### Abstract
Our study used the log likelihood ratio methodology proposed by Even and Noble (2) to test the market efficiency of both point spread betting and totals betting for consecutive National Basketball Association (NBA) seasons from 2000–01 to 2007–08. It was motivated by recent contradictory evidence that both support and reject opportunities to exploit inefficiencies in NBA gambling by Paul and Weinbach (9, 11) as well as other evidence suggesting that these opportunities fade as the market responds to new information (12).
Based on the results of over 10,000 games in eight consecutive NBA seasons, betting the over on the total points per game is a fair bet, indicating an efficient market. For the higher totals (totals 211-220), the winning percentage on betting the over was above 52.38% (the percentage necessary to cover commissions) in eight of 10 cases, but the null hypothesis of a fair bet could not be rejected. The results for point spread betting also showed strong support for an efficient market in NBA gambling, with one exception: betting the home underdog was profitable for underdogs of 10 points or more. However, this was only true for a very small sub-sample and the inefficiency fades in the most recent sample period.
The few cases of big home underdogs beating the spread are consistent with the model of spread betting where bookmakers exploit the uninformed investor’s home favorite bias, shade the point-spread and maximize profits by betting on the underdog (7,6). Informed bettors may also bet the underdog but will not drive the point spread to the true value but only to the point where the probability of winning is no more than 52.38% (11). While bookmaker’s point shading activity is constrained by the action of informed bettors, the persistence of profit opportunities in a very small sub-sample can be explained by betting market constraints such as low limits on bets and the relative volume of bets placed by informed and uninformed bettors (9).
**Key Words:** point spreads, totals, National Basketball Association, NBA, gambling
### Introduction
Studies of market efficiency in sport betting are similar to those in the financial markets for good reason. Both markets involve many market participants and large sums of money, both involve informed and uninformed traders, market frictions, asymmetric information, and, as the weight of the evidence shows, both are heavily influenced by market psychology. In both markets, however, claims of abnormal returns and profitable strategies always raise a red flag. Like the anomalies literature in financial markets, claims of exploitable inefficiencies must be validated with out-of-sample tests to confirm that these inefficiencies are not confined to specific periods, or are driven by a few outliers in the data, or are simply artifacts of extensive data mining. Sport betting provides a unique test for market efficiency since the payoffs are known with certainty in advance of the outcome and the final outcome is determined when the game is played. This is not the case with equity investing (1).
The market for sports betting consists of a market maker, called a bookmaker or sports book, and a bettor. The bookmaker establishes the lines at which betting commences and then moves the line as bets are wagered on both sides of the line. Bettors typically pay the bookmaker $11 to win $10, providing the bookmaker a commission profit if money on both sides of the bet are balanced. Because of this commission, commonly called the “vig” or “juice”, bettors must win 52.38% of their bets to break even. A winning percentage greater than 52.38% insures a profit for the bettor. Recent evidence using data on dollars wagered has rejected the claim that bookmakers strive to balance the dollar on both sides of a wager and lends support to the argument that bookmakers attempt to set the line to accurately reflect actual game outcomes (6,7,11).
In the sports gambling world, an over/under or totals wager is a bet that is won or lost depending upon the combined score of both teams in a game. A bookmaker will predict the combined score of the two teams and bettors will bet that the actual number of points scored in the game will be higher or lower than that combined score. For example, in an NBA game of the Miami Heat versus the San Antonio Spurs the over/under for the score of the game was set at 195. A bet on the under wins the wager if the combined score at the end of the game is 194. If the combined score is 196 or more, then the over bet wins. If the combined score equals 195, then it is a tie and the bettor’s money is returned.
### Data And Methodology
This study was designed to test for the presence of exploitable inefficiencies in NBA sport gambling. Recent research in NBA gambling has produced evidence of over betting the over in totals betting, and over betting the favorite by uninformed bettors in point spread betting. The research also claims that there are profitable opportunities in betting the big underdog. This study tests those claims by examining both totals betting and point spread betting using updated data.
The data for studying the totals and point spread markets for National Basketball Association games was taken from the Gold Sheet, a well-known handicapping company, for eight NBA seasons 2000-01 through 2007-08. The data included all games from these years, both regular season and playoffs, except for games where totals or point spreads were not posted. Table 1 shows the summary statistics for the 10,325 games included in the sample. Five of the games had no line posted for the over/under and 175 games were ties. The average or mean actual total score for our sample of NBA games was 192.72 points and the average or mean over/under total for the sample was 192.27 total points per game.
The log likelihood ratio methodology proposed by Even and Noble (2) was used to test for market efficiency for the over/under betting market in the NBA. From the perspective of the over bettor, the value of the unrestricted log likelihood function (Lu) takes the form
> Lu = n[ln(q)] + (N – n)ln(1 – q) (1)
where N is the total number of NBA games where the over bettor or under bettor won the bet. The n is the number of games where the over covers the bet, and q is the proportion of games where the over covers the bet. If the betting market is efficient and a fair bet, then q = 0.5.
This creates the restricted log likelihood function (Lr), which was obtained by substituting 0.5 for q in Equation 2. The log likelihood ratio statistic for the null hypothesis that q = 0.5 is
> 2(Lu – Lr) = 2{n[ln(q) – ln(0.5)] + (N – n)[ln(1 – q) – ln(0.5)]} (2)
where q is the actual percentage of overs winning the over/under bet from our sample. To test for profitability, where the bettor must win enough to offset the commission or vigorish of the bookmaker, the test ratio changes into
> 2(Lu – Lr ) = 2{n[ln(q) – ln(0.524)] + (N – n)[ln(1 – q) – ln(0.476)]}. (3)
### Results And Discussion
#### Totals Betting
In a 2004 study, covering the seven NBA seasons from 1995-96 through 2001-2002, Paul et al.(8) found that, for all games, a bet on the underdog won about 50% of the time, as is expected in an efficient market. However, for the high scoring games (games above 200), they found a pattern of over betting the over, and this pattern increased as game totals increased. For every one point increase from 200 to 210, the winning percentage of the under bet was greater than 50%. In eight of those totals the winning percentage was greater than 52.38%, enough to cover the vigorish, and in five of those totals, the null hypothesis of a fair bet was rejected. However, none of the totals in their study produced a result that rejected the null of no profitability when accounting for commissions. Taking the contrarian bet, and betting against market sentiment, was not profitable. In a later study, using data on actual dollar amounts wagered, Paul and Weinbach (11) found that overs received a much higher percentage of bets compared to unders, but here again it was shown that informed bettors pushed the total to where it was not profitable to bet the under.
The results found the opposite of the 2004 study (8) for the high scoring games. For all games in the eight seasons from 2000-01 through 2007-08, a bet on the underdog still won about 50% of the time. However, a bet on the over won more often than a bet on the under for high scoring games. The game results, and the log likelihood test of efficiency, are reported in Table 2. For game totals between 200 and 210, the winning percentage of the over bets hover right around 50%, indicating an efficient market. When we extended the testing to higher totals (211-220) the percentage of over winners was more than the commission breakeven point (52.38%) for eight of the 10 totals. However, in no instance was the log likelihood ratio large enough to reject the null hypothesis of a fair bet.
Point Spread Betting and Betting the Underdog
When an NBA gambler bets the point spread of an NBA game he is not interested in who wins the game, only the final score. For example, if the point spread for a National Basketball Association game reads
> Heat -4 Pacers +4
The (-) before the 4 indicates that the Heat is the point spread favorite. The (+) indicates that the Pacers are the point spread underdog. If one bets on the Heat, the Heat would have to win by a total of five points for the bettor to win. If one bets on the Pacers, the Pacers would have to win outright or lose by no more than three points for the bettor to win. A four point victory by the Heat (four point loss by the Pacers) would equal a tie and the money bet by the NBA gambler is returned to him.
Prior evidence suggests that there are systemic bettor misperceptions in the NBA point spread gambling market. In a 2005 study Paul and Weinbach (9) presented evidence from the 1995-96 through 2001-2002 seasons that favorites are over bet by uninformed bettors. In that study, a strategy of betting big underdogs rejected the null hypothesis of a fair bet, and betting big home underdogs not only rejected a fair bet was also profitable. Levitt (7) provides us with a model where bookmakers do not attempt to balance the dollars wagered, but rather they shade the point spread to exploit uninformed bettor bias and then take positions on the opposite side, betting the big underdog. Informed bettors may attempt to exploit this inefficiency by also betting the big underdog but will only bet to the point where it is profitable to do so, meaning that they may bet on the underdog and push the point spread only to where there is no less than 52.38% chance of winning the bet. Other studies (6, 11), using data on actual dollars wagered, have found that a majority of dollars are wagered on the stronger or favorite team by uninformed bettors.
This study examined the NBA betting market on point spreads for the seasons 2000-01 through 2007-08 to see if this underdog anomaly persists. It used the closing line on point spreads for NBA games for the same seasons that we examined in the over/under analysis performed in the previous section of the paper. For the market to be efficient the actions of the informed bettors should offset any bias shown by uninformed bettors and the bookmakers closing line should equal the actual game score outcome. Recent studies have shown that the betting public removes biases in sport book’s opening lines in NBA betting by game time (3-5).
Table 3 is a summary of the data for point-spread betting. The sample contained 10,325 games with five of the games posting no closing line to bet on and 90 games posting a closing line of zero. This is called a push and these games were not included when betting favorites and underdogs. There were 141 ties which indicate that the difference in the score (underdog – favorite) was equal to the closing point spread. The average closing line based on the favorite score minus the underdog score was 5.89 and actual difference in score in the NBA games in the sample was 5.38. For the entire sample of games the underdog won 49.86% of the games, indicating that a strategy of betting the underdog was a fair bet, based on the log likelihood ratio test.
The results in Table 4 indicate that the betting public appears to over bet the heavy favorite by a slight margin, but, unlike the study by Paul and Weinbach (9), we found that the winning percentage of betting the big underdog (10 points or more) hovered around 50% and thus we failed to reject the null hypothesis of a fair bet. The same result occurred for the sub-sample of games for seasons 2000-01 through 2003-04 and for the sub-sample of games for seasons 2004-05 through 2007-08. In all of these cases the null hypothesis of a fair bet could not be rejected.
The results for the small sample of games involving the home underdog of 10 points or more had significant results for both a fair bet and profitability. For the entire sample of games (50 games over the entire seasons) the null hypothesis of a fair bet was rejected at a 10% significance level. For the small sample of games in the earlier sub-period (25 games) we found that a bet on the home underdog also rejected the null hypothesis of no profitability.
### Conclusion
This study found that gambling markets for both point spread betting and totals betting for NBA seasons spanning from 2000–01 to 2007–08 are efficient. Based on the results of over 10,000 games in eight consecutive NBA seasons, betting the over on the total points per game is a fair bet. Although for higher totals (211-220) the winning percentage on betting the over was above 52.38% (the percentage necessary to cover commissions), in eight of 10 cases the null hypothesis of a fair bet could not be rejected. The results for point spread betting also showed strong support for an efficient market in NBA gambling, with one exception: betting the home underdog was profitable for underdogs of 10 points or more. However, this was only true for a very small sub-sample and the inefficiency fades in the most recent sample period.
### Applications In Sports
Many fans enjoy wagering on their favorite sport whether it is NBA basketball or another sport. Gambling can be fun and can enhance the excitement of the game by adding a financial component. The evidence suggests that the average bettor is biased toward high scores and prefers betting on the favorite. However, utilizing this knowledge and betting on the underdog will probably not be a profitable strategy for a fan wagering on NBA games because of the actions of informed (professional) gamblers. The informed gambler will bet on the underdog until it is not profitable for him to do so. This activity drives the point spread to a level where a fan cannot make a profit on an underdog bet after accounting for commission. Therefore, the average gambler should focus on having fun and not count on making a profit when gambling on NBA games.
### Tables
#### Table 1
NBA Seasons 2000-01 Through 2007-08 Summary Statistics for Over/Under Betting for All NBA Games
| Totals | Actual game | |
|---|---|---|
| Mean | 192.27 | 192.72 |
| Median | 191 | 192 |
| Total games | 10,325 | |
| Games with no line | 5 | |
| Ties | 175 | |
| Over wins | 5,059 | |
| Under wins | 5,086 | |
| Winning % for betting overs | 49.87% | |
| Log likelihood | 0.07 |
#### Table 2
Winning Percentages for Betting the Overs
| Point level | Over/Under winners | Winning % of betting the over | Log likelihood ratio for fair bet |
|---|---|---|---|
| 200 | 1252-1234 | 50.36 | 0.13 |
| 201 | 1139-1131 | 50.18 | 0.03 |
| 202 | 1022-1027 | 49.88 | 0.01 |
| 203 | 919-914 | 50.14 | 0.01 |
| 204 | 801-796 | 50.16 | 0.02 |
| 205 | 699-695 | 50.14 | 0.01 |
| 206 | 621-625 | 49.84 | 0.01 |
| 207 | 542-547 | 49.77 | 0.02 |
| 208 | 470-474 | 49.79 | 0.02 |
| 209 | 415-401 | 50.86 | 0.24 |
| 210 | 66-339 | 51.91 | 0.52 |
| 211 | 321-290 | 52.54 | 1.57 |
| 212 | 282-246 | 53.41 | 2.46 |
| 213 | 239-214 | 52-76 | 1.38 |
| 214 | 210-183 | 53.43 | 1.86 |
| 215 | 186-156 | 54.39 | 2.63 |
| 216 | 162-136 | 54.39 | 2.63 |
| 217 | 139-127 | 52.26 | 0.54 |
| 218 | 114-102 | 52.78 | 0.67 |
| 219 | 93-88 | 51.38 | 0.14 |
| 220 | 80-71 | 52.98 | 0.53 |
Note. The log likelihood test statistics have a chi-square distribution with one degree of freedom.
Critical values are 2.706 (for an α = 0.10), 3.841 (for an α = 0.05), 6.635 (for an α = 0.01).
* is significant at 10%.
** is significant at 5%.
*** is significant at 1%.
#### Table 3
Closing Line Betting Seasons 2000-01 Through 2007-08
| Total games | 10,325 |
| Average closing line (favorite – dog) | 5.89 |
| Average actual score difference (favorite – dog) | 5.38 |
| Games with no point spread line | 5 |
| Ties | 141 |
| Pushes | 90 |
| Neutral sites | 2 |
| Favorite wins | 5,058 |
| Underdog wins | 5,029 |
| Winning % for underdog | 49.86 |
| Log likelihood ratio | 0.01 |
#### Table 4
Betting the NBA Underdog Seasons 2000-01 Through 2007-08
| Seasons | Wins for underdog | Winning % | Log likelihood ratio fair bet | Log likelihood ratio no profitability |
|---|---|---|---|---|
| Point spread betting for all games | ||||
| 2000-01 thru 2007-08 | 5029 | 49.86 | 0.08 | NA |
| 2000-01 thru 2003-04 | 2448 | 49.62 | 0.28 | NA |
| 2004-05 thru 2007-08 | 2581 | 50.08 | 0.01 | NA |
| Betting underdog by +10 points or more | ||||
| 2000-01 thru 2007-08 | 689 | 52.08 | 2.28 | NA |
| 2000-01 thru 2003-04 | 319 | 51.45 | 0.52 | NA |
| 2004-05 thru 2007-08 | 370 | 52.63 | 1.95 | NA |
| Betting home underdog by +10 points or more | ||||
| 2000-01 thru 2007-08 | 50 | 59.52 | 3.07* | 1.72 |
| 2000-01 thru 2003-04 | 25 | 69.44 | 5.59** | 4.33** |
| 2004-05 thru 2007-08 | 25 | 65.79 | 0.08 | NA |
| Betting road underdog by +10 points or more | ||||
| 2000-01 thru 2007-08 | 639 | 51.57 | 1.23 | NA |
| 2000-01 thru 2003-04 | 294 | 50.34 | 0.03 | NA |
| 2004-05 thru 2007-08 | 345 | 52.67 | 1.87 | NA |
Note. The log likelihood test statistics have a chi-square distribution with one degree of freedom.
Critical values are 2.706 (for an α = 0.10), 3.841 (for an α = 0.05), 6.635 (for an α = 0.01).
* is significant at 10%.
** is significant at 5%.
*** is significant at 1%.
NA – not applicable
### References
1. Brown, W., Sauer, R. (1993). Fundamentals or noise? Evidence from the professional basketball betting market. Journal of Finance, 48, 1193–1209.
2. Evan, W. E., & Noble, N. R. (1992). Testing efficiency in gambling markets. Applied Economics, 24, 85-88.
3. Gandar, J., Zuber, R., O’Brien, T., & Russo, B. (1988). Testing rationality in the point spread betting market. Journal of Finance, 43, 995-1007.
4. Gandar, J., Dare, W., Brown, C., Zuber, R. (1998). Informed traders and price variations in the betting market for professional basketball games. Journal of Finance, 53, 385–401.
5. Gandar, J, Zuber, R. & Lamb, R. (2000). The home field advantage revisited: a search for the bias in other sports betting markets. Journal of Economics and Business, (53) 4, 439-453.
6. Humphreys, B. (2010). Point spread shading and behavioral biases in NBA betting market. Rivista Di Diritto Economia Dello Sport, 13-26.
7. Levitt, S. (2004). Why are gambling markets organized so differently? The Economics Journal, 114, 223-246.
8. Paul, R., Weinbach, A., Wilson, M. (2004). Efficient markets, fair bets, and profitability in NBA totals 1995–1996 to 2001–2002. The Quarterly Review of Economics, 44, 624–632.
9. Paul, R. J. & Weinbach, A. P. (2005). Bettor misperceptions in the NBA, Journal of Sports Economics, (6) 4, 390-400.
10. Paul, R. J. & Weinbach, A. P. (2007). Does Sportsbook.com set pointspreads to maximize profits? Tests of the Levitt model of sportsbook behavior. Journal of Prediction Markets, (1) 3, 209-218.
11. Paul, R. J. & Weinbach, A. P. (2008). Price setting in the NBA gambling market: Tests of the Levitt model of sportsbook behavior. International Journal of Sports Finance, (3) 3, 2-18.
12. Wever, S., & Aadland, D. (2010). Herd Behavior and the Underdogs in the NFL. Applied Economics Letters, (forthcoming).
### Corresponding Author
Kevin Sigler, PhD
601 S. College Road
Cameron School of Business
University of North Carolina-Wilmington
Wilmington, NC 28403
<siglerk@uncw.edu>
910-962-3605
William Compton is Associate Professor of Finance in the Cameron School of Business, UNCW Kevin Sigler is Professor of Finance in the Cameron School of Business, UNCW
