted in. In contrast, we attempt to account for
experimental data by empirically separating wealth factors from non-wealth factors in
players’ preferences.
unobserved.
Non-wealth factors in players’ preferences are typically
The gist of our approach is to utilize players’ observable socioeconomic
characteristics as proxies for these unobservables.
We justify the correlation between
players’ socioeconomic characteristics and their social preferences in our context when
we come to the regression models.
It should be emphasized that our approach is not new and, indeed, it is the standard
practice adopted by applied econometricians when they use naturally-occurring data to
test empirically the validity of some theoretic prediction of a wealth maximization or pure
self-interest model.
wealth
maximizing
Naturally-occurring data result from the combined effect of players’
behavior
and
their
non-wealth
considerations.
Applied
econometricians must fully recognize this fact, since they typically incorporate
socioeconomic variables in their regression equations in order to isolate the pure effect of
wealth factors (implied by the pure self-interest model) from the overall effect observed
in the data. Most experiments, including lab and field, do not collect data in regard to
subjects’ socioeconomic characteristics.
The reason is perhaps that experimental
subjects are often rather homogeneous in their socioeconomic attributes (say, student
subjects have more or less the same age and the same education).
This may explain why
applied econometricians’ standard practice is not popular in the field of experimental
economics.
It is interesting to note that previous ultimatum game experiments on the effect of
the size of stakes almost all employ student subjects.
4
Indeed, all the studies surveyed by
Camerer (2003) use student subjects in their experiments.
It thus becomes important to
know whether “real” people may behave differently from student subjects when stakes
are raised.
In this paper we run an experiment on ultimatum games with representative subjects,
in the sense that our subjects are randomly sampled from the adult population of a nation
with a full probability. Previous experiments on ultimatum games are conducted either
with student subjects or subjects who are not representative of an economy.
These
experiments usually suffer from the constrained subject pool and the selectivity of the
subject pool. 6
The representativeness of our sample frees us from these shortcomings.
More importantly, we have via survey the collection of a range of standard
socioeconomic characteristics of the subjects such as age, sex, education, personal
income, marital status, etc.
This collection of varied socioeconomic characteristics
across subjects potentially allows us to control for subjects’ non-wealth attributes in their
preferences and hence obtain a cleaner effect of monetary stakes.
It also enables us to
see the role of socioeconomic characteristics or, more generally, non-wealth attributes in
explaining players’ behavior. 7
In the remainder of this introductory section, we briefly summarize the main
findings from our experiment.
Offering more than half of the “pie” to responders is a hyperfair offer and hence it
may be viewed as an outliner.
Similarly, rejecting a hyperfair offer may also be viewed
6
See Fehr et al. (2003), Harrison and List (2004), List (2006), and List and Levitt (2006) for details.
7
To our knowledge, Fehr et al. (2003) is the only experimental paper that uses subjects who are representative of a
nation (the German population). They combine surveys with experiments in studying trust and trustworthiness.
5
as an outliner. 8
In our data, raising stakes substantially reduces the number of
“outliners” in both offers and rejections.
This result is consistent with Smith and
Walker’s (1993, p. 245) finding: “rewards reduce the variance of the data around the
predicted outcomes.”
It is also consistent with Camerer and Hogarth’s (1999, p. 31)
finding: “Incentives often reduce variance by reducing the number of extreme outliners,
probably caused by thoughtless, unmotivated subjects.”
A notable exception to Camerer’s (2003) conclusion that raising stakes has little
impact on offers and rejections is the work of Slonim and Roth (1998), who show for
experienced play of student subjects that: (i) responders’ rejection rates decrease as stakes
increase, and (ii) proposers’ offers move closer toward the subgame-perfection prediction
of the standard theory as stakes increase. However, for inexperienced or one-shot play,
Slonim and Roth fail to detect significant difference between low and high stakes
proposals or between low and high stakes rejection frequencies. 9
We extend their
finding from students subjects to a representative sample, showing that the effect of
higher stakes features significantly in players’ offer and rejection behavior as the standard
economic theory predicts even for inexperienced or one-shot play.
This significant
effect remains after the control for subjects’ socioeconomic characteristics.
Henrich et al. (2002) find in a cross-cultural study that there exists a close
relationship between subjects’ experimental behavior and their everyday life. Hoffman
et al. (1998, p. 350) vividly put it: “A on-shot game in the laboratory is part of a life-long
sequence, not an isolated experience from one’s reputation norm.”
8
Perhaps, the
Henrich et al. (2002) show that the Ache headhunter of Paraguay and the Lamelara whalers of Indonesia offer
more than half of the “pie” on average in the ultimatum game. However, this is unusual for other societies as
noted by Camerer (2003, p. 71). A robust regularity of experimental ultimatum games considered by Fehr and
Schmidt (1999, p. 826) is: “There are virtually no offers above 0.5.” According to Fehr and Schdmit (1999,
Proposition1), it is a dominant strategy for the responder to accept any hyperfair offer.
9
List and Cherry (2000) has a similar finding. It should be noted that both Slonim and Roth (1998) and List and
Cherry (2000) employ students subjects in their experiments.
6
difference between our finding and Slonim and Roth’s (1998) simply reflects the fact that
everyday life led by student subjects differs from everyday life led by our nation-wide
representative subjects.
An important finding from our experiment is that socioeconomic characteristics
dominate responders’ behavior when stakes are low, whereas monetary stakes dominate
responders’ behavior when stakes are high. More precisely, we find that when stakes are
low, subjects’ socioeconomic characteristics feature statistically significantly in
explaining responders’ behavior, whereas monetary stakes have little explanatory power.
The opposite occurs when stakes become high. To discuss the external validity of
findings in laboratory experiments, Levitt and List (2006) consider a simple model in
which utility is additively separable in the morality (non-wealth) and wealth arguments.
Our finding suggests that non-wealth arguments outweigh wealth arguments when stakes
are low, whereas the opposite is true when stakes are high.
Gender and age are the two socioeconomic variables that stand out significantly in
explaining players’ behavior in our data.
Consistent with Eckel and Grossman (2001),
we find that women reject less often than men, but there is no gender difference in offer
behavior. We also find that age exerts a lifecycle effect on players’ behavior when
stakes are low: those who are young and old behave closer toward the
subgame-perfection prediction of the standard theory than those who are in the middle
age.
This result with adult subjects complements the previous experimental findings in
Murnighan and Saxon (1998) and Harbaugh et al. (2000) with children subjects. Both
studies find that the youngest children behave closer toward the subgame-perfection
prediction of the standard theory than older children and adult population.
When stakes are high, neither gender nor age features significantly in explaining
players’ behavior in our data.
Responders’ behavior is overwhelmed by monetary
incentives, and education is the only socioeconomic variable that survives capable of
explaining proposers’ behavior.
7
Burks et al. (2001) experimentally compare student and non-student subjects
(employees of a book and magazine distribution business) in high-stake (US$100)
ultimatum games. They do not find different behavior across the two groups. 10
By
contrast, student subjects behave differently from non-student subjects in our
representative sample.
Specifically, we find that students’ responses are completely
consonant with the subgame-perfection prediction of the standard theory: they always
accept whatever is offered.
However, due to the small sample of student subjects in our
data, the finding is suggestive but not conclusive.
2. Experimental design
Subjects
During the middle of year 2005, Center for Survey Research of Academia Sinica at
Taiwan launched a survey, in an attempt to understand people’s acquaintance with and
their attitude toward the gene technology in the Taiwan economy. 11
The size of the
survey was targeted to be 2000 adult individuals (20 years old or older).
These
individuals were randomly sampled from the stratified population of the economy to
ensure their representative. (NOTE) We appended our experiment on ultimatum games to
the survey, and asked interviewers to conduct the experiment after all survey questions
had been answered.
Due to financial constraints, only 800 individuals out of the
targeted size of the survey were randomly chosen to play the ultimatum game.
Of these
800 individuals, the actual number participates in the game was 791, consisting of 397
proposers and 394 responders.
Procedures
10
They do find different behavior across the two groups in the so-called “dictator game.”
11
Center for Survey Research of Academia Sinica routinely conducts many surveys each year.
8
The present study is mainly to see how financial incentives would affect observed
behavior in ultimatum bargaining. Thus, our treatment variable is the size of stakes,
which is varied between two amounts: NT$200 (low stake) and NT$1000 (high stake). 12
Of 800 chosen individuals, 400 individuals (200 pairs) were designated to play the
low-stake game, while the other half the high-stake game.
Subjects were randomly
assigned to be either a proposer or a responder, and randomly matched in pairs in ex ante.
Each subject took part in a one-shot game.
The average hourly wage rate in Taiwan at
the time of the experiment was around NT$100. It took less than half an hour for our
subjects to complete the game.
Prior to conducting experiments, we held an interviewer meeting. The purpose was
to explain the ultimatum game to the interviewers and let them be familiar with the
game. 13
We ran a practice game between the interviewers and answered all questions
raised by the interviewers.
The procedure of running the experiment was as follows.
First, each subject was
given a written instruction sheet explaining the rules of the ultimatum game. The content
of the instruction is standard, except that it was emphasized in the instruction that the
money involved was real rather than hypothetical, and that the subjects’ decision should
be made privately and kept secret from the interviewer. 14
The interviewer read the
instruction to the subject and then asked whether the rules of the game were well
understood.
If not, the interviewer read the instruction again until the subject
understood the rules of the game.
Second, we implemented the experiment sequentially:
in the first stage, the proposer’s proposal was elicited; in the second stage, the responder
was informed of the proposer’s proposal and then the responder’s response was elicited.
12
NT$ denotes New Taiwan dollar and the exchange rates were around NT$32+ per US$.
13
The interviewers have been trained in regard to survey skills elsewhere.
14
See Levitt and List (2006) for a discussion on the advantage and disadvantage of keeping anonymity between
the experimenter and the subjects in experiments.
9
We describe these two stages separately in the following.
Each subject who played the role of the proposer was given an envelope and a
decision card.
The subject was asked: (i) to fill in his or her pre-assigned ID number in
the decision card, and (ii) to mark a choice in the decision card among six alternatives:
{(180, 20), (160, 40), (140, 60), (120, 80), (100, 100), (80, 120)}
in the case of the low stake (NT$200);
{(900, 100), (800, 200), (700, 300), (600, 400), (500, 500), (400, 600)}
in the case of the high stake (NT$1000);
where the first entry in a vector denotes the amount of money offered to the proposer,
while the second entry denotes the amount of money offered to the responder. For
example, the vector (180, 20) means that a proposer allocated NT$180 to him or herself
and NT$20 to his or her responder in the case of NT$200.
After marking the choice, the
subject him or herself put the decision card into the given envelope and sealed it with his
or her own signature. Even with the signature, we believe that the subjects’ identities
largely remained anonymous to each other because of the large size of our sampled
population.
The envelope sealed by the proposer together with a new envelope were given to the
ex ante matched subject who played the role of the responder.
The responder was asked:
(i) to open the envelope, (ii) to fill in his or her pre-assigned ID number in the decision
card, and (iii) to mark either “yes” or “no” choice to the proposer’s proposal in the
decision card.
After marking the choice, the subject him or herself put the decision card
into the given new envelope and sealed it with his or her own signature.
Academia Sinica is a well-known and well-established research unit in Taiwan.
The subjects knew that Academia Sinica was responsible for both the survey and the
experiment.
However, to enhance the credibility of our experiment, it was announced in
the instruction sheet that we would open all the envelopes publicly on a particular day
(July 21, 2005), and that the money earned by the subjects from the game would be
10
mailed to them immediately after that day. We filmed the whole process of opening the
envelopes and established a website (http://www.sinica.edu.tw/as/survey) for the subjects
to check the results of the experiment.
3. Results
Characteristics of the sample
Table 1 summarizes the definitions and descriptive statistics of socioeconomic
characteristics of our completed cases in the experiment.
In our sample, 51% are males,
64% are parents with at least one child, and 7.4% are students.
Since our targeted
survey respondents are adults, the mean age of the sample is 41 years old.
About 49%
of samples are at age 20-40, 44% are between 40-60, and the rest of 7% are elder than 60
years old. The mean schooling year of the sample is 12 years. The distribution of
educational attainment in Table 1 indicates that majority (34%) of the sample are senior
high school graduates, who are followed by college graduates (23%) and junior college
graduates (19%).
[Insert Table 1 about here]
Offer and (conditional) acceptance pattern
Proposers were asked to choose between 6 discrete allocations which representing 6
different fractions of money split between the proposer and the responder. For example,
the 1st choice (90/10) in Table 2 means that a proposer chose to keep 90% of the money
for herself and offered 10% to her responder. Similar interpretation applies to all other
choices.
Thus, the 5th choice (50/50) implies that there is an equal split of money
between the proposer and the responder.
[Insert Table 2 about here]
Table 2 summarizes the offer and the (conditional) acceptance frequencies in our
sample.
Of the six choices, the equal split of the money (5th offer choice) is the
dominant choice by proposers. Their frequencies are as high as 70% for both stake=200
11
and stake=1000.
Table 2 shows that a few proposers chose the 6th choice.
This is an offer in which
the fraction of money split allocated to responders is above 50%. 15
As noted in the
Introduction, a robust regularity of experimental ultimatum games is: “There are virtually
no offers above 0.5” (Fehr and Schmidt, 1999, p. 826). Thus, it seems reasonable to
deem the “higher-than-50%” hyperfair offer as an outliner. We see from Table 2 that the
number of the outliners is reduced substantially from 18 (9.14%) to 4 (2.00%) as the size
of stakes increases from 200 to 1000.
Table 2 reports that the mean conditional acceptance rates for low (=200) and high
(=1000) stakes are 89% and 94%, respectively.
There is thus a 5% difference on
average in the conditional acceptance rate between the low and the high stake.
Note that
the outliner offer 40/60 receives a rejection rate as high as 22% in the case of the low
stake, while it receives no rejection at all in the case of the high stake. According to the
social-preference theory proposed by Fehr and Schdmit (1998, Proposition1), it is a
dominant strategy for the responder to accept any hyperfair offer.
Thus, it seems
reasonable to deem the rejection of the “higher-than-50%” hyperfair offer as an outliner
as well.
Once again, we see that higher incentives substantially reduce the number of
players who have outlying behavior.
Smith and Walker (1993) survey 31 experimental studies which report data on the
effect of increased monetary rewards.
They conclude: “in virtually all cases rewards
reduce the variance of the data around the predicted outcome” (p. 245). Camerer and
Hogarth (1999) review 74 experiments with different degrees of performance-based
financial incentives.
One of their main findings is that “Incentives often reduce variance
by reducing the number of extreme outliners, probably caused by thoughtless,
unmotivated subjects” (p. 31).
15
The substantial reduction in the number of the
Higher-than-50% ultimatum offers are also observed in Slonim and Roth (1998).
12
“outliners” from Stake=200 to Stake=1000 in our data is consistent with these previous
findings.
Overall, our data on ultimatum offers and acceptances are comparable with those
found in previous lab experiments summarized by Camerer (2003, Tables 2.2 and 2.3).
Figure 1 plots the offer and the acceptance data of Table 2.
[Insert Figure 1 about here]
Student versus non-student sample
In this study we employ a representative sample, in which both student and
non-student subjects are included. This is different from most of previous experiments
that use either student or non-student subjects only. Table 3 summarizes the offer and
the (conditional) acceptance frequencies in our sample according to whether subjects are
students or not.
a student.
student.
Note that the responder corresponding to a student proposer may not be
Similarly, the proposer corresponding to a student responder may not a
Figure 2 plots the data of Table 3. It is interesting to observe that: (i) student
subjects choose low (high) offers with lower (higher) frequencies than non-student
subjects, and they never choose the hyperfair offer; and (ii) student subjects always
accept whatever is offered, while non-student subjects do not. However, due to the
small sample of our student subjects in our data, the finding is suggestive but not
conclusive. 16
[Insert Table 3 and Figure 2 about here]
Responder behavior
From Figure 1, we see sizable discrepancies of acceptance rates between the high
and the low stake in some given offers. However, due to the existence of non-economic
factors, this may not imply a causal impact of the stake on acceptance rates.
Since responders either accept or reject a given offer, the significance of factors
affecting responder behavior can be investigated by a Probit regression. The dependent
16
Ordered probit regressions fail to detect behavioral difference in offers between student and non-students.
13
variable for such regression is the probability of acceptance (Accept), which is a latent
variable.
Empirically such latent variable is defined as a dummy variable: y=1 if a
responder accepts her corresponding proposer’s offer and y=0 otherwise.
We define the explanatory variable Offer as the fraction of money allocated to
responders, which consists of 10%, 20%, 30%, 40%, 50% and 60% for the corresponding
choices 1 to 6.
Other thing being equal, a responder will derive a higher utility from
receiving an offer with high fractions of money allocated to her than an offer with low
fractions.
Therefore, the relationship between Offer and Accept is expected to be
positive.
Stake is another explanatory variable in our regression model.
It consists of two
values, Stake=200 as the low stake while Stake=1000 as the high stake. The empirical
definition for Stake is a dummy variable: Stake=1 for the 1000 stake and Stake=0 for the
200 stake. Facing the same fraction of money offered, most sensible theories predict
that the larger the size of stakes the higher the probability that a responder will accept the
offer.
Both variables, Offer and Stake, are related to the amount of money received by a
responder. Thus, they represent economic or wealth factors that may affect responders’
decision to accept in our regression model.
In addition to these two economic variables,
we consider subjects’ socioeconomic characteristics, including age, gender, education
level, and the status as a parent to be other factors that may affect responders’ decision.
These observable socioeconomic variables are used to control for subjects’ unobservable
non-economic or non-wealth factors in their preferences. We now briefly justify the
correlation between subjects’ socioeconomic characteristics and their social preferences.
Experiments with children indicate that fairness tastes are not innate, but learned
through socialization. 17
17
It does not seem unreasonable to extrapolate this finding to
See Camerer (2003, Section 2.3.4).
14
adult people since experience (captured by Age) may teach people to form different social
preferences over fairness.
Note that the impact of Age on subjects’ conditional
acceptances is allowed to take a quadratic form in our Probit regression (Age2 denotes
age square). This allowance takes into account the lifecycle possibility that responders’
response to an offer when they are young may differ from that when they are old.
Evidence from social and behavioral science tends to support the hypothesis that
men are more individually-oriented (selfish) while women are more socially-oriented
(selfless). 18
This gender difference in social preferences may lead to gender differences
in rejection or offer behavior in ultimatum games.
We define Gender=1 for the male
and Gender=0 for the female.
Fairness norms may be learned through experience, but they may also be embodied
through formal education.
Indeed, education is perhaps the most important instrument
of embodying fairness norms into people’s preferences for most societies.
A person’s
years of schooling or education attainment, defined as Education, may thus reflect her
degree of the embodiment of fairness norms.
The effect of family characteristics on responder’s behavior is examined in this study
via the variable Parent.
Those peoples who have children might have stricter moral
attitude toward a windfall than those who do not have children.
This is perhaps because
most parents would like to teach their children to behave “good” in some sense.
Thus,
negative impact of Parent on Accept may be expected.
[Insert Table 4 about here]
Age, Education and Income may be highly correlated.
Education, the higher may be the Income.
For example, the higher the
Table 4 reports the correlations between these
three socioeconomic variables for both proposers and responders.
[Insert Table 5 about here]
18
See Eckel and Grossman (1998) for a brief review on the issue.
15
Table 5 reports our Probit estimates of responders’ behavior. In the full sample
case, we present two regression results: the first regression only includes economic
variables, while the second regression includes both economic and socioeconomic
variables. Economic variables, Stake and Offer, in the 1st regression are shown to be
significant with the expected positive signs.
Thus, as the standard economic theory will
predict, both economic variables exert positive impacts on responders’ acceptance rates.
The size of such marginal impacts in elasticity are 0.0302 and 0.1701 for Stake and Offer
respectively, which implies that an 1% increase of Stake (Offer) would resulted in an
increase of 0.0302% (0.1701%) of Accept.
The model fitting statistics for the 1st
regression is significant in LR chi-square test and 0.088 in Pseudo R-square.
Results from the 2nd regression with the inclusion of socioeconomic variables show
that Gender and Parent are the two significant variables other than Stake and Offer.
The
model fitting statistics for the 2nd regression show an increase in significance of LR
chi-square test and in value of Pseudo R-square from 0,088 to 0.1464, which means a
70% increase in model explaining power with the inclusion of socioeconomic variables.
Note that such inclusion of socioeconomic variables causes some reduction in magnitudes
of the estimated parameters of economic factors (Stake and Offer) in the 2nd regression,
as compared to the 1st regression. Nevertheless, both Stake and Offer remain significant
after controlling for socioeconomic factors.
Statistically, estimated parameters of
economic factors after controlling for socioeconomic variables can be regarded as “pure”
marginal effects of economic factors.
It is evidenced from Table 4 that Offer has a much
larger magnitude in elasticity than those of Gender and Parent, while Stake has more or
less the same magnitude as those of Gender and Parent.
We can conclude that Stake and Offer as well as other two socioeconomic variables
(Gender and Parent) are all significant factors which could affect responders’ decisions.
It is interesting to observe that women tend to reject less often than men. This result is
16
consistent with Eckel and Grossman (2001), but not with Solnick (2001). 19
To investigate whether the impact of socioeconomic variables could vary by stakes,
we also run Probit regressions using sub-samples of low and high stakes separately.
Results from these two sub-samples are shown in Table 5. In the low stake sample, the
economic variable Offer is no longer significant, while those socioeconomic variables
including Age and Gender become significant. In the high stake sample, by contrast,
none of socioeconomic variables is significant while the economic factor Offer is the
dominant and only significant variable.
It is plausible to conclude on the basis of this
finding that responders’ acceptance decision only depends upon fractions of money
offered when stakes are high, whereas socioeconomic variables play major and
significant role in explaining responders’ decision when stakes are low.
Rabin (1993) proposes the so-called “fairness equilibrium” to account for the role of
fairness in game theory.
In this equilibrium, the fairness term will become less and less
important as monetary stakes get larger and larger, and will disappear eventually if stakes
are high enough.
Our empirical finding here is consistent with the theoretic implication
of Rabin’s fairness equilibrium.
Stigler (1981), p. 176) claims: “[When] self-interest and ethical values with wide
verbal allegiance are in conflict, much of the time, most the time in fact, self-interest
theory … will win.”
As far as responders’ behavior is concerned, our finding suggests
that Stigler’s claim was right when stakes become high.
Proposer behavior
In our ultimatum game experiment, each proposer picks only one choice among 6
discrete choice alternatives.
These 6 choice alternatives can be represented by an
ordered variable, with values of 1 to 6, according to their fractions of money split. For
19
Solnick’s (2001) experimental design is the so-called “strategy method,” while Eckel and Grossman’s (2001) is
the so-called “game method.” Our experimental methodology is the same as Eckel and Grossman’s.
17
instance, the first choice, coded by a value of 1, represents a fraction of 10% of money
offered to responders. Similarly, the second choice (coded 2) represents a fraction of
20%, while the third (coded 3), forth (coded 4), fifth (coded 5) and sixth (coded 6)
choices represent 30%, 40%, 50% and 60% of money offered to responders.
With such
ordered discrete response data, it is natural to employ the Ordered Probit model to
identify determinants of proposer behavior.
The dependent variable is an ordered
response discrete variable representing proposers’ decision to those 6 offers, and the
explanatory variables are economic variable (Stake) and socioeconomic variables as
defined earlier.
The increase in value of such ordered response variable implies the
tendency of a more generous offer by proposers. Most sensible theories predict that a
proposer will tend to be less generous when stakes are high than when they are low.
[Insert Table 6 about here]
Table 6 reports our Ordered Probit estimates of proposer behavior for both full and
sub-samples.
In the full sample case, we run two regressions, the 1st one with only
Stake variable and the 2nd one including Stake and other socioeconomic variables.
Results from both regressions show that Stake is the only significant variable in
explaining proposers’ decision.
The negative sign of estimated coefficient of Stake
indicates that proposers will tend to pick choices that allocate lower fractions of money to
responders in the high stake than in the low stake.
This result is consistent with the
prediction of the standard theory.
[Insert Figure 3 about here]
Figure 3 plots the predicted offer probability and its cumulative distribution function
(CDF) from our Ordered Probit model.
Since the curve representing the CDF in the low
stake strictly lies below that in the high stake, it is apparent that the CDF in the low stake
first-degree stochastically dominates the CDF in the high stake. 20
20
The dominant result holds for the observed offers in Figure 1 as well.
18
In other words,
proposers tend to offer low (high) offers with higher (lower) probabilities in the high
stake than in the low stake.
Smith and Walker (1993) argue that the failure of rational
models in explaining data can be attributed to low cost of deviations from the rational
prediction. Our finding here is consonant with Smith and Walker’s (1993) remark:
“increased financial rewards shift the central tendency of the data toward the predictions
of rational model” (p. 245).
Even though not a single socioeconomic variable is significant at 10% level in the
full-sample regression, the model fitting statistics with the inclusion of socioeconomic
variables still perform better than those with the exclusion of socioeconomic variables.
Table 6 also reports results of the ordered Probit regressions for the sub-samples of
low and high stakes separately.
Model fitting statistics for both models are poor.
Nevertheless, it is interesting to observe that Age is the only socioeconomic variable that
features significantly when stakes are low, while Education is the only socioeconomic
variable that features significantly when stakes are high. This result suggests in the
bargaining environment that socioeconomic characteristics that feature significantly for
offer behavior in low stakes may differ from those in high stakes.
The positive sign of
Education implies that proposers with higher educational attainment tend to pick choices
more favorable to responders.
This may have to do with social learning of treating
others fairly through education.
Overall behavior
Putting together the lifecycle impacts of Age from Tables 5 and 6, we detect an
interesting behavior pattern: those who are young and old offer less and reject less often
than those who are in the middle age.
This implies that those who are young and old
behave closer toward the subgame-perfection prediction of the standard theory than those
who are in the middle age.
It thus seems that experience teaches people to move away
from the prediction of the economic model in the earlier stage of their life, but return
19
toward it in the later stage.
The turning points are around 39 and 43 years of old,
respectively, for responders and proposers. It should be emphasized that this lifecycle
pattern holds only if stakes are low.
When stakes are high, responders’ behavior is
overwhelmed by monetary incentives, and education is the only socioeconomic variable
that survives capable of explaining proposers’ behavior.
It is worth noting that we detect significant difference between low and high stakes
proposals and rejection frequencies in an experiment with inexperienced or one-shot play.
Slonim and Roth (1998) show for experienced play of student subjects that: (i)
responders’ rejection rates decrease as stakes increase, and (ii) proposers’ offers move
closer toward the subgame-perfection prediction of the standard theory as stakes increase.
For inexperienced or one-shot play, however, they fail to detect significant difference
between low and high stakes proposals or between low and high stakes rejection
frequencies.
Henrich et al. (2002) find in a cross-cultural study that there exists a close
relationship between subjects’ experimental behavior and their everyday life. Hoffman
et al. (1998, p. 350) vividly put it: “A on-shot game in the laboratory is part of a life-long
sequence, not an isolated experience from one’s reputation norm.”
Perhaps, the
difference between our finding and Slonim and Roth’s (1998) simply reflects the fact that
everyday life led by student subjects differs from everyday life led by our nation-wide
representative subjects.
4. Conclusion
The Rabin-Fehr-Schmidt-Bolton-Ockenfels approach attempts to account for
experimental data by theoretically building an extended model in which both wealth and
non-wealth factors in players’ preferences are counted in.
In contrast, we attempt to
account for experimental data by empirically separating wealth factors from non-wealth
factors in players’ preferences, and isolating the pure effect of wealth factors from the
20
overall effects observed in the experimental data.
preferences are typically unobserved.
Non-wealth factors in players’
The gist of our approach is to use players’
observable socioeconomic characteristics as proxies for these unobservables.
This
approach is not new and, indeed, it is the standard practice adopted by applied
econometricians when they use naturally-occurring data to test empirically the validity of
some theoretic prediction of a wealth maximization or pure self-interest model.
We
justify in our context why different socioeconomic characteristics tend to be associated
with different social preferences between players.
It is found that: (i) raising stakes substantially reduce the number of “outliners” on
both offers and rejections; (ii) higher stakes exert a significant impact on players’ offer
and rejection behavior as the standard economic theory predicts even for inexperienced or
one-shot play; and (iii) socioeconomic characteristics dominate responders’ behavior
when stakes are low, whereas monetary stakes dominates responders’ behavior when
stakes are high.
Smith and Walker (1993) survey experimental studies which report data on the effect
of increased monetary rewards.
They remark (p. 245):
[Increased] financial reward shifts the central tendency of the data toward the
predictions of rational models, …[and] rewards reduce the variance of the data
around the predicted outcomes.
This remark seems somewhat at odds with the conclusion reached by Camerer (2003) ten
years later in the context of ultimatum games (see the quotation in the Introduction).
Our experimental study on the effect of the size of monetary stakes attests to the validity
of Smith and Walker’s remark.
21
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24
Table1. Descriptive statistics of the sample
Definition
Mean
Standard
Deviation.
0.512
0.500
0.636
0.481
0.074
0.263
Dummy;
Gender
1 for male
0 for female
Dummy;
Parent
1 for Yes
0 for No
Dummy;
Student
1 for Yes
0 for No
Income
NT$/ month
28,574.3
29,906.2
Age
Years of age
40.882
12.670
Age distribution
20-40
40-60
60~
49.18%
43.49%
7.33%
100.00%
Education
Years of schooling
12.242
3.565
Education distribution
Elementary & below
Junior high school
Senior high school
Junior college
College & above
11.90%
12.03%
34.18%
18.48%
23.42%
100.00%
25
Table 2. Frequencies of offers and (conditional) acceptances by stakes
Stake=200
Offer
choices
Offers
Nos.
Stake=1000
Acceptances
(%) Nos.
(3.55%)
Offers
(%)
Nos.
4
(57.14%)
14
Acceptances
(%) Nos.
1(90/10)
7
2(80/20)
3(70/30)
4(60/40)
5(50/50)
6(40/60)
4
19
12
137
18
(2.03%)
3
(9.64%) 18
(6.09%) 10
(69.54%) 126
(9.14%) 14
(75.00%)
(94.74%)
(83.33%)
(94.03%)
(77.78%)
12
13
18
139
4
Total
197 (100.00%) 175
(88.83%)
200 (100.00%) 188
26
(7.00%)
11
(%)
(78.57%)
(6.00%)
7 (58.33%)
(6.50%) 12 (92.31%)
(9.00%) 18
(100%)
(69.50%) 136 (97.84%)
(2.00%)
4 (100.00%)
(94.00%)
Percentage
120%
100%
R1000
80%
R200
60%
40%
20%
P200
P1000
0%
1
2
3
4
P200:Proposer at stake=200
P1000:Proposer at stake=1000
5
6
Choices
R200:Responder at stake=200
R1000:Responder at stake=1000
Figure 1. Frequencies of offers and (conditional) acceptances by stakes
27
Table 3. Frequencies of offers and (conditional) acceptances by students and non-students
Offers
Acceptances
Offer
choices
Students
(%)
Non-students
(%) Students
(%)
Non-students
(%)
1(90/10)
2(80/20)
3(70/30)
4(60/40)
5(50/50)
6(40/60)
1 (3.10%)
0 (0.00%)
1 (3.10%)
4 (12.50%)
26 (81.30%)
0 (0.00%)
20
16
31
26
250
22
(5.50%)
(4.40%)
(8.50%)
(7.10%)
(68.50%)
(6.00%)
1 (100.00%)
0 (0.00%)
3 (100.00%)
2 (100.00%)
19 (100.00%)
2 (100.00%)
14 (70.00%)
10 (62.50%)
27 (93.10%)
26 (92.90%)
243 (95.70%)
16 (80.00%)
Total
32 (100.00%)
365
(100.00%)
27 (100.00%)
336 (91.60%)
28
Percentage
120.00%
100.00%
▲
80.00%
60.00%
40.00%
20.00%
0.00%
1
2
3
4
Student proposer
Proposer-student
5
6
Non-student
proposer
Proposer-non-student
Responder-non-student
Non-student
responder
Responder-student
Student responder
Figure 2. Frequencies of offers and (conditional) acceptances by students and
non-students
29
Choices
Table 4. Correlations between Age, Education, and Income
Proposers
Responders
Age
Age
1
Education -0.5506
Income
0.1503
Education
1
0.246
Income
Age
Education
Income
1
1
-0.4773
0.069
1
0.3143
1
30
Table 5. Probit estimates of responder behavior (dependent variable y=1 if accept, y=0
otherwise)
Full sample Full sample Full sample 200-sample 1000-sample
(1)
(2)
(3)
Stake
0.434
0.3913
0.394
(0.204)*** (0.2146)* (0.214)*
0.0302
0.0228
0.0231
2.815
Offer
2.8311
2.792
1.558
4.219
(0.675)*** (0.7053)*** (0.701)*** (1.083)
(1.080)***
0.1701
0.1432
0.1425
0.0995
0.1035
Age
-0.0901
-0.093
-0.221
-0.016
(0.0762)
(0.074)
(0.117)*
(0.139)
-0.4274
-0.4435
-1.2505
-0.0398
Age2
0.0013
0.001
0.003
0.0003
(0.0009)
(0.001)
(0.001)*** (0.001)
0.2810
0.292
0.7613
0.0367
Gender
-0.3762
-0.394
-0.537
-0.357
(0.2197)* (0.212)*
(0.287)*
(0.357)
-0.0219
-0.0232
-0.0398
-0.001
Education
0.0096
0.012
-0.005
0.032
(0.0371)
(0.035)
(0.050)
(0.055)
0.0130
0.017
-0.008
0.0218
Parent
-0.5659
-0.572
-0.359
-1.099
(0.3412)* (0.340)*
(0.400)
(0.828)
-0.0373
-0.038
-0.298
-0.0332
Income
-0.0042
(0.0355)
-0.0014
InD
0.3894
(0.5516)
0.0021
0.036
Constant
1.8941
1.931
4.917
0.747
(0.321)
(1.5670)
(1.494)
(2.335)**
(2.691)
394
Observations 394
394
194
200
2
19.12*** 32.45***
31.78***
15.76***
24.39***
LR χ
2
Pseudo R
0.088
0.1459
0.1464
0.1268
0.2686
Note:
1. Standard errors are in parentheses; boldfaces denote elasticity.
2. Statistical significance at the *10%, **5% and ***1% level.
3. The unit of income is NT$10,000/month; lnD=1 if income is missing, lnD=0 if income
is available.
31
Table 6. Ordered Probit estimates of proposer behavior (dependent variable y=1 if
offer=90/10, y=2 if offer=80/20, y=3 if offer=70/30, y=4 if offer=60/40, y=5 if
offer=50/50, and y=6 if offer=40/60)
Full sample Full sample Full sample 200-sample 1000-sample
(1)
(2)
(3)
-0.3589
Stake
-0.3592
-0.3597
(0.1187)*** (0.1194)*** (0.1193)***
Age
0.0576
0.0512
0.1212
-0.033
(0.0410)
(0.0390)
(0.0532)*** (0.0589)
Age2
-0.0006
-0.0005
-0.0014
0.0004
(0.0005)
(0.0005)
(0.0006)*** (0.0007)
Gender
0.0009
-0.0061
-0.0671
0.0462
(0.1232)
(0.1191)
(0.1694)
(0.1713)
Education
0.03214
0.0327
-0.0158
0.076
(0.0224)
(0.0213)
(0.0321)
(0.0295)***
Parent
-0.0804
-0.0426
-0.3544
0.2825
(0.1928)
(0.1928)
(0.2686)
(0.2831)
Income
-0.0013
(0.030)
InD
-0.2176
(0.2856)
λ1
λ2
λ3
λ4
λ5
-1.8211
(0.1254)
-1.5189
(0.1104)
-1.1317
(0.0988)
-0.2717
(0.8011)
0.0360
(0.8009)
0.4286
(0.8010)
-0.3757
(0.7490)
-0.0685
(0.7474)
0.323
(0.7463)
0.1154
(1.0253)
0.3366
(1.0226)
0.9147
(1.0199)
-0.9322
(1.1148)
-0.5577
(1.1126)
-0.2769
(1.1112)
-0.8678
(0.0945)
1.4442
(0.1142)
0.6952
(0.8010)
3.0292
(0.8139)***
0.5893
(0.7460)
2.9194
(0.7610)***
1.1503
(1.0200)
3.3153
(1.0434)***
0.0213
(1.1102)
2.6872
(1.1367)***
397
397
197
200
9.25***
15.49***
14.96***
5.33
7.55
0.0106
0.0178
0.0172
0.0127
0.0175
Observations 397
LRχ
2
Pseudo R2
Note:
1. Standard errors are in parentheses.
2. Statistical significance at the *10%, **5% and ***1% level.
3. The unit of income is NT$10,000/month; lnD is the same as that in Table 4.
32
Probability
120.00%
100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
Choices
1
Predict-200
2
3
4
Predict-1000
CDF-200
5
6
CDF-1000
Figure3. Predicted offer probability and its cumulative distribution by stakes
33