An empirical examination of the parimutuel sports lottery market versus the bookmaker market.
Chung, Jaiho ; Hwang, Joon Ho
1. Introduction
Market mechanisms under which sports betting takes place can be
categorized into two major types: a bookmaker system and a parimutuel
market system. In a bookmaker-based system, the bookmaker posts odds and
charges a commission (also called "vigorish" or
"juice") for placing a bet. In this type of market, bettors
(or "punters") are promised a fixed payoff according to the
odds that are posted at the time they make their bets. In the parimutuel
betting system, there are no bookmakers, and the payout from a specific
outcome is inversely related to the aggregate amount wagered on that
outcome relative to the aggregate amount bet across all outcomes.
Therefore, the payout will not be determined until all bets are
submitted. In this study, we use a unique data set in the
parimutuel-type Korean sports lottery market and examine the joint
efficiency of the parimutuel sports lottery market and the bookmaker
market.
Sports betting markets have been used in many studies as a way of
testing market efficiency (see Sauer 1998; Vaughan Williams 1999; and
Paton, Siegel, and Vaughan Williams 2009 for reviews). A widely
documented anomaly against market efficiency in the sports betting
market is the favorite-longshot bias. The favorite-longshot bias, first
shown in the horse race betting market, states that horses with lower
odds (higher probability of winning) tend to win more frequently than
projected by their odds, while horses with higher odds (lower
probability of winning) tend to win less frequently than indicated by
their odds. In a parimutuel market structure, this suggests that
favorites are underbet while longshots are overbet (see Ziemba and
Hausch 1984; Thaler and Ziemba 1988; and Paton, Siegel, and Vaughan
Williams 2009 for reviews). (1)
Most horse tracks in the world operate as a parimutuel system (the
system is called the totalisator in the United Kingdom, Australia, and
New Zealand), (2) whereas horse tracks in the United Kingdom, Ireland,
and Australia also employ bookmaker-based systems. Researchers have
examined market efficiency between these two different types of market
structures for horse tracks. Gabriel and Marsden (1990, 1991), Bruce and
Johnson (2000), and Peirson and Blackburn (2003) compare returns in the
British horse track for bets made in the parimutuel ("tote")
market and bets made with the bookmaker. Gabriel and Marsden (1990,
1991) examine winning bets in two markets under the situation where tote
bettors had restricted information on current odds due to the absence of
mechanical or electronic tote boards during the year of their sample
period. They find that the level of winnings from successful bets in the
parimutuel market is larger than that in the bookmaker market. Bruce and
Johnson (2000) examine all bets made in the two markets under full odds
information available for the tote bettors and find that the
favorite-longshot bias is strong in the bookmaker market but not in the
parimutuel market. Peirson and Blackburn (2003) also find similar
results in British racetracks and note that profit maximization by
bookmakers and efficient behavior by bettors can cause the difference
between the two markets.
For sports betting other than horse racing, most markets are
bookmaker based, with the exception of betting pools and some sports
lotteries. Therefore, almost all previous studies on sports betting
outside of horse racing are based on a bookmaker market, and none of
these studies examines efficiencies across different market structures.
(3)
In this study, we use a unique data set in the parimutuel-type
Korean sports lottery market to ascertain whether there is a significant
difference in winning payoffs between this parimutuel sports lottery
market and the bookmaker-based market. In the Korean sports lottery
market, odds are determined by the parimutuel method, and current odds
are readily available on its website, which allows participants to
monitor odds on a real-time basis. Among the various betting
opportunities that are offered by the Korean sports lottery system, we
focus on "soccer special triple" and "soccer special
double," where participants who pick the correct final scores of a
predetermined three- or two-game soccer lottery win the share of the
betting pool. For instances where the Korean sports lottery features
games in the English Premier League, the odds on the final score of
these three (or two) games are available on major bookmakers, such as
the online betting website of William Hill PLC. Using data from 34
sports lotteries for 86 English Premier League matches between 2005 and
2008, we compare the return on winning bets placed in the
parimutuel-type Korean sports lottery market to the corresponding return
on a bet placed in the U.K. bookmaker-based market.
We find notable differences in the realized return between these
two markets. For example, a $1.00 bet on the final score of 1:0, 2:0,
and 0:3 for matches between Bolton and Manchester United, Arsenal and
Wigan, and Middlesbrough and Aston Villa, played on November 25, 2007,
paid an amount of $6605.20 in the Korean sports lottery market, compared
to $2028.00 in the U.K. bookmaker market. However, another $1.00 bet on
the final score of 2:l, 1:1, and 1:2 for matches between Arsenal and
Tottenham, Fulham and Wigan, and Middlesbrough and West Ham on December
22, 2007, paid only $92.90 in Korea's sports lottery market,
compared to $585.00 in the U.K. bookmaker-based market. These
discrepancies in winning payouts between two markets are surprising
since current odds on both markets are accessible to participants in the
Korean sports lottery market. We find that for low-payout outcomes
(games that turn out as many have expected), the Korean sports lottery
market returns lower payouts compared to the corresponding payoffs in
the bookmaker-based market. The reverse relationship is found for
high-payout outcomes (games that do not turn out as many have expected).
Overall, our results suggest that the two markets are not jointly
efficient.
Our main contributions are twofold. First, to the best of our
knowledge, this is the first study to provide an empirical examination
of a parimutuel sports lottery market relative to a bookmaker market.
(4) Some sports lotteries, such as Pro-line, operated by the Ontario
Lottery and Gaming Corporation in Canada, are organized as
bookmaker-based, fixed-odds systems and do not have a corresponding
parimutuel market to compare against. For parimutuel-type sports
lotteries such as The New Football Pools or Brittens National Pools in
the United Kingdom or Sports Action in the United States, the lack of
information on current odds or on the final payoff data make it
impossible to compare the return against a bookmaker-based market. (5)
The sports lottery market in Korea, which provides current odds
information to all participants and ex post payout information for their
events, provides a unique opportunity for us to examine the
parimutuel-type sports lottery market against the bookmaker market.
Second, given that the sports lottery, which we investigate,
operates as a parimutuel market, this is also the first study to
empirically test the joint efficiency of the parimutuel market and
bookmaker-based market outside the realm of horse tracks. (6) In horse
racing, studies (for example, Crafts 1985 and Shin 1992) note that
private inside information is believed to be prevalent, and Shin (1991,
1992) argues that the bookmakers worsen the odds for longshots to guard
against bettors with inside information, thereby causing the
favorite-longshot bias in the bookmaker market. In comparison to horse
race betting, betting on the final score of a soccer game would see many
fewer cases of insider traders. For example, even if someone has inside
information on a player's physical condition, the insider trader is
unlikely to predict the final score of the game with high accuracy
because of the larger proportion of the unpredictable component
("luck") present in the correct score betting. Therefore, it
is interesting to see whether the favorite-longshot bias would still
exist in the bookmaker market relative to the parimutuel market when
bookmakers are faced with less threat of privately informed bettors. Our
results suggest that the bias still exists in the bookmaker market.
Our study is not without limitation. Since we compare the
parimutuel market and the bookmaker market on the basis of only winning
bets, we do not intend to test the market efficiency in each of the two
markets. Rather, we test the joint efficiency of these two markets and
look only for the relative extent of bias across these two markets.
The article is organized as follows: Section 2 introduces the
Korean sports lottery market and the U.K. bookmaker market. Section 3
explains the different market structures and mechanisms between the two
markets, while section 4 compares the payoffs from the Korean sports
lottery market against those from the U.K. bookmaker-based market.
Section 5 analyzes these results with respect to the existence of the
favorite-longshot bias. Section 6 provides some additional analyses,
including the comparison against betting exchanges and tests of
alternative specifications of the model, and section 7 concludes the
article.
2. The Korean Sports Lottery Market and the U.K. Bookmaker Market
Korean Sports Lottery Market
In Korea, the only government-sanctioned and legalized sports
betting markets outside of on-track betting are Sports Toto, which is
structured as a parimutuel market, and Sports Proto, which is structured
as a bookmaker market. (7) In this study, our reference to Korean sports
lottery refers to the parimutuel-type Sports Toto.
Participants in the Korean sports lottery can make their bets
offline through licensed betting offices, which are equipped with online
terminals, or directly purchase online through the Korean sports lottery
website (www.betman.co.kr). (8) To bet online, the participant must
register online for free using his national identification number and
fund his account through electronic bank wire. In contrast, the identity
of the participant is not checked when betting through licensed offline
betting offices.
Participants of the Korean sports lottery can purchase multiple
entries within the maximum wager limit of 100,000 Korean Won
(approximately US$100). Real-time odds are available on the Korean
sports lottery website. Unlike in horse tracks where odds are rounded
down, odds in the Korean sports lottery market are rounded off to the
first decimal point. (9) At the end of the event, participants who
correctly predict the outcome of the games offered in the lottery share
the prize pool in proportion to their bets, net of taxes. The prize pool
consists of 50% of the total amount of money wagered for the event. If
there are no winners for a specific round of sports lottery, the prize
money is carried forward and added to the prize pool for the next event.
In our analysis of the Korean sports lottery market, there was one
occasion where there were no winning ticketholders. We exclude that
event and the following one from our sample of data.
Among the different types of bets offered in Sports Toto, we focus
on betting events called soccer special triple and soccer special
double, where the corresponding odds of these events can also be found
in overseas bookmaker markets. In soccer special triple (double),
participants try to pick the correct final score (out of six available
choices of 0, 1,2, 3, 4, and 5 or more goals for each team) of a
predetermined three (two) soccer matches.
Soccer special triple and soccer special double are offered once or
twice a week for the Korean K-League games and the English Premier
League matches during their respective seasons. Betting was also offered
for international matches, such as games in the 2006 World Cup and Euro
2008. We use only English Premier League games as our sample of analyses
and exclude Korean K-League and international matches. This is because
the established bookmaker market that we use in this study is based in
the United Kingdom and would therefore provide the best benchmark for
games in the English Premier League for our analyses of the sports
lottery market. Our choice of the U.K. bookmaker's odds as the
benchmark is not without potential problems, as these odds may be biased
and not represent a perfect indicator of the likelihood of each outcome.
However, given that our sample consists of games in which the U.K.
bookmaker is likely to have an informational advantage in setting the
odds, the bookmaker's odds in the United Kingdom can provide a
reasonable forecast of the outcome. (10) Our objective is to examine how
the odds that are generated from the parimutuel sports lottery market
differ from those posted by the bookmaker and examine the relative
extent of the bias between the two markets.
With regard to the above objective, we exclude games in the Korean
domestic league because the U.K. bookmaker may have informational
disadvantage. We exclude events that feature international matches
because all of them include a game in which the Korean national team is
playing, where a skewed betting behavior due to bettors' sentiment
is likely to occur in the Korean betting market.
An average soccer special triple lottery in our sample attracted
154,427 participants for an average bet amount of 4907 Korean Won
(approximately US$4.91) per participant. The size of soccer special
triple was smaller, attracting 14,588 participants on average, with the
average bet amount of 4697 Korean Won (approximately US$4.70) per
participant.
U.K. Bookmaker Market
To find the corresponding odds for the same event in the bookmaker
market, we use a major U.K.-based bookmaker, William Hill PLC. The
website of William Hill PLC (http://www.williamhillplc.com) states that
the company employs over 16,000 people in the United Kingdom and Ireland
and is licensed and regulated in the United Kingdom. Its shares have
been trading on the London Stock Exchange since 2002, and the company is
included in the Financial Times and London Stock Exchange 250 index. Its
website attracts customers from over 150 countries and currently has
over 300,000 active online customers. It also has over 2250 offline
betting shops across the United Kingdom and Ireland, making it the
largest operator in the United Kingdom in terms of the number of betting
shops. Therefore, bettors in the U.K. bookmaker market can place their
wagers through the bookmaker's website or through betting shops. To
bet online, participants can open an account at the bookmaker's
website free of charge and fund their account using methods such as bank
wire, credit card, or third-party payment service such as PayPal,
NETeller, or Moneybookers. Similar to the case of the Korean sports
lottery, the identity of the participant is typically not checked when
betting takes place at offline betting shops.
The maximum amount of winning by a customer varies depending on the
type of event. For wagers placed for games in the English Premier
League, the maximum winning per day is 1 million [pounds sterling].
Therefore, much more money can be wagered at the U.K. bookmaker's
market compared to the Korean sports lottery market, where the maximum
wager is 100,000 Korean Won (approximately US$100 or 65 [pounds
sterling]). (11)
3. Comparison of Market Structures
Market Operating Hours
Betting in the Korean sports lottery market opens at 9:30 A.M.
(Korean standard time) on Thursday for the upcoming weekend matches and
closes 10 minutes before the start of the first match. The corresponding
U.K. bookmaker market opens earlier than the Korean sports lottery
market, typically about one week before the start of the match. Once the
Korean sports lottery market is open for these matches, both markets
stay open until the Korean sports lottery market closes. The U.K.
bookmaker market closes just before the start of each match. In our
article, both the Korean sports lottery market odds and the U.K.
bookmaker market odds represent final odds in their respective market.
There may be some concern about comparing odds based on the final odds
in each market because of the time gap between the closing time of the
Korean sports lottery market and that of the U.K. bookmaker market.
These concerns can be addressed by considering the following. First,
bookmaker odds may change during the last 10 minutes before the start of
the first match. However, after most information becomes available to
the public, such as the weather, starting lineup of players, and so
forth, there would be little change in the U.K. bookmaker odds within
that time frame. A second concern may be that, if there is a big time
gap between the closing time of the Korean sports lottery market and the
kickoff time of the second match (and third match in the case of the
three-game lottery), then the final odds from the Korean sports lottery
market may not incorporate all relevant information for these second or
third matches. However, for the three-game lottery, 32 matches out of 54
matches (59.26%) in our sample had the same kickoff times for the first,
second, and third matches offered in the sports lottery. As for the
two-game lottery, 22 out of 32 matches (68.75%) had the same kickoff
times between the first and the second match. For some lotteries that
featured matches with different kickoff times, the second and third
matches are within only a couple of hours from the first match offered
in the sports lottery. Therefore, we believe the above concern will not
be driving our results.
Tax and the Calculation of After-Tax Payoffs
The quoted odds in the Korean sports lottery market are stated in a
pre-tax payout amount. For quoted odds greater than 100, a 22% tax is
automatically levied on the difference between the winning payouts and
the bet amount. The actual winning payoff is paid out after the tax has
been taken out. For example, if the participant of the Korean sports
lottery market places a wager of 1000 Korean Won and wins the lottery
with quoted odds of x, which is greater than 100, then her after-tax
payoff will be (1000x - 1000) x (1 - 0.22) + 1000. Therefore, the
corresponding after-tax winning payout odds can be calculated as
follows:
(1000x- 1000) x (1 0.22) + 1000/1000 = 0.78x + 0.22. (1)
The next equation summarizes the relationship between the after-tax
odds and the quoted odds in the Korean sports lottery market:
After-tax odds = 0.78 x quoted odds + 0.22; if quoted odds > 100
After-tax odds = quoted odds; if quoted odds [less than or equal
to] 100. (2)
Contrary to the tax situation in the Korean sports lottery market,
bettors of William Hill PLC do not pay any tax on their winnings.
Instead, only the bookmaker is charged with tax, called gross profits
tax (GPT), which is based on the net revenue of the bookmaker.
Therefore, from the standpoint of the bettors, the following payout
equation applies to the outcomes in the U.K. bookmaker market:
After-tax odds = quoted odds. (3)
Readers can refer to Paton, Siegel, and Vaughan Williams (2002) for
more information on betting taxation in the United Kingdom.
Participants' Access to the Other Market
In general, for participants in each market, the ability to
participate in the other market is difficult due to the following
restrictions. For the Korean bettors, they are restricted from betting
in the overseas market due to government restrictions. Specifically,
they cannot fund their account using conventional methods such as credit
cards or bank wire since the Korean government blocks transactions to
foreign gambling sites. For the U.K. bettors, they are restricted from
betting in the Korean market because the Korean sports lottery is closed
to participation by foreigners or anyone accessing from a foreign IP
address.
Finding a way to participate in the other market is not impossible,
though. One way of participating in the other market is by placing a
local representative who works as an agent in the other country.
However, as this would incur a large transaction cost, we believe it is
generally the case that bettors do not participate in the other market.
Since prices in different markets are likely to converge with each
other if bettors are allowed to participate in both markets without any
trading cost or restriction, we think that our setting where bettors are
not able to participate in the other market presents a better
environment in which to compare the prices determined in each market.
4. Results
Table 1 shows the results for the Korean sports lottery, which
featured games in the English Premier League. We retrieve data from
www.betman.co.kr, and the final sample consists of 34 sports lotteries
for 86 English Premier League matches between 2005 and 2008. (12) Even
with a pre-tax payout ratio of only 50%, a sizable amount of money is
bet on the sports lottery market in Korea. (13) As can be seen from
Table 1, the three-game lottery market, with an average total bet amount
of 755,898,461 Korean Won (approximately US$755,898), constitutes a much
bigger market than the two-game lottery market. We conjecture that the
greater participation for the three-game lottery market compared to the
two-game lottery market is because investors are drawn to the higher
payout, in spite of the lower probability of winning. Table 1 shows that
the mean (median) payout on the winning ticket is 6114.6 (532.4) times
the amount bet for the three-game lottery and 234.0 (60.0) times the
wagered amount for the two-game lottery.
To examine whether any discrepancies exist in the winning payoffs
between the parimutuel sports lottery market in Korea and the bookmaker
market in the United Kingdom, we construct the after-tax payoffs from
winning bets in each market. For the Korea sports lottery market, we
need to account for the 22% tax that is levied on winnings with quoted
odds greater than 100. Therefore, as illustrated in Equation 2, the
after-tax payoff on the winning bet i in the Korean sports lottery
market is as follows:
[s.sub.i] = 0.78 x [p.sub.i] + 0.22 if quoted odds [p.sub.i] >
100
[s.sub.i] = [p.sub.i] if quoted odds [p.sub.i] [less than or equal
to] 100. (4)
We retrieve corresponding final odds of the English Premier League
games from the website of William Hill PLC. In the bookmaker market,
odds for the final score are posted for each game. Therefore, if we note
[m.sub.ki] as the posted odds for the kth soccer match, the
corresponding odds for the winning bet i are as follows: (14)
[m.sub.i] = [m.sub.1i] x [m.sub.2i] x [m.sub.3i] for the three-game
bet
[m.sub.i] = [m.sub.1i] x [m.sub.2i] for the two-game bet. (5)
Table 2 shows the average winning payoffs and the accompanying
standard deviations from the two markets. For a $1.00 bet, the average
payout in the three-game (two-game) Korean sports lottery market is
$3948.00 ($87.30) greater than the corresponding payout in the U.K.
bookmaker market. The Wilcoxon matched-pairs signed-rank tests indicate
that the differences in the average winning payouts between two markets
are not significant at conventional levels. This is surprising given
that the Korean sports lottery market pays out only 50% of the total bet
amount and further levies 22% tax for winning payouts with quoted odds
of over 100, which applies to all cases in our three-game lottery
sample. By comparison, as we show later, the average commission charged
by the bookmaker is 28.1%. While these differing circumstances lead us
to expect significantly higher winning payouts in the U.K. bookmaker
market, our results show that this is not the case. Rather, the results
in Table 2 show that the sports lottery market yields higher, although
statistically insignificant, average winning payouts compared to
identical bets made with the bookmaker.
The comparison of winning payouts between the sports lottery market
and the bookmaker market reveals drastic differences between the two
markets once we divide our sports lottery sample into two groups: one
where the after-tax winning payout is less than the median and the other
where the after-tax winning payout is greater than the median. For the
former subset, winning payouts from the sports lottery market are
significantly less than the corresponding payouts from the bookmaker
market for both the three-game lottery (panel A) and the two-game
lottery (panel B). For winning payouts greater than the median payout,
the sports lottery market exhibits significantly higher return than the
bookmaker market in the case of the three-game lottery. For the two-game
lottery, the difference is statistically insignificant at conventional
levels. These results, taken as a whole, suggest that participants in
the Korean sports lottery market tend to bet too much (little) on
outcomes with higher (lower) probabilities of occurrence than the amount
suggested by the bookmaker's odds. Given the 50% payout rate in the
Korean sports lottery market, the way to rationalize participation in
this market is to expect for a customer to bet on a high-odds outcome in
hopes of winning a fortune. If this rationale holds, outcomes that are
higher (lower) than average should result in relatively lower (higher)
payoffs than the corresponding payouts in the bookmaker market.
Interestingly, however, our results show that the opposite is true.
To better gauge the relationship between the odds set in the Korean
sports lottery market against those set in the U.K. bookmaker market, we
run a regression analysis where the dependent variable is the winning
payout in the Korean sports lottery and the right-hand side variable is
the corresponding payout from the U.K. bookmaker market. Since our
sample consists of games in the English Premier League, under the null
hypothesis that the bookmaker's odds are not biased, these odds
should be the best unbiased estimates of the odds that are set in the
Korean sports lottery market.
We therefore estimate the following regression:
[s.sub.i] = [alpha] + [beta] x [m.sub.i] + [[epsilon].sub.i]. (6)
In Equation 6, [[epsilon].sub.i] is assumed to be an N(0,
[[sigma].sub.2]) error term, and [s.sub.i] and [m.sub.i] represent the
winning payoffs from the sports lottery market and the corresponding
payoffs from the bookmaker market, respectively, where the specific
construction of each variable was explained in Equations 4 and 5.
The regression results are shown in Table 3. The [R.sup.2] of the
regression is only 31% for the three-game lottery market and 85% for the
two-game lottery market. (15) This shows that the odds quoted by the
U.K. bookmaker are a better predictor of the odds in the Korean sports
lottery market for the two-game lottery market, in which the pool is
smaller, than in the larger three-game lottery market.
For both the two-game and three-game markets, the slope of the
regression is greater than unity, along with the negative intercept.
(16) Consistent with the result of Table 2, the slope of the regression
implies that for outcomes with smaller payouts, winning payoffs in the
Korean sports lottery market are less than the corresponding payoffs
from the bookmaker market in the United Kingdom. However, for outcomes
with larger payouts, winning payoffs in the Korean sports lottery market
are greater than the corresponding payoffs from the bookmaker market in
the United Kingdom. The negative intercept of the regression shows that
the aggregate payout in the Korean sports lottery market is less than
the payout in the U.K. bookmaker market. In the Korean sports lottery
market, the payout ratio is 50% of the total wager pool, and an
additional 22% tax is levied on winnings with quoted odds greater than
100. In the bookmaker market, on the other hand, tax is not charged to
the winning ticketholder. Rather, tax is levied to the bookmaker, who in
turn charges a commission that is incorporated into the quoted odds.
To examine whether the two-game lottery and the three-game lottery
show different betting behavior (for example, participants in the
three-game lottery may overbet longshots in hopes of winning a big
jackpot, compared to participants in the two-game lottery), we tested
for the difference in the slope of our regression between the two types
of lottery using the Wald test. Results show that the difference in the
slope coefficient between the two types of lottery is not statistically
significant at conventional levels. Therefore, although the three-game
lottery market attracts more participants than the two-game market, as
shown in Table 1, we do not find support for the different betting
behavior between the participants of the two-game lottery and those of
the three-game lottery.
So far, we have compared the winning payout of the Korean sports
lottery market against the corresponding payout of the bookmaker using
its quoted odds. However, since bookmakers charge a certain amount of
commission as noted above, the "fair" odds for a bet should be
greater than the quoted odds. As an example of the difference between
quoted odds and fair odds, let us consider the case where the bookmaker
posts odds of 1.90 for event A and odds of 2.00 for event B, where
events A and B are mutually exclusive and span the universe of all
outcomes. Then the fair odds are
1/1.9 + 1/2/1/1.9 = 1.95
for event A and
1/1.9 + 1/2/1/2 = 2.05
for event B. Therefore, a clever parimutuel bettor will compare the
expected payout in the parimutuel market to the fair odds in the
bookmaker market and place a bet for an outcome whose payout in the
sports lottery market is greater than the fair odds that are generated
from the bookmaker market. If the payout difference between the two
markets varies between smaller payouts and larger payouts even after the
bookmaker's commission has been accounted for, the result will
provide stronger evidence that two markets are not jointly efficient.
Therefore, in our analysis below, we take into account the average
bookmaker's commission of 28.1% that is charged on the final-score
betting. The procedure for calculating the average bookmaker's
commission is illustrated in the Appendix. The use of the average
commission for all possible outcomes is under the null hypothesis that
the bookmaker's odds are not biased across different odds of
final-score bets, and therefore the same amount of commission is charged
across different outcomes. (17) We then make the comparison based on
this measure of the after-commission fair payout, which is calculated as
1.281 times the posted odds for each game. Therefore, in Equation 6, if
[m.sub.ki] is the posted odds for the kth soccer match, the
corresponding fair odds in the bookmaker market for the winning bet i
are as follows:
[m.sub.i] = [m.sub.1i] x (1.281) x [m.sub.2i] x (1.281) x
[m.sub.3i] x (1.281)
= [m.sub.1i] x [m.sub.2i] x [m.sub.3i] x [(1.281).sup.3] for the
three-game lottery
[m.sub.i] = [m.sub.1i] x (1.281) x [m.sub.2i] x (1.281) =
[m.sub.1i] x [m.sub.2i] x [(1.281).sup.2] for the two-game lottery. (7)
The results in Table 3 show that even with the fair odds of the
bookmaker, the estimated slope is greater than unity, along with a
negative intercept. This implies that for outcomes with higher (lower)
probabilities of occurrence, bettors in the Korean sports lottery market
overbet (underbet) these outcomes relative to the fair probabilities
that are suggested by the bookmaker's odds even after accounting
for their commissions.
5. Favorite-Longshot Bias
Our comparison between the parimutuel sports lottery market in
Korea and the bookmaker market in the United Kingdom allows us to
compare the extent of the favorite-longshot bias between the two
markets. Since we are examining only winning bets, we do not intend to
assess the absolute magnitude of the favorite-longshot bias for each
market. Nonetheless, we can compare the relative extent of the bias
between two markets. Specifically, our empirical results can contribute
to the following aspects of the favorite-longshot bias.
First, as for the comparison between different market structures,
previous studies have found that the favorite-longshot bias exists, with
some exceptions, for both the parimutuel and bookmaker markets. The
evidence for the parimutuel market is shown in studies such as Ali
(1977) and surveyed in Thaler and Ziemba (1988) for racetracks in the
United States, where longshots are found to be overbet relative to
favorites. More recently, Ottaviani and Sorensen (2008a) theoretically
show that the extent of the bias depends on the amount of private
information in the market. For the bookmaker market, one of the widely
accepted theories for observing the favorite-longshot bias is that
bookmakers trim the odds on longshots relative to favorites to protect
themselves from the existence of informed bettors (Shin 1991, 1992).
Vaughan Williams and Paton (1997) find empirical support for the above
argument in U.K. horse tracks.
Our results in Tables 2 and 3 show that for higher-odds
(lower-odds) outcomes, bookmakers are posting odds that are lower
(higher) than those found in the parimutuel market. Therefore, for our
sample of soccer betting, we find that the favorite-longshot bias is
relatively more severe in the bookmaker market when compared to the
parimutuel market. Even though bookmakers are less likely to encounter
bettors with inside information in the case of correct-score betting in
soccer games compared to the case of horse racing, our results imply
that bookmakers are still reducing the odds on longshots. Therefore, our
results suggest that the threat of traders with inside information is
not the sole reason why bookmakers exhibit the favorite-longshot bias.
As for the parimutuel market, it exhibits a reverse
favorite-longshot bias relative to the bookmaker market. Ottaviani and
Sorensen (2008a) show in their model that when signals contain little
information and there is aggregate uncertainty about the final
distribution of bets due to noise, there will be a reverse
favorite-longshot bias in the parimutuel market. As bets made on
predicting the final scores of two- or three-game soccer matches are
less likely to contain material information compared to other types of
bets, such as bets on a team to win for a single soccer match or bets in
horse racing, our results show empirical support for Ottaviani and
Sorensen (2008a).
Second, with regard to cross-country comparison, there is evidence
that horse tracks in Europe show a strong favorite-longshot bias,
whereas Asian markets show a reverse favorite-longshot bias: see Busche
and Hall (1988) for evidence in the Hong Kong racetrack betting market,
Busche (1994) for Hong Kong and Japanese racetracks, and Coleman (2004)
for a survey. Our results show that this differing pattern in the
betting behaviors for European and Asian participants extends to betting
markets outside of horse racing. As mentioned by Busche and Hall (1988)
and Coleman (2004), it is puzzling why participants in different
countries demonstrate different betting behaviors. These can be due to
different factors such as the market power of market makers, patterns of
gambling behavior, cultural factors, or attitudes toward risk. (18) For
example, Asian bettors, compared to European bettors, may be relatively
more risk averse or have a tendency to underestimate the chances of low
probability events, which results in Asian bettors overbetting the
favorites and thereby causing the reverse favorite-longshot bias. (19)
While uncovering the exact reason for this diversity is beyond the scope
of this article, we believe it is an interesting topic for future
research.
6. Additional Tests
Betting Exchanges
Our categorization of the betting market follows from studies like
Coleman (2004), which categorize betting markets into bookmaker-based
markets and parimutuel markets. However, sports betting exchanges can be
considered another category of betting markets. In the betting exchange
market, participants in the market can act as bookmakers by submitting
their own prices through the Internet; see Smith, Paton, and Vaughan
Williams (2006) for a study on betting exchanges. In this subsection, as
an additional analysis, we examine the winning payoffs between the
sports lottery market and the betting exchange market.
Data on betting exchanges are from Betfair.com, one of the major
betting exchanges in the United Kingdom. Data from Betfair.com have
time-stamped odds for games played in 2007 and 2008. (20) We extracted
odds for each game at two different time periods: one for the last
transaction odds before the closing time of the Korean sports lottery
and the other for the last transaction odds before the starting time of
the match.
In Betfair.com, since odds are available for scores of three or
fewer goals by both teams (other outcomes are under the "any
unquoted" category of bets), there are no corresponding odds for
outcomes with high scores of more than three goals by either team. As a
result, we are able to collect betting exchange samples for 33 matches.
As for the correlation coefficient between the betting exchange
odds and the bookmaker's final odds, it is 0.95 when we use the
betting exchange odds right before the start of the match. Second, the
correlation coefficient of the two markets is 0.94 when we use the last
transacted betting exchange odds before the Korean sports betting market
closes. Therefore, the odds at the bookmaker market and those at the
betting exchanges are highly correlated with each other.
To examine the relationship in the winning payoff odds between the
sports lottery market and the betting exchange market, we follow a
similar method as the case of the bookmaker market. Specifically, for
each winning payoff odds in the Korean sports lottery market, we
construct corresponding odds in the betting exchange market, as in
Equation 6.
We then estimate the following regression:
[s.sub.i] = [alpha] + [beta] x [n.sub.i] + [[epsilon].sub.i] (8)
where [[epsilon].sub.i] is assumed to be an N(0, [[sigma].sup.2])
error term, and [s.sub.i] and [n.sub.i] represent the winning payoffs
from the sports lottery market and the corresponding payoffs from the
betting exchange market, respectively.
The result of the regression is provided in Table 4. For our
analysis between the sports lottery market and the betting exchange
market, we do not distinguish between the two- and three-game lottery
markets because of the small number of betting exchange data available
that corresponds to the three-game lottery market. This is because for
us to find the corresponding betting exchange data for the three-game
sports lottery, all three games must end with three goals or less. In
the first column of the results, the payoff from the betting exchange
market is calculated using betting exchange odds for the last
transaction before the Korean sports lottery closing time. In the second
column of the results, the payoff from the betting exchange market is
calculated using betting exchange odds for the last transaction before
the start of the match.
Results in Table 4 show that the intercept of the regression is
significantly negative, but the coefficient of the slope is not
significantly different from 1. The negative intercept reflects the low
payout ratio of the Korean sports lottery market compared to the U.K.
betting exchange market. More importantly, since the slope coefficient
is not statistically different from unity, the previously noted
favorite-longshot bias in the relationship between the Korean sports
lottery market and the U.K. bookmaker market is not observed in the
relationship between the Korean sports lottery market and the U.K.
betting exchange market. The result suggests that the favorite-longshot
bias (reverse favorite-longshot bias) present in the U.K. bookmaker
market (Korean sports lottery market) relative to the Korean sports
lottery market (U.K. bookmaker market) is more likely to be caused by
the difference in the market structure between the parimutuel market and
the bookmaker market, rather than by the difference in the location of
the betting market.
Alternative Specifications of the Regression
For the robustness of our results, we run alternative
specifications of the regression model. (21) The relationship of the
winning payoff between the Korean sports lottery market and the U.K.
bookmaker market may be nonlinear as opposed to linear and begin at the
origin rather than having a negative intercept. If participants in the
sports lottery market overbet longshots in hopes of winning a big
jackpot, the odds for higher payoff outcomes would be lower in the
sports lottery market compared to the bookmaker market. This implies
that for higher odds, odds in the sports lottery market will increase at
a decreasing rate relative to odds in the bookmaker market. Therefore,
in the nonlinear regression of the sports lottery odds against the
bookmaker odds, we add a logarithm term interacted with a
greater-than-median payoff dummy. Conversely, for the lower payout
outcomes, odds in the sports lottery market may increase at an
increasing rate relative to odds in the bookmaker market. Therefore, we
allow for a square term interacted with a less-than-median payoff dummy.
The specification of the nonlinear regression model is as follows:
[s.sub.i] = [alpha] + [[beta].sub.0] x [m.sub.i] + [[beta].sub.1] x
[D.sub.1] x [m.sub.i.sup.2] + [[beta].sub.2] x [D.sub.2] x
log([m.sub.i]) + [[epsilon].sub.i]. (9)
In the above equation, [s.sub.i] is the winning payout from the
Korean sports lottery market, as shown in Equation 4, and [m.sub.i] is
the corresponding payoff from the U.K. bookmaker market, as shown in
Equation 5. In panel A of Table 5, [D.sub.1] is a dummy variable that
takes a value of 1 for outcomes with payoffs greater than the median
payoff in our sample. [D.sub.2] is a dummy variable that takes a value
of 1 for outcomes with payoffs less than the median payoff in our
sample. In Model 1 and Model 2, we consider only the nonlinear term, and
in Model 1 and Model 3, we restrict the intercept to be zero. Results in
panel A show that the coefficients of the logarithm of the bookmaker
odds for less-than-median payouts are significant in Models 1 and 2.
However, their significance disappears once we include the linear term
in Models 3 and 4. Therefore, once we account for the linear
relationship between the Korean sports lottery market and the U.K.
bookmaker market, there is little additional explanatory power of
nonlinear terms.
In panel B of Table 5, we test for the different betting behavior
between the two lottery markets, in which participants in the three-game
lottery may overbet longshots in hopes of winning a big jackpot.
Therefore, [D.sub.1] in Equation 9 represents the three-game lottery
dummy, and [D.sub.2] represents the two-game lottery dummy. Results in
panel B are very similar to those of panel A. That is, the relationship
between the Korean sports lottery market and the U.K. bookmaker market
seems to follow a linear relationship. Also, we do not find support for
the possibly different betting behavior between the participants in the
two-game lottery and those in the three-game lottery, even if we account
for nonlinearity.
7. Conclusion and Discussion
This study examined the winning payoffs from a sports lottery
market in a parimutuel wagering form. Apart from the fact that the
sports lottery market has not been empirically examined, previous
research on the comparison between parimutuel markets and bookmaker
markets was confined to the horse racing market. In studies of horse
tracks, the threat of private inside information faced by the bookmaker
has been one of the widely accepted explanations for the
favorite-longshot bias that is observed in the bookmaker market.
In our sample of correct-score betting of English Premier League
soccer games that is offered by the Korean sports lottery market,
participants can monitor real-time odds in the sports lottery market and
have access to odds that are quoted by the bookmaker at the same time
for the same event. This presents a unique opportunity for comparing the
parimutuel sports lottery market against the well-established
bookmaker-based system for events where inside information is likely to
be less pervasive than in horse racing.
We find that for winning bets with relatively low payouts, the
Korean sports lottery market yields lower payouts compared to
corresponding payouts in the U.K. bookmaker market. The reverse is true
for winning bets with relatively high payoffs. When we regress the
winning payoff resulting from the Korean sports lottery market against
the corresponding payoff from the U.K. bookmaker-based market, the slope
of the regression is significantly greater than unity. Overall, our
results suggest that bookmakers still exhibit favorite-longshot bias
even when the threat of inside traders is smaller than in horse racing.
Conversely, participants in the parimutuel sports lottery market in
Korea show a reverse favorite-longshot bias relative to the U.K.
bookmaker market.
As a final note, since our study considers only winning bets, the
results are not conclusive but only suggestive. We hope that our study
can serve as a building block for studying the efficiency of the sports
lottery market. A further test of market efficiency of each market by
using bets that span all combinations of outcomes and analyzing the
distribution of betting ratios over various outcomes is left for
possible future research.
Appendix: Calculation of Bookmakers' Commission for Correct
Final-Score Betting
To calculate the fair odds in the bookmaker market, we need to
incorporate the bookmaker's commission into the quoted odds.
However, we cannot directly compute the bookmaker's commission
because the quoted odds on final scores do not span the universe of all
possible outcomes. (22) However, under the null hypothesis that the
bookmaker's odds are not biased, we can assume that the commission
is the same for all outcomes. Then we can make use of the odds on the
over/ under total goals that are scored in a game. First, by using the
odds on the over/under total score, we calculate the probability that
the final score will be less than or equal to the specified total goals.
At www.williamhillplc.com, odds are available for less than two goals
scored, exactly two goals scored, and more than two goals scored for
each game.
Let us denote the odds on less than two goals scored as U, the odds
on exactly two goals scored as E, and the odds on more than two goals
scored as O. The probability that the total goals scored will be two or
less is
(1/U)+(1/E)/ (1/U)+(1/E)+(1/O).
Now, there is a finite number of cases of the final score where the
total goals scored is less than or equal to two. If we define [S.sub.i]
as the quoted odds of the final score being i [member of] {(1, 0), (2,
0), (0, 0), (1, 1), (0, 1), (0, 2)} and denote the bookmaker's
fractional commission by C, then the fair odds of the final score being
i is [S.sub.i] x (1 + C) and the corresponding fair probability of the
final score being i is 1/[[S.sub.i] x (1 + C)].
To find the value of C, we can solve for C in the following
equation:
[summation over (i)] 1/[S.sub.i] x (1 + C) =
(I/U)+(1/E)/(1/U)+(1/E)+(1/O).
Using 20 matches between August 16, 2008, and August 25, 2008, we
find that the average commission C = 28.1%.
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Jaiho Chung * and Joon Ho Hwang ([dagger])
* Business School, Korea University, Anam-dong, Seongbuk-Gu, Seoul
136-701, Korea; E-mail
[email protected].
([dagger]) Business School, Korea University, Anam-dong,
Seongbuk-Gu, Seoul 136-701, Korea; E-mail
[email protected];
corresponding author.
The authors would like to thank Leighton Vaughan Williams for
organizing the Symposium on Gambling, Prediction Markets and Public
Policy. We would also like to thank David Paton, Donald Siegel, Peter
Norman Sorensen, two anonymous referees, and seminar participants at the
Symposium on Gambling, Prediction Markets and Public Policy at
Nottingham Trent University and the Research Workshop on Prediction
Markets at the University of Copenhagen. The financial support from the
Korea University Business School is gratefully acknowledged.
(1) On the other hand, there is also some evidence of a reverse
favorite-longshot bias for betting markets in some Asian horse tracks
(Busche and Hall 1988) and in other sports, such as baseball and hockey
(Woodland and Woodland 1994, 2001).
(2) In the United Kingdom, the brand of the totalisator is the
"tote." In Australia, Tabcorp Holdings, through the
privatization of the Victorian Totalisator Agency Board (TAB) in 1994,
is the major operator of the totalisator. We thank the referee for
providing us with this information.
(3) With regard to studies that examine sports betting markets
outside of horse tracks, Zuber, Gandar, and Bowers (1985); Gandar et al.
(1988); Dare and MacDonald (1996); and Gray and Gray (1997) investigate
the National Football League (NFL) betting market; Camerer (1989) and
Brown and Sauer (1993) study the National Basketball Association (NBA)
betting market; Woodland and Woodland (1994) examine the Major League
Baseball (MLB) betting market; and Woodland and Woodland (2001)
investigate the National Hockey League (NHL). There are also numerous
studies on soccer betting, including Pope and Peel (1989); Forrest and
Simmons (2000); Forrest, Goddard, and Simmons (2005); and Graham and
Stott (2008).
(4) Metrick (1996) analyzed 24 cases of National Collegiate
Athletic Association (NCAA) basketball tournament betting pools, which
function similarly to a sports lottery. However, he did not examine the
efficiency of the betting pool against the bookmaker market.
(5) In Europe, many countries operate their own sports lottery (for
example, Svenska Spel in Sweden), most of which are governed by the
European State Lottery and Toto Association. For a study of football
(soccer) pools in the United Kingdom, see Fon'est (1999).
(6) As for theoretical models that compare the parimutuel and
bookmaker markets, Ottaviani and Sorensen (2005) model the equilibrium
in each of the two markets under the presence of privately informed
bettors. In doing so, they extend Shin (1991, 1992) by introducing
expost competition among bookmakers. Koch and Shing (2008) argue that
the coarse grid odds offered by the bookmaker worsen the odds for the
longshot, whereas odds are almost continuous in the parimutuel market;
this results in the favorite-longshot bias in the bookmaker market but
not in the parimutuel market. Vaughan Williams and Paton (1998) propose
a general theory of explaining the bias using transaction costs, the
extent of public information, and the consumption benefit from betting.
Onaviani and Sorensen (2008b) provide an overview of the theoretical
explanations for the bias, including those based on the misestimation of
probabilities, market power of informed bettors and uninformed
bookmaker, preference for risk, heterogeneous beliefs, and limited
arbitrage by informed bettors.
(7) As for the two types of lottery, Sports Yoto and Sports Proto,
the football fixtures offered therein and the rules of the lottery are
different, thereby making it impossible to conduct any market efficiency
comparison between the two types of sports lottery.
(8) As of September 2008, the website is accessible only in Korea
and is closed to access from foreign IP addresses. When we asked the
customer representative the reason for the limited access, we were told
that it was for the protection of user information and website security.
Also mentioned were plans for opening up the website to access from
outside of Korea; however, the exact date of opening was not determined
at the time of the discussion. A reading of the bulletin board of users
suggests that participants who reside outside of Korea are able to
indirectly access the site using Korean proxy servers.
(9) Busche and Walls (200l) argue that the rounding down of odds in
horse tracks, known as breakage, can be a cause of the favorite-longshot
bias because breakage imposes a proportionately larger cost on favorites
than on longshots. We do not have such a problem in our sample as odds
are rounded off rather than rounded down.
(10) Direct evidence that the Korean sports lottery participants
are using information from the U.K. bookmaker market comes from the user
bulletin board of the Korean sports lottery website. In the bulletin
board, we were able to observe numerous posting of the current U.K.
bookmakers' odds information by the website users.
(11) A scan of the online user forum revealed that some
participants are betting with greater stakes, such as 1 million Korean
Won (approximately US$1000 or 650 [pounds sterling]). This is possible
because the identity of the purchaser is not recorded when wagering
takes place through an offline vendor, and therefore the participant can
place multiple wagers across different vendors.
(12) No lotteries were offered in 2006 for matches in the English
Premier League. For the two-game lottery, it started offering games in
the English Premier League from 2007.
(13) The pre-tax payout ratio of 50% implies that the expected
pre-tax return for all bets at the aggregate level is 50%. For an
individual bettor, the participation decision will depend on his or her
assessment of the expected return after observing other
participants' selections and the resulting odds in the sports
lottery pool. For example, if the bettor observes that a specific
outcome is underbet by a significant amount, the bettor will decide to
participate in the sports lottery. We thank the referee for pointing
this out.
(14) Bettors in the bookmaker market can make this form of bet,
which is known as "trebles" or "parlay," where
multiple bets must all win for the winnings to be collected.
(15) By way of comparison, Gabriel and Marsden (1990) show an
[R.sup.2] of 0.193 in their study of horse tracks.
(16) If we restrict the intercept to be zero in the linear
regression, the coefficients of the slopes are 2.82 for the three-game
lottery and 2.57 for the two-game lottery, and both are still
significantly greater than unity.
(17) The bookmaker's commission can vary for different types
of bets. However, for the specific type of bet (the correct final score
bets) we are examining, the null hypothesis is that the bookmaker's
commission is the same across different odds. We thank the referee for
pointing this out.
(18) We thank the referee for pointing this out.
(19) Snowberg and Wolfers (2008) argue that the favorite-longshot
bias is based on bettors' perception of overestimating the chances
of low-probability events. More recent studies on explaining the
different degrees of favorite-longshot bias include Peel and Law (2009)
and Bruce et al. (2009). Paton, Siegel, and Vaughan Williams (2009)
provide a nice review of these recent research studies.
(20) We thank Adrian Murdock and Michael Robb at Betfair.com for
providing us with the data.
(21) We thank the referee for suggesting the analyses on the
alternative specifications of the regression.
(22) This is because no odds are available for unusually high
scores such as 7-4, 5-5, or 8-0, even though they have non-zero chances
of occurrence.
Table 1. Results of the Korean Sports Lottery
Game 1
Year Date Home Team Away Team Score
Panel A: Three-game lottery
2005 Aug 13 Everton Man Utd 0 2
2005 Sep 18 Liverpool Man Utd 0 0
2007 Oct 20 Arsenal Bolton 2 0
2007 Oct 27 Man Utd Middlesbrough 4 1
2007 Nov 03 Arsenal Man Utd 2 2
2007 Nov 12 Tottenham Wigan 4 0
2007 Nov 25 Bolton Man Utd 1 0
2007 Dec 02 Portsmouth Everton 0 0
2007 Dec 09 Middlesbrough Arsenal 2 1
2007 Dec 16 Derby Middlesbrough 0 1
2007 Dec 22 Arsenal Tottenham 2 1
2008 Jan 20 Fulham Arsenal 0 3
2008 Jan 31 Chelsea Reading 1 0
2008 Feb 03 Portsmouth Chelsea 1 1
2008 Feb 10 Derby Tottenham 0 3
2008 Feb 24 Fulham West Ham 0 1
2008 Mar 02 Birmingham Tottenham 4 1
2008 Mar 22 Tottenham Portsmouth 2 0
Mean
Median
Minimum
Maximum
Panel B: Two-game lottery
2007 Oct 20 Arsenal Bolton 2 0
2007 Oct 27 Man Utd Middlesbrough 4 1
2007 Nov 03 Arsenal Man Utd 2 2
2007 Nov 12 Tottenham Wigan 4 0
2007 Nov 25 Bolton Man Utd 1 0
2007 Dec 02 Portsmouth Everton 0 0
2007 Dec 09 Middlesbrough Arsenal 2 1
2007 Dec 16 Derby Middlesbrough 0 1
2007 Dec 22 Arsenal Tottenham 2 1
2008 Jan 30 Fulham Arsenal 0 3
2008 Jan 31 Chelsea Reading 1 0
2008 Feb 03 Portsmouth Chelsea 1 1
2008 Feb 10 Derby Tottenham 0 3
2008 Feb 24 Fulham West Ham 0 1
2008 Mar 02 Birmingham Tottenham 4 1
2008 Mar 22 Tottenham Portsmouth 2 0
Mean
Median
Minimum
Maximum
Game 2
Year Home Team Away Team Score
Panel A: Three-game lottery
2005 Aston Villa Bolton 2 2
2005 Blackburn Newcastle 0 3
2007 Fulham Derby 0 0
2007 Tottenham Blackburn 1 2
2007 Fulham Reading 3 1
2007 Man Utd Blackburn 2 0
2007 Arsenal Wigan 2 0
2007 Wigan Man City 1 1
2007 Tottenham Man City 2 1
2007 Portsmouth Tottenham 0 1
2007 Fulham Wigan 1 1
2008 Tottenham Sunderland 2 0
2008 Everton Tottenham 0 0
2008 Tottenham Man Utd 1 1
2008 Everton Reading 1 0
2008 Liverpool Middlesbrough 3 2
2008 Fulham Man Utd 0 3
2008 Middlesbrough Derby 1 0
Mean
Median
Minimum
Maximum
Panel B: Two-game lottery
2007 Fulham Derby 0 0
2007 Tottenham Blackburn 1 2
2007 Fulham Reading 3 1
2007 Man Utd Blackburn 2 0
2007 Arsenal Wigan 2 0
2007 Wigan Man City 1 1
2007 Tottenham Man City 2 1
2007 Portsmouth Tottenham 0 1
2007 Fulham Wigan 1 1
2008 Tottenham Sunderland 2 0
2008 Everton Tottenham 0 0
2008 Tottenham Man Utd 1 1
2008 Everton Reading 1 0
2008 Liverpool Middlesbrough 3 2
2008 Fulham Man Utd 0 3
2008 Middlesbrough Derby 1 0
Mean
Median
Minimum
Maximum
Game 3
Year Home Team Away Team Score
Panel A: Three-game lottery
2005 Middlesbrough Liverpool 0 0
2005 Man City Bolton 0 1
2007 Middlesbrough Chelsea 0 2
2007 Sunderland Fulham 1 1
2007 Middlesbrough Tottenham 1 1
2007 Bolton Middlesbrough 0 0
2007 Middlesbrough Aston Villa 0 3
2007 Reading Middlesbrough 1 1
2007 Bolton Wigan 4 1
2007 Fulham Newcastle 0 1
2007 Middlesbrough West Ham 1 2
2008 Reading Man Utd 0 2
2008 Man Utd Portsmouth 2 0
2008 Blackburn Everton 0 0
2008 Middlesbrough Fulham 1 0
2008 Newcastle Man Utd 1 5
2008 Middlesbrough Reading 0 1
2008 Newcastle Fulham 2 0
Mean
Median
Minimum
Maximum
Panel B: Two-game lottery
2007
2007
2007
2007
2007
2007
2007
2007
2007
2008
2008
2008
2008
2008
2008
2008
Mean
Median
Minimum
Maximum
After-Tax
Winning Total
Year Payout Bet Amount
Panel A: Three-game lottery
2005 2086.5 1,551,239,000
2005 5664.7 2,132,140,000
2007 198.5 515,507,900
2007 286.4 488,752,200
2007 1577.6 509,572,800
2007 2266.1 579,813,500
2007 6605.2 494,523,000
2007 635.8 487,055,400
2007 9620.4 611,740,700
2007 429.0 536,675,200
2007 92.9 551,584,600
2008 199.1 831,048,200
2008 188.1 889,009,500
2008 306.4 738,088,100
2008 132.4 662,848,100
2008 10,677.6 561,243,200
2008 68,745.1 810,836,000
2008 351.3 654,494,900
Mean 6114.6 755,898,461
Median 532.4 595,777,100
Minimum 92.9 487,055,400
Maximum 68,745.1 2,132,140,000
Panel B: Two-game lottery
2007 42.7 47,547,000
2007 75.0 55,017,700
2007 136.3 64,872,000
2007 171.9 71,677,600
2007 82.3 63,293,000
2007 68.0 60,920,500
2007 94.4 77,529,000
2007 51.9 64,517,500
2007 16.2 68,218,800
2008 36.8 80,604,700
2008 35.2 82,657,400
2008 41.4 74,071,900
2008 34.1 70,411,800
2008 321.4 55,490,300
2008 2505.1 87,345,000
2008 30.8 73,911,800
Mean 234.0 68,630,375
Median 60.0 69,315,300
Minimum 16.2 47,547,000
Maximum 2505.1 87,345,000
This table shows the results for the two types of Korean sports
lottery, "soccer special triple" (three-game lottery) and "soccer
special double" (two-game lottery). In the three (two)-game Korean
lottery, participants need to pick the correct final score for three
(two) predetermined football matches. For our study, we limit
lotteries to those that feature English Premier League games. The
after-tax winning payouts are stated in multiples of the bet amount,
after the deduction of a tax of 22% for winnings whose quoted odds
were greater than 100. The total bet amount is shown in Korean Won,
where 1000 Korean Won are equivalent to approximately US$1. Panel A
shows the results for the three game lottery. Panel B shows the
results for the two-game lottery.
Table 2. Comparison of Average Winning Bets
Number of
Observations Sports Lottery
Panel A: Three-game lottery
All events 18 (54 games) 6114.6 (15,993.0)
Lotteries with win 9 (27 games) 242.7 (108.3)
payouts less than
the median win
payout of 532.4
Lotteries with win 9 (27 games) 11,986.6 (21,585.5)
payouts greater
than the median
win payout of
532.4
Panel B: Two-game lottery
All events 16 (32 games) 234.0 (610.4)
Lotteries with win 8 (16 games) 36.1 (10.4)
payouts less than
the median win
payout of 60
Lotteries with win 8 (16 games) 431.8 (841.9)
payouts greater
than the median
win payout of 60
Bookmaker Difference
Panel A: Three-game lottery
All events 2166.6 (3150.3) 3948.0 (-14,477.5)
Lotteries with win 535.6 (309.7) -292.9 *** (311.1)
payouts less than
the median win
payout of 532.4
Lotteries with win 3797.6 (3874.0) 8188.9 ** (20,120.5)
payouts greater
than the median
win payout of
532.4
Panel B: Two-game lottery
All events 146.6 (167.1) 87.3 (460.7)
Lotteries with win 58.6 (10.5) -22.5 ** (13.7)
payouts less than
the median win
payout of 60
Lotteries with win 234.6 (205.0) 197.2 (653.5)
payouts greater
than the median
win payout of 60
This table shows the average after-tax winning payout stated as a
multiple of the bet amount for the Korean sports lottery market and
for the corresponding payoff from the bookmaker market of William
Hill PLC. Panel A shows the results for a three-game Korean sports
lottery, where participants need to pick the correct final score for
three predetermined football matches that are played in the English
Premier League. Panel B shows the results for a two-game lottery.
Standard deviations are in parentheses. The significant differences
are indicated at 5% and 1% levels by ** and ***, respectively, using
a Wilcoxon matched-pairs signed rank test.
Table 3. Regression of Winning Bets in the Parimutuel Sports Lottery
against Corresponding Bets in the Bookmaker Market
Right-Hand Side Variable:
Corresponding Payout from the
Dependent Variable: Bookmaker Market
Winning Odds from the
Sports Lottery Market Posted Odds Fair Odds
Panel A: Three-game lottery
Slope 2.83 ** (1.05) 1.34 ** (0.50)
Intercept -9.06 (3954.19) -9.06 (3954.19)
[R.sup.2] 0.31 0.31
F value 7.19 ** 7.19 **
Panel B: Two-game lottery
Slope 3.37 *** (0.38) 2.05 *** (0.23)
Intercept -260.36 *** (82.11) -260.36 *** (82.11)
[R.sup.2] 0.85 0.85
F value 80.37 *** 80.37 ***
This table shows the results of an ordinary (cast squares regression
where the dependent variable is the after-tax payout from the
winning sports lottery ticket and the independent variable is the
corresponding payoff from the bookmaker market of William Hill PLC.
In the first column of the results, the payoff from the bookmaker
market is calculated using the quoted odds. In the second column of
the results, we use fair odds = (quoted odds) * (1.281) to account
for the 28.1% average commission charged by bookmakers for the
correct final-score betting. The procedure for calculating the
average commission is shown in the Appendix. Panel A shows the
results for the three-game Korean lottery, where participants need
to pick the correct final score for three predetermined football
matches that are played in the English Premier League. Panel B shows
the same results for the two-game lottery. Standard errors are in
parentheses. F values are for testing the null hypothesis of
intercept = 0 and slope = I. Significant coefficients and F values
are indicated at 5% and 1% levels by ** and ***, respectively.
Table 4. Regression of Winning Bets in the Parimutuel Sports Lottery
against Corresponding Bets in the Betting Exchange Market
Right-Hand Side Variable: Corresponding
Payout from the Betting Exchange
Dependent Variable:
Winning Odds Odds at Sports
from the Sports Lottery Closing
Lottery Market Time Odds at Kickoff
Slope 1.12 (0.09) 1.07 (0.09)
Intercept -258.00 ** (109.98) -240.68 * (125.44)
[R.sup.2] 0.89 0.85
F value 172.83 *** 127.45 ***
This table shows the results of an ordinary least squares regression
where the dependent variable is the after-tax payout from the
winning sports lottery ticket, and the independent variable is the
corresponding payoff from the betting exchange market of
Betfair.com. In the first column of the results, the payoff from the
betting exchange market is calculated using betting exchange odds
for the last transaction before the Korean sports lottery closing
time. In the second column of the results, the payoff from the
betting exchange market is calculated using betting exchange odds
for the last transaction before the start of the match. Standard
errors are in parentheses. F values are for testing the null
hypothesis of intercept = 0 and slope = I. Significant coefficients
and F values are indicated at 10%, 5%, and I% levels by *, **, and
***, respectively.
Table 5. Alternative Specifications for the Regression of Winning
Bets in the Parimutuel Sports Lottery against Corresponding Bets in
the Bookmaker Market
Right-Hand
Side Variables Model 1 Model 2
Panel A: Nonlinear
relationship based
on the level of
payout
Bookmaker odds
[(Bookmaker odds). 0.06 (2.57) 0.84 (3.01)
sup.2] x (greater-
than-median payout
dummy)
Log (bookmaker odds) 2.61 *** (0.88) 3.13 ** (1.33)
x (less-than-median
payout dummy)
Intercept -1613.51 (3100.37)
Adjusted [R.sup.2] 0.17 0.11
F value 4.44 ** 2.96 *
Panel B: Nonlinear
relationship based
on the type of
lottery market
Bookmaker odds
[(Bookmaker odds). 0.49 (2.19) 0.96 (2.38)
sup.2] x (greater-
than-median payout
dummy)
Log (bookmaker odds) 2.45 *** (0.86) 2.97 ** (1.31)
x (less-than-median
payout dummy)
Intercept -1613.31 (3002.08)
Adjusted [R.sup.2] 0.15 0.09
F value 4.05 ** 2.62 *
Right-Hand
Side Variables Model 3 Model 4
Panel A: Nonlinear
relationship based
on the level of
payout
Bookmaker odds 2.70 *** (0.86) 2.68 *** (0.89)
[(Bookmaker odds). -0.50 (2.29) -0.29 (2.70)
sup.2] x (greater-
than-median payout
dummy)
Log (bookmaker odds) 0.24 (l.09) 0.39 (1.49)
x (less-than-median
payout dummy)
Intercept -427.04 (2786.81)
Adjusted [R.sup.2] 0.35 0.29
F value 7.03 *** 5.54 ***
Panel B: Nonlinear
relationship based
on the type of
lottery market
Bookmaker odds 2.76 *** (0.85) 2.74 *** (0.87)
[(Bookmaker odds). 0.06 (1.93) 0.21 (2.12)
sup.2] x (greater-
than-median payout
dummy)
Log (bookmaker odds) 0.12 (1.05) 0.30 (l.43)
x (less-than-median
payout dummy)
Intercept -501.15 (2673.29)
Adjusted [R.sup.2] 0.35 0.29
F value 6.98 *** 5.50 ***
This table shows the results of alternative specifications of
regression where the dependent variable is the after-tax payoff from
the winning sports lottery ticket, and the independent variable is
the corresponding payoff from the bookmaker market of William Hill
PLC. The specification of the nonlinear regression model is
[s.sub.i] = [alpha] + [[beta].sub.0] x [m.sub.i] + [[beta].sub.1] x
[D.sub.1] x [m.sup.2.sub.i] + [[beta.sub.2] x [D.sub.2] x
log([m.sub.i]) + [[epsilon].sub.i], where [s.sub.i] is the winning
payout from the Korean sports lottery market as shown in Equation 4,
and [m.sub.i] is the corresponding payoff from the U.K. bookmaker
market as shown in Equation 5. In panel A, [D.sub.1] is a dummy
variable that takes a value of 1 for outcomes with payoffs greater
than the median payoff in our sample. [D.sub.2] is a dummy variable
that takes a value of 1 for outcomes with payoffs less than the
median payoff in our sample. In panel B, [D.sub.1] represents the
three-game lottery dummy and [D.sub.2] represents the two-game
lottery dummy. Standard errors are in parentheses. Significant
coefficients and F values are indicated at 10%, 5%, and 1% levels by
*, **, and ***, respectively.
