1. Investment Under Uncertainty: An Experimental Application to Educational Investments
by
Daria Voskoboynikov1
Faculty Advisor: Professor Barry Sopher
Department of Economics
Rutgers, The State University of New Jersey
April 11, 2016
1
I would like to thank the Aresty Research Center and the SAS Scholarship Office for funding this research via the
Alan Marc Schreiber Memorial Scholarship and Research Award. I would also like to thank my thesis advisor,
Professor Sopher, and the faculty of the Economics Department for their guidance and support throughout the year.
2. 1
Abstract
With each decade, the decision to enroll in college becomes riskier. Due to the growing
uncertainty of the costs and benefits of a college education for a given individual, educational
investment decisions can be treated as investments under uncertainty. Modeled after the concept
of option value in education, questions in this study examine participants’ ability interpret signals
and maximize their payoffs, subject to varying levels of information. After conducting the
analysis of participants’ responses to these questions, I conclude that with access to useful
information, participants can better infer their profit-maximizing prospects and act accordingly,
particularly when prospects are not favorable.
3. 2
1. Introduction
Pressure on young Americans to attend and finish college is high and rising. This is not
surprising, as many studies show that college graduates on average face lower unemployment
rates and higher earnings throughout their lifetimes relative to those of their peers who do not
attend college (Oreopoulos and Petronijevic). Though there are economic and noneconomic
benefits to earning a college degree, the cost-benefit analysis facing prospective college students
is increasingly dauntingly complex and involves a great deal of uncertainty. The cost of
attending college is skyrocketing, and students are borrowing more than ever to finance their
investment in an undergraduate degree (Turner; Bound et al.). Although the risk associated with
an investment in a college degree is growing, college enrollment continues to increase
dramatically. In fall 2013, total undergraduate enrollment was 17.5 million students, an increase
of 46 percent from 1990, and this number is projected to continue to grow (NCES). Is this
efficient? As colleges become less selective in their admissions process in order to admit more
students, students who previously would not have attended college are now enrolling, widening
the ability gap between high school and college graduates (Oreopoulos and Petronijevic).
Presumably due to this phenomenon, as well as the growing number of high-paying technical
jobs now requiring a college education, the relative returns (in terms of earnings) to a college
degree are rising on average, but this is not true across all college graduates.
Earning a college degree requires a great amount of resources – time, financial means,
and mental effort – but these resources are likely not allocated efficiently. For the 2012-2013
academic year, the average cost of a year of education in a four-year institution in the United
States was approximately $23,872 per year (NCES). Thus, an average four-year education costs
about $96,000. Though many schools offer generous financial aid packages, the fact remains that
4. 3
many students find themselves deeply in debt. Do the benefits of a college education outweigh
this cost for these graduates? According to the Bureau of Labor Statistics, in 2014, about 273,000
college graduates were working at or below the federal minimum wage of $7.25 per hour. Many
college graduates were also underemployed, meaning that they were working at a job that did not
require a college degree. The underemployment rate for recent graduates was 44 percent, while
the overall underemployment rate for all college graduates was at about 33 percent (Abel et al.).
Furthermore, the quality of jobs held by the underemployed is decreasing, with many college
graduates working in low-wage jobs or working part-time (Abel et al.). High underemployment
among recent graduates is not so unusual, as many graduates take time to find jobs that fit their
educational background; however, an increased underemployment rate suggests that it has
become more difficult for recent graduates to find jobs that utilize their degrees over the past
decade (Abel et al.). Evidently, there are many people who find themselves in a position in
which they have incurred high costs to obtain their degree, and the benefits of it are very low or
potentially negative. According to Cunha, Heckman, and Navarro (2005), college graduates who
find themselves in this position have stated that had they had better information regarding their
costs and their expected wages, many of them would have modified their actions accordingly.
Due to the complexity of its cost-benefit analysis and the growing risks of unemployment and
underemployment, an investment in education can be treated as an investment under uncertainty.
With better information regarding their costs and benefits, people should be able to better assess
whether they should attend college or not.
This study used a laboratory experiment to address the question of education enrollment
and continuation as an investment under uncertainty. Of course, there are many reasons why
someone would attend college: an interest in a specialization, a chance to increase future wages,
5. 4
pressure from relatives or peers, or an opportunity for a better life; none of these factors were
considered in this study, as societal pressures of this nature cannot be replicated in a laboratory.
Though there was no mention of education in the experiment, the questions themselves were
modeled after the enrollment and continuation decisions that students have to make. In the
experiment, all participants had to respond to signals that carry information regarding one’s
ability to maximize their payoffs, much like current students receive signals regarding their
academic strength and interpret them in order to gain a better understanding of their future
potential income. Though the questions cannot address educational decisions directly, the
findings of this study can be useful in interpreting students’ education investment decisions in so
far as they are investment decisions made under uncertainty. The questions in this study convert
these complicated life decisions into simple profit-maximizing decisions. Without the context of
college and all the pressures and expectations that come with it, do people generally make good
or bad investment decisions? I predict that on average, participants will be able to make
reasonable investment choices and maximize their profits subject to their constraints.
2. Literature Review
2.1 Decision Making Under Uncertainty
Contrary to the analysis in this paper, most economic models assume perfect information.
As discussed, however, this assumption does not hold in educational investment. In such cases,
expected utility is particularly useful in explaining people’s decisions.
The Von Neumann-Morgenstern Expected Utility Theorem states that when consumer i
faces a set of alternatives, certain and uncertain, there exists a utility function 𝑢𝑖 such that the
consumer’s preferences are represented and it satisfies the expected utility property (Serrano &
6. 5
Feldman). The utility function assigns utility numbers to all the alternatives, and for pairs of
alternatives X and Y, 𝑢𝑖(𝑋) > 𝑢𝑖(𝑌) if and only if consumer i prefers X to Y, and 𝑢𝑖(𝑋) = 𝑢𝑖(𝑌)
if and only if consumer i is indifferent between them. In any risky alternative L with outcomes X,
Y, and Z and their corresponding probabilities 𝑝 𝑥, 𝑝 𝑦, and 𝑝𝑧, the utility of the risky alternative is
the expectation of the utilities of its possible outcomes. This can be represented by the following
equation: 𝑢𝑖(𝐿) = 𝑝 𝑋 𝑢𝑖(𝑋) + 𝑝 𝑌 𝑢𝑖(𝑌) + ⋯ + 𝑝 𝑍 𝑢𝑖(𝑍) (Hirshleifer & Riley). Using this
equation, it is possible to evaluate how people with different risk preferences would make
different decisions under uncertainty. Risk-averse consumers, the majority of individuals, are
willing to pay to avoid risk (Holt). The utility function for risk-averse consumers is concave (i.e.
𝑢𝑖(𝑥) = √ 𝑥). People can also be risk-loving and willing to pay for more risk, with convex
utility functions (i.e. 𝑢𝑖(𝑥) = 𝑥2
). Lastly, people can be risk-neutral if they neither avoid risk
nor seek it. Utility functions for risk-neutral individuals are upward-sloping straight lines (i.e.
𝑢𝑗(𝑥) = 𝑥). Assuming that all people are expected utility maximizers, individuals will respond
differently in the experiment based on their risk preferences. This understanding of expected
utility can be used to explain why individuals would choose different investments given the same
options, which is directly applicable to this study.
2.2.1 Educational Investment
Beginning with the work of Gary Becker in 1964, economists applied investment theory
to the study of individuals’ education decisions. According to Becker (1964), individuals must
weigh the returns to college investment against the costs, both direct and indirect. If the
difference between the benefits and the costs is greater than the present value of a prospective
student’s lifetime earnings without attending college, the individual would choose to enroll.
7. 6
Although relative returns to college education are rising on average, returns for each individual
remain uncertain. As individuals consider various college options, they must form expectations
about the costs and returns to a college education and assess the likely variation in the prediction
of both parameters. Variation in cost derives from uncertainty about one’s own ability and
various characteristics of the college experience, while variation in the returns comes from
uncertainty about future demand and supply conditions of the labor market (Turner). If
individuals based their education investment decisions on this static model, then one could
assume that for those who choose to attend college, the present value of their lifetime earnings
without a college degree does not exceed the difference between the benefits and the costs of an
investment in higher education; in other words, their investment is optimal. It is very likely,
however, that many people do not achieve the optimal level of education as prescribed in this
model, seeing that many graduates find themselves underemployed and / or deeply in debt.
Becker’s theory loses predictive power due to its failure to account for incomplete
information and uncertainty. Despite its salience in investments, uncertainty has historically
received relatively little attention within the study of education (Stange). One information
problem arises because the costs and benefits of a college education vary greatly from person to
person. In addition, individuals may not have an exact idea what their costs and benefits will be
prior to enrollment (Oreopoulos and Petronijevic). Lack of cost and benefit information leads to
an uncertain payoff from a college education, making investment very risky. In fact, about 80
percent of potential wage variation is a reflection of that uncertainty (Chen). Many statistics
confirm that college graduates enjoy higher lifetime wages than high school graduates on
average, but with colleges relaxing their admission standards, signal noise may prompt
prospective low-ability students to overestimate the benefits to a college education (Hendricks
8. 7
and Leukhina). In addition, aggregate statistics on the returns to investing in a college degree
ignore the large differences in earnings by field and choice of major (Altonji). Individuals could
make more efficient investment decisions if they had better information regarding the expenses
and the benefits associated with a college education; however, these are difficult to estimate prior
to enrollment.
Before students attend college, they may not have relevant information regarding various
financial aspects or their ability to succeed. Psychic costs depend on ability, and people can only
refine their expectations of these costs over time (Stange). In a static model, people are assumed
to weigh short-term costs against future benefits and choose the level of education that
maximizes welfare; however, this framework abstracts from uncertainty, assumes that all
relevant parameters are known, does not account for systematic mistakes in prospective students’
assessments of the investment problem, and is inconsistent with high numbers of college
dropouts when the marginal earnings gained from graduating are high (Stange; Turner). In the
educational investment model developed in Becker (1964), individuals are assumed to maximize
lifetime utility. Utility is a function of lifetime earnings and the cost of education, 𝑈 =
𝑙𝑛 𝑦(𝑆) − 𝑐(𝑆), where 𝑐(𝑆) is some increasing and convex function of years of schooling. If
𝑦(𝑆) and 𝑐(𝑆) are continuous and differentiable, then the optimal level of education satisfies the
first-order condition (𝑑𝑦𝑖(𝑆𝑖
∗
)/𝑑𝑆)(1/𝑦𝑖(𝑆𝑖
∗
)) = (𝑑𝑐𝑖(𝑆𝑖
∗
)/𝑑𝑆). Theoretically, at the optimal
level, the benefit of an additional year of schooling would only offset the additional cost;
however, this does not seem to be the case empirically because the returns to college are
nonlinear with substantial degree or “sheepskin effects” (Stange). A diploma results in an
increase of about 28 percent in earnings on top of each additional year of schooling (Oreopoulos
and Petronijevic). Although the present discounted value of lifetime earnings minus costs jumps
9. 8
discretely at four years of college, it is unrelated to education level until that time (Stange). If
psychic costs are constant, very few people should drop out and forgo higher earnings; however,
there are evidently many individuals whose schooling level puts them on the flat part of the
earnings production function, and in fact, 28 percent of high school graduates drop out before
finishing their fourth year of college (Stange).
Although rewards to completing a college degree have increased substantially on average
over the past few decades, the rise in incentives for collegiate completion has not been
accompanied by an increase in the share of students making the transition from college
enrollment to college completion. Between 1970 and 1999 alone, completion rates for
individuals aged 23 years old fell by 25 percent. Completion rates at older ages remain mostly
stagnant, implying an increase in the time to complete a degree, which causes an increase in the
cost of schooling and a delay in the benefits, as well as a potential decrease in the benefits
(Turner). Prospective students’ predictions about success in college may be inconsistent with
actual academic prospects, and due to this information problem, students may make mistakes in
enrolling in college even when it is predictable that the likelihood of a positive return is very low
(Turner). Recent studies at the intersection of economics and psychology suggest the presence of
a “belief perseverance” or “overconfidence bias” that captures the reluctance of individuals to
abandon college after receiving poor grades in high school; in addition, Rosenbaum (2001)
suggests that one explanation for high college attrition is the mismatch between expectations
formed in high school (which encourage a “college-for-all” social norm) and the realities related
to the academic requirements for degree completion. Evidently, the static model of investment
does not explain high dropout rates well, and thus a model incorporating option value needs to be
examined.
10. 9
2.1.2 Option Value
Uncertainty in education investments creates option value, which arises because students
have the option, but not the obligation to continue college after enrollment. Individuals derive
value in an investment when they have the option to opt out of it, as is the case in educational
investments when students drop out. The ability to delay irreversible investment expenditures
can profoundly affect the decision to invest, which is why colleges have students pay their tuition
throughout the years of attendance instead of as one lump sum during enrollment (Dixit &
Pindyck). This system makes investing in education more appealing because if needed, it is
possible to stop investing in it. Enrollment in college reveals an individual’s collegiate aptitude,
the non-persistent shocks to the relative cost (or benefit) of schooling (i.e. getting sick), and labor
market opportunities associated with an increase in education, though the specific realization is
unknown ex ante (Stange). It also gives time for students’ career orientations to develop, which
is heavily influenced by the education process (Bayirbag). Conditioning educational investment
decisions on this information increases welfare and rationalizes dropping out. Once enrolled,
more than half of college entrants can predict whether they will graduate with at least 80 percent
probability (Hendricks and Leukhina). In order to account for updates in information, Stange
(2012) analyzes educational investments via a simple dynamic model with two stages, which
correspond to the first and second halves of college. This model allows individuals to acquire
information regarding their costs and benefits in the first half before deciding whether or not to
pursue the second half. Through this analysis, Stange (2012) concludes that option value
accounts for 14 percent of the total value of the opportunity to attend college, which increases to
32 percent for those closer to the enrollment margin. For moderate-ability students, the
information they receive after enrollment is more valuable because their decision is most
11. 10
uncertain. Approximately 60 percent of this value is derived from the information received in the
first year of college (Stange). The information received in the first year or two of college can act
as a signal to students of their potential success after college. These findings are important
because the “ability to make decisions sequentially increases both enrollment and dropout, but
also closes a quarter of the welfare gap between the first-best scenario (individuals maximize
welfare ex post) and the static one (individuals commit to outcomes ex ante)” (Stange, 51). The
impact of information and uncertainty on educational investments must be examined closely due
to its economic implications: Altonji (1993) finds that while ex ante returns to schooling are
positive, some individuals would not choose to invest ex post, implying that once the outcome to
their educational investment was revealed, these individuals realized that their investment was
not optimal. In addition, Cunha, Heckman, and Navarro (2005) concludes that 30 percent of
people would change their educational investments and schooling decisions if they had perfect
information regarding their costs and benefits. Due to the sequential revelation of information
present in educational investments, utilizing a two-stage model that accounts for option value
may increase welfare among students and lead to more accurate predictions of educational
investments.
3. The Experiment
Participants in this study were asked a series of incentivized questions using Qualtrics. In
each question, participants were asked to make a costly decision that affected their payoff2
. Each
decision “counted” in such a way that one decision was randomly selected to determine each
2
The question of whether to pay participants or not is one of several issues that divides research in experimental
economics from similar work by psychologists. In choices involving risk, however, it is “dangerous to assume that
real incentives are not needed” to elicit truthful responses (Holt). It is therefore standard practice in economics
experiments to pay participants based on their responses.
12. 11
participant’s payoff at the end of the experiment. In this between-subjects study3
, participants
were randomly selected into either a low or high information group by picking an index card out
of a deck. The experiment consisted of two parts. In Part One, participants answered twenty
questions regarding various one-shot investment opportunities in order to get a baseline
understanding of participants’ risk preferences. In Part Two, there were 36 potentially two-round
questions modeled after the concept of option value.
3.1 Part One Question Design
In each of the twenty questions in Part One, participants had the choice between a certain
amount of $10 and a random payoff drawn from a given beta distribution (also known as the
“investment option”). The distributions presented to participants varied in mean, support, and
direction of skewness. The mean ranged from eight to twelve; the support was either six or ten;
and the skewness did not vary in magnitude, but in direction: very skewed to the right (𝛼 =
2, 𝛽 = 5) or very skewed to the left (𝛼 = 5, 𝛽 = 2). Each of these elements interacted with the
others for a total of twenty questions.
In each question, participants were given the expected mean4
of the distribution as well as
a graph of the distribution with an accompanying chart that outlined the probability of attaining
payoffs within a given range. The graph and chart of question 13 is provided below:
3
This experiment was conducted between-subjects in order to gather more data in a shorter period of time, and also
to avoid issues such as sequence effects.
4
In order to simplify the experiment for participants as much as possible, the word “average” was used instead of
“expected mean.” Short quizzes were given at the end of instructions for each part to ensure that participants
understood the concepts necessary for them to make informed decisions for each question.
13. 12
Graph 1: Example of Part One Question Distribution
The table below summarizes the parameters of each beta distribution in Part One:
Table 1: Part One Questions’ Beta Distribution Parameters
Question
Number5 Mean Support 𝜶 𝜷
Lower
Bound
Upper
Bound
Variance Skewness
6 8 6 2 5 $6.26 $12.26 0.918 0.596
2 9 6 2 5 $7.26 $13.26 0.918 0.596
5 10 6 2 5 $8.26 $14.26 0.918 0.596
13 11 6 2 5 $9.26 $15.26 0.918 0.596
9 12 6 2 5 $10.26 $16.26 0.918 0.596
16 8 10 2 5 $5.10 $15.10 2.551 0.596
14 9 10 2 5 $6.10 $16.10 2.551 0.596
12 10 10 2 5 $7.10 $17.10 2.551 0.596
17 11 10 2 5 $8.10 $18.10 2.551 0.596
15 12 10 2 5 $9.10 $19.10 2.551 0.596
11 8 6 5 2 $0.90 $10.90 0.918 -0.596
3 9 6 5 2 $1.90 $11.90 0.918 -0.596
1 10 6 5 2 $2.90 $12.90 0.918 -0.596
19 11 6 5 2 $3.90 $13.90 0.918 -0.596
8 12 6 5 2 $4.90 $14.90 0.918 -0.596
7 8 10 5 2 $3.74 $9.74 2.551 -0.596
18 9 10 5 2 $4.74 $10.74 2.551 -0.596
10 10 10 5 2 $5.74 $11.74 2.551 -0.596
4 11 10 5 2 $6.74 $12.74 2.551 -0.596
20 12 10 5 2 $7.74 $13.74 2.551 -0.596
5
Though all questions were randomized, these question numbers refer to the way in which each particular
distribution was identified in STATA and throughout analysis.
14. 13
There were questions in which it was advantageous to select a $10 payoff or a random
payoff drawn from the distribution, and there were also questions in which the profit maximizing
choice was not so obvious. The goal of Part One was to get a baseline understanding of
participants’ risk preferences before estimating the models addressing option value in Part Two.
3.2 Part Two Question Design
Part Two contained the questions of high interest in this experiment. Designed to model
the concept of option value in educational investments, each of the 36 questions in Part Two had
the potential to enter a “second round.” Based on a participant’s actions in the “first round,” a
follow-up question could be triggered. Essentially, participants in the first round must make a
decision between a certain amount of $10 and a random payoff drawn from a distribution
(similar to the questions in Part One, with key differences described later). If participants select
the $10 option, that question is closed and they move on to the next question; however, if they
select the distribution option, a second round question is triggered, in which participants learn
more about their prospects of a high payoff via a monetary signal. In the second round,
participants are given a monetary signal that is randomly drawn from the same distribution from
which their payoff would be drawn. This signal is extremely valuable in most cases, though in
some, it is not completely informative. At that point, participants could either continue with their
decision to receive a payoff from the distribution, or they could “opt out” and still receive a
certain payment of $8. The certain amount is reduced slightly in the second round to ensure that
participants are seriously considering their profit-maximizing potential in the first round. They
may not simply choose a distribution payoff in the first round and receive a free signal in the
15. 14
second. If they choose the investment option, not following through with it comes at a slight
cost.
The key difference in Part Two questions is that in the first round, participants do not
know precisely from which distribution their payoff will be drawn. All participants see a “mixed
distribution,” which is constructed with equal parts of two underlying distributions, Distribution
A and Distribution B. In addition, only half of participants will see the underlying distributions -
the other half must make their investment decision based solely on the mixed distribution graph.
Those with more information do not only benefit from knowledge of the underlying distributions
themselves, but also from their advantage in interpreting a monetary signal. A monetary signal
from a given distribution will be less informative to a participant who does not see the
underlying distributions.
Distributions A and B had varying means, levels of skewness, and directions of
skewness. Distribution A had means six, eight or ten and was always skewed to the right in one
of two magnitudes: very skewed (𝛼 = 2, 𝛽 = 5) or less skewed (𝛼 = 2, 𝛽 = 3). Distribution B
had means ten, twelve, and fourteen, and was always skewed to the left in one of two
magnitudes: very skewed (𝛼 = 5, 𝛽 = 2) or less skewed (𝛼 = 3, 𝛽 = 2). All underlying
distributions in Part Two had a support of six. Each Distribution A interacted with each
Distribution B for a total of 18 combinations. For each combination of Distribution A and
Distribution B, there were two questions: one in which a participant’s signal comes from
Distribution A and another in which the participant’s signal comes from Distribution B.
Participants were told the mean of the mixed distribution, and those who saw the underlying
distributions were given their means as well. Below is an example of a mixed distribution and its
underlying distributions:
16. 15
Graph 2: Example of Part Two Question Mixed Distribution
Graph 3: Example of Part Two Question Underlying Distributions
17. 16
A participant who sees the underlying distributions would be completely informed by
signals such as $7.69 (the payoff is definitely coming from Distribution A) or $15.10 (the payoff
is definitely coming from Distribution B); thus, having that information can be extremely
valuable. In fact, for the two questions using these distributions, these were the signals received
by participants. The table below summarizes the parameters of each distribution in Part Two:
19. 18
This design will help to study the impact of information on participants’ investment decisions.
This experiment was designed to represent the concept of option value. In Round One,
participants have some idea of their prospects. Choosing the distribution in the first round is
analogous to enrolling in college. In Round Two, participants receive signals that contain
pertinent information regarding their profit maximizing prospects. A year or two into college,
students already have much better information regarding their potential income after graduation
than when they initially enrolled. By this time, students have received many signals regarding
their future income potential: guidance from mentors, professors, or deans, internships they were
accepted to or rejected from, or simply their GPAs. All of these are examples of signals that
college students receive throughout their education that can influence their decisions either to
stay and graduate or cut their losses and drop out. In the second round, opting out of a payoff
from the distribution is representative of dropping out of college, whereas the decision to
continue with the investment option is representative of graduating college. Though this
experiment was designed based on the concept of option value in education, there are far more
factors that go into a decision to attend college and graduate or not; however, the results of this
study may offer some insight into how students make these decisions and what can influence
them.
3.3 Experiment Logistics
This experiment was held in the Gregory Wachtler Experimental Economics Laboratory
in Scott Hall at Rutgers University. Participants were recruited via the online Economics Sona
System. All participants must be over the age of 18, but there were no other exclusion
restrictions. I explained the study to the participants immediately prior to beginning the
20. 19
experiment. Key points of the assent form were read aloud, participants’ questions were
addressed, and copies of the assent form were made available to all participants. Each of the 91
participants was present for one of seventeen 45-minute sessions. To ensure random selection
into a condition, participants pulled an index card from a deck, which indicated their participant
number and also assigned them to a condition. All participants within a condition received the
same set of questions in a randomized order. Survey demographics (i.e. GPA, college major,
gender) were collected and all research data was anonymous. This data will be kept for the
minimum of three years, as required by the Institutional Review Board.
At the end of the experiment, I used MS Excel to randomly select a question from the
survey to determine each participant’s payoff. For each question, a participant either selected the
certain value or a random payoff drawn from a particular distribution. Those who selected the
certain amount were paid that amount plus $5. For those who chose to receive a random payoff
drawn from the distribution, I used the beta.inv6
function in excel in order to draw random
payoffs. Participants were paid their randomly drawn payoff plus $5.
4. Data Description
The data in this experiment was collected using Qualtrics in seventeen sessions over three
days. There were 91 participants in this study (n = 91). For each participant, demographic
information on major, GPA, and gender was collected. The major options were Economics, Math
/ Physical Sciences, Social Sciences (excluding Economics), and Other. There were five
categories for GPA: below 2.00, 2.01 - 2.50, 2.51 - 3.00, 3.01 - 3.50, and 3.51 - 4.00.
6
The beta.inv function returns the inverse of the beta cumulative probability density function. It requires four
parameters of the beta distribution: α, β, lower bound, upper bound. Entering RAND() in lieu of a numerical
probability returns a random draw from the beta distribution.
21. 20
The analysis in this experiment used a linear probability model. Therefore, each value for
each variable was converted into a dummy variable in order to run a saturated regression.
Though all values are labeled as a dummy variable, one dummy in each category was included in
the constant term (i.e. at least one dummy from each category was equal to 0). The dummy
variables of the demographic statistics are described in the tables below:
Table 3: Major Distribution of Participants
Dummy Variable
( = 1 if true)
Major
Number of
Participants
Percentage
Major1 Economics 21 23.08%
Major2
Math / Physical
Sciences
7 7.69%
Major3
Social Sciences
(excluding Economics)
6 6.59%
Major4 Other 57 62.64%
Total 91 100.00%
Table 4: GPA Distribution of Participants
Dummy Variable
( = 1 if true)
GPA
Number of
Participants
Percentage
GPA1 3.51 - 4.00 44 48.35%
GPA2 3.01 - 3.50 27 29.67%
GPA3 2.51 - 3.00 15 16.48%
GPA4 2.01 - 2.50 3 3.30%
GPA5 Below 2.00 2 2.20%
Total 91 100.00%
Table 5: Gender Distribution of Participants
Dummy Variable
( = 1 if true)
Gender
Number of
Participants
Percentage
Male Male 53 58.24%
Female Female 38 41.76%
Total 91 100.00%
22. 21
4.1 Part One Data Description
Distributions in Part One had means ranging from eight to twelve. They took on two
levels of support, six and ten. The skewness of the distributions was always equal in magnitude,
but varied in direction. The data also included variance, of which there were two levels. These
properties were calculated using the following formulas as functions of 𝛼, 𝛽, and / or the lower
bound (𝑙), and the upper bound (𝑢) of the distribution:
Mean Support Skewness Variance
𝛼𝑢 + 𝛽𝑙
𝛼 + 𝛽
𝑢 − 𝑙
2(𝛽 − 𝛼)√𝛼 + 𝛽 + 1
(𝛼 + 𝛽 + 2)√𝛼𝛽
𝛼𝛽(𝑢 − 𝑙)2
(𝛼 + 𝛽)2(𝛼 + 𝛽 + 1)
Dummy variables were created for each value of each variable, including session number.
Though all values are labeled as their own dummy variable, at least one from each category was
included in the constant term. The dummy variables relevant to Part One analysis (in addition to
demographic information) are summarized in the table below. This table contains all variables
considered throughout analysis, but not all of them were included in the final regression.
23. 22
Table 6: Definitions of Dummy Variables Used in Part One
Dummy Variable
( =1 if true)
Description Frequency
C
Participant chose
the distribution option
787 / 1820
Mean1 Mean = 8 4 / 20
Mean2 Mean = 9 4 / 20
Mean3 Mean = 10 4 / 20
Mean4 Mean = 11 4 / 20
Mean5 Mean = 12 4 / 20
Support1 Support = 6 10 / 20
Support2 Support = 10 10 / 20
Variance1 Variance = 0.918 10 / 20
Variance2 Variance = 2.551 10 / 20
Skewness1 Skewness = -0.596 10 / 20
Skewness2 Skewness = -0.596 10 / 20
Session1 – Session17 One for each session
The dependent variable throughout the experiment is whether or not the participant chose the
distribution. Variable 𝑐 = 1 if the participant chose the distribution option and 0 if not. The table
below summarizes the share of responses selecting the distribution option versus the $10 option
based on mean:
Table 7: Shares of Investment Decisions Based On Mean in Part One
Mean = 8 Mean = 9 Mean = 10 Mean = 11 Mean = 12
Investment Option 3.57% 5.77% 26.65% 84.07% 96.15%
$10 Option 96.43% 94.23% 73.35% 15.93% 3.85%
As shown in this chart, as the mean of the distribution increased, the proportion of respondents
choosing the investment option increased, with a dramatic jump occurring from mean = 10 to
mean = 11.
24. 23
4.2.1 Part Two – Round One Data Description
The underlying distributions in Part Two each took on three means: Distribution A had
means 6, 8, and 10; Distribution B had means 10, 12, and 14. There were two levels of variance
and skewness per underlying distribution. Mixed distributions had means ranging from eight to
twelve. Variances of the mixed distributions were categorized into three variables based on low,
medium, and high variances. Mixed distribution means and variances were calculated using the
formulas below:
Mean Variance
𝐸(𝑍) = 𝜇 𝐴 + 𝜇 𝐵 𝑉𝑎𝑟(𝑍) = 0.5𝜇 𝐴
2
+ 0.5𝜎𝐴
2
+ 0.5𝜇 𝐵
2
+ 0.5𝜎 𝐵
2
− 𝜎 𝑍
2
Treatment is the variable used to specify whether the participant saw the underlying
distributions or not. The variable Order refers to the order in which Part Two questions were
presented. Due to the conditional two-round nature of Part Two, each condition had two separate
surveys, one of which displayed questions in the order opposite to the other. This was done to
avoid order effects. IsBimodal is a variable that indicates whether or not a mixed distribution is
bimodal. The variable LessRiskAverse was arbitrarily constructed using data from Part One.
Using participants’ responses to Part One questions in which the expected mean of the
distribution was 10, if a participant chose the distribution option 50 percent of the time or more,
he or she was categorized as less risk-averse relative to the other participants. Below is a table
summarizing all the dummy variables considered in Part Two – Round One analysis:
25. 24
Table 8: Definitions of Dummy Variables Used in Part Two – Round One
Dummy Variable
( = 1 if true)
Description Frequency
CR1
Participant chose
the distribution option
1,339 / 3,276
Meanp11 Distribution A Mean = 6 1,092 / 3,276
Meanp12 Distribution A Mean = 8 1,092 / 3,276
Meanp13 Distribution A Mean = 10 1,092 / 3,276
Meanp21 Distribution B Mean = 10 1,092 / 3,276
Meanp22 Distribution B Mean = 12 1,092 / 3,276
Meanp23 Distribution B Mean = 14 1,092 / 3,276
Variancep11 Variance of Dist. A = 0.918 1,638 / 3,276
Variancep12 Variance of Dist. A = 1.440 1,638 / 3,276
Variancep21 Variance of Dist. B = 0.918 1,638 / 3,276
Variancep22 Variance of Dist. B = 1.440 1,638 / 3,276
Skewnessp11 Skewness of Dist. A = 0.286 1,638 / 3,276
Skewnessp12 Skewness of Dist. A = 0.596 1,638 / 3,276
Skewnessp21 Skewness of Dist. B = -0.286 1,638 / 3,276
Skewnessp22 Skewness of Dist. B = -0.596 1,638 / 3,276
Meanmix1 Mixed Distribution Mean = 8 364 / 3,276
Meanmix2 Mixed Distribution Mean = 9 728 / 3,276
Meanmix3 Mixed Distribution Mean = 10 3,276 / 3,276
Meanmix4 Mixed Distribution Mean = 11 728 / 3,276
Meanmix5 Mixed Distribution Mean = 12 364 / 3,276
Variancemix1 Mixed Distribution Variance < 4 1,092 / 3,276
Variancemix2 4 < Mixed Dist. Variance < 10 1,456 / 3,276
Variancemix3 Mixed Distribution Variance > 10 728 / 3,276
Treatment Saw Underlying Distributions 1,584 / 3,276
Order Order of Part Two Questions 1,620 / 3,276
IsBimodal Mixed Distribution is Bimodal 2,548 / 3,276
LessRiskAverse
Demonstrated to be less risk averse
in Part One
1,044 / 3,276
(29 participants)
Session1 – Session17 One for each session
The table below shows the total number of responses selecting the distribution option
versus the $10 option based on underlying distribution means in Part Two – Round One:
26. 25
Table 9: Investment Decisions by Underlying Distribution Mean in Part Two – Round One
Distribution A
& Distribution B
Investment Option $10 Option
Frequency Percentage Frequency Percentage
Mean = 6
& Mean = 10
20 / 364 5.49% 344 / 364 94.51%
Mean = 6
& Mean = 12
72 / 364 19.78% 292 / 364 80.22%
Mean = 6
& Mean = 14
136 / 364 37.36% 228 / 364 62.64%
Mean = 8
& Mean = 10
27 / 364 7.42% 337 / 364 92.58%
Mean = 8
& Mean = 12
101 / 364 27.75% 263 / 364 72.25%
Mean = 8
& Mean = 14
247 / 364 67.86% 117 / 364 32.14%
Mean = 10
& Mean = 10
120 / 364 32.97% 244 / 364 67.03%
Mean = 10
& Mean = 12
287 / 364 78.85% 77 / 364 21.15%
Mean = 10
& Mean = 14
329 / 364 90.38% 35 / 364 9.62%
The table below also shows the total number of responses selecting the distribution option versus
the $10 option in Part Two – Round One; however, it summarizes the information based on the
mean of the mixed distribution.
Table 10: Shares of Investment Decisions Based On Mixed Distribution Mean in Part Two –
Round One
Mixed Distribution Investment Option $10 Option
Frequency Percentage Frequency Percentage
Mean = 8 20 / 364 5.49% 344 / 364 94.51%
Mean = 9 99 / 728 13.60% 629 / 728 86.40%
Mean = 10 357 / 1,092 32.69% 735 / 1,092 67.31%
Mean = 11 534 / 728 73.35% 194 / 728 26.65%
Mean = 12 329 / 364 90.38% 35 / 364 9.62%
27. 26
As seen in the charts above, as the means of the distributions increase, participants are
more likely to select a payoff from the distribution.
4.2.2 Part Two – Round Two Data Description
In previous analysis, the dependent variable equaling one indicated that the participant
chose the investment option. In Part Two – Round Two, however, when the dependent variable
(cr2) equaled one, it indicated that the participant chose the certain amount of $8. In other words,
when cr2 = 1, participants changed their initial decisions from Round One, presumably after
learning some new information about their profit-maximizing prospects.
Most of the variables were already considered in Part Two – Round One analysis;
however, there are five new variables: DefA, ProbA, ProbB, DefB, and BFOSDA. Essentially,
DefA, ProbA, ProbB, and DefB are a way of categorizing what the participant is able to infer
from a particular signal in a given question. As an example, if a signal fell outside of the supports
for Distribution B, the payoff must then necessarily come from Distribution A. In those cases,
DefA = 1. The opposite is true for DefB. ProbA and ProbB both attempt to capture the more
nuanced case in which a given signal was within the supports of each distribution, but it was
more likely to come from one than the other.
The variable BFOSDA equals one when Distribution B first order stochastically
dominates A. Stochastic ordering is a concept that arises in situations in which one lottery can be
ranked as superior to another for a broad class of decision-makers. Lottery B has first-order
stochastic dominance over lottery A if for any outcome x, B gives at least as high a probability of
receiving at least x as does A; and for some x, B gives a higher probability of receiving at least x.
In notation form, 𝑃(𝐵 ≥ 𝑥) ≥ 𝑃(𝐴 ≥ 𝑥) for all x, and for some x, 𝑃(𝐵 ≥ 𝑥) > 𝑃(𝐴 ≥ 𝑥). In
28. 27
terms of the cumulative distribution functions of the two lotteries, B dominating A means that
𝐹𝐵(𝑥) ≤ 𝐹𝐴(𝑥). These conditions hold in 28 out of 36 questions in Part Two; thus, in these 28
questions, Distribution B first order stochastically dominates Distribution A. This is important to
note because that then clarifies that in at least 28 questions, drawing a random payoff from
Distribution A is objectively less favorable to drawing a random payoff from Distribution B.
This has important implications for participants who learn about their prospects of receiving a
random payoff from either distribution via a signal.
In the remaining eight questions, the left tail of Distribution B is lower than that of
Distribution A, so it was important to test whether or not Distribution A could be second order
stochastically dominant to Distribution B. This would indicate that Distribution A is more
favorable to Distribution B for risk-averse participants. For Distribution A to be second order
stochastically dominant to Distribution B, 𝐸[𝑢(𝐴)] ≥ 𝐸[𝑢(𝐵)] must hold; however, this
condition is only satisfied in four of eight questions. For the questions that do satisfy this
condition, it must also be the case that in terms of cumulative distribution functions 𝐹𝐴 and 𝐹𝐵,
the area under 𝐹𝐴 is less than or equal to that under 𝐹𝐵 for all values of x. This condition does not
hold in the remaining questions. Thus, in these eight questions, it is really not clear which
distribution is favorable, and consequently, receiving a signal indicating that the randomly drawn
payoff would come from either one of these distributions may not be of particular use when
making a decision in Part Two – Round Two.
The variables used in Part Two – Round Two, in addition to demographic variables, are
summarized in the table below:
29. 28
Table 11: Definitions of Dummy Variables Used in Part Two – Round Two
Dummy Variable
( = 1 if true)
Description Frequency
CR2
Participant chose
the $8 option
178 / 1,160
Meanmix1 Mixed Distribution Mean = 8 364 / 3,276
Meanmix2 Mixed Distribution Mean = 9 728 / 3,276
Meanmix3 Mixed Distribution Mean = 10 3,276 / 3,276
Meanmix4 Mixed Distribution Mean = 11 728 / 3,276
Meanmix5 Mixed Distribution Mean = 12 364 / 3,276
Treatment Saw Underlying Distributions 1,584 / 3,276
Order Order of Part Two Questions 1,620 / 3,276
LessRiskAverse
Demonstrated to be less risk averse in
Part One
1,044 / 3,276
(29 participants)
DefA
Signal informed that the payoff was
definitely coming from Distribution A
910 / 3,276
ProbA
Signal informed that the payoff was most
likely coming from Distribution A
728 / 3,276
ProbB
Signal informed that the payoff was most
likely coming from Distribution B
819 / 3,276
DefA
Signal informed that the payoff was
definitely coming from Distribution B
819 / 3,276
BFOSDA
Distribution B first order stochastically
dominates Distribution A
2,548 / 3,276
Session1 – Session17 One for each session
The table below shows the total number of responses selecting the distribution option
versus the $10 option based on underlying distribution means in Part Two – Round Two:
30. 29
Table 12: Investment Decisions by Underlying Distribution Mean in Part Two – Round Two
Distribution A
& Distribution B
Investment Option $8 Option
Frequency Percentage Frequency Percentage
Mean = 6
& Mean = 10
14 / 20 70.00% 6 / 20 30.00%
Mean = 6
& Mean = 12
37 / 72 51.39% 35 / 72 48.61%
Mean = 6
& Mean = 14
78 / 135 57.78% 57 / 135 42.22%
Mean = 8
& Mean = 10
25 / 27 92.59% 2 / 27 7.41%
Mean = 8
& Mean = 12
79 / 101 78.22% 22 / 101 21.78%
Mean = 8
& Mean = 14
201 / 247 81.38% 46 / 247 18.62%
Mean = 10
& Mean = 10
119 / 120 99.17% 1 / 120 0.83%
Mean = 10
& Mean = 12
285 / 287 99.30% 2 / 287 0.70%
Mean = 10
& Mean = 14
322 / 329 97.87% 7 / 329 2.13%
The table below also shows the total number of responses selecting the distribution option versus
the $10 option in Part Two – Round Two; however, it summarizes the information based on the
mean of the mixed distribution.
Table 13: Shares of Investment Decisions Based On Mixed Distribution Mean in Part Two –
Round Two
Mixed Distribution Investment Option $8 Option
Frequency Percentage Frequency Percentage
Mean = 8 14 / 20 70.00% 6 / 20 30.00%
Mean = 9 62 / 99 62.63% 37 / 99 37.37%
Mean = 10 276 / 356 77.53% 80 / 356 22.47%
Mean = 11 486 / 534 91.01% 48 / 534 8.99%
Mean = 12 322 / 329 97.87% 7 / 329 2.13%
31. 30
As seen in the charts above, participants generally continue with the initial investment
decision that they made in Round One, and this behavior increases as the distribution means
increase. It is therefore important to examine the cases in which participants switch their
investment decision and opt out to receive $8.
5. Econometric Analysis
In order to estimate the effect of information on participants’ decisions, I used a linear
probability model with random effects for a binary response 𝑌, specified as
𝑃(𝑦 = 1 | 𝑥) = 𝛽0 + 𝛽1 𝑥1 + ⋯ + 𝛽 𝑘 𝑥 𝑘 + 𝑢𝑖 + 𝜀𝑖𝑡
where 𝑃(𝑦 = 1 | 𝑥) is the probability that the investment option (a random payoff from the
distribution) is chosen. The only exception to this is in Part Two – Round Two, in which 𝑦 = 1
indicates that the participant chose to opt out and receive $8. The data was organized as a panel
of observations and it allowed for random individual effects. Suppressing the i (for individual)
and t (for question number) notation, in this analysis, 𝑥j, where j =1 to k, is always a binary
explanatory variable, so 𝛽𝑗 is the difference in the probability that the investment option was
chosen when 𝑥𝑗 = 0 and 𝑥𝑗 = 1, holding all other variables fixed. In other words, all coefficients
𝛽𝑗 represent the marginal effect, in probability units, of going from 𝑥𝑗 = 0 to 𝑥𝑗 = 1. The error
term in this analysis is 𝑒𝑟𝑟𝑜𝑟 = 𝑢𝑖 + 𝜀𝑖𝑡, a sum of the individual effect, 𝑢𝑖, and the idiosyncratic
error component, 𝜀𝑖𝑡. The data is not truly panel data, since the “time” dimension is really just
the set of all questions that subjects responded to; however, conceiving of this as panel data
allows the group of all responses by a given individual to be treated as dependent within the
group, but independent of other groups (i.e., other individuals). Standard errors are corrected by
32. 31
“clustering” the data by individual ID number, so that the standard errors allow for correlation
within participants. Using this method returned cluster-robust standard errors.
In Part One analysis, the regression included variables for mean of the investment option,
variance, demographic information, and session number. Skewness and support were not
included in the model because they are collinear with other variables. It is expected that as the
mean of the investment option increased, participants were more likely to invest. As variance
increases, participants would be less likely to invest. It is not certain what to expect to find for
major and GPA; however, it is possible that there may be a gender difference because when
stakes are small, as in this experiment, women are generally more risk averse than men (Holt
2005). In addition, session number will likely have no impact on investment decisions.
Analysis of Part Two – Round One will be very similar to that of Part One. For this
section, there are two separate models: one in which the means and variances of the underlying
distributions are regressors and another in which the means and variances of the mixed
distributions are regressors. A wide gap between the two models is unlikely. Both sets of
expected mean variables will interact with the treatment variable, in order to understand the
impact of additional information. Participants in the treatment condition may be more likely to
invest as the means of underlying distributions increase relative to other participants, due to their
knowledge of more specific information. Treatment will also be included in the regression,
though it may not make any participant more or less likely to invest on its own. Order effects are
not expected therefore Order will likely have no impact on participants’ choices. IsBimodal and
LessRiskAverse are included as well. Though the effect of IsBimodal is uncertain, the variable
LessRiskAverse will have a very positive impact on investing. Other variables in the regression
33. 32
include major, GPA, gender, and session, whose impacts will be similar to their impact in Part
One analysis.
In Part Two – Round Two analysis, it is already known that the investment option
seemed attractive to participants so variables that categorize the signal and its impact are
important in this last set of regressions. In Part Two – Round Two, Y = 1 if participants changed
their decision from Round One and opted out of the investment option for a certain value of $8.
As in Round One analysis, there will be two separate models: one with the means and variances
of the underlying distributions and another with the means and variances of the mixed
distributions. As in Round One, a wide gap between the two models is not expected. Order will
not have an effect on the results. The variable LessRiskAverse may have a negative effect in this
regression, since less risk-averse participants are likely to continue with the riskier investment
option. To have a better understanding of participants’ investment behavior when signals are
informative, the relatively more nuanced ProbA and ProbB are part of the constant term while
DefA and DefB are in the model. The coefficient on DefA may be positive – only because the
signals indicating DefA are generally low and may be discouraging. The coefficient on
DefA*Treatment may also be high and positive. For someone who sees the underlying
distributions, a signal indicating that the payoff is definitely coming from Distribution A is
usually unfavorable, which would cause participants to prefer the certain amount of $8. DefB is
likely closer to zero or slightly negative. DefB usually has higher signals, which would
encourage participants to continue with the investment option. In addition, for participants who
see the underlying distributions, a signal indicating that the random payoff is definitely coming
from Distribution B would encourage participants to continue with the investment option, more
so than if the signal was more likely to come from Distribution B or more likely to come from
34. 33
Distribution A .Therefore, the coefficient on DefB*Treatment will likely be slightly negative. It
is not clear what effect BFOSDA will have, but it is included to control for first order stochastic
dominance. Demographic variables and session variables were also included in the regression.
6. Results
1. Test for Individual Coefficients in the Part One Model
The following table shows the marginal effects of observing a different value in each
variable relative to the dummy variable of that category that is included in the constant term.
Male participants are included in the constant term along with Mean1, Variance1, Major1,
GPA1. In this model, a higher individual coefficient indicates a higher probability of selecting a
random payoff from the distribution. Testing for individual coefficients in this model returns the
following results:
Table 14: Individual Coefficients in the Part One Model
Estimated Coefficient
(Robust Standard Error)
P-value
Mean2
0.0220
(0.0122)
0.072
Mean3
0.231
(0.0308)
0.000
Mean4
0.805
(0.0289)
0.000
Mean5
0.926
(0.0228)
0.000
Variance2
-0.0297
(0.0119)
0.013
Major2
0.0218
(0.0371)
0.556
Major3
-0.00891
(0.0406)
0.826
Major4
0.0558
(0.0252)
0.027
GPA2
0.0411
(0.0233)
0.078
0.0815 0.000
35. 34
GPA3 (0.0222)
GPA4
0.124
(0.0495)
0.012
GPA5
0.0136
(0.0391)
0.728
Female
0.0199
(0.0173)
0.249
Constant
-0.0132
(0.0350)
0.707
𝒖𝒊 0.0514
𝜺𝒊𝒕 0.296
𝝆 0.0293
In the results above, it is evident that as the mean of the distribution increases,
participants are more likely to choose the investment option. The marginal effect of going from
mean = 8 to mean = 9 is very small, but participants are more likely to invest when the mean of
the distribution is $10. If all participants were risk-neutral, there would be a much greater
probability of investment at mean = 10. Evidently, most participants are slightly risk averse. As
the mean of the distribution increases from 10 to 11, there is a 57.4 percent increase in the
probability that a participant would select the investment option. The probability of investment
jumps another 12.1 percent when mean increases from 11 to 12. When variance increases, the
probability of selecting the investment option decreases, but this difference is very small.
Generally people of different majors behave similarly. Only major category “Other” was
significantly different from Economics majors, but the difference is not great. Surprisingly, for
GPAs within the range of 2.01 to 4.00, as GPA decreases, the probability of selecting the
investment option increases. In this analysis, this trend does not continue for participants with
GPAs below 2.00, but this result is inconclusive because the sample size is small. Perhaps, there
is something particular about participants with low GPAs that also makes them less risk averse.
Another possibility is that participants with lower GPAs were worse than participants with
36. 35
higher GPAs at accessing their overall risk. In trials, interacting Major and GPA did not offer
any new results, but this is perhaps because the sample size was not big enough or diverse
enough on these parameters. Contrary to results found in the literature, the results of this
experiment did not demonstrate that females are more risk averse than males. Though Session
variables were always included in the regressions, their coefficients are not reported because they
did not add anything to the interpretation of the results. Other than a few interesting findings in
the demographic variables, the results were generally as expected: investment increased as the
mean of the distribution increased and participants were generally more risk-averse.
2. Test for Individual Coefficients in the Part Two – Round One Models
For Part Two – Round One, there were two models, one in which means and variances of
the underlying distributions are regressors (Model One) and another in which the means and
variances of the mixed distributions are regressors (Model Two). The results of Model One are
used to draw conclusions because its parameterization is more specific than that of Model Two;
however, Model Two results are provided to serve as a comparison. In these models, higher
coefficients indicate a higher probability of selecting a random payoff from the distribution in the
first round. Along with non-bimodal distributions, as well as male and more risk averse
participants, Meanp11, Meanp21, Meanp11*Treatment, Meanp21*Treatment, Meanmix1,
Meanmix1*Treatment, Variancemix1, Major1, and GPA1 are included in the constant term.
Testing the individual coefficients in these two models returns the following results:
38. 37
(0.0827) (0.0827)
Major4
0.0308
(0.0436)
0.480
0.0308
(0.0436)
0.480
GPA2
-0.00782
(0.0482)
0.871
-0.00782
(0.0482)
0.871
GPA3
-0.00340
(0.0422)
0.936
-0.00340
(0.0422)
0.936
GPA4
0.123
(0.0683)
0.073
0.123
(0.0683)
0.073
GPA5
-0.119
(0.0866)
0.170
-0.119
(0.0866)
0.170
Female
0.00499
(0.0293)
0.864
0.00499
(0.0293)
0.865
Constant
-0.240
(0.115)
0.037
-0.0664
(0.112)
0.554
𝒖𝒊 0.150 0.150
𝜺𝒊𝒕 0.389 0.365
𝝆 0.142 0.144
Since Model One is more finely calibrated than Model Two, it will be used to draw
conclusions about investment decisions in Part Two – Round One. The results of the two models
are not very different from one another, but Model One offers more specific information.
Generally, as the means of the underlying distributions increased, the probability of choosing the
investment option increased. Interactions between Treatment and the means of the underlying
distributions were positive only for higher means of Distribution B, with no significance for the
other variables. A higher variance increased the probability of selecting the investment option by
2.2 percent, but this result is only significant at the 10 percent level. Treatment on its own was
not significant, as expected. Its effect was captured in the interaction terms with means. Order
was also not significant, indicating that there were no order effects. The coefficient on IsBimodal
is positive, though there is no natural interpretation for this result. Speculatively, participants
may be intrigued by the bimodality or perhaps hopeful that their payoff could be close to that of
the higher value of mode. The coefficient on LessRiskAverse is positive and significant,
meaning that participants who were less risk averse in Part One, were also 13 percent more likely
39. 38
to choose the investment option in Part Two – Round One. This trait influences their investment
decisions throughout the experiment. People of different majors and GPAs do not respond
significantly differently from each other with the exception of participants with GPAs between
2.01 and 2.50, who are more likely to choose the investment option. Gender is not significant in
these results as well.
3. Test for Individual Coefficients in the Part Two – Round Two Model
Since CR2 = 1 if participants chose to receive $8, and 0 if they chose to continue with the
investment option, a higher coefficient on a variable indicates a higher probability of opting out
to receive $8. Meanmix1, ProbA, ProbB, Major1, and GPA1 are all included in the constant
term. Testing for individual coefficients in this model returns the following results:
Table 16: Individual Coefficients in the Part Two – Round Two Model
Estimated Coefficient
(Robust Standard Error)
P-value
Estimated Coefficient
(Robust Standard
Error)
P-value
Meanp12
-0.214
(0.0308)
0.000
. .
Meanp13
-0.535
(0.0438)
0.000
. .
Meanp22
0.0722
(0.0191)
0.000
. .
Meanp23
0.0385
(0.0231)
0.096
. .
Meanmix2 . .
0.00361
(0.0675)
0.957
Meanmix3 . .
-0.0268
(0.0657)
0.683
Meanmix4 . .
-0.187
(0.0626)
0.003
Meanmix5 . .
-0.415
(0.0693)
0.000
Treatment
0.0204
(0.0155)
0.189
0.0144
(0.0160)
0.369
Order
-0.0161
(0.0177)
0.362
-0.00795
(0.0179)
0.658
LessRiskAverse
-0.0249
(0.0231)
0.282
-0.0286
(0.0234)
0.221
DefA 0.0139 0.753 0.152 0.001
41. 40
indicating that there were no order effects. Interestingly, LessRiskAverse was not significant in
this regression, likely because more risk-averse participants did not make the initial investment,
so these observations represent a less risk-averse subset of all participants. LessRiskAverse may
also be non-significant because risk was not as influential as the signal and its implications.
When participants received a lower signal, categorized as DefA, the probability of opting out
increased relative to ambiguous signals in ProbA and ProbB; however, this effect was not found
in the model with underlying distribution-specific means. The coefficient on DefB was very
negative in both models, indicating that the probability of continuing with the investment
decisions increased relative to ambiguous signals in ProbA and ProbB. When the signal was
high, access to underlying distributions did not have any impact – DefB*Treatment was not
significant; however, when the signal is low, participants who have access to underlying
distributions are much more likely to opt out and receive $8 than those who do not see the
underlying distributions. In this way, information makes a large and important difference in
participants’ investment decisions, particularly when profit-maximizing prospects are low. The
coefficients on BFOSDA are very different in the two models, likely because controlling for
underlying distribution means is sufficient for Distribution B to first order stochastically
dominate Distribution A in many questions. Though it would have been interesting to study the
impact of the interaction terms of BFOSDA with Treatment and DefA / DefB, these variables
were always omitted due to collinearity. Majors and GPAs were not significant, with the same
exception for GPA4 as in previous analysis. Participants with GPAs between 2.01 and 2.50 were
more likely to continue their initial investment decisions relative to participants with GPAs
between 3.51 and 4.00. Gender is not significant in these results as well.
42. 41
7. Discussion
This experiment demonstrated that people can generally make reasonable investment
decisions. Without the context of the natural world and the pressure that comes along with
college, participants were able to base their investment decisions on their risk preferences and
the information provided to them. Most participants in the study were slightly risk-averse, as is
true of the general population. In Part One and Part Two – Round One, an increase in expected
mean of the distribution lead to an increase in the probability of selecting the investment option.
Participants attempt to maximize their expected utility throughout the experiment. In addition,
being less risk-averse in Part One was an important indicator as to how participants responded in
Round One of Part Two – such participants were more likely to choose the investment option.
Analysis of Part Two – Round Two returned several interesting results. For example,
when participants received a high signal, they were much more likely to stick to their initial
investment option. Seeing the underlying distributions or not did not impact these decisions.
Depending on the model used, a low signal on its own could increase the probability that a
participant would opt out to receive $8; however, undisputedly, when the signal was low, and
participants had information about the underlying distributions, this increased the probability that
participants would opt out by at least 17 percent. This perhaps suggests that receiving a low
signal is not enough to get individuals to back out of a bad deal – participants also need to have
better information to correctly assess their profit-maximizing prospects. There were many
participants who received a low signal but did not have the context of the underlying
distributions. Therefore, these participants did not opt out of the investment option as much as
they should have. This may carry important implications for students entering and dropping out
of school. With better access to information and clear signals of profit-maximizing prospects,
43. 42
students may have a greater aptitude to plan their educational investments, particularly the
students who are not doing so well. It is thus extremely important that students receive accurate
signals of their success after college in addition to other pertinent information so that they may
interpret their signals correctly and plan accordingly.
It is of particular relevance today to discuss the overwhelming amount of uncertainty that
may be found in educational investment decisions. Efforts have been made to alleviate some of
the uncertainty in the college investment decision, the College Scorecard being one example.
The College Scorecard gives very basic information and statistics on a university or college for
anyone to see, but the information is so broad that it is not particularly helpful or informative for
prospective students. Nevertheless, it is the first of many steps in the right direction. Enrollment
in college is increasing, as are student debt and college dropout rates. This should be a topic of
great concern, not only to students or prospective students, but also to economists, politicians,
and many others. Aggregate student debt is currently in the trillions of dollars. This amount of
debt, based on the growing underemployment rate of college students, is likely extremely
inefficient; however, the growing cost of college is not only an issue in so much as it acts as a
barrier to entry for those who could benefit from it, but in addition to that, there is an entire
generation of students who will not be able to consume as many goods and services in the future
because they will be burdened with paying back their student loans. This could potentially be a
huge lag on the economy, and thus these phenomena in educational investments deserve a great
deal of attention.
In future research, it would be interesting to examine how the results would change if
participants would be endowed with their money first, and then they would have to pay in order
to enter the lottery option, just as students pay money to go to college. In addition, a 2x2 study
44. 43
could be done, in which half the participants take a survey with questions labeled as college
investment decisions. Potentially, because there is such a push for all students to go to college,
participants in the labeled condition would be more likely to choose the investment option
because the actual profit-maximizing prospects would be less salient. There is still much research
to do in the area of understanding how people make investments in education. By examining this
topic carefully, it is possible that students could be guided to make better educational investment
decisions.
45. 44
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Altonji, Joseph G. "The Demand for and Return to Education When Education Outcomes Are
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47. 46
Appendix A: Sample Questions from the Experiment
Part One
You have the choice between receiving $10 or receiving a payoff determined from the following
distribution:
The average payoff of this distribution is $10.
Do you choose to receive $10 or to receive a payoff from this distribution?
1) Receive $10
2) Receive a payoff from the distribution
48. 47
Part Two – Underlying Distributions Unknown Condition
First Round
In the following question, you have the choice between $10 and a random payoff drawn from
this distribution:
The average of this distribution is $11. If you choose to receive a payoff from this distribution,
you will receive a signal in the next question.
Do you choose to receive $10 or a random payoff drawn from the distribution?
1) Receive $10
2) Receive payoff from the distribution
49. 48
Second Round
You have chosen to receive a payoff from the distribution.
The average of this distribution is $11.
Your signal from the distribution is $8.47. You may use it as a “check” to see if you would still
like to receive a payoff from the distribution. This signal is not necessarily the payoff you would
receive if you choose to receive a payoff from the distribution.
If you like, you may choose a certain value of $8 as your payoff instead of drawing your payoff
randomly from the distribution.
Do you choose to receive $8 or a random payoff pulled from your distribution?
1) Receive $8
2) Receive payoff from the distribution
50. 49
Part Two – Underlying Distributions Known Condition
First Round
In the following question, you have the choice between $10 and a payoff generated from one of
two distributions represented in this mixed distribution:
The average of this mixed distribution is $11. It is equally constructed from the following two
distributions:
The average payoff from Distribution A is $10.
The average payoff from Distribution B is $12.
Your payoff will be drawn from one of these distributions. If you decide to choose a payoff from
this distribution, you will receive a signal in the next question that may help you determine from
51. 50
which distribution your payoff will be randomly selected.
Do you choose to receive $10 or a random payoff drawn from the distribution?
Second Round
You have chosen to receive a payoff from the mixed distribution.
The average of this mixed distribution is $11. It is equally constructed from the following two
distributions:
The average payoff from Distribution A is $10.
The average payoff from Distribution B is $12.
Your signal from the distribution is $8.47. You may use this number to help determine from
52. 51
which distribution your payoff will be randomly selected. This signal is not necessarily the
payoff you would receive if you choose to receive a payoff from your distribution.
If you like, you may choose a certain value of $8 as your payoff instead of drawing your payoff
randomly from the distribution.
Do you choose to receive $8 or a random payoff drawn from your distribution?
53. 52
Appendix B: Assent Form
You are invited to participate in a research study that is being conducted by Daria
Voskoboynikov, who is a student in the Economics Department at Rutgers University. The
purpose of this research is to interpret people’s choices under uncertain costs and payoffs.
This research is anonymous. Anonymous means that I will record no information about you that
could identify you. This means that I will not record your name, address, phone number, date of
birth, etc. If you agree to take part in the study, you will be assigned a random code number that
will be used on each test and the questionnaire. Your name will appear only on a list of subjects,
and will not be linked to the code number that is assigned to you. There will be no way to link
yours responses back to you. Therefore, data collection is anonymous.
The research team and the Institutional Review Board at Rutgers University are the only parties
that will be allowed to see the data, except as may be required by law. If a report of this study is
published, or the results are presented at a professional conference, only group results will be
stated. All study data will be kept for three years.
There are no foreseeable risks to participation in this study.
Participating in the survey will involve answering a series of incentivized questions. The
questions will ask about your investment decisions under uncertain conditions. You will be paid
$5 for your participation plus an additional amount, which will depend upon the decisions you
make. The specifics of decisions you will have to make and how they translate into your
payment will be explained in detail by the administrator of the experiment before the experiment
begins. The experiment will last approximately one hour and your final payment can vary
anywhere from about $10 to $20 depending upon the decisions made. I expect that the total
payment will be $15 on average for completing the entire study.
Participation in this study is voluntary. You may choose not to participate, and you may
withdraw at any time during the study procedures without any penalty to you. You are free to
leave whenever you wish but unless you stay until the end you will be paid nothing except your
initial $5 fee for showing up.
If you have any questions about the study or study procedures, you may contact myself at:
Daria Voskoboynikov
9152 RPO Way
New Brunswick, NJ 08901
Tel: 201-663-3365
E-mail: dariavosko@gmail.com
54. 53
You may also contact my advisor, Professor Barry Sopher, at:
Department of Economics, Rutgers University
301A New Jersey Hall
75 Hamilton Street
New Brunswick, NJ 08901
Tel: 732-932-7363 or 732-932-7850.
Email: Sopher@econ.rutgers.edu
If you have any questions about your rights as a research subject, you may contact the IRB
Administrator at Rutgers University at:
Rutgers University, the State University of New Jersey
Office of Research Regulatory Affairs
Institutional Review Board for the Protection of Human Subjects
Liberty Plaza / Suite 3200
335 George Street, 3rd Floor
New Brunswick, NJ 08901
Phone: 732.235.9806
Email: humansubjects@orsp.rutgers.edu
You will be given a copy of this assent form for your records.
By participating in this study/these procedures, you agree to be a study subject.
55. 54
Attachment C: Advertisement for Recruitment
We use an online system for recruiting. The following is the type of notice we send out to
departmental listserves at Rutgers (for those departments who have agreed to let us post to their
listserves):
A number of experiments about decision making in economic settings will be conducted
this year in the Gregory Wachtler Experimental Economics Laboratory in Room 107 Scott Hall
on College Avenue Campus. The experiments vary from individual decision making with
uncertainty (e.g., investing) to small group strategic decision making (e.g., game theory
experiments) to large market experiments (e.g., information markets to forecast the outcome of
an election). Participants are paid an initial set fee of $5 simply for showing up on time for an
experimental session, plus an additional payment that depends on the details of the particular
experiment. The typical experimental session lasts from 45 minutes to an hour and a half, and
typical total earnings vary from $15 to $40. The experiments are interesting and fun--they are not
required class activities and you do not receive a grade for participating, just money. If you
would like to be contacted in the future about such experiments, please go to the following
website to register: http://rutgers-econ.sona-systems.com
Please register using your Scarlet Mail email address when you request an account on our
system (lower left screen). If you do not normally use your Scarlet Mail address, please set your
Scarlet Mail account to forward mail to whatever address you do use, as our system is configured
to only accept Scarlet Mail addresses. Once you have registered, you will be able to sign up
online to participate in experiments. You will also receive email notices when new sessions are
posted on the site.
Investments: In this experiment, we are interested in how people decide how much to
invest when earnings are uncertain. Participants will answer a series of questions about
investment choices. At the end, one of the questions will be selected at random, and participants
will receive the payoff associated with that question. The experiment will last approximately 45
minutes. Participants receive $5 simply for arriving on time for the experiment, plus an
additional payment that depends on the choices made and on the question selected at the end of
the experiment. We expect that the average participant will earn about $10, including the initial
$5 payment for showing up on time. The exact amount earned will depend upon the question
chosen at the end of the experiment, and may range from $10 to $20.
