Does Cumulative Advantage Increase Inequality in the Distribution of
Success? Evidence from a Crowdfunding Experiment ∗
Rembrand Koning
Stanford GSB
Jacob Model
Stanford GSB
September 29, 2015
Abstract
The diffusion of online marketplaces has increased our access to “social information” – records of past behavior and opinions of consumers. One concern with this
development is that social information may create cumulative advantage dynamics
that distort marketplaces by increasing inequality in the distribution of success. Critically, this argument assumes that products that are likely to succeed disproportionately
benefit from social information. We challenge this assumption and argue that cumulative advantage processes can aggregate to have nearly any effect on distribution of
success, even decreasing the level of inequality in some cases. We assess these claims
using archival data and a field experiment in a crowdfunding marketplace. Consistent
with prior work, randomized changes to social information generate cumulative advantage. However, our treatments did not change the distribution of success. Products
benefited equally from our treatments regardless of their predicted likelihood of success. Our treatments still affected marketplace dynamics by weakening the relationship
between predicted and achieved success.
∗
Both authors contributed equally, order is alphabetical. Special thanks to DonorsChoose for making this
research possible. This work has benefited from feedback provided by participants at CAOSS, AOM, and
ASA. Generous funding was provided by Stanford’s Center on Philanthropy and Civil Society and Center
for Social Innovation.
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1
Introduction
One of the seminal insights of social science is that social information, past behavior and opinions of others, powerfully shapes our attitudes, beliefs and actions. The
rise of online platforms has made social information in the form of popularity, ratings
and reviews increasingly accessible. However, concomitant with this profusion of social information, there is a growing controversy as to whether it facilitates or distorts
marketplaces (Zuckerman, 2012; Muchnik, Aral and Taylor, 2013). On the one hand,
such information may provide informative signals about difficult to observe aspects of
products and services (Zhang and Liu, 2012). On the other hand, the use of social information is subject to many well-documented biases that may undermine or even pervert
its potential signaling value (Cialdini, 1993; Simonsohn and Ariely, 2008). Small and
arbitrary initial differences in social information may lead to large differences in success
due to self-reinforcing “cumulative advantage” dynamics (Banerjee, 1992; DiPrete and
Eirich, 2006). Indeed, a large body of archival (Simonsohn and Ariely, 2008; Chen,
Wang and Xie, 2011; Burtch, Ghose and Wattal, 2013) and experimental (Salganik,
Dodds and Watts, 2006; Tucker and Zhang, 2011; Muchnik, Aral and Taylor, 2013;
van de Rijt et al., 2014) research has found evidence that changes in social information
lead to cumulative advantage processes. Scholars have used this evidence to argue that
social information in marketplaces inherently magnifies inequality in the distribution
of success (Salganik, Dodds and Watts, 2006; Tucker and Zhang, 2011; Muchnik, Aral
and Taylor, 2013; van de Rijt et al., 2014).
Prior to any differences in social information, products in a marketplace already
differ in their expected success. For instance, we take for granted that some products
auctioned on eBay are more or less likely to reach their reserve price based on factors
such as appearance and the reputation of the seller. Past research has largely sidestepped these incoming differences by employing sophisticated statistical controls or
using randomization to account for pre-existing differences (cf. Salganik, Dodds and
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Watts, 2006). However, in order to understand how social information changes the
distribution of success in marketplaces, we need to know how incoming differences in
expected success interact with changes in social information.
Consider the case of two new books being sold in an online marketplace such as
Amazon. One that is from a first-time author and one that is from a well-known
author. A positive change in social information to both books, such as a good review,
will create a wider gap in sales if the well-known author benefits more from the review.
Alternatively, the same good review may plausibly benefit the first-time author more
than a well-known author as the latter already has some established reputation and
audience (Kovács and Sharkey, 2014). In this case, the change in social information will
reduce the gap in success between these two books. This example illustrates how the
direction of the effect of cumulative advantage on inequality is not obvious (Allison,
Long and Krauze, 1982). Without an understanding of how cumulative advantage
processes interact with a product’s expected success, we can say little about the effects
of social information on inequality of success in a marketplace.
In this paper, we investigate how changes to social information interact with a
product’s expected success and explore the implications of these interactions for the
distribution of success in online marketplaces. Specifically, we couple archival data
from a online fundraising marketplace with an field experiment to examine the interaction between a product’s expected success and cumulative advantage processes.
The marketplace, DonorsChoose, is a two-sided marketplace that helps public school
teachers find donors willing to fund school supplies. DonorsChoose facilitates this
by enabling teachers to post fundraising appeals (called “projects”) and by providing
modern search tools to donors to find projects that appeal to them.
One significant challenge is that the expected success of a product is typically not
directly observable in field settings. We build on the approach taken in Salganik,
Dodds and Watts’s (2006) seminal study of online culture markets in which Salganik
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and colleagues created multiple online markets with the identical set of products. The
authors experimentally manipulated the presence or absence of social information (in
their context, popularity) and tracked the success of products in the marketplace. In
their design, the success of products in marketplaces without social information served
as a direct measure of expected success in the absence of social information.
Although we cannot observe the same DonorsChoose project in multiple conditions, we take a conceptually similar approach to Salganik and colleagues by creating a
counterfactual measure of expected success for projects that received our experimental
treatments. We estimate expected success by generating predictions for each project
using a Random Forest algorithm trained on the characteristics and fundraising outcomes of over 400 thousand past projects. With this measure of expected success,
we then randomly contributed $5 or $40 to 320 projects and tracked their progress towards their fundraising goals. This randomization enables us to test whether exogenous
changes in social information lead to cumulative advantage dynamics. Critically, we
test whether these dynamics disproportionately benefit projects that were (un)likely to
succeed by interacting our randomized treatments with our measure of expected success. This approach enables us to assess how arbitrary changes to social information
change the distribution of funding outcomes.
Consistent with prior work on social information in online marketplaces, we find
that our randomly assigned contributions result in cumulative advantages that increase the probability that a project reaches its funding goal. Surprisingly, this effect
is remarkable homogeneous across projects with different levels of predicted success;
projects with characteristics that made them very likely to succeed benefited just as
much as projects with characteristics that made them unlikely to succeed. In this way,
our intervention introduced more noise into the fundraising process by reducing the
correlation between a project’s expected and realized success. The utility of this noise
depends on the goals of the marketplace. Increasing the unpredictability of success
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reduces the chance that projects with the most popular characteristics succeed. However, a little noise also creates greater opportunities for novel, innovative and risky
projects to reach their funding goals.
2
Social Influence, Cumulative Advantage and
Inequality
Cumulative advantage processes are considered a key driver of inequalities in society
and markets (DiPrete and Eirich, 2006). Initial advantages become magnified through
self-reinforcing dynamics resulting in a distribution of success that is highly unequal.
Scholars have suggested two mechanisms that produce cumulative advantage processes:
resource accumulation and information cascades. In the first case, small initial differences in the distribution of resources become magnified as those with initial resource
advantages can channel these into additional resources and skills (Merton, 1968). An
example of this is Merton’s pioneering work on cumulative advantage in academic research. This work takes a broad view of resources to include status garnered through
awards. Award winners leverage their increased status to get additional funding for
their work and so attract more talented students and collaborators (Merton, 1968).
In this sense, Merton’s mechanism is one of resource accumulation —award winners
can use their initial advantage to improve the quality of their work. Even if the initial allocation of the award were arbitrary, the emergent performance differences are
merited.
An alternative mechanism is information cascades (Banerjee, 1992). In these models, the assumption is that the quality of a product or service is difficult to observe
directly. Consumers must rely on other observable signals such as social information
that have some correlation, however weak, with quality (Podolny, 2005). For products
and services, common examples of social information include information on popular-
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ity, endorsements or reviews. This information tends to be self-reinforcing; a marginal
increase in the popularity or number of endorsements a product receives makes it more
attractive to subsequent consumers and reviewers. The core difference between Merton’s cumulative advantage mechanism and the information cascade model is that in
the latter, success can become decoupled from a product’s underlying quality. The
primary concern regarding cumulative advantage in Merton’s example of resource accumulation is that arbitrary and negligible initial differences allow certain scientists
to receive a disproportionate amount of success. This resulting inequality reflects real
differences in quality. That said, the arbitrary processes that lead to the inequality
may be deemed unfair. The concern of information cascades scholars is quite different;
their concern is that the resulting inequality in success may be largely uncorrelated
with the underlying features and quality of a product or service.
The expanding role of social information in our everyday lives has increased concerns
over the decoupling of social information from underlying quality (Muchnik, Aral and
Taylor, 2013; Colombo, Franzoni and Rossi-Lamastra, 2014). For example, business
owners can manipulate their online reputations by purchasing and then reviewing their
own products, or more nefariously, by paying others to give perfect reviews. This is
a commonplace activity; one scholar estimated that upwards of one-third of online
reviews are fake.1 Even if they are eventually identified and removed, fake reviews or
ratings may influence other consumers, leading them to alter their ratings, reviewing
or purchasing behavior. Once a process of cumulative advantage begins, it may be
difficult to correct. Even beyond outright fraud, natural variation in marketplace
activity may generate cumulative advantage processes. For example, differences in the
timing of initial endorsements or purchases, which may be effectively random, could
lead to cumulative advantages. These sources of advantage may be even more difficult
to address than outright fraud. Ideally, outcomes in marketplaces would be relatively
1
http://www.nytimes.com/2012/08/26/business/book-reviewers-for-hire-meet-a-demand-for-onlineraves.html
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robust to these perturbations.
However, research suggests that some marketplaces may not be especially robust to
these effects. In their pioneering studies of online culture markets, Salganik and colleagues examined how such arbitrary differences in social information change a product’s success in a simplified online marketplace created by the researchers (Salganik,
Dodds and Watts, 2006; Salganik and Watts, 2008). They randomly assigned participants into marketplaces that either displayed the popularity of a product —in their
setting, the number of downloads of a song— or suppressed this information. Product success, as measured by the number of downloads of a song, was more variable
in marketplaces with popularity information than those without it. In the popularity conditions, the songs that arbitrarily received a few initial downloads achieved
greater market share than the same song in a marketplace without social information.
This study demonstrates that markets with social information create cumulative advantages through endogenous processes that disproportionately benefit the most appealing
songs. In turn, this increases inequality within the marketplace.
In a follow-up study, Salganik and colleagues also show that these endogenous
processes are not destiny. In the same marketplace, they directly manipulated the
popularity of songs by reversing the endogenous ranking of songs (Salganik and Watts,
2008). The impact of this manipulation of social information varied across products
within the same marketplace, as measured by their success in marketplaces without
social information. It strongly benefited the songs with the lowest levels of expected
success. In contrast, songs with the highest levels of expected success did regain position in the download rankings but only after hundreds of subsequent downloads. The
combination of these effects implies that in this second experiment, the social influence
manipulation (i.e., the ranking inversion) likely reduced the inequality in popularity.
In sum, the results of Salgnik and colleagues’ studies suggest that social information
can generate cumulative advantage processes that have the potential to magnify or
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reduce inequality in success, even within the same marketplace.
A flourishing body of research has extended this work by examining whether changes
to social information produce cumulative advantage effects in field settings. For example, Zhang and Liu (2012) found evidence in an observational study of a peer-to-peer
lending platforms that changes in social information lead to cumulative advantage effects. However, it is difficult to casually identify social influence effects with observation
data. Changes to social information are usually endogenous to a product’s expected
success; people tend to consume, review or rate products that are likely to be appealing and succeed in the marketplace. The combination of the difficulty of measuring
a product’s appeal coupled with selection processes makes it extremely challenging to
determine how much success is due to changes in social information.
Several recent studies have adopted a field experiment approach and to address
these concerns by randomizing changes to social information. This research stream
has largely reached similar conclusions (cf. Burtch, Ghose and Wattal, 2013) with
studies finding that changes to social information through exogenous positive endorsements or contributions leads to cumulative advantage in a wide variety of settings
including wedding services marketplaces (Tucker and Zhang, 2011), online social news
aggregators (Muchnik, Aral and Taylor, 2013), online petitions, online reviews, and
crowdfunding platforms (van de Rijt et al., 2014). Therefore, we expect as our baseline
hypothesis:
Hypothesis 1: Exogenous endorsements of or contributions to products produce cumulative advantage effects
If changes to social information produce cumulative advantage, what are the implications for the distribution of success in markets? Scholars have tended to erroneously
equate evidence of cumulative advantage with evidence for increased inequality of success (van de Rijt et al., 2014). However, the relationship is not deterministic because
changes in the inequality of success implies that success of products relative to one
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another—not just the absolute levels of success of a given product—change. The research design of randomly assigning changes to social information in a market only
allows a researcher to estimate changes to absolute levels of success. It does not allow
a researcher to draw conclusions on how these effects shape the distribution of success
across products.
Consider the simplest case where the cumulative advantage effects are identical
across all products (e.g., a 10% increase in success rates). This produces a general
shift in absolute success rates but relative success is unchanged. However, when cumulative advantage effects systematically vary across different types of products, the
implications are more complex. If products with high levels of expected success benefit
more from changes to social information than products with low levels of expected
success, this would increase the inequality of success. Conversely, inequality in the distribution of success may decrease if products with low levels of expected success benefit
more from changes to social information than products with high levels of expected
success. Of course more complex relationships are possible, such as products with high
and low levels of expected success benefiting more than those with average levels of
expected success. Overall, these cases illustrate how cumulative advantage effects may
aggregate to have nearly any effect—or no effect at all—on the level of inequality in
the distribution of success.
To illustrate this more concretely, imagine two musicians who are trying to raise
similar amounts of money to produce a new album. These musicians decide to list
their fundraising efforts on a crowdfunding platform, a popular type of online marketplace. These platforms aggregate individual contributions towards a publicly stated
fundraising goal. They also commonly feature social information, typically the number
and size of prior contributions. The first musician is an established one and is likely to
reach her fundraising goal because she can rely on her established fan base for support.
The second musician is an inexperienced musician and is less likely to reach his goal.
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Imagine that each one of them randomly receives an initial contribution of $50. The
question for scholars of cumulative advantage is which album benefits more from this
contribution?
One possibility suggested by the cumulative advantage literature is a “successbreeds-success” dynamic where the experienced musician disproportionately benefits.
Potential contributors may prefer--perhaps because they are risk averse—to fund an
album that they perceive as very likely to succeed. In this case, the first contribution
will lead to stronger cumulative advantage effects for the established musician that, in
turn, will create a wider gap in fundraising success than one would expect without this
contribution. An alternative possibility suggested by the research on status and market
uncertainty is that the inexperienced musician may disproportionately benefit from the
contribution. The rationale is that it is likely that potential contributors have more
uncertainty regarding the success of the inexperienced musician relative to the established musician (Podolny, 2005; Azoulay, Stuart and Wang, 2013). If the contribution
decreases the level of uncertainty regarding the quality of the inexperienced musician’s
fundraising appeal more than the experienced musician, then the contribution would
disproportionately help the inexperienced musician. This mechanism would suggest
that the first contribution would reduce the gap in realized success relative to expected
success.
These mechanisms are meant to be illustrative rather than exhaustive. Either one
or even a combination of each, is plausible. Moreover, the functional form of the
interaction of expected success and the $50 contribution may change the implications
for inequality as well. For instance, there could be ceiling or floor effects. If an
extremely well-known musician like Paul McCartney were to try to fundraise for an
album on a crowdfunding platform, an initial contribution would likely have little effect
on the distribution of success. Understanding these considerations is important, but
the focus of our study is not to quantify the size or form of these mechanisms. Instead,
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we are concerned with how the effects produced by changes in social information,
regardless of the mechanism, aggregate to affect the distribution of success.
While we shown that cumulative advantage may affect the distribution of success
in myriad of ways, prior work has suggested that cumulative advantage processes tend
to increase inequality in the distribution of success (Salganik, Dodds and Watts, 2006;
Tucker and Zhang, 2011; Muchnik, Aral and Taylor, 2013; van de Rijt et al., 2014).
Therefore, we expect:
Hypothesis 2: Exogenous endorsements of or contributions to products benefit products with high levels of expected success more than products with low levels of expected
success
We explicitly examine whether cumulative advantage leads to increased inequality
by testing if social information has differential benefits across products and services
with different levels of expected success. In order to map differential benefits from
cumulative advantage to changes in inequality in success, one needs a counterfactual
estimate of expected success. Following Salganik and colleagues, we propose that a
product’s expected level of success without an exogenous change to social information provides a reasonable way to create a counterfactual (Salganik, Dodds and Watts,
2006). This counterfactual enables us overcome two measurement issues. First, it provides a basis for measuring cumulative advantage. Deviation from expected success
can be used to quantify cumulative advantage effects. Second, it enables one to locate
products on the distribution of expected success prior to being affected by social information. This placement enables us to draw conclusions on how a change in social
information may affect the shape of the distribution of success. In the following section, we describe the philanthropic marketplace we use in order to demonstrate our
methodology.
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3
Study Context
Our study takes place on one of the largest and oldest crowdfunding platforms: www.donorschoose.org
(referred to as DonorsChoose henceforth). Since its founding in 2003, DonorsChoose
has raised over $300 million dollars for classrooms by acting as an online marketplace
that connects teachers in need of school supplies with donors interested in supporting
public education. Teachers list a project in the marketplace by completing a standardized template which includes the supplies they want for their classroom, the total cost
of the supplies and a brief description of how the supplies will be used. The projects
range from basic classroom supplies such as textbooks to more novel requests such as
educational technology. The majority of funding requests range from a few hundred
up to one thousand dollars.
Potential donors can search through tens of thousands of projects using standard
internet searching features such as keywords and facets. The barriers to entry for
support are extremely low: one can contribute little as one dollar. Figure 1 presents a
screenshot taken at the time of our experiment of a randomly selected project’s website.
DonorsChoose provides very detailed information, most of which they verify through
third-party sources. This information includes time-invariant details like the shipping
costs, the level of poverty of the school and a text description of the project created by
a teacher. It also includes dynamically updated social information such as the number
of previous donations, the total amount raised so far and text messages of support
from prior donors. When making a donation, donors can either reveal their identity or
make an anonymous donation. They are also prompted to write a message for other
potential donors to view, though this step is optional. Immediately after a donation,
the project page is updated to reflect the donated amount, number of past donors and
any new messages of support written by the donor.
DonorsChoose operates in a similar manner to other prominent crowdfunding sites
such as Kickstarter or IndieGoGo. Projects are successful if they achieve their funding
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goal. When a project reaches its goal, DonorsChoose ships the supplies directly to
the school to be used. Projects cannot receive donations above their fundraising goal
and are immediately closed to donations once it reaches its goal. If the project fails to
meet its funding goal after five months of listing, DonorsChoose removes the project
from the website and donors are refunded the amount they donated to use towards
other projects in the marketplace. This marketplace, which over 200,000 teachers and
two million donors have participated in, is a prototypical example of a philanthropic
crowdfunding platform (Agrawal, Catalini and Goldfarb, 2011; Burtch, Ghose and
Wattal, 2013; Meer, 2014).
We selected a philanthropic crowdfunding platform for several reasons. First, there
are clear metrics of success on the platform: whether a product is funded and how
quickly it reaches its fundraising goal. Second, there is variation in outcomes and expected levels of success. Roughly two-thirds of projects are funded in the marketplace
and this varies greatly based on geography, subject matter and other project characteristics. Third, social information is very prominent in this marketplace. Information
on the number of donors and a progress bar, which displays how much money has
already been contributed, is displayed at the top of a project’s website (see Figure 1).
Donors to a project also have the option of writing short messages that are listed on
the project page. Overall, the platform is designed to communicate social information
to inform the decisions of potential donors.
Fourth, philanthropic crowdfunding is a place where social information should matter because “quality” of charitable causes is often difficult to directly observe (Hansmann, 1987). In particular, on DonorsChoose the beneficiaries of the donations, the
teacher and the students, are typically distinct groups from the donors.2 This makes
it difficult to assess the potential use of the pedagogical resources in the classroom
2
DonorsChoose tracks donations made by teachers who posted the projects These are rare events.
Through our qualitative examination of donor comments and conversations with DonorsChoose, it is clear
that the majority of donors are direct beneficiaries
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and, once acquired, to monitor their use. The lack of direct observability creates the
necessary preconditions for social information to lead to cumulative advantage effects;
it is possible for social information to influence evaluations such that there is a decoupling of evaluations from a product’s underlying features. Indeed, a large literature in
behavioral economics on matching and seed donations show that initial or matching
contributions lead to increased rates of subsequent donations (e.g., List and Reiley,
2002; Karlan and List, 2012). Based on this work and research on crowdfunding platforms (van de Rijt et al., 2014), we expect that cumulative advantage operates in this
setting. This is critical to our study because in order for social information to affect the
distribution of funding outcomes, it needs to be able to produce cumulative advantage
effects.
Beyond theoretical considerations, understanding cumulative advantage in the philanthropic domain is important in its own right. Individual philanthropy is economically significant as individuals are estimated to have donated over $250B to charities
in 2014.3 According to Gallup surveys, the vast majority of Americans —over 80 %
overall and over 95 % of Americans with incomes above 75 thousand dollars— are estimated to donate to charity in a given year.4 Online giving, in particular, is becoming
increasingly important. Online donations grew at an estimated 8.9% in 2014 compared
to the 2.1% increase of donations overall.5 As online donations and crowdfunding platforms become an increasingly important way for charities to fundraise, it is critical to
understand how social information shapes these decisions and affects the fundraising
process.
In addition, our particular setting is well-suited for our approach. DonorsChoose
has collected extremely detailed data on all the donation activity on its website since its
inception. They have the exact time (up to the nearest second), amount and source for
3
Giving USA 2015
http://www.gallup.com/poll/166250/americans-practice-charitable-giving-volunteerism.aspx
5
Blackbaud Charitable Giving Report: How Nonprofit Fundraising Performed in 2014
4
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every donation on the website. This data enables us to construct a linked longitudinal
set of donations for every project and every donation on DonorsChoose. We leverage
this rich historical data to construct accurate predictions on the expected success of a
project based on its time-invariant features when it is first listed on the marketplace.
While attractive for theoretical reason and empirical reasons, there is a reasonable
concern that studying donation decisions may not be generalizable to choice decisions
in other marketplaces, such as consumer purchasing decisions. Scholars have suggested
that donations are motivated by factors such as social signaling (Bénabou and Tirole,
2006; Karlan and McConnell, 2014) and the intrinsic utility or “warm glow” of the act
of donating rather than aspects of the cause (Andreoni, 1990). While these factors
certainly affect some donations on this platform, there are several reasons to believe
that a large subset of donors on DonorsChoose are basing their decisions on more
technical aspects of a project and are making decisions which are more analogous to
traditional models of choice. First, one of the main attractions of DonorsChoose as a
resource to donors is the ability to search and compare projects through keyword search
and faceting (hence the name of the platform). Donors also have many alternatives
to support public education such as school fundraisers, parent-teacher organizations,
scholarship funds and a plethora of education nonprofits. If a donor wanted to support
public eduction for warm glow or social signaling reasons, these are widely available
options that do not involve the costs of searching through the platform. Thus, we
would expect some degree of sorting amongst donors.
Second, empirical analyses of DonorsChoose suggest donors are sensitive to aspects
of the projects. For instance, Meer (2014) and our own analysis of DonorsChoose have
found that donors are less likely to fund projects that have high levels of so-called
“overhead costs” such as sales tax and shipping costs. In addition, Meer (2014) found
that projects with higher levels of competition from similar projects are less likely
to be funded, suggesting that donors are choosing between projects. These results
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are consistent with recent research on donations more generally, which suggests that
some donors are concerned with how their charitable dollars are being used (Gneezy,
Keenan and Gneezy, 2014). In sum, the empirical evidence suggests that donors on
DonorsChoose are making decisions using criteria, such as competition and price, that
we would expect would affect choice behavior in other marketplaces.
4
Data and Methods
Identifying the effect of cumulative advantage is difficult even with the detailed data
that DonorsChoose collects. Without some form of randomization, naturally occurring
or induced by a third-party, one cannot confidently separate social influence effects from
unobserved heterogeneity (Manski, 1993; Shalizi and Thomas, 2011). For example, one
project in our data very quickly attracted numerous donors from across the country
and was funded withing hours of being listed, far less time than a typical project.
It was a request for a set of books from the very popular book series, The Hunger
Games. The timing of the project also set it up for success as a popular movie based
on the book had been released shortly before the project was listed. As a result, fans
of this series from around the country were searching the marketplace for The Hunger
Games-related projects to support. Based on our reading of the messages left by the
donors, it is likely that most were making their decisions to support the project based
on their interest in the series and not based on the activities of prior donors. However,
this relationship would be difficult to discern statistically as the flurry of donations
would induce a correlation between early donations and eventual fundraising success.
This is certainly an extreme example, but it illustrates how difficult it is for an analyst
to differentiate social influence from difficult to observe aspects of products.
We follow recent studies of social influence and sidestep the problem of unobserved
heterogeneity by taking an experimental approach. Specifically, we donated $7,200
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in $5 and $40 increments to 320 randomly selected projects on DonorsChoose over
a 30-day period. We randomized which projects received our anonymous donations
to isolate the causal effects of an initial contribution on funding outcomes. We also
account for the fact that in this marketplace our donation simultaneously increases
the probability of a project being funded by reducing the amount a project needs to
reach its funding goal by either five or forty dollars. We control for this “mechanical”
reduction by using a non-parametric modeling strategy to isolate the social influence
effect.
Our experimental approach allows us to test if social influence leads to cumulative advantages. This alone does not inform us on how cumulative advantage process
affect the inequality of outcomes. This is because we cannot say if projects with a
greater chance of being funded disproportionately benefit from changes to social information. Randomization has washed away such differences. To overcome this hurdle,
we construct a counterfactual measure of a project’s expected success in the absence
of our randomized donation. The interaction between this predicted level of success
and the randomized contribution enables us to assess if the change in unpredictability
varies across projects in different parts of the distribution of success. If projects likely
to succeed disproportionately benefit then we have evidence for success-breeds-success
inequality dynamics.
We take advantage of the fine-grained historical data available on DonorsChoose
to build a measure of expected success. Specifically, we use data on the characteristics and outcomes of hundreds of thousands of past projects to create an estimate of
the probability a project would be funded without our intervention. If our exogenous
changes to social information through randomized donations lead to success-breedssuccess dynamics, then projects with a higher predicted probability of success should
benefit more than other projects. Conversely, if projects with lower predicted probabilities of success disproportionately benefit, then our randomized donations may actually
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be reducing inequality in outcomes.
The next section describes how we generate these estimates.
4.1
Predicting Success
We used data on all projects posted between January 1st, 2005 and August 21st,
2012 to train a prediction algorithm that estimates a newly posted project’s chance of
success. We excluded data from before 2005 as the website was significantly smaller
and the information concerning each project is much less detailed. This window leaves
us with 426,790 projects after excluding a handful of outlier projects that requested
either zero or more than ten-thousand dollars. These 426,790 projects comprise our
training set.
This data provides a large number of project characteristics with which we can
predict funding success. For each project, we know the timing of when a project
comes online, the sales tax amount, fulfillment costs, vendor shipping charges, number
of students who benefit, resource type, subject matter and if the project is eligible
for matching funds. For each teacher, we know the teacher’s salutation (a proxy for
gender), the grade-level they teach, if the teacher is part of Teach for America, and
if the teacher is a New York Teaching Fellow. For each school, we know the location
of the school, the school poverty level, if it is a charter school, a year-round school, a
magnet school, an New Leaders for New Schools school, and if the school is part of the
KIPP system.
Each of these variables may independently affect the probability that a project will
reach its funding goal. In addition, it is likely that complex functions and combinations
of these variables matter as well. For example, on average projects that request iPads
may be less successful than projects that request that request textbooks. However,
this difference may be even greater for low poverty schools than high-poverty schools.
Donors may see technology at wealthier schools as especially unappealing. One might
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also imagine that teacher characteristics have important interactions. Perhaps gender
stereotypes matter and male teachers are more successful at fundraising for sports
equipment projects than their female counterparts. Overall, we want to use a prediction
method that is able to account for potentially complex interactions between the project,
teacher and school characteristics.
We account for these potential interactions in a computationally tractable manner using a widely used machine learning algorithm, Random Forests (Breiman, 2001;
Friedman, Hastie and Tibshirani, 2008). Random Forests are an extension of Classification and Regression Trees (CART). Unlike standard regression analysis, the CART
algorithm automatically determines which covariates to include and how to include
them. It does this by scanning over the set of variables and picking the single variable
that best predicts the outcome of interest when split into two groups. Within each of
these groups, the algorithm again selects the single variable that best subdivides these
groups when split. It iterates in this manner until it there are less than a pre-specified
number of observations in each group.
The tree allows for the modeling of complex interactions since each of the subgroups may split on different variables. For example, the tree may first split projects
into high-poverty schools that are likely to be funded and low-poverty schools that are
unlikely to be funded. Then the algorithm would try to split each of these groups.
For the low-poverty group, the split may be on the fundraising goal, indicating that
donors are price sensitive. For the high-poverty group, the next split may be on vendor
shipping charges, indicating that donors to this group are averse to funding overhead
costs.
Despite the theoretical and interpretative clarity of CARTs, they often have poor
predictive performance and are unstable. The Random Forests algorithm addresses this
issue by averaging the predictions of many CARTs trained on different bootstrapped
samples of the data and by considering a randomly selected set of variables at each
19
split in each tree. This second step decorrelates the trees, which results in less variability when making predictions using the entire forest of trees. Random Forests have
performed extremely well in benchmark tests of machine learning prediction techniques
(Friedman, Hastie and Tibshirani, 2008; James et al., 2013). We use the “randomForest” package in R to build a predictive model of project success and treat the 426,790
projects described above as our training set.
In the training sample, 65.5% of projects reach their funding goal. Therefore,
a naive algorithm that always predicts every project ends up funded would yield a
classification error rate of 34.5%. We improve on this error rate by fitting a Random
Forest using 28 variables (see Figure 2 for the complete list). Instead of including
cities or zip codes for each school, we include the longitude and latitude of the school,
allowing the Random Forest to inductively partition geographical variation. After
an initial exploratory analysis, we found that building a forest with 250 trees and
considering five randomly selected variables at each split yielded the greatest predictive
performance. The performance of the model is evaluated using the out-of-bag error
rate, a procedure conceptually similar to cross-validation. The final error rate is 24.2%,
a 30% decrease from naively guessing that every project is funded. The model exhibits
few false negatives. The Random Forest incorrectly classifies projects that are funded
as unfunded only 8.7% of the time. In contrast, for projects that do not reach their
funding goal, the algorithm is only correct 53.6% of the time. In sum, the Random
Forest greatly improves upon the naive prediction and does an especially good job at
predicting which projects are likely to be winners.
We also assessed the performance of the algorithm against our qualitative understanding of what factors matter on the platform. We first examined which variables the
algorithm determined were most important when predicting whether a project would
be funded. Figure 2 shows the importance of each variable in fitting the Random Forest, from most important at the top to least important at the bottom. Intuitively, this
20
graphs shows which variables lead to the greatest gains in classification accuracy. While
variable importance tends to be biased towards variables that are continuous or have
more categories, the plot nonetheless provides a way to check for which variables matters for project success. Consistent with our expectations, the amount requested is the
most important variable, followed by date posted, geography, and then the project’s
subject. This accords with other analyses of this platform (Meer, 2014) and more
generally, with analyses of descriptive statistics of crowdfunding platforms (Mollick,
2014).
To further unpack what the algorithm is doing, we randomly selected projects from
each decile of predicted funding probability and present the differences and similarities
in Table 1. A cursory glance at this table would suggest that larger projects are less
likely to get funded. But this relationship is not deterministic. For example, the
project in the 5th row only requests $194.02 but the predicted probability of funding is
27.6%. However, the project in the 20th row is the most likely to get funded (95.6%)
and requests an additional $201.84 for a total of $395.86. That said, reducing the
amount requested for the project in the 1st row from $816.54 to $250 would increase
the predicted probability of funding from 5.6% to 48.28%. Moving the date of posting
for the project in the 15th row from March to June reduces the funding probability
from 77.2% to 66.4%. This difference suggests that it may be harder to raise money
in the summer months when most schools are not in session. For the project in the
16th row, removing the secondary focus subject reduces the predicted probability of
funding from 97.6% to 66.8%, suggesting that having a secondary focus subject may
be beneficial. In contrast, for the project in the 18th row adding a secondary focus
subject lowers the predicted probability of funding from 88.8% to 82.4%. It may be
that secondary focus subject interacts with other project characteristics in ways that
would likely be difficult to capture without the aid of the Random Forest algorithm.
21
4.2
Experimental Design
Our experiment made contributions to 320 randomly selected DonorsChoose projects
as soon as they were listed in the marketplace: 160 five dollar donations and 160
forty dollar donations. These are the 10th percentile and 65th percentile of donation
amounts on the site, respectively. We worked with the Chief Technology Officer of
DonorsChoose to create a customized data feed for all newly listed projects in a given
day. We only include projects that have not received any prior contribution in the few
hours since they went live. Over 99% of projects were included in our feed. We also
restricted our sample to projects with a primary or secondary subject as “Literacy,”
“Literature & Writing,” or “ESL”. This second selection criterion reduces the natural
variation that occurs across categories and increases our statistical power (Gerber and
Green, 2012).
The selection procedure worked as follows. For each of the twenty days on which
we made contributions, we first generated a list of all projects that fit the criteria
described above. We avoid altering macro-level market dynamics by only donating to
a small number of the listed projects. Donating to a large number of projects in one day
might change the probability of funding for both the treated and untreated projects.
Therefore, we restrict the number of projects that we contribute to on any given day in
order to ensure that we can meet the stable unit treatment value assumption (SUTVA).
This assumption would be violated if we dramatically changed the dynamics of the
marketplace (Morgan and Winship, 2007). Specifically, we limit the number of projects
we donate to at most 16 on any given day, a small fraction of active projects.
We donated to projects on twenty days from August 22nd to September 18th, 2012.
To maintain strict comparability between our treatment and control groups, we only
analyze projects listed on days in which we made a donation. Figure 3 shows the
number of projects we donated to, and the total number of projects at risk at receiving
our treatment. Our donations were anonymous and all contributions appear visually
22
identical to potential donors to minimize potential donor identity effects (Karlan and
List, 2012). We worked with DonorsChoose to ensure that our initial donation did not
immediately alter the project’s search rank.6 (Ghose and Yang, 2009; Ghose, Ipeirotis
and Li, 2014).
We construct a control group out of the 2,651 projects that were at risk for being
selected into treatment but were randomly excluded. We cannot include the entire
sample of non-treated projects in our models because the number of projects and
funding probability vary greatly across days. From our prediction modeling, we know
that timing effects are important and thus including the entire sample may lead us to
bias our estimates by improperly weighting some days more than others. To address
this issue, one can include fixed effects for each day to control for inter-day variation in
funding probability or sample projects proportional to the number of projects treated
on each day. We chose to randomly sample projects in proportion to the number
treated on that day.7 Specifically, we sample 5 control projects per 1 treatment project
to maximize our power. We have to drop two days where there are too few potential
control projects to sample. This procedure leaves us with 144 projects that receive the
$5 treatment, 144 the $40 treatment, and 1,440 randomly selected projects serving as
our control group.
5
Results and Analysis
We begin our analysis by comparing our treatment and control groups. Table 2 presents
summary statistics for the control, $5 treatment, and $40 treatment groups. The vari6
By the design of the search algorithm, the amount contributed does not alter the search rank until the
project is nearing its 5 month time frame for funding Even at this point, the primary characteristics used to
rank search are the school characteristics, time remaining and amount outstanding to raise. Checking the
search order qualitatively during our experiment revealed no difference between our treatment and control
groups. Finally, the effects observed from our treatments occur before the algorithm meaningfully alters
search results.
7
Results are qualitatively unchanged using the fixed-effect approach.
23
ables “Total Project Amount”, “Number of Students Reached”, and the “Random
Forest Predicted Probability of Funding” are time invariant and are measured prior
to our treatment donations. In Table 2 the italicized variables “Days to Funding”
and “Project is Funded” are measured after our treatment. Correlations between
these variables are presented in Table 3. The pre-treatment variables show no statistically significant differences between groups and provide first-order evidence that our
randomization procedure was successful. More formally, Table 4 regresses the three
pre-treatment variables on our treatment conditions. If our randomizations are unrelated to project characteristics, then it should be the case that the coefficients on our
treatment variables should be very close to zero. Indeed, the coefficients are small, and
all insignificant. This gives us confidence that our procedure resulted in randomized
treatment assignments.
The effects of our intervention on the post-treatment variables are presented in
Table 2. We find that our randomized donations alter the time to funding and the
probability that a project is funded. Projects in the $40 treatment group have a
funding rate of 84%, 10% higher than the control group rate of 73%. They also reach
their funding goal on average 13 days faster than the 70 days it takes the control group.
Unexpectedly, projects in the $5 treatment arm appear to perform somewhat worse
than the control being funded at a 72% rate as compared to the control rate of 74%.
Moreover, projects that received the $5 donations take on average 9 days longer to be
funded than projects in the control group.
We examine the hazard of funding by fitting separate Kaplan-Meier survival curves
by experimental condition in order to better understand how our treatments altered
funding dynamics. This approach also enables us to account for the right-censoring that
occurs when DonorsChoose removes projects that have been on the site for 5 months
(approximately 150 days). Given enough time, some (or perhaps all) of these projects
would be fully funded. Analyzing funding rates allows us to account for this censoring.
24
Figure 4 plots the probability of funding by experimental condition. The x-axis is the
number of days since posting. Consistent with Table 2, the $40 treatment appears to
have a greater hazard rate of funding than the control group and this difference appears
to grow larger with time. This increase is consistent with models of social influence
leading to cumulative advantages. The funding probability for projects that received
the $5 treatment appears slightly lower, but this decrease does not appear to change
with time.
While the $40 Kaplan-Meier curve is suggestive of cumulative advantage, it does
not take into account that our donation reduces the amount of outstanding money a
teacher has to raise to reach their goal, which we refer to as the “amount outstanding”.
For instance, projects in the $40 treatment arm have $40 less to raise. Therefore, even
if donors were unaffected by our treatment, projects in the $40 treatment arm should
still be more successful at reaching their funding goals, ceteras paribus. We account
for the reduction in the amount outstanding by controlling for it in a non-parametric
way. Specifically, we fit Generalized Additive Models with coefficients for each of our
treatments and with a penalized regression spline in the amount outstanding. We set
the number and location of knots for the spline by minimizing the error rate using
cross-validation.
Table 5 present linear probability models of funding and Table 6 presents hazard
models of funding. Model 1 and Model 5 replicate the summary statistic and KaplanMeier analysis presented above in regression form. The results are consistent with
the visual evidence in Figure 4. The probability of funding for the projects in the $40
treatment condition increases by 9.7 percent (SE = 0.038) and the hazard of funding by
0.29 (SE = 0.096). While the $5 treatment is negative, the coefficient size is small and
statistically insignificant. Model 2 and Model 6 control for the mechanical reduction
by incorporating a spline in the amount outstanding. Model 6, the Cox model, also
25
includes strata for each quintile of the amount outstanding.8
We find some evidence for cumulative advantage in Models 2 and 6. Once we
account for the mechanical reduction in the amount outstanding, the coefficient for
the $40 treatment drops from 0.097 to 0.069 and is only significant at the 10% level.
In Model 6, the hazard drops from 0.291 to 0.217 but remains significant at the 5%
level. Taking these results together, it appears that roughly one-third of our effect
occurs because of the mechanical reduction in the amount outstanding and the other
two-thirds by changing the hazard rate of future donations. The $5 donations do not
appear to have any meaningful effect on outcomes. Overall, Hypothesis 1 is largely
supported for our $40 treatment.
Next, we investigate Hypothesis 2 by testing whether these cumulative advantage
effects change the distribution of outcomes. We estimate these changes by interacting
our treatments with our measure of predicted success. We begin by first including
normalized predicted funding probability in Models 3 and 7 in Tables 5 and 6. The
predicted funding probability coefficient is highly significant in the linear probability
and Cox models. To get a sense of the effect size, it is useful to compare the coefficient
in our regression to the overall variability of our measure. A one standard deviation
increase corresponds to a 15% increase in predicted probability. In the linear probability model, the coefficient on our normalized measure is 14.5%. This implies that a
one standard deviation increase in the predicted probability of funding (15% higher)
leads to a 14.5% increase in the actual probability of funding in our data. This provides strong evidence that our predicted probability measure is capturing differences
in the likelihood of project success. Moreover, we increase our power by capturing a
substantial amount of project heterogeneity with the inclusion of the predicted funding
8
One concern with fitting Cox models is meeting the proportional hazards assumption. In unreported
analyses, we find that larger projects are less likely to get funded in the early days than projects that request
a smaller amount, though this difference dissipates over time. To account for this, we stratify models 6-8 on
quintiles of project size. Testing the model with these strata reveals that we meet the proportional hazards
assumption.
26
probability variable in Models 3 and 7. The magnitude and statistical significance of
our $40 treatment increases in both models. This provides further evidence that our
$40 treatment leads to cumulative advantage.
Models 4 and 8 in Tables 5 and 6 interact our treatment indicators with the predicted probability of funding. We find little evidence that projects that have higher
predicted probabilities of funding benefit more from a randomized $40 treatment. The
coefficient on the interaction term is small and statistically insignificant in both models. The main effect of the $40 donation does not change in magnitude nor significance.
Thus, we find little evidence for cumulative advantage distorting the baseline chance of
funding success. All projects appear to benefit equally from arbitrary and exogenous
variation in social information.
A potential concern with our analysis is that the functional form of the interaction
effect may be non-linear. Our assumption of linearity may be masking underlying effects that at the tails of the distribution. We account for these possibilities by again
using Generalized Additive Models with a flexible spline specification. Specifically, we
fit models in which we interact our $40 treatment with a spline of predicted funding
probability. This allows us to see if our treatments have larger or small effects at different points in the predicted funding probability distribution. Since the $5 treatment has
no effect thus far, we drop these observations from this analysis to ease interpretation.
As above, we determine the location and number of knots using cross-validation and
minimization of the the out-of-fold error rate.
Interpreting the implications of these non-linear interactions using coefficient estimates directly is extremely difficult. In lieu of regression tables, we present the results
of these non-linear interactions by plotting the marginal effects in Figure 5. The xaxis in both plots is the standardized predicted funding probability. The black lines
represent changes in funding probability for projects in the control group. The blue
line changes in the funding probability for the the projects in the $40 treatment group.
27
The light shaded areas are the 95 percent confidence intervals. While the spline in
the linear probability model is slightly non-linear at the tails, the Cox hazard model
is perfectly linear. For both groups and in both models, the realized funding probability increases linearly with the predicted funding probability. We find no evidence
for non-linear interaction effects. In short, we find no evidence for Hypothesis 2. Our
treatments are constant no matter a project’s expected level of success; the unlikely to
succeed and the very likely to succeed appear to equally benefit.
In summary, we find that our $40 treatment leads to cumulative advantage. However, we find little evidence that our treatment varies across projects with different
predicted levels of success. These results taken together suggest that cumulative advantage operates in our setting but that random variation in social information plays
no direct role creating wider differences in success than one would expect. Instead, it
seems that exogenous changes in social information induced by our treatments benefits projects across the distribution of expected success in a relatively equally way.
This suggests that exogenous changes in social information may simply lead to more
unpredictable outcomes, even if the distribution of success in expectation is relatively
similar.
6
Discussion
Our study has shown that the existence of cumulative advantage induced by exogenous changes does not necessarily increase inequality of success. This existence proof
is an important corrective to the assumption made by many scholars that cumulative
advantage, often created by social influence processes, inherently increases the inequality of success in marketplaces (Salganik, Dodds and Watts, 2006; Muchnik, Aral and
Taylor, 2013; van de Rijt et al., 2014). Rather than assume a direct mapping between
cumulative advantage and changes in the inequality of success, we provide a replicable
28
methodology for assessing the existence and strength of this link. Quantifying these
effects is necessary in order to design marketplaces that balance the benefits of social
information with the potential costs of the distortions such information may introduce.
Though we did not find evidence that exogenous changes to social information
produce differential cumulative advantage effects, there are other mechanisms through
which cumulative advantage processes may exacerbate inequalities. For example, products that have an innate appeal may be more likely to receive an initial review, endorsement or contribution. Receiving earlier support could provide an initial advantage
for these products over comparable products. In addition, some products in certain
marketplaces may be more likely to receive large contributions, which prior research
(along with our results) suggests may be more likely to attract subsequent customers
(List and Reiley, 2002). Future research should investigate the role of these endogenous
processes. However, our analysis greatly reduces concerns that arbitrary and exogenous early differences in social information deterministically lead to increases in the
inequality of success. In our setting, products that are unlikely to succeed and that are
likely to succeed equally benefit from changes to social information that is uncorrelated
with underlying features of a product.
By focusing on the inequality of success, our approach also sidesteps the contentious
debate over whether the “wisdom of crowds” exists when social information in a marketplace leads to potentially interdependent rather than independent judgments (Zhang
and Liu, 2012). This is an important debate to have, but it ignores the fact that marketplaces are often designed in ways that shepherd crowds towards particular goals. For
example, crowdfunding platforms like Kickstarter and IndieGoGo curate and promote
products that would otherwise be difficult to discover. DonorsChoose explicitly highlights projects that serve high-poverty public schools precisely because one of its goals
as a philanthropic marketplace is to help direct capital to needy students. In practice,
these platforms tend to reduce the inequality of success by promoting products that
29
are aligned with the goals of the marketplace designers.
An important limitation to our study is the generalizablilty of its context. While we
argue that the behavior of donors on DonorsChoose is similar to what we would expect
in other marketplaces, we cannot be certain without replication in other contexts. For
instance, it is possible that our interventions affect only aspects of evaluation that are
particular to a philanthropic context, such as social signaling or warm glow, rather
than more general perceptions of project’s features. Although we cannot rule out these
explanations with our current study, future research should attempt to study these
possibilities. Moreover, understanding both commercial and philanthropic motivations
is particularly timely as new products and services are increasingly combining the two
(Battilana and Lee, 2014). For example, consider the case of so-called “rewards-based”
crowdfunding platforms such as Kickstarter and IndieGoGo. While some contributors
use these platforms to buy a product or service (i.e., the contributor’s reward is the
actual product or service being developed), many contributors support the overall
endeavor and do not receive the actual products and services created. Instead, they
typically receive recognition or trinkets as their rewards, which is very similar to what
donors normally receive in exchange for charitable contributions.
7
Conclusion
One of the key findings of our study is that social information may increase unpredictability of success in marketplaces. But when would unpredictability be desirable
and when might it be detrimental? Our view is that the value of unpredictability is
that it is one way to promote diversity of success in a marketplace. Diversity is not
always a good thing. For instance, unpredictability is especially harmful in marketplaces where consumers have common goals, a consensus on what constitutes “quality”
and a consensus on how to measure it. One example of this type of marketplace would
30
be peer-to-peer lending. Lenders have similar goals; they are looking for the best risk
adjusted rate of return they can get. A situation in which social influence increases the
funding of poor performing loans is bad for the consumers of these marketplaces and,
in the long run, may jeopardize the viability of these platforms if lenders lose money.
It is also likely bad for people who are taking the loan, as failing to repay a loan may
hurt their credit rating or ability to get future loans.
Alternatively, many marketplaces have consumers with diverse motivations and
where assessments of quality are more varied (Zuckerman, 2012). These marketplaces
may benefit from unpredictability as it leads to diversity in the distribution of success.
The chance of an unpredicted success may encourage risk-taking, innovation, and exploratory strategies (March, 1991). The diversity of success may be an explicit goal
for marketplaces seeking to foster innovation, a goal of many crowdfunding platforms.
Concerns about decoupling success from the underlying appeal of a product may be
muted because the single ideal or metric of “quality” may not exist. In sum, the
greater levels of unpredictability created through social information may, on average,
help products with less inherently popular characteristics. But it may also enable less
appealing but more innovative products to succeed.
31
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8
Figures and Tables
Figure 1: Example of the DonorsChoose website at the time the experiment took place.
35
Variable Importance
Tota Project Amount
●
Day in Year
●
School Longitude
●
School Latitude
●
Secondary Focus Subject
●
Primary Focus Subject
●
Vendor Shipping Charges
●
Number Students Reached
●
Day of Week
●
Sales Tax
●
Year
●
Secondary Focus Area
●
Resource Type
●
Grade Level
●
Fulfillment Costs
●
Primary Focus Area
●
Double Impact Match Eligible
●
Teacher Mr, Mrs, or Ms
●
School Poverty Level
●
Almost Home Match Eligible
●
TFA Teacher
●
Charter School
●
Year Round School
●
Magnet School
●
NLNS School
●
NYTF Teacher
●
Promise School
●
KIPP School
●
0
5000
36
10000
15000
20000
MeanDecreaseGini
Figure 2: Random Forest variable importance. Variables at the top of the list are more
important in predicting a project’s probability of success than variables at the bottom of the
list.
Table 1: Two Randomly Selected Projects from each Decile of Predicted Funding Probability
PID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
P(Funding)
Project Title
Date
Project Amt
Primary Subject
0.056
05-12
816.54
Literature & Writing
A New Projector Would Make Presentations-Kids Brighter!
0.068
04-16
377.72
Literature & Writing
Continuing To Expand Our Electronic Library
0.136
05-06
1, 899.96
Mathematics
Teaching Through Technology
0.156
03-15
412.50
Mathematics
Becoming Mathematicians With Technology
0.276
02-18
194.02
Special Needs
Create And Learn
0.292
08-09
661.49
Environmental Science
Geologists In The Making
0.316
07-21
210.02
Literature & Writing
Stay Gold, Ponyboy
0.320
03-30
379.85
Literature & Writing
Learning and Growing through Music
0.404
06-16
440.60
Literature & Writing
Help Us Make Writing Wondrous!
0.452
03-14
252.47
Literacy
iCan Learn with iPods!
0.536
05-10
735.32
Music
Keyboards to Keep Kids Learning Part 2!
0.572
06-15
434.45
Literacy
We Are Greatly in Need of General School Supplies
0.656
02-23
777.27
Literature & Writing
New Technology for Our Classroom 2
0.696
06-12
142.32
Literacy
Keeping Our Classroom Colorful in the New School Year!
0.772
03-07
598.26
Special Needs
Technology for Special Needs Students
0.796
02-19
296.83
Literacy
Where Are the Books?
0.816
05-25
339.35
Music
Help Us Learn Guitar!
0.888
03-11
612.59
Literacy
Empowering First Graders
0.928
03-16
250.84
Special Needs
Read and Explore
0.956
08-11
395.86
Literacy
Fill Our bookshelves!
37
Secondary Subject
Shipping
Students
Literacy
12
175
Literacy
12
27
Other
158
24
31.29
100
Mathematics
0
30
Applied Sciences
0
70
0
24
Mathematics
12
18
Literacy
0
31
Mathematics
12
21
0
120
0
30
12
75
0
21
47.21
9
0
18
0
10
0
27
12
13
0
26
ESL
Literature & Writing
Special Needs
Early Development
Literacy
Table 2: Summary Statistics by Experimental Condition
Variable
N
Mean
St. Dev.
Min
Median
Max
Control Projects
Total Project Amount
Number Students Reached
Predicted Probability
Days to Funding
Project is Funded
1,440
1,440
1,440
1,440
1,440
534.80
68.1
0.66
69.9
0.74
393.55
111.4
0.15
56.1
0.44
133.47
5
0.22
1
0
434.77
30
0.67
49
1
4,512.88
999
0.96
150
1
$5 Treated Projects
Total Project Amount
Number Students Reached
Predicted Probability
Days to Funding
Project is Funded
144
144
144
144
144
495.18
61.8
0.66
78.7
0.72
297.88
96.8
0.15
56.2
0.45
131.72
7
0.27
2
0
435.63
30
0.65
58
1
2,216.21
999
0.94
150
1
$40 Treated Projects
Total Project Amount
Number Students Reached
Predicted Probability
Days to Funding
Project is Funded
144
144
144
144
144
491.74
74.5
0.67
56.7
0.84
391.95
145.1
0.15
52.2
0.37
134.95
12
0.29
1
0
418.36
29.5
0.69
39
1
3,297.28
999
0.92
150
1
Training Set Projects
Italicized variables are measured post treament.
Table 3: Correlations
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Total Project Amount
Number Students Reached
Predicted Probability
Number of Natural Donors
Days to Funding
Reached Funding Goal
Forty Dollar Treatment
Five Dollar Treatment
(1)
(2)
(3)
(4)
(5)
(6)
(7)
0.08
-0.63
-0.09
0.18
-0.14
-0.03
-0.03
-0.07
-0.02
-0.01
0.02
0.02
-0.02
0.09
-0.32
0.28
0.02
-0.02
-0.26
0.38
-0.01
-0.02
-0.78
-0.07
0.05
0.06
-0.02
-0.09
38
(8)
Table 4: Balance Tests
Dependent variable:
PFP
Log(Project Amount)
Log(Num Students Reached)
(1)
(2)
(3)
$5 Treatment
−0.008
(0.013)
−0.046
(0.051)
0.004
(0.075)
$40 Treatment
0.010
(0.013)
−0.088
(0.051)
0.013
(0.075)
Constant
0.663∗∗
(0.004)
6.098∗∗
(0.015)
3.705∗∗
(0.023)
1,728
3
−843.173
−0.001
1,728
3
−1,513.84
0.001
1,728
3
−2,195.23
−0.001
Observations
Model D.F.
Log Likelihood
Adjusted R2
∗
p<0.05; ∗∗ p<0.01
Linear Regression Models.
Predicted Funding Probability (PFP).
Note:
39
Table 5: Linear Probability Models
Is the project funded?
(1)
(2)
(3)
(4)
$5 Treatment
−0.028
(0.038)
−0.033
(0.037)
−0.020
(0.036)
−0.019
(0.036)
$40 Treatment
0.097∗
(0.038)
0.069
(0.037)
0.087∗
(0.036)
0.084∗
(0.037)
0.145∗∗
(0.015)
0.141∗∗
(0.015)
Predicted Funding Probability (PFP)
$5 Treatment × PFP
0.022
(0.036)
$40 Treatment × PFP
0.027
(0.036)
Constant
Project Amount Splines
Observations
Estimated Model D.F.
Log Likelihood
Adjusted R2
Note:
0.744∗∗
(0.011)
0.746∗∗
(0.011)
0.744∗∗
(0.011)
0.744∗∗
(0.011)
No
Yes
Yes
Yes
1,728
3
−1,007.198
0.003
1,728
6.87
−974.833
0.042
1,728
6.17
−934.405
0.085
1,728
8.12
−935.990
0.085
∗
p<0.05; ∗∗ p<0.01;
Linear probability models with penalized splines.
All continuous variables standardized.
Predicted Funding Probability (PFP).
40
Table 6: Cox-Proportional Hazard Models
Days till project is funded
(5)
(6)
(7)
(8)
$5 Treatment
−0.138
(0.103)
−0.186
(0.103)
−0.145
(0.104)
−0.149
(0.105)
$40 Treatment
0.291∗∗
(0.096)
0.217∗
(0.097)
0.270∗
(0.097)
0.239∗
(0.102)
0.388∗∗
(0.045)
0.374∗∗
(0.047)
Predicted Funding Probability (PFP)
$5 Treatment × PFP
0.032
(0.106)
$40 Treatment × PFP
0.160
(0.107)
Project Amount Quintile Strata
Project Amount Splines
Observations
Estimated Model D.F.
Log Likelihood
Note:
No
No
Yes
Yes
Yes
Yes
Yes
Yes
1,728
2
-8,936.613
1,728
3.17
-6,813.302
1,728
5.01
-6,773.990
1,728
7.91
-6,772.463
∗
p<0.05; ∗∗ p<0.01;
Cox-proportional hazard models with penalized splines.
All continuous variables standardized.
Predicted Funding Probability (PFP).
41
Count of new projects
300
Condition
200
40
5
0
100
0
9−18
9−17
9−16
9−15
9−14
9−13
9−12
9−11
9−10
9−09
9−08
9−07
9−06
9−05
9−04
9−03
9−02
9−01
8−31
8−30
8−29
8−28
8−27
8−26
8−25
8−24
8−23
8−22
Date
Figure 3: Number of projects per day posted during experimental intervention period by
condition.
42
1
0.8
0.6
0.4
Exp Condition
0
0.2
Control
$5
$40
0
50
100
150
Days since project posted
Figure 4: Kaplan-Meier Curves showing the probability after x days that a project is fully
funded by condition.
43
Cox-Proportional Hazard
0
Change in Hazard of Funding
-1
0.2
0.0
-0.2
-0.4
-2
-0.6
Change in Probability of Funding
1
0.4
0.6
Linear Probability Model
-3
-2
-1
0
1
2
-3
Standardized Predicted Funding Probability
-2
-1
0
1
2
Standardized Predicted Funding Probability
Figure 5: Testing the Linearity of the Interaction Effect using Generalized Addative Models.
44