Selling to biased believers: strategies of online lottery ticket vendors.
Lien, Jaimie W. ; Yuan, Jia
I. INTRODUCTION
Do sellers in the marketplace take advantage of the belief biases
of their consumers? Models of markets typically assume that sellers are
profit maximizing, consumers are utility maximizing, and both sides of
the market have accurate, unbiased beliefs. However, an increasing
collection of evidence shows that the presence of cognitive and
strategic heterogeneity can account for many stylized facts in the
marketplace which appear puzzling when confined to the framework of
classical models.
We examine the behavior of sellers in a popular online market which
provides an unusual opportunity to identify seller exploitation of
biased beliefs held by buyers: the Chinese collective lottery betting
market. In this market, shares of ticket "portfolios" for
China's national lottery are sold, commissions are charged by the
sellers, and the success rates of sellers are made public. As in most
state-run lotteries, success is completely random, with known
probabilities that are independent of previous outcomes. These objective
probabilities allow us to cleanly detect deviations from rational
beliefs among buyers, and assess the corresponding responses by sellers
in the market, to these demand shocks of buyers. We find evidence that
sellers cater to the biased beliefs of lottery ticket buyers, by
tailoring certain features of their portfolio product.
Specifically, collective lottery portfolio sellers adjust their
sales strategies in response to demand fluctuations that are driven by
buyers' biased beliefs about the correlation of past and future
lottery outcomes. Lottery ticket buyers tend to purchase tickets with
numbers which have not recently won, in accordance with previous
evidence of the Gambler's Fallacy in lottery sales (Clotfelter and
Cook 1993; Terrell 1994). Ticket buyers also tend to purchase tickets
from those ticket sellers who have recently experienced an exogenous
increase in their win rates, in accordance with previous evidence of the
Hot-Hand Fallacy (Croson and Sundali 2005; Guryan and Kearney 2008). (1)
We find combined evidence that lottery portfolio sellers take
advantage of buyers' belief in the Hot-Hand Fallacy in the
following ways: (1) Directly following a previous win in their
portfolio, sellers increase their commission rate charged to the buyers,
to be collected as a percentage of the lottery winnings. Sellers are
thus able to collect more commission money from the buyers conditional
on winning. (2) Sellers offer portfolios of larger value (and more
ticket coverage) directly following a large win in their portfolio.
Using this strategy, sellers increase their chances of holding a winning
ticket in the current round, as well as being able to collect more
commission money conditional on winning. (3) Sellers self-invest less in
their own portfolios after a large win, when their popularity among
buyers in the market is high. By purchasing a lower share of their
portfolios on their own, sellers are able to obtain a larger expected
return without investing their personal money.
We also find some evidence consistent with lottery ticket sellers
adapting to buyers' belief in the Gambler's Fallacy. First,
sellers generally tend to offer lottery number portfolios which are
numerically dissimilar to recent winning tickets, thus appealing to
buyers with Gambler's Fallacy beliefs. Although the more
numerically dissimilar the portfolio is to the previous winning ticket,
the less the seller himself invests in it. Sellers who have proposed
very similar numbers compared to the previous winning ticket, on the
other hand, self-invest more, thus entering the lottery with higher
expected returns under the pari-mutuel prize structure. This
self-investment pattern is not consistent with the alternative story
that sellers themselves subscribe equally to the Gambler's Fallacy
compared to buyers.
Our paper is unique to the existing evidence in this area in that
we are the first, to our knowledge, to study sellers' responses to
consumers' probabilistic belief biases, specifically the
Gambler's Fallacy and Hot-Hand Fallacy. In field data, biased
beliefs are typically difficult to observe and quantify. Clotfelter and
Cook (1993) and Terrell (1994) find evidence for the Gambler's
Fallacy in lottery customers' number choices. De Bondt and Thaler
(1985) and Barber, Odean, and Zhu (2009) find that investors are more
likely to purchase stocks with strong recent performances. In the
horserace-betting market, Griffith (1949), Ali (1977), Asch, Malkiel,
and Quandt (1982), and Hausch, Ziemba, and Rubinstein (1981) find that,
on average, bettors tend to overvalue "long shots" and
undervalue favorites, known as the favorite-long shot bias.
Our study has novelties over the existing evidence on belief biases
in the field. First, in our setting (as in Clotfelter and Cook 1993 and
Terrell 1994), the lottery game probability structure is simple and
transparent. As the probabilities, as well as the information structure
of the lottery game, are transparent and simple, we can be fairly
confident that the demand variations in our data are due to the
non-standard beliefs. Second and importantly, we focus on assessing
sellers' responses to consumers with such beliefs, and the data
allows us to observe how sellers choose to respond to a market of biased
buyers. The market structure in our data has the peer-to-peer feature
that anyone can become a seller, similar to the sellers on eBay,
Craigslist, or other online retail communities. Thus it is very likely
that sellers' observed choices reflect their individual decisions
as online entrepreneurs, in contrast to cases where a firm's
actions may be the result of institutional policies or group-based
decision-making. We find that sellers exhibit best-response tendencies
toward behaviorally biased agents, consistent with increasing their net
expected gains in the lottery market structure.
We also contribute to the small but growing literature empirically
documenting heterogeneity in behavioral biases among agents in markets.
Brown, Camerer, and Lovallo (2012a, 2012b) propose that movie-goers are
limited in their strategic thinking when they decide which movies to
see. The evidence lies in the fact that movie studios consistently earn
greater box-office revenues by withholding low quality movies from
critics (aka a "cold opening"). In a separate paper, Brown,
Camerer, and Lovallo (2012b) estimate a structural level-k model to the
movie industry data to show that less sophisticated consumers can
account for this pattern. For a review of recent advances in structural
models of non-equilibrium strategic behavior, see Crawford, Costa-Gomes,
and Iriberri (2013).
Other recent papers examine different degrees of strategic
sophistication on the seller side (see Goldfarb et al. 2011 for a
survey). In particular, Goldfarb and Xiao (2011) examine the entry
behavior and personal characteristics of telecommunications managers
after market liberalization in the United States in 1996. They find that
CEOs with high quality economics or business training entered markets
with lower competition, and argue that this is evidence that some
managers are better in strategic situations than others. Malmendier and
Shanthikumar (2014) distinguish between strategic and belief-based
motives for inflated security recommendations by examining the
within-analyst empirical relationship between forecasts and
recommendations. They find that analysts affiliated with a firm's
underwriter tend to produce lower forecasts for the firm, which are
easier to outperform, and they subsequently recommend the firm more
strongly to investors.
In a paper quite related to ours, Levitt (2004) finds that
bookmakers in sports betting markets systematically exploit
bettors' biases and achieve higher profits by doing so. A
distinction is that the "home bias" studied in Levitt (2004)
can be fully accounted for using a rational model with correct belief
specifications, while we focus on the lottery structure's ability
to reveal inaccurate beliefs. An additional difference is that while
bookmakers can be fairly considered professionals, our sellers are
non-professional at the task of selling lottery portfolios. Thus, it is
perhaps surprising that our sellers have the ability to generate similar
behavior as professional bookmakers.
We employ a reduced-form approach in estimating the sellers'
responses to buyers' biased beliefs, focusing primarily on the
direction of sellers' responses rather than the magnitude. The
advantage of this approach is that it allows us to straightforwardly
test whether sellers on average significantly respond to the biased
beliefs of buyers. The disadvantage is that we cannot very precisely
estimate sellers' magnitude of response, as we do not attempt to
specify or estimate sellers' objective function. Indeed, we find
the reduced-form approach most appropriate for our setting, given that
the online marketplace and the motivations of participants in it, may be
quite complex to model. While a structural approach also has merits, it
may be substantially more difficult, while possibly not yielding
substantially different insights compared to a simple reduced-form
analysis.
The remainder of the paper proceeds as follows: Section II
describes the background of the lottery game and the peer-to-peer
market, including the data; Section III briefly describes facts about
lottery buyers' purchase behavior; Section IV describes our
hypotheses for sellers' responses to buyers with belief biases;
Section V presents the empirical results; Section VI concludes and
discusses directions for future work.
II. FIELD SETTING
A. The SSQ Lottery Game
SSQ lottery is one of the biggest Chinese national lottery games in
China. The gaming rules of the SSQ are similar to those of other popular
lotteries, such as Powerball in the United States and LottoMax in
Canada. SSQ stands for (Shuang Se Qiu), which means dual-colored balls
in Chinese. An individual ticket is sold for 2 Yuan (about $0.32 USD).
It requires players to pick numbers from two groups of numbers. In the
first group, players need to pick six numbers from the range 1 to 33,
which are called the red numbers. In the second group, players need to
pick one number from the range 1 to 16, which is called the blue number.
To win first prize, the player needs to match all seven numbers randomly
drawn as the winning number combination.
The SSQ has six levels of prizes, which are shown in Table 1. The
first prize is shown in the first row, and requires matching all the
drawn numbers, red and blue. The second prize requires the matching of
all six red numbers except for the blue one. The first and second prizes
are parimutuel as the final reward depends on the number of winners and
the prize pool for each payout, whereas the third to sixth prizes are
non-parimutuel fixed prizes.
B. The Online Lottery Betting Game
Taobao (sometimes referred to as "China's Amazon")
is the largest online shopping service provider in China. In 2010,
Taobao had over 370 million registered customers and generated over 400
billion Yuan in sales (over 60 billion U.S. dollars) with an annual
growth rate of over 100%. Besides general e-business, Taobao also
provides a platform called the "Taobao Lottery" for online
lottery gambling in China, and one of its main services is to sell the
SSQ tickets online such that potential lottery buyers can purchase SSQ
tickets together
Any registered Taobao user can independently purchase a lottery
ticket from the Taobao Lottery store online. However, the site is more
than just an online lottery store, also providing a platform for group
or "collective lottery purchases" in a peer-to-peer market. As
the leading online retailer for consumer goods in China, the traffic and
transactions volume on the Taobao website is likely to be among the
world's largest.
In many lottery games, it is common for family, friends, or
co-workers to pool their money together to buy a certain number of
lottery tickets. In the event of a win, those who have contributed money
to the pool receive a share of the winnings. Recent studies show that
collective lottery purchases are common in many countries. Guillen,
Garvia, and Santana (2012) found that 12% of U.S. players, 22% of
players in the United Kingdom, and 33% of players in Spain collectively
purchase lottery tickets on a regular basis. Humphreys and Perez (2013)
found that in Spain sociological factors such as employment and gender
significantly predict participation in collective lottery purchase.
The market in Taobao Lottery is a formalized version of the usual
collective betting arrangement, with institutional rules set and
enforced by the website. Collective bettors do not need to know one
another, and they do not need to know the seller. Similar to other
online peer-to-peer retail websites such as eBay or Craigslist,
users' real identities are kept confidential. The online lottery
marketplace automates the necessary transactions so that trust between
participants is not a requirement for the transaction. There are a few
possible reasons why an individual would purchase lottery tickets from
the online market. Perhaps the most obvious reason is risk-sharing, as
in the traditional collective betting arrangement. Consumers wish to
"invest" in a larger set of number combinations, without
having to fully use their own money to do so. The online marketplace
provides a formal platform for them to do so, and a key motivation is
that they do not need to find interested betting partners and coordinate
with those bettors on their own. The anonymity and automated structure
of the marketplace provides an additional attractive draw for consumers.
If a win does occur, consumers in the online marketplace do not need to
arrange the sharing of winnings with their betting partners on their
own--the exchange is reduced to a transaction, without any personal
complications in the arrangement. The motivation for seller
participation is similar, but with the additional draw that they can
earn commissions from buyers when a win occurs.
We now describe the market structure and transaction procedure. In
the marketplace, there are low barriers to entry to becoming a seller.
The only prerequisite is having a Taobao account. A lottery portfolio is
a collection of lottery numbers chosen by a Taobao user (the
"seller"). The seller announces the number of shares in a
package and the corresponding price per share (Total Cost/Total Shares).
For example, the seller can propose a package which consists of two
lottery tickets, such that the total cost is 2 Yuan; 1 Yuan for each
ticket. Meanwhile, this same user can divide this package into 100
shares, so that more people can participate in the package. Therefore,
there will be 100 shares for this lottery package and each share will be
worth 0.02 Yuan. In this example, at most 100 lottery gamblers can take
part by purchasing shares in this portfolio.
[FIGURE 1 OMITTED]
Besides the lottery number selection, total portfolio size, total
shares, and price per share, the seller also has to reveal the number of
shares self-purchased (i.e., the seller's own investment in this
lottery portfolio). The seller also states a commission fee: the
percentage of the total winning prize that the seller collects before
the prize is divided among the investors according to their shares in
this package. For example, suppose that a lottery package wins a total
payout of 100 Yuan and the commission fee of the seller is 3%. Before
the others share the prize, the proposer will first receive 3 Yuan of
the winnings, leaving 97 Yuan to be shared among the rest of the
shareholders. Figure 1 illustrates how the Taobao collective lottery
purchase system works.
Taobao Lottery also requires that sellers self-invest either at
least 1 % of the shares in their own proposed portfolio, or the same
percentage as the commission rate they charge buyers, whichever is
larger. This ensures that sellers have some of their own money at stake
in the portfolios that they are offering on the market. Sellers can
self-invest on up to 100% of the portfolio, where investing 100% is
equivalent to a solo purchase. Sellers are also allowed to propose more
than one portfolio in each round. In practice there are about 10,000
sellers participating in each lottery round, with the large majority of
them posting a single portfolio.
Portfolios can be open for sale directly after the previous round
of the SSQ lottery is over. Once a portfolio is posted online, the
seller cannot change it. Similarly once buyers purchase shares, they
cannot request a refund. Selling of shares closes 3 hours before the
actual SSQ lottery drawing. If by that time, a seller's portfolio
is not 100% sold out, the portfolio is canceled and all investors have
their money automatically refunded to their account.
Once a portfolio is sold out, Taobao receives the money invested in
it. Taobao then purchases the tickets on behalf of the seller, and is
responsible for distributing the winnings among investors in the event
of a win. (2) Taobao thus eliminates any concerns from buyers about the
trustworthiness or reliability of the seller.
III. BELIEF BIASES OF BUYERS
We focus on two main belief biases of buyers in the lottery market:
the Gambler's Fallacy and the Hot-Hand Fallacy. Both fallacies have
been previously documented independently in the field among lottery
players. Clotfelter and Cook (1993) and Terrell (1994) documented the
Gambler's Fallacy in number picks for fixed prize lotteries and
pari-mutuel lotteries, respectively. Specifically, they found that
players were unlikely to choose a particular number in their ticket if
that number had recently been drawn on a winning ticket. Guryan and
Kearney (2008) found that lottery retailers that had recently sold a
jackpot winning ticket experienced a growth in lottery sales immediately
afterwards, consistent with the predictions of a Hot-Hand Fallacy, or
what they describe as a "Lucky Store Effect."
These biases of lottery players have been previously documented in
the literature and can be taken as given in our analysis of seller
behavior. To confirm the validity of the belief biases, we also find
strong empirical support for the Gambler's and Hot-Hand Fallacies
by buyers in our data. The detailed results for the Taobao Lottery
market are reported in Lien, Yuan, and Zheng (2014), which focuses on
buyers' beliefs, including the theoretical relationship between the
HotHand Fallacy and the Lucky Store Effect. While the patterns that we
document here classify technically as a Lucky Store Effect (a seller is
more attractive to buyers after just one "win"), for
simplicity, we refer to it throughout this paper as the Hot-Hand
Fallacy. (3) We have discussed only the behavior of buyers briefly, as
our primary objective in the current paper is to study sellers'
responses to the buyers. To introduce these buyer biases, we first
explain our main measures of the Gambler's Fallacy and Hot-Hand
Fallacy.
A. Buyers' Susceptibility to the Gambler's Fallacy
As documented by the previous literature, lottery players tend to
avoid picking numbers which have won in the preceding round. The
Gambler's Fallacy says that they do this because they believe those
numbers are unlikely to be drawn again soon, contrary to the independent
nature of the draws in each round. To measure the similarity of a
lottery portfolio to the winning lottery ticket, we create a Similarity
Index which summarizes how similar or different a chosen ticket is to
the most recent previously winning ticket. Our approach differs slightly
from Clotfelter and Cook (1993) and Terrell (1994) because we need to
consider entire tickets or portfolios of tickets offered by the seller
as the objects of comparison, rather than individual numbers on a
ticket. To accommodate this difference, we use an indexing approach.
Each lottery ticket in SSQ consists of seven numbers with six red
balls drawn without replacement from the integers in the range [1,33],
plus one blue ball drawn from the range of integers [1,16]. We use the
following notation to denote a lottery number combination for a single
ticket i, where r denotes red numbers and b denotes the blue number:
[Ticket.sub.i] = {[r.sub.1], [r.sub.2], [r.sub.3], [r.sub.4],
[r.sub.5], [r.sub.6]| [b.sub.1]} .
Let the winning numbers in lottery round t be denoted by the vector
w:
{[w.sup.t.sub.1], [w.sup.t.sub.2], [w.sup.t.sub.2],
[w.sup.t.sub.3], [w.sup.t.sub.4], [w.sup.t.sub.5], [w.sup.t.sub.6]|
[w.sup.t.sub.7]}.
Let I(x) be an indicator function. If the event in parentheses is
true, the value is one; and the value is zero otherwise.
Then, we define [S_Index.sub.i,t] as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The Similarity Index consists of two components. One component is
the similarity measure on the red numbers chosen, in which we count the
number of matched number picks between the previous winning ticket and a
seller's ticket, where the ordering of numbers does not matter. The
second component is the similarity on the blue number, which is merely
an indicator for whether the blue number on the seller's ticket
matches the previous winning blue number. We use equal weighting on
these two components. (4)
As an example, if the chosen blue number [b.sub.7] is the same as
the winning blue number [w.sup.t.sub.7], in round t, by the definition
above, this similarity contributes to one half of the total index.
Second, on the red ball division, according to the above definition, if
any of {[w.sup.t.sub.1], [w.sup.t.sub.2], [w.sup.t.sub.3],
[w.sup.t.sub.4], [w.sup.t.sub.5], [w.sup.t.sub.1]} shows up in the
ticket {[r.sub.1], [r.sub.2], [r.sub.3], [r.sub.4], [r.sub.5],
[r.sub.6]}, that number contributes 1/12 [= (1/2) x (1/6)]) to the total
index. It is clear that [S_Index.sub.i,t] for the hypothetical purchased
lottery ticket [Ticket.sub.i] = {[w.sup.t.sub.1], [w.sup.t.sub.2],
[w.sup.t.sub.3], [w.sup.t.sub.4], [w.sup.t.sub.5], [w.sup.t.sub.6]|
[w.sup.t.sub.7]} is equal to 1.
When a lottery portfolio consists of several different lottery
tickets, we take the average of the Similarity Index across tickets in
the portfolio as the portfolio's Similarity Index. (5)
Buyer's Gambler's Fallacy: Buyers in the online lottery
market are less likely to purchase lottery portfolios when the portfolio
numbers are more similar to the winning ticket in the previous round.
B. Buyers' Susceptibility to the Hot-Hand Fallacy
To detect the Hot-Hand Fallacy, we require a measure of lottery
sellers' success rates. According to the literature, players will
gravitate toward sellers who have previously won, because they believe
that those sellers are particularly lucky (i.e., somehow having a higher
theoretical return rate than other sellers).
In Taobao Lottery, buyers can observe each seller's
information such as the total wager and past performance, as well as
package-specific information, such as total shares, share price,
commission rate, and so forth. However, note that the return rate of the
seller in any period is a transient shock, which cannot predict any
future performance of the seller. Therefore, if lottery players'
beliefs are rational, their purchase behavior should not be affected by
sellers' past wins or current return rate. The return rate of a
seller j in the previous round is defined as follows:
[WinRate.sub.j] = (Total [Win.sub.j]/Total [Wager.sub.j])
where [TotalWin.sub.j] the total amount of money won by seller j in
the previous round and [TotalWager.sub.j] is the total lottery
investment by seller j in the previous round.
Buyer's Hot-Hand Fallacy: Buyers in the online lottery market
are more likely to purchase lottery shares from sellers who have won in
the previous round.
C. Data
Our data is based on 4,529,730 observations over 25 rounds of SSQ
lottery games. We observe each seller's portfolio on the market,
including the exact numbers chosen for each of these rounds. We also
observe several other variables important for our analysis, displayed in
Table 2.
The sample summary statistics for these and other variables are
shown in Table 3. In each round over 2 million lottery portfolios are
put on the market by over 41,000 sellers. The mean value for win rate of
sellers is 58% for the whole sample and around 23% excluding the single
jackpot win in our data, which is consistent with the theoretical
expected rate of the lottery game. For details on the calculation of the
prize odds and structure in the SSQ lottery, we refer the reader to Yuan
and Gao (2014). The summary statistics show that portfolio size and
number of tickets included per portfolio vary widely. There is also
substantial variation in self-investment behavior and commissions
charged. Approximately 77% of lottery portfolios in our data
successfully sell out.
Before our regression analysis, we want to first examine the
prediction power of the past WinRate on the future WinRate.
Theoretically, as WinRate is a random shock, there should be no serial
correlation present. However, to test the validity of this claim, we
check whether past lottery performance has any statistically significant
or economically significant impact on future performance. Therefore, we
regress the current WinRate on the past four rounds of WinRate and the
regression results are shown in Table 4. We can see that the [R.sup.2]
is zero and the coefficients are all insignificant and extremely small,
which confirms the independent nature of lottery draws and further
provides the foundation for the following empirical analysis.
D. Empirical Evidence for the Hot-Hand Fallacy and Gambler's
Fallacy of Buyers
The results in this section providing empirical evidence for
buyers' belief fallacies are borrowed from Lien, Yuan, and Zheng
(2014), which discusses buyers' beliefs in greater detail. We use a
standard Tobit model with a cutoff value of 100% to analyze how lottery
players choose lottery portfolios given the return rate (WinRate) of the
sellers and the Similarity Index. We focus on the fraction of the
lottery portfolio which has been purchased at the time the market
closes. This serves as a proxy for buyers' preferences over the
lottery portfolios in the market among buyers.
The regression model for buyers' behavior is as follows:
[PROGRESS.sup.*] = [[beta].sub.0] + [[beta].sub.1] WinRate
+ [[beta].sub.2] Similarity Index + [[beta].sub.3] COMMISSION
+ [[beta].sub.4] SIZE + [[beta].sub.5] SHARES + [[beta].sub.6]
PRICE
+ [[beta].sub.7] SELFBUY + [[beta].sub.8] TIMEEXPOSE + [epsilon]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where PROGRESS represents the sales progress of a portfolio, in
other words the proportion of shares in this portfolio that are sold at
the time the market closes. PROGRESS* is the latent variable which we
cannot observe, due to the fact that once a portfolio is completely sold
out, no further shares can be sold. WinRate and S_Index are the
variables defined above. COMMISSION is the seller's chosen
commission fee, SIZE is the total amount of money to be collected if the
lottery portfolio is completely sold out, SHARES represents the total
number of shares in the portfolio, PRICE is the value for each single
share, SELFBUY measures the percentage of the shares purchased by the
seller himself and TIMEEXPOSE measures how early the portfolio is put up
online for sale. These variables are controlled for in the regression
since buyers' decisions may also depend on these features. (6) In
the empirical analysis, including all the subsequent regressions on the
sellers' behavior, we also include round dummy variables to control
for roundspecific factors.
In the Taobao online lottery data, buyers are significantly more
attracted to ticket sellers who have experienced an increase in their
WinRate in the previous round, all else equal, in accordance with the
Hot-Hand Fallacy. Buyers are significantly more attracted to lottery
portfolios which have a low Similarity Index, and avoid lottery
portfolios which have high similarity to the previous winning ticket; in
accordance with the Gambler's Fallacy.
As the focus of the current paper is on seller behavior, the basic
empirical results of this regression are relegated to Tables S1 and S2,
Supporting Information, which show the results of the Tobit
specification discussed in Section III.C, taken from Lien, Yuan, and
Zheng (2014). The objective is to test whether the coefficient on
WinRate is significantly positive, consistent with the Hot-Hand Fallacy;
and whether the coefficient on Similarity Index is significantly
negative, consistent with the Gambler's Fallacy. We see that across
specifications including several control variables, the Hot Hand and
Gambler's Fallacies remain robust. Table S2 checks the robustness
of these results under a Probit specification. (7) As the regression
results in the Tables S1 and S2 show, a high WinRate is significantly
positively associated with higher sales progress of a portfolio, and a
high Similarity Index is significantly negatively associated with sales
progress. As both WinRate and Similarity Index are independent of the
current lottery round's performance, the differences in sales
success of these portfolios can be attributed to buyer's biased
beliefs. We note that the coefficients on portfolio features in the
buyer regression are in the directions as expected. The identification
of the belief fallacies from the buyers' side provides the
foundation for us to further study the sellers' behavior.
IV. SELLERS' RESPONSES
Sellers in the market must make several choices in putting their
portfolios online for sale. In each round, the winning lottery numbers
and the winning return rates of the sellers in the previous round will
shift their decisions. They must decide on their commissions to be
earned in the event of a win, the portfolio size and number of shares
(thus jointly determining the share price), how much to self-invest in
their own portfolio, and the actual numbers in their lottery portfolio.
In a competitive market such as this one, with no barriers to
entry, if all participants in the market had unbiased beliefs about
probabilities, none of these choices should make any difference in the
sales success of sellers. We hypothesize in accordance with a rational
model of seller behavior, that sellers will respond to the
aforementioned buyer beliefs in their decisions about portfolio feature,
in specific ways in order to increase their expected profits.
Each of the seller's choices can alter expected profits in the
following ways, holding all other factors and choices constant.
1. Commissions are, by definition, the proportion of the
buyers' share of the winnings that are paid to the seller.
Holding all else constant, increasing the commission rate increases
the seller's revenue and profits.
2. The portfolio size is simply how much money has been invested in
the lottery in the seller's portfolio.
Holding all else constant, increasing the portfolio size increases
the expected profits of a seller through the expected commission money
on his customers' shares, and through the increased probability of
holding a winning ticket (assuming that increasing portfolio size also
results in greater purchase of unique tickets).
3. Self-investment is the proportion of a seller's own lottery
portfolio that he himself purchases. (8)
Holding all else constant, a seller can increase his expected
profits by reducing his self-investment. This is because the expected
commissions earned from his customers' purchased lottery shares are
a marginal cost-free method of increasing expected revenues--sellers can
earn an extra expected profit without putting any additional of their
own money into the portfolio.
With these three ways of increasing expected profits in mind,
Subsections IV.A and IV.B specify precisely how we expect sellers to
take advantage of the Hot-Hand and Gambler's Fallacies of buyers to
increase their expected profits. The key is that sellers choose
portfolio features to take advantage of their level of popularity on the
demand side, in a manner consistent with responding to the belief
fallacies held by buyers.
A. Response to the Hot-Hand Fallacy
Recall that a lottery ticket buyer who subscribes to the Hot-Hand
Fallacy will tend to buy from a seller who has won a large prize in the
previous round in the lottery game, as he or she believes that such a
seller stands a higher chance of winning this round than other sellers
who have not won.
This change in the buyers' demand over ticket sellers'
services is induced by the randomness of the previous lottery
round's outcome. Faced with an increase in willing customers, we
expect the sellers to adopt the following strategies, directly after
experiencing a large lottery win in the previous round, in order to take
advantage of this anticipated increase in popularity of their own ticket
portfolio.
Hot-Hand Response Hypothesis: Sellers charge higher commission
rates, offer larger portfolio size, and self-invest less directly
following a large win.
B. Response to the Gambler's Fallacy
Recall that a lottery ticket buyer who subscribes to the
Gambler's Fallacy tends to buy from sellers whose portfolios offer
numerically dissimilar lottery tickets compared to the previous winning
ticket, as he or she believes that the numbers on the previous winning
ticket are less likely to appear again in this time's winning
ticket.
We wish to point out a key difference between exploitation of the
Hot-Hand Fallacy, and exploitation of the Gambler's Fallacy in this
lottery market setting. In the case of the Hot-Hand Fallacy,
sellers' ability to take advantage of it depends on sellers'
attractiveness to the lottery buyers, which by the SSQ lottery
procedure, is beyond sellers' control. In the case of the
Gambler's Fallacy, however, sellers can choose which lottery
tickets to include in their portfolio.
Thus, in addition to the three variables sellers can determine in
the case of reacting to the Hot-Hand Fallacy (commission, portfolio
size, self-investment), sellers can additionally decide what kind of
numbers to include in their portfolio, conditional on the previous
winning ticket numbers. We hypothesize that sellers will tend to offer
lottery portfolios which are dissimilar to the recent winning tickets.
The other three portfolio features chosen by the seller are hypothesized
to follow the same direction as the predicted reaction to the Hot-Hand
Fallacy, the effects increasing in the degree to which a portfolio
caters to Gambler's Fallacy beliefs, by avoiding recent winning
numbers.
Gambler's Fallacy Response Hypothesis: Sellers tend to offer
lottery portfolios which are numerically dissimilar to the winning
tickets in the previous round. They tend to charge a higher commission,
offer larger portfolio sizes, and self-invest less on these
Gambler's Fallacy targeted portfolios
We now take these hypotheses to the data, to test whether holding
all else equal, sellers adopt these specific strategies after the
results generated by previous lottery outcomes, when faced with a
population of buyers holding biased beliefs.
V. RESULTS
We use an ordinary least squares (OLS) regression to test the
Hot-Hand and Gambler's Fallacy Response Hypotheses, with the
exception of the hypothesis about sellers' lottery number choices,
which we test by tabulating the popularity of previously winning lottery
numbers in the market.
In each of the OLS regressions, we assume that sellers have
rational expectations that buyers' belief fallacies are strongest
with respect to the most recent lottery that has occurred. There may be
time dynamics in terms of biased beliefs with respect to lottery
outcomes in a more distant past, but our analysis treats these as noise.
A key feature in the regressions is that in the lottery market,
wins are randomly assigned because the lottery is a game of pure chance.
This random assignment of wins is implemented across the sample of
sellers in our data, and the portfolio characteristics they selected.
Thus, the coefficient on the winning indicator variable can be
reasonably interpreted as a "causal" effect on the portfolio
characteristics, although the mechanism for the causality is not evident
from the OLS regression results alone. This requires the additional
evidence on buyer's responses to lottery outcomes in the
marketplace, which can be found in Tables S1 and S2 (and is consistent
with previous literature on lottery player biases in the non-portfolio
setting).
Our primary objective in the regressions is to check whether
lottery market events which would induce biased beliefs by buyers, are
systematically associated with adjustments made by sellers to their
portfolios. We do not attempt to account for the bulk of variation in
sellers' decisions on these portfolio variable choices here.
However, since we observe several features of the sellers'
portfolio, we do include these features as control variables wherever
possible, to ensure our results are robust to variation in these other
variables.
A. Hot-Hand Response Results
Hot Hand Response Hypothesis: Sellers charge higher commission
rates, offer larger portfolio size, and self-invest less directly
following a large win.
We begin with the part of the hypothesis on sellers'
commission rates. We regress sellers' current commission rates on
an indicator variable for whether his or her previous lottery
round's rate of return exceeded 200%. Portfolio-specific traits
such as portfolio size, self-investment ratio of the seller, shares
offered, price, and similarity index are included as control variables
to ensure that the relationship between commission and large wins is
robust to these other potentially influencing portfolio features.
As the results in Table 5 show, a "big win" in the
previous period is associated with a significantly higher commission
rate set the current period, of magnitude between 0.56 and 0.66 percent.
Accounting for heterogeneity in the portfolio control variables
increases the size of the effect. Another way to interpret it is that if
a seller's winning rate increases from 0 to 200%, which is highly
possible given the huge standard deviation of the winning rate shown in
the Table 3, he will increase his commission by around 0.5% (in absolute
value) thanks to the biased belief of the buyers, even though his
winning rate is pure luck. We also consider a threshold win rate value
of 100% (such that a buyer would earn back his investment if such a win
rate were to be realized again), and a specification containing the
continuous variable WinRate instead of a threshold variable. The results
are robust to these alternative specifications, and the robustness check
results are provided in Tables S3 and S4.
We now turn to the part of the hypothesis on sellers'
portfolio size, or the total monetary value of the portfolio offered, in
response to the HotHand Fallacy. To test this hypothesis, we regress the
monetary value of the portfolio offered on the indicator variable for
whether the previous round's rate of return exceeded 200%. We
include commission rates, self-investment ratio of the seller, price,
and similarity index as control variables, as before. As Table 6 shows,
large wins in the immediately previous round are associated with
significantly larger monetary portfolios in the next round.
As in the regressions in Table 5, the assignment of large wins is
randomly assigned. The relationship between large wins and seller
response can be interpreted as: large wins induce sellers to increase
their volume of tickets sold in the next round. To be precise, if the
winning rate increases by 200%, the seller will increase the size of the
lottery package by around 50 RMB. As all portfolios must sell out in
order to be implemented, sellers must believe that more customers will
be willing to purchase their shares this period, after they have won
previously. Thus sellers' attempt to sell higher volumes of tickets
is a response to customers' revealed incorrect belief that those
sellers who won last time are more likely to win again this time.
Finally, we turn to the part of the hypothesis proposing that
sellers will self-invest less in their own portfolio after experiencing
a win in the previous round. Table 6 shows the results of regressing the
self-investment ratio on the indicator for a previous large win, and the
control characteristics of portfolio size, shares, price and Similarity
Index. All else equal, sellers also invest less in their own portfolio
after experiencing a large win, as seen from the coefficient estimates
for BigWin Indicator. To be precise, if the winning rate increases by
200%, the seller will decrease his self-investment ratio by around 2%.
These results are quite significant and robust to the inclusion of other
control variables in the regression.
B. Gambler's Fallacy Response Results
To test sellers' responses to the Gambler's Fallacy, we
first address the issue of portfolio selection. Recall that sellers
choose their own numbers for each ticket in the portfolio, after the
previous round of winning numbers has been realized. The Similarity
Index, our measure of (un)attractiveness of a portfolio, is thus a
choice made by sellers, not an exogenous outcome as in the case of
WinRate under the Hot-Hand Fallacy. To test the Gambler's Fallacy
Response Hypothesis, we first examine trends in number selection
choices, then regress the sellers' other choice variables on the
Similarity Index of the portfolio, along with other portfolio control
variables.
We conduct a simple aggregate analysis to determine how sellers, on
average, react to previous winning numbers in their choice of numbers.
Suppose in round 0, there is a winning lottery ticket. Figure 2 plots
the trend of Similarity [Index.sub.i,0], or how similar the lottery
ticket i in round t is to the winning number combination in round 0. The
horizontal axis plots rounds from 10 rounds prior to the winning ticket
to 15 rounds after the winning ticket. The vertical axis represents the
average Similarity Index in the SSQ lottery market at each time period.
[FIGURE 2 OMITTED]
Figure 2 shows clearly that prior to the revelation of the winning
numbers at time 0, the Similarity Index is quite stable around the 0.12,
which is equivalent to the average Similarity Index when numbers are
completely randomly drawn. When the winning numbers are revealed to the
public at time 0, we observe a sharp drop in the Similarity Index of
lottery portfolios in the market, meaning that sellers, on average,
began offering portfolios which were less similar to the winning ticket
from time 0 onward. Eventually as time goes on, the Similarity Index
value returns to the original average level. This closely mirrors the
findings of Clotfelter and Cook (1993) in their analysis of time trends
in number picks for the Maryland Pick-3 game.
We now examine the relationship between commission rates and
portfolio Similarity Index in Table 7. This part of the Gambler's
Fallacy Response Hypothesis is not strongly confirmed in the data, as
seen by the coefficients and standard errors in the top row (Similarity
Index). In the first row of Table 7, we can see that the coefficients
are in the predicted direction (low similarity, higher commissions), but
are not statistically significant in any of the specifications. In other
words, sellers do not seem to strongly manipulate their commissions
based on how similar the portfolios are to the previous winning ticket.
This result is in contrast to the Hot-Hand Response result, where
sellers do increase the commission when they have just won in the
previous round.
One interpretation is that although buyers do not like similar
lottery numbers, their willingness to pay for dissimilar numbers is much
smaller than their willingness to pay for a store with a higher previous
winning rate. In other words, buyers may not be willing to pay a higher
price (i.e., the commission) for this easily manufactured portfolio
feature, and so the commission response from the seller side is weaker
here.
We now turn to the relationship between portfolio size and
Similarity Index. As Table 8 shows, we find some tentative evidence that
sellers attempt to sell more tickets for portfolios catering to the
Gambler's Fallacy belief, but not as precisely as in the Hot-Hand
response result. The Gambler's Fallacy Response Hypothesis suggests
that the coefficient on Similarity Index should be significantly
negative, as sellers offering very dissimilar portfolios may try to
exploit their popularity by offering larger portfolios in the market.
The coefficients are in the predicted directions, but are at the margin
of 10% significance. We take this as evidence that sellers generally do
exhibit this tendency, but not as sharply as the tendency to take
advantage of the Hot-Hand Fallacy using the portfolio size variable.
Finally, we turn to the last part of the Gambler's Fallacy
Response Hypothesis--that sellers will self-invest less on portfolios
catering to the Gambler's Fallacy. We in fact see this hypothesis
strongly confirmed in the data. The regression is identical to that used
to test the self-investment portion of the Hot-Hand Response Hypothesis
in Table 9. However, this time we focus attention on the coefficients
for Similarity Index. The regression shows that more similar portfolios
do have significantly higher rates of seller self-investment.
One way to understand this result is through the cost-free increase
in expected profits, which sellers can obtain by leaving a greater
fraction of their portfolios (portfolio size held fixed) for buyers to
purchase. By self investing less in the popular (i.e., less similar)
portfolios, sellers can achieve this objective. At the same time,
self-investing more in the less popular (i.e., more similar) portfolios
gives sellers a chance to exploit the pari-mutuel prize structure of the
lottery, as less popular number combinations will yield a higher prize
conditional on winning. These two forces likely reinforce one another to
explain the positive relationship between portfolio Similarity Index and
seller self-investment.
VI. CONCLUSION
In this paper, we examined seller behavior in the Chinese national
lottery market, a marketplace where buyers have predictably biased
beliefs. Specifically, lottery ticket buyers subscribe to the Hot-Hand
Fallacy, tending to buy tickets from sellers who have sold winning
tickets in the previous round; buyers also subscribe to the
Gambler's Fallacy, tending to buy ticket portfolios which are
numerically dissimilar to winning numbers in the previous round. These
two belief fallacies on the part of buyers, make their behavior
predictable to lottery portfolio sellers, and we show that sellers
respond to this behavior by tailoring the features of their lottery
ticket portfolios in order to increase their expected profits. We
provide the first evidence, to our knowledge, of sellers' responses
to these belief biases by consumers.
We find evidence which is consistent with the following claim:
Sellers respond to their increase in popularity resulting from
buyers' belief fallacies in three main ways which increase their
expected profits, holding market conditions and sellers' other
choices constant: (1) By setting commissions high following a winning
lottery outcome, to take advantage of buyers' Hot-Hand Fallacy; (2)
by selling a higher volume of tickets as evidenced by their opening of a
larger portfolio size, following a winning lottery outcomes, to take
advantage of buyers' Hot-Hand Fallacy; (3) by self-investing less,
in order to gain an increase in expected revenue at zero cost (all else
constant). We find that sellers implement this self-investment strategy
both in the case of an exogenous previous win, and in the case of
choosing to open a dissimilar (i.e., more popular) portfolio. This
provides solid evidence that sellers tend to adjust their
self-investment in response to both the Hot-Hand and the Gambler's
Fallacies.
We find weaker evidence in our data, that sellers adjust their
portfolio size and commissions in response to the market popularity
generated by opening dissimilar ticket portfolios. One interpretation of
these weaker results is that although buyers do not like similar lottery
numbers, their willingness to pay for dissimilar numbers is much weaker
than their willingness to pay for placing their money with a previously
"lucky" seller. This may be related to the fact that choosing
a dissimilar portfolio is a relatively easy task, while obtaining a
previous win in the lottery game is difficult. Sellers may anticipate
that buyers will not be as drawn to a numerically attractive portfolio
as they are to a previously winning seller, and thus adjust their
portfolios less, accordingly.
A key advantage of the SSQ lottery market we examine in studying
these behaviors, is that there is no informational or skill advantage,
as each lottery round is independent of all previous outcomes. Thus the
Hot-Hand and Gambler's Fallacies of buyers can be well-identified
as inaccurate beliefs. Without these biased beliefs of buyers, sellers
would have little incentive to choose any of their choice variables
(i.e., lottery number picks, commission rate, portfolio size,
self-investment) systematically in response to previous outcomes. Yet,
we find that sellers set these variables systematically, in ways
consistent with increasing their returns subject to the buyers'
beliefs, implying some degree of awareness of these biased beliefs in
the marketplace.
We anticipate several directions for future research. First, in
this paper we have presented reduced-form evidence for our main
hypotheses about seller reactions to biased believers. Another approach
may be a structural one which attempts to specify the sellers' (and
buyers') objective functions and estimates the relevant parameters.
We have also limited our analysis to seller behavior in the aggregate.
As aggregate level analyses have limitations in studying individual
behavioral patterns, future work may examine sellers' behavior at
the individual level. While our paper focuses on behavior in the lottery
market setting, it would be useful to explore using other sources of
field data, the degree to which the patterns of seller behavior detected
carry over to other settings. Finally, it would be meaningful to explore
whether sellers exploit buyers' other types of biased beliefs such
as (for example) a belief in lucky numbers, in similar ways.
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SUPPORTING INFORMATION
Additional Supporting Information may be found in the online
version of this article:
TABLE S1. Dependent Variable: Sales Progress
TABLE S2. Dependent Variable: Sold Out Indicator
TABLE S3. Dependent Variable: Commission
TABLE S4. Dependent Variable: Commission
JAIMIE W. LIEN and JIA YUAN *
* We thank Vincent Crawford, Brad Humphreys, Zhigang Li, Guang-zhen
Sun, and Jie Zheng for helpful comments. We also thank our editor,
Robert Rosenman, and three anonymous referees for comments which helped
improve the paper. All errors are our own. Jaimie W. Lien acknowledges
financial support from the National Science Foundation of China
(#71303127 and #71203112), the Ministry of Education, China
(#20130002120030), and Tsinghua University (Project #2012z02182 and
#2012z02181). Jia Yuan acknowledges financial support from the
University of Macau (#MYRG046-FBA12-YJ).
Lien: Department of Decision Sciences and Managerial Economics, The
Chinese University of Hong Kong, Hong Kong; Department of Economics,
School of Economics and Management, Tsinghua University, Beijing, China.
E-mail jaimie.lien.tsinghuasem@gmail.com
Yuan: Department of Business Economics, Faculty of Business
Administration, University of Macau, Macau. E-mail jiayuan@umac.mo
doi: 10.1111/ecin.12198
Online Early publication March 5, 2015
(1.) The biased beliefs of buyers in our specific collective
lottery portfolio market are examined in detail in Lien, Yuan, and Zheng
(2014), and we also provided a brief summary of their results in the
current paper as a background for understanding sellers' responses.
(2.) For providing this service Taobao receives a commission from
the official lottery authority, and they have the additional benefit of
advance cash flow when the lottery investment is made.
(3.) Specifically, Lien, Yuan, and Zheng (2014) show that a
cross-sectional extension of the representativeness bias framework
presented in Rabin (2002) and Rabin and Vayanos (2010), can generate the
Lucky Store Effect after just a sin gle lottery win by a seller, rather
than over streaks of wins as assumed by the Hot-Hand Fallacy. They
suggest that when considering a class of belief fallacies in which
decision-makers believe that "winners in the past will win
again," the Lucky Store Effect can be considered a special case of
a generalized Hot-Hand Fallacy.
(4.) We believe this is a reasonable model of players'
impression of the relative importance of red and blue numbers due to the
format of the SSQ game which highlights this point. Our results are
robust to different weightings of the red and blue ball components.
(5.) The Similarity Index may not be the perfect indicator in the
sense that if difference among lottery packages is too tiny, some people
may not be sensitive to small differences in the index. However, it is
able to capture the difference if several winning numbers are involved.
(6.) In the robustness check, we further add the jackpot size and
the number of winning tickets in the previous round in the regressions,
and the results remain robust.
(7.) To check the robustness of the regression results, we also
implement a Probit model. These results can be found in Table 2 in
Tables SI and S2. The regression coefficients are of the same sign
direction as the Tobit model, which suggests that the regression results
are robust.
(8.) Recall that by the lottery market regulations, the seller must
self-invest in no less than 1 % of his own portfolio, and at least as
high a percentage as the commission rate he charges.
TABLE 1
SSQ Prize Policies
Winning Conditions
Number of
Red Balls Blue
Matched Ball
Award Level (out of 6) Matched? Prize Distribution
First prize 6 Yes If the rollover money from the
last jackpot is less than 100
million RMB, then the grand
prize jackpot winners will
split the rollover from the
previous draw and the 70% from
the "high prize pool." If the
prize is more than 5 million
RMB, each winning ticket will
only be worth 5 million RMB.
If the rollover money from the
last jackpot is at least 100
million RMB or more, there is
a two-part prize package. The
winners split the rollover
money from the previous draw
and 50% from the "high prize
pool," as well as 20% from the
"high prize pool." With each
prize, a maximum of 5 million
RMB is paid (total of 10
million RMB).
Second prize 6 No To split the 30% of "high
prize pool"
Third prize 5 Yes Fixed amount of 3,000 Yuan per
winning lottery ticket
Fourth prize 5 No Fixed amount of 200 Yuan per
4 Yes winning lottery ticket
Fifth prize 4 No Fixed amount of 10 Yuan per
3 Yes winning lottery ticket
Sixth prize 2 Yes Fixed amount of 5 Yuan per
1 Yes winning lottery ticket
0 Yes
TABLE 2
Variable Definitions
Similarity index Similarity Index of a lottery portfolio, as
defined in Section III.A
WinRate Winning rate of the seller, as defined in Section
III.B
Commission Commission rate charged by seller, as a percentage
of the total winnings of the portfolio
Size Total amount of money in the lottery portfolio
Shares Total number of shares in the portfolio
Price The price of a single share
Self-investment The percentage of shares purchased by the seller
Time expose The time prior to lottery draw when portfolio is
available for purchase
TABLE 3
Summary Statistics
Variable Minimum Mean Median
Similarity index 0 .098 0.083
WinRate (all) 0 .58 0
Excluding jackpot 0 .229 0
Commission 0 .0575 0.08
Size (all) 8 774.32 18
Successful only 8 53.46 14
Shares 1 911.67 50
Price 0.2 3.91 0.5
Self-investment 0.01 .55 0.6
Number of tickets in 4 387.16 9
each package
Number of buyers for 1 9.76 5
each package
Portfolios in each round 18899 20132 20260
Time expose (hours) 0.01 25.67 24.02
Sold-out portfolios 77%
Total number of sellers
Standard
Variable Deviation Maximum Observations
Similarity index 0.082 1 283,083
WinRate (all) 35.41 14,930.84 248,523
Excluding jackpot 1.87 376.25 248,460
Commission 0.0437 0.10 301,982
Size (all) 8035.93 937,770 301,982
Successful only 400.31 58,848 233,481
Shares 13859.6 2,491,060 301,982
Price 87.55 12,400 301,982
Self-investment 0.27 1 301,982
Number of tickets in 4017.96 468,885 301,982
each package
Number of buyers for 46.39 6,118 301,982
each package
Portfolios in each round 1015 22,242 301,982
Time expose (hours) 19.532 70.49 301,982
Sold-out portfolios 301,982
Total number of sellers 41,418
TABLE 4
Correlation of Previous Return Rates on Future
Performance
WinRate
WinRate (Lag 1) -0.00012 (0.0022)
WinRate (Lag 2) -0.00019 (0.0040)
WinRate (Lag 3) -0.00004 (0.0040)
WinRate (Lag 4) -0.00016 (0.0040)
N 84,579
Adjusted [R.sup.2] -0.000
Note: Standard errors in parentheses.
TABLE 5
Dependent Variable: Commission (a)
(1) (2) (3) (4)
Big Win 0.567 *** 0.548 *** 0.643 *** 0.626 ***
indicator (0.071) (0.071) (0.069) (0.069)
Size (in [10.sup.-4]) 3.87 *** 5.43 *** 2.66 ***
(.236) (.230) (.335)
Self-investment 4.797 *** 4.800 ***
(0.0431) (0.0431)
Shares (in [10.sup.-4]) 2.99 ***
(.263)
Price
Similarity index
Time expose
N 218,709 218,709 218,709 218,709
Adjusted [R.sup.2] .000 .002 .055 .056
(5) (6) (7)
Big Win 0.657 *** 0.663 *** 0.658 ***
indicator (0.069) (0.069) (0.069)
Size (in [10.sup.-4]) 3.52 *** 3.51 *** 3.48 ***
(.336) (.336) (.336)
Self-investment 4.864 *** 4.864 *** 4.938 ***
(0.0431) (0.0431) (0.0441)
Shares (in [10.sup.-4]) 2.47 *** 2.47 *** 2.46 ***
(.263) (.263) (.263)
Price -0.00980 *** -0.00980 *** -0.00974 ***
(0.000393) (0.000393) (0.000393)
Similarity index -0.157 -0.165
(0.113) (0.113)
Time expose 0.00388 ***
(0.000496)
N 218,709 218,709 218,709
Adjusted [R.sup.2] .058 .058 .059
Notes: Standard errors in parentheses; ** p < .05, *** p < .01.
(a) Ordinary Least Squares, Big Win Indicator = 1 if WinRate >200%.
TABLE 6
Dependent Variable: Portfolio Size (a)
(1) (2) (3)
Big Win indicator 50.28 *** 48.48 *** 44.70 ***
(6.494) (6.491) (6.476)
Commission 3.164 *** 4.669 ***
(0.193) (0.198)
Self-investment -135.9 ***
(4.096)
Price
Similarity index
Time expose
N 218,709 218,709 218,709
Adjusted [R.sup.2] .000 .001 .006
(4) (5) (6)
Big Win indicator 41.03 *** 41.89 *** 41.33 ***
(6.462) (6.478) (6.477)
Commission 4.996 *** 4.995 *** 4.960 ***
(0.198) (0.198) (0.198)
Self-investment -144.6 *** -144.6 *** -135.8 ***
(4.095) (4.095) (4.192)
Price 1.162 *** 1.162 *** 1.168 ***
(0.0363) (0.0363) (0.0363)
Similarity index -19.94 -20.81 *
(10.49) (10.49)
Time expose 0.448 ***
(0.0460)
N 218,709 218,709 218,709
Adjusted [R.sup.2] .011 .011 .012
Notes: Standard errors in parentheses; p < .05, *** p < .01.
(a) Ordinary Least Squares, Big Win Indicator = 1 if WinRate>200%.
TABLE 7
Dependent Variable: Commission (a)
(1) (2) (3) (4)
Similarity index -0.100 -0.0933 -0.0511 -0.0621
(0.116) (0.116) (0.113) (0.113)
Size ([10.sup.-4]) 3.90 *** 5.46 *** 2.64 ***
(.236) (.230) (.335)
Self-investment 4.792 *** 4.795 ***
(0.0431) (0.0431)
Shares ([10.sup.-4]) 3.04 ***
(.263)
Price
Big Win indicator
Time expose
N 218,709 218,709 218,709 218,709
Adjusted [R.sup.2] .000 .001 .055 .055
(5) (6) (7)
Similarity index -0.0822 -0.157 -0.165
(0.113) (0.113) (0.113)
Size ([10.sup.-4]) 3.49 *** 3.51 *** 3.484 ***
(.337) (.336) (.336)
Self-investment 4.858 *** 4.864 *** 4.938 ***
(0.0431) (0.0431) (0.0441)
Shares ([10.sup.-4]) 2.53 *** 2.47 *** 2.46 ***
(.263) (.263) (.263)
Price -0.00974 *** -0.00980 *** -0.00974 ***
(3.93x (3.93x (3.93x
[10.sup.-4]) [10.sup.-4]) [10.sup.-4])
Big Win indicator 0.663 *** 0.658 ***
(0.0699) (0.0699)
Time expose 0.00388 ***
(0.000496)
N 218,709 218,709 218,709
Adjusted [R.sup.2] .058 .058 .059
Notes: Standard errors in parentheses; ** p< .05, *** p < .01.
(a) Ordinary Least Squares, Big Win Indicator = 1 if WinRate >200%.
TABLE 8
Dependent Variable: Portfolio Size (a)
(1) (2) (3)
Similarity index -17.27 -16.95 -17.93
(10.52) (10.52) (10.49)
Commission 3.188 *** 4.697 ***
(0.193) (0.198)
Self-investment -136.5 ***
(4.096)
Price
Big Win indicator
N 218,709 218,709 218,709
Adjusted [R.sup.2] -.000 .001 .006
(4) (5) (6)
Similarity index -15.20 -19.94 -20.81 *
(10.46) (10.49) (10.49)
Commission 5.022 *** 4.995 *** 4.960 ***
(0.198) (0.198) (0.198)
Self-investment -145.1 *** -144.6 *** -135.8 ***
(4.095) (4.095) (4.192)
Price 1.166 *** 1.162 *** 1.168 ***
(0.0363) (0.0363) (0.0363)
Big Win indicator 41.89 *** 41.33 ***
(6.478) (6.477)
0.448 ***
(0.0460)
N 218,709 218,709 218,709
Adjusted [R.sup.2] .011 .011 .012
Notes: Standard errors in parentheses; ** p < .05, *** p < .01.
(a) Ordinary Least Squares, Big Win Indicator = 1 if WinRate > 200%.
TABLE 9
Dependent Variable: Self-Investment (a)
(1) (2) (3)
Similarity index 0.0284 *** 0.0319 *** 0.0303 ***
(0.00614) (0.00617) (0.00615)
Big Win indicator -0.0234 *** -0.0215 ***
(0.00375) (0.00374)
Size ([10.sup.-5]) -3.28 ***
(.115)
Shares ([10.sup.-6])
Price ([10.sup.-4])
Time expose
N 196.888 196.888 196,888
Adjusted [R.sup.2] .001 .001 .005
(4) (5) (6)
Similarity index 0.0304 *** 0.0306 *** 0.0322 ***
(0.00615) (0.00614) (0.00599)
Big Win indicator -0.0213 *** -0.0222 *** -0.0185 ***
(0.00374) (0.00373) (0.00364)
Size ([10.sup.-5]) -2.93 *** -3.31 *** -2.96 ***
(.168) (.168) (.164)
Shares ([10.sup.-6]) -3.76 ** -1.40 -.891 **
(1.31) (1.31) (1.28)
Price ([10.sup.-4]) 6.61 *** 5.85 ***
(.249) (.243)
Time expose -0.00245 ***
(0.0000246)
N 196.888 196,888 196,888
Adjusted [R.sup.2] .005 .009 .056
Notes: Standard errors in parentheses; ** p < .05, *** p < .01.
(a) Ordinary Least Squares, Big Win Indicator = 1 if WinRate>200%.
