Point shaving in college basketball: a cautionary tale for forensic economics.

Bernhardt, Dan ; Heston, Steven

The study of crime and corruption has significant interest to economists. Of central interest is the actual extent to which individuals engage in illegal behavior. The problem is that the very laws that discourage corruption (Shleifer and Vishny 1993) also make it difficult to measure illegal activity, as criminals conceal their activities to avoid punishment.

Accordingly, economists have turned to indirect approaches to measure illegal activity. For example, Persico (2002) and Anwar and Fang (2006) use patterns in arrest records to detect racial profiling by police. Jacob and Levitt (2003) use the distribution and correlation of test scores to identify cheating by teachers. Porter and Zona (1993, 1999), McAfee and McMillan (1992), and Pesendorfer (2000) exploit implications of auction theory to uncover illegal bid rigging by auction cartels from the distribution of bids. Muelbroek (1992) documents the effect of insider trading on stock returns, and Christie and Schultz (1994) use the distribution of even-and odd-eighth quotes to detect collusion by NASDAQ market makers.

Researchers have also turned to sports to investigate corruption, exploiting the fact that sports settings provide abundant clean data. (1) Duggan and Levitt (2002) document match throwing in sumo wrestling by exploiting the highly nonlinear payoff structure. Wolfers (2006) and Gibbs (2007) conclude from asymmetric distributions in winning margins around the spread that point shaving--the illegal practice by favored teams of attempting to win basketball games by less than the point spread in order to yield profits for gamblers who bet on the underdog--is pervasive throughout college and professional basketball.

The danger with indirect methods is that, by their very nature, their identification of illegal activity hinges on subtle assumptions. For example, in auctions, it is crucial to identify competitive equilibrium bidding strategies. So, too, a suspicious price or point spread pattern is not sufficient to prove corruption or to identify its source. For example, Yermack (1997) concludes from negative stock returns prior to (unobserved) executive stock option grants and positive stock returns after that firms strategically disseminate bad news before the grants and good news after. Lie (2005) argues that this pattern reflects strategic backdating of the grants, and Heron and Lie (2007, 2009) distinguish their backdating interpretation by showing the return pattern falls off sharply following a 2002 SEC disclosure requirement. Indeed, many firms are now under investigation for illegal backdating. Our article's investigation of point shaving in college basketball highlights the necessity of identifying appropriate counterfactuals and robustness checks to validate findings. Betting in basketball typically involves a point spread handicap for the favored team. A wager on the favored team wins only if the favorite wins by more than the point spread, while a wager on the underdog pays off if that team wins the game outright or loses by less than the point spread. Because players care primarily about winning, while gamblers care about the winning margin, there is scope for profitable point shaving by strong favorites who are very likely to win.

Wolfers (2006) argues that gambling-related corruption is manifest in men's NCAA college basketball, and Gibbs (2007) makes the same claim for professional NBA basketball. Both papers begin with the key premise that since "the deviation of [winning margins] from the spread is a forecast error," then the margin of victory should be symmetrically distributed around the point spread. Such symmetry would imply that if there is no point shaving, then a team favored by S points should be as likely to win by less than S points, as to win by between S and 2S points. Wolfers finds that 46.2% of strong favorites--teams favored to win by more than 12 points--win, but fail to cover the spread, while only 40.7% win by between S and 2S points and concludes that between 1989 and 2005 5.5% of games with strong favorites featured point shaving. Wolfers' estimate implies systemic corruption, as either the very best teams that comprise the bulk of strong favorites shave points or else far more than 5.5% of teams sometimes shave points. Such pervasive levels of criminal activity, if true, would mandate massive policing in college sports and radical policy reform. Reflecting the extent of public interest in this issue, Wolfers' findings were the subject of feature articles in 14 newspapers, including the New York Times, Chicago Tribune, USA Today, Sports Illustrated and Barrons, as well as National Public Radio and CNBC TV, and highlighted by Ayres (2007) in "The Economist's Voice." The Atlantic Monthly highlighted the findings by Gibbs (2007).

There is historical precedent for point shaving. One of the largest scandals in sports history was a 1951 point shaving scandal at City College of New York, the defending national champion. Ultimately, seven schools and 32 players, including the Sporting News player of the year, were implicated for fixing 86 games. A second scandal in 1961 led to the arrest of 37 players from 22 schools for fixing 43 games. (2) More recent point shaving scandals emerged at Tulane and Boston College in the 1980s and Northwestern and Arizona State in the 1990s. These scandals led some schools to drop their major college basketball programs (CCNY, New York University, Long Island University) and others to suspend their programs for years (Kentucky, following its national championship in 1951, and Tulane).

Our article develops methods that allow us to distinguish whether patterns in winning margins are due to massive corruption or to the primitive data-generating process of the game of basketball itself. To do this, we use a sample of every Division I NCAA basketball game over the past 16 seasons, providing a definitive data set with tens of thousands of games. For games where lines are set, we obtain both the opening line quoted by bookmakers and the closing line quoted just before the game begins. We then use movements in the line to identify games where extensive gambling on point shaving is implausible. In addition to games with betting activity, our sample contains thousands of games for which active betting markets do not exist. These games provide a natural control for the effects of betting incentives.

We focus on a category of strong favorites who win 95% of their games. By shaving points, players on these teams can profit from bets on their opponents without jeopardizing their chances of victory. Consistent with Wolfers, we find that a team favored by S > 12 points is far more likely to win their games by a few points less than the point spread than a few points more than the spread--teams won by less than S 45.13% of the time, but won by (strictly) between S and 2S points only 38.97% of the time, yielding a "probability discrepancy" of 6.16%.

While this is suspicious, the pattern might alternatively emanate from skewness in the scoring process of basketball. To establish corruption, one must correlate the statistical pattern to other indicators of point shaving incentives and activity. We do this in three ways.

First, we exploit movements in point spreads. Just as in stock markets, where buy orders tend to drive share prices up, and sell orders tend to drive prices down (Bernhardt and Hughson 2002; Easley, Kiefer, and O'Hara 1997; Glosten and Harris 1989), if far more money is bet on the underdog than the favorite, the closing line will be less than the opening line. In stock markets, prices move because order flow may be from informed traders (Glosten and Milgrom 1985; Kyle 1985) and to compensate liquidity providers for holding unbalanced positions (Biais 1993; Ho and Stoll 1983). So, too, betting lines move to reflect information arrival (3) and to balance dollars bet on each side, which reduces a bookmaker's exposure to risk. Criminals who fix basketball games profit by betting significant amounts on the underdog that drive the point spread down. For example, point spreads for the four fixed Arizona State games in 1994 fell by an average of 2 points. By separating games with negative point spread movements from games with positive point spread movements, we control for the flow of gambling money.

We compute the difference in the "probability discrepancy" for games where the spread falls between open and close versus those where it does not. This difference-in-differences approach controls for features of the distribution of winning margins intrinsic to the game of basketball, isolating the portion due to point shaving. We find a statistically insignificant difference between the two probability discrepancies. Restricting attention to larger declines in the point spread or to narrower intervals around the line does not alter this conclusion.

This analysis precludes extensive point shaving on which there is large-scale gambling that moves the line. However, there remains the possibility of small-scale gambling on point shaving that does not alter point spreads. This possibility leads us to investigate patterns in games where bookmakers do not set gambling lines, where betting activity is insignificant. Bookmakers do not set lines for games between nonelite Division I teams that would not stimulate enough betting interest (e.g., Winthrop vs. Gardner-Webb) to justify the costs of determining the spread. Moreover, in games that would draw little recreational gambling, bookmakers would still draw substantial betting from informed professional gamblers were they to get the point spread wrong and hence lose money due to this strong adverse selection.

Without a liquid market for gambling in these games, there are no incentives to shave points. We first estimate expected winning margins and document that they have almost identical efficiency properties as point spreads--it almost appears that bookmakers use regression methods to set point spreads. We then compute probability discrepancies relative to the estimated winning margins. Comparing games with and without gambling, we find qualitatively and statistically identical patterns in the frequencies with which strong favorites do and do not cover the spread. If anything, the probability discrepancies are greater for games without gambling. We conclude that the asymmetric distribution of winning margins around the point spread also holds in games where point shaving is implausible and hence is not indicative of an epidemic of gambling-related corruption. (4)

This leaves the question--What accounts for the patterns in winning margins? What is the appropriate counterfactual? We show that strategic efforts by teams to maximize the probability of winning the game better reconciles the patterns in winning margins for all teams. In particular, for both strong favorites and weaker favorites (teams favored by 12 points or less), we find substantial "excess probability" relative to a normal probability density on moderate winning margins and probability "deficits" on both high winning margins and losing margins. Indeed, the excess probability on a winning margin of between 1 and 12 points for both strong favorites and weaker favorites is exactly the same, 3.4%. However, these "excess" probabilities on moderate winning margins have opposing implications vis-a-vis the point spread. For weaker favorites, high probabilities of moderate winning margins imply negative probability discrepancies relative to the point spread: there is more than one-quarter percentage point more probability mass on winning margins of S + J than on S - J for each of J = 1,..., 8, the opposite of what point shaving predicts. In contrast, for strong favorites, the high probabilities on moderate winning margins underlie the greater probability mass on winning by less than the spread that drove past point shaving conclusions.

In sum, our analysis emphasizes the caution with which one must evaluate indirect methods that seek to distinguish legal from illegal activity--before looking at the findings, one must evaluate the methods and recall that indirect reduced-form estimates are only as good as their assumptions.

I. EMPIRICAL ANALYSIS

The Logical Approach of Las Vegas, Nevada, provided us opening and closing lines for 44,546 NCAA Division I basketball games from 1990 to 2006 in which one team was favored at the closing line. Our data come from Las Vegas sportsbooks because they were the only legal commercial U.S. market for college basketball betting. An industry of professional bettors, arbitrageurs, scalpers, and competitive bookmakers ensures that point spreads are virtually identical across Vegas, offshore, and illegal domestic sportsbooks. Market participants compare lines using real-time odds services such as Don Best. Consequently, point spreads stay within a half a point of each other and move within seconds. (5)

In these games, the favorite won 21,346 times, the underdog won 21,376, and 993 were "pushes," where the winning margin exactly matched the closing point spread. Dropping pushes, 49.96% of favorites beat the spread. Table 1 presents the results from regressing winning margins on opening and closing lines. These results reveal that, individually, opening and closing lines are essentially mean unbiased, and when we include both opening and closing lines, the coefficient of one on the closing line and zero on the opening line indicates that the closing line subsumes the opening line and that the movements in the line reflect and capture information arrival. The actual game produces substantial unanticipated information, with an estimated residual standard error in the margin of victory of about 11 points.

These seeming "market efficiency" results mask the fact that strong favorites--teams favored by more than 12 points--performed less well, winning 4,388 games and losing 4,701, with 217 games pushed. (6) Dropping pushes, only 48.28% of strong favorites beat the spread. This bias is slight: reducing the point spread by l/2--the smallest amount possible--more than reverses the bias, with strong favorites beating the reduced spread 50.37% of the time. Among strong favorites, the average point spread was 17 points, which was also their average margin of victory, and strong favorites won 94.62% of the games outright.

Figure 1a shows the distribution of winning margins for weak and moderate favorites--teams favored by 12 points or less--around the closing line, and Figure lb shows this distribution for strong favorites. The qualitative features of both figures mirror those that Wolfers (2006) uncovers. In particular, winning margins for strong favorites are far more likely to fall just short of the spread than to just exceed the spread, yet strong favorites are more likely to win by a large "blowout" margin than the estimated normal distribution predicts. There were 9,306 games with a strong favorite, and 34,909 games with a weak or moderate favorite. The strong favorite won without covering the closing spread 45.13% of the time, but won by (strictly) between S and 2S points only 38.97% of the time, generating a probability discrepancy of 6.16% (SE, 0.95%). Under the maintained premises that (1) deviations from the line are forecast errors and (2) the forecast error distribution is symmetric about zero, it follows that 6.16% is the appropriate estimate of point shaving, as had a team not shaved, it still should have failed to cover the spread half of the time. This probability discrepancy estimate would indicate incredibly widespread point shaving and mandate a radical reform of college basketball, including a massive increase in policing.

[FIGURE 1 OMITTED]

Still, the pattern of weaker favorites in Figure la suggests caution: there is distinctly more probability mass slightly above the closing spread of S than below, the opposite of what point shaving predicts. For example, there is 3.01% more probability mass on winning margins between S and S + 8 than between S- 8 and S, and this probability discrepancy holds pointwise. Hence, a more subtle argument is required to argue that the data are consistent with point shaving. In particular, one must argue that all teams care primarily about winning; and that the risk of losing is high enough for moderate favorites that only strong favorites point shave; and that the asymmetric distribution in winning margins about the spread for weaker favorites is driven by some other force, unrelated to the point shaving incentives for strong favorites.

The estimate of point shaving for strong favorites hinges on the validity of the twin premises that (1) deviations from the line are forecast errors and (2) the forecast error distribution should be symmetric about zero. Some experts, including Joe Duffy (basketball's self-proclaimed "premier handicapper"), Thomas Paskus (Principal Research Scientist, NCAA, letter, Sports Illustrated, 04/17/06), and Steven Levitt (AEA meetings, 2006), have argued that the first premise is wrong because of the strong favorite bias and that this accounts for the finding of rampant point shaving. However, we find that the strong favorite bias is far too small to account for this finding--the line is close to being median unbiased, so that deviations from the line are "close" to being forecast errors. In particular, if we reduce the point spread by 1/2 point--which over corrects for the strong favorite bias--and recompute statistics, we still find a significant 3.37% probability discrepancy. (7)

It follows that much more than the strong favorite bias underlies the patterns in winning margins. However, one cannot yet conclude that there is an epidemic of point shaving--it may be that the second symmetry premise is invalid, that is, that the nature of the game of basketball itself may lead to an asymmetric distribution of forecast errors. To determine whether this is so, we begin by distinguishing games where point shaving is plausible from those where it is implausible and compare the statistical patterns in these two populations.

To do this, we first exploit the information contained in changes in the price of a bet on a favorite, that is, in the difference between the opening and the closing lines. Betting lines move to reflect information arrival and to balance dollars bet on each side, which reduces a bookmaker's exposure to risk. In particular, when far more money is bet on the underdog than the favorite--as is likely when gamblers exploit point shaving--the closing line will be less than the opening line, that is, the price of a bet on the underdog will rise. Conversely, when the closing line does not exceed the opening line, substantial betting on point shaving is implausible. By computing the difference in probability discrepancies for these two scenarios, we plausibly control for factors intrinsic to the game of basketball itself, isolating a residual that can be attributed to point shaving. ======== It is useful to evaluate methodologies with reference to Arizona State, the last team caught shaving points. (8) In each of the four games that players admitted to fixing, the closing line was less than the opening line, with the spread falling by an average of 2 points. However, in only one of those four games was Arizona State favored by at least 12 points, and there were also two games where Arizona State was a strong favorite and won by less than the spread, but testimony indicated that these games were not fixed. Thus, it seems that focusing on games where the spread falls reduces both type 1 and type 2 errors.

For strong favorites, we identify 4,350 games in which the closing spread was wider than the opening spread, and 4,956 games in which the spread either remained unchanged or fell. Figure 2a presents the distribution of winning margins relative to the spread when the point spread increased, and Figure 2b presents the distribution when the point spread did not increase. Comparing these figures reveals qualitatively identical statistical patterns. In games where the spread fell between open and close--where more money was likely bet on the underdog--the favorite won without covering the spread 45.12% of the time and won by (strictly) between S and 2S only 38.48% of the time, yielding a probability discrepancy of 6.64% (SE, 1.30%). In games where the spread did not fall between open and close--where point shaving is implausible--the favorite won without covering the spread 45.15% of the time and won by (strictly) between S and 2S only 39.54% of the time, yielding a probability discrepancy of 5.61% (SE, 1.40%). The difference in probability discrepancies is 6.64%-5.61% = 1.03%, which is less than the standard error (1.91%) of the estimated probability discrepancy difference. This indicates that the suspicious patterns in winning margins are unrelated to the movements in the point spread that we would expect to find with extensive gambling on point shaving.

One can argue that the reason we do not find evidence of point shaving is that the strategic impact of point shaving should be concentrated on outcomes near the line and that considering wider intervals introduces noise. Strategic considerations suggest that the impact of point shaving should be narrowly concentrated. An 18-point favorite up by three points with 5 min left is unconcerned about beating the spread and has no need to distort behavior; if it is up by 33 points, it cannot affect the outcome versus the point spread. In contrast, if their lead is close to the line, intentionally poor play can sharply raise the probability of failing to cover the point spread. Accordingly, we consider narrower intervals around the line. The final column of Table 2 shows that our findings are not altered, as the difference in probability discrepancies is always less than its standard error. In fact, the point estimate of the difference in probability discrepancies is only 0.37% for 6-point intervals around the spread, where the impact of point shaving should be the greatest.

[FIGURE 2 OMITTED]

One can also argue that the reason we fail to uncover point shaving is that most movements in the line between open and close are due to noise. (9) Possibly, we would find evidence of point shaving in games where the line moved sharply between open and close in the direction of the underdog. Table 3 reveals that this is not so--there is insignificantly different levels of probability discrepancies for larger changes in the point spread. Furthermore, none of the estimated differences in probability discrepancies (between the first and the subsequent lines) is close to being statistically significant.

One can contemplate increasing power by augmenting the sample to include moderate favorites. Teams favored by more than 5 points, but no more than 12, win 79% of the time and such spreads leave scope for successful point shaving. Indeed, in three of the four games where Arizona State players shaved points, the closing point spread was in this range. Inspecting Table 4 reveals that adding this sample works against the hypothesis of point shaving. In particular, for this sample, the probability discrepancy is not significantly different from zero. Moreover, the probability discrepancy is greater for games where the spread grew between open and close (1.3%)--where more money was bet on the favorite--than for games where the spread did not fall (0.2%) or fell by at least half a point (1.2%).

In sum, probability discrepancies following movements in point spreads provide no support for point shaving. Still, this analysis does not preclude the possibility of point shaving where gamblers place their money sufficiently cleverly or bet on such a small scale that the line is not moved. For example, if players gamble their own money, the small scale of their wagers might not affect the point spread.

These possibilities lead us to consider games in which no Las Vegas point spread was set, so that it is not possible to profit from point shaving. Absent an active market for gambling, gamblers have no incentive to provide inducements to teams to shave points, and players themselves cannot gain by gambling on their own actions.

Almost all games without lines are between teams that do not attract enough betting interest to make it cost-effective for bookmakers to post point spreads. (10) More generally, in games that would draw little public interest gambling, were lines set, most money would be bet by informed gamblers: bookmakers would not only fail to cover their costs but also lose money to gamblers--as in Glosten and Milgrom (1985), adverse selection leads to market breakdown.

In lieu of the point spread against which to benchmark winning margins, we use the end-of-year power ratings of team strength estimated by Jeff Sagarin and published in the USA Today newspaper Web site for the period 1998-2006. (11) Dare and MacDonald (1996) describe these statistical methods for estimation, which are based on the approach of Harville (1977, 1980).

Our estimate of the winning margin is given by the difference in team ratings adjusted for home court advantage, where applicable. We identify 4,723 games without betting lines featuring strong favorites, that is, games in which one team is favored by more than 12 points using the Sagarin estimates.

We first verify the efficiency of adjusted team ratings, regressing the actual winning margin on our estimated winning margin, that is, on the difference in team ratings adjusted for home court advantage. Table 5 reveals estimates that are extremely close to those in Table 1 for the closing line--so close that one could plausibly conclude that bookmakers determine point spreads by running regressions. Note particularly that the residual standard error of 10.67 in Table 5 is nearly identical to the residual standard error of 10.87 in Table 1. (12) We conclude that adjusted team ratings are a very good proxy for expected winning margins.

We next look at the 19,715 games where there are both point spreads and Sagarin estimates of winning margins--consistent with the regression findings, we find that the correlation between the spread and the estimated winning margin is 92.2%. These games included 4,404 games with a strong favorite of more than 12 points. In this sample, the probability discrepancy versus the spread was 5.89%, while it was 5.99% using the Sagarin estimates of winning margins. These two discrepancy measures are statistically indistinguishable and reinforce the fact that Sagarin estimates are very good measures of winning margins.

We now consider games without quoted spreads. Figure 3 shows the distribution of winning margins relative to the estimated winning margin for games without quoted spreads. Comparing Figures 1 and 3 reveals that the distribution of winning margins around the spread is essentially identical to that around the estimated winning margin. The patterns are qualitatively identical--the distribution of "forecast errors" around the expected winning margin is not affected by whether the games draw betting interest. The strong favorite won by less than the estimated spread S in 2,293 games and won by between S and 2S points in 2,014 games, yielding an estimated probability discrepancy of 5.91%. (13) This estimate is insignificantly less than our estimate of 6.16% using closing lines. Furthermore, comparing Table 6 with Table 2 reveals that in narrower intervals of 6 or 12 points around the estimated spread, where impacts of point shaving should be concentrated, we estimate higher probability discrepancies in nonlined games than in games where bookmakers accept wagers. This evidence is sharply inconsistent with significant point shaving.

II. WHAT EXPLAINS THE PATTERNS IN WINNING MARGINS?

A full forensic accounting should provide a coherent explanation for the data and in particular account for the different asymmetric distributions of winning margins around the spread for strong favorites versus weak favorites. We now do this.

We first observe that the within-game strategic incentives to win (14) skew winning margins in the way that we see in the data for both strong favorites and weaker favorites. Consider a game in which one team is up by a few points with 3 min to play. To maximize its chances of winning, each team should engage in strategic game management. In particular, the team that is ahead should "manage the clock," holding the ball to reduce the number of opportunities that the other team has to score, thereby raising the probability of winning. But clock management also reduces the variance of score changes, reducing the probability of covering a large spread, but not a small one. Conversely, a team that is slightly behind may foul repeatedly in order to get the ball back, and shoot quickly and from distant three-point range in order to increase variance and raise the probability of winning, albeit at the extent

of losing by more on average. (15) In contrast, when games are not close--there is no need to engage in strategic game management; and, indeed, a team that is far ahead may substitute backups, reducing the probability of beating a large point spread, but not a small one.

[FIGURE 3 OMITTED]

We now document that strategic game management by favorites works and that it provides a common explanation for the skewed distributions in winning margins vis-a-vis the point spread for both strong favorites and weaker favorites. Figure 4a compares the frequency by which strong favorites win by a given amount with the fitted normal probability density, and Figure 4b is the analogue for weaker favorites. These figures reveal that all favorites are far more likely to win by small to moderate margins than the normal frequency and are far less likely to win by large margins. Indeed, the "excess" probability on winning margins of 1-12 points is the same for strong favorites as it is for weaker ones (3.4% for both samples). Crucially, these common "excessively high" probability levels on moderate winning margins have opposite implications vis-a-vis the spread. Most sharply, for strong favorites, the "excess" probability on moderate winning margins of 1-17 points is 6.68%, (16) which exceeds the probability discrepancy vis-a-vis the spread of 6.16%. In particular, note that the average spread for strong favorites is 17.2 points, and many are favored by well over 20 points. This implies that the probability discrepancy vis-a-vis the spread for strong favorites is generated by the excess probability on small to moderate winning margins and not on larger winning margins that are close to very large spreads. Conversely, for weaker favorites, the high probabilities of moderate winning margins imply negative probability discrepancies relative to the spread: there is more than one-quarter percentage point more probability mass on winning margins of S + j than on S-j for each of j = 1,..., 8, the opposite of what point shaving predicts.

[FIGURE 4 OMITTED]

In sum, the simplest and most plausible explanation for the patterns in winning margins, and winning margins against the spread for both moderate favorites and strong favorites, is that all teams are driven by a common desire to maximize the probability of winning.

III. CONCLUSIONS

Economists must often resort to indirect methods and inference to uncover the level of illegal activity in the economy. Methodologically, our article highlights the care with which one must design indirect methods in order to distinguish legal from illegal behavior. We first show how a widely reported interpretation of the patterns in winning margins in college basketball can lead a researcher to conclude erroneously that there is an epidemic of gambling-related corruption. We uncover decisive evidence that this conclusion is misplaced and that the patterns in winning margins are driven by factors intrinsic to the game of basketball itself.

Our approach is multipronged. We first use point spread movements to identify betting patterns, showing that while the market is efficient and point spread movements contain information, that information is unrelated to the suspicious scoring patterns. In particular, the patterns in games where the spread increases--where more money is bet on favorites--are the same as in those where the spread shrinks. We then show that the same scoring patterns are manifest in games where there is no betting, indicating that the patterns are not related to information or betting activity.

Finally, we document substantial "excessive" probability mass on small to moderate winning margins and show that this can explain both the "excessive" probability mass below the spread for strong favorites and the "excessive" probability mass above the spread for weaker favorites. That is, we provide evidence that the distinct asymmetric patterns in winning margins of both strong favorites and weak favorites visa-vis the spread are driven by a common desire to maximize the probability of winning.

From a policy perspective, our findings against point shaving are important. History suggests that future discoveries of point shaving are inevitable--just last year, Tim Donaghy was found to have bet on the outcomes of many NBA games that he refereed, and this year, a player from the University of Toledo has been charged with point shaving. However, our analysis reveals that such incidents do not reflect widespread corruption and that costly significant changes in policy--fanned by past and likely future media alarm--would be unwarranted.

doi: 10.111 l/j.1465-7295.2009.00253.x

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(1.) See Szymanski (2003) for a survey of how the design of sports contests affects incentives.

(2.) See Joe Goldstein, "Explosion: 1951 Scandals Threaten College Hoops," "Explosion II: The Molinas Period," espn.go.com/classic/for historical reviews of these scandals.

(3.) Brown and Sauer (1993) show point spread movements are informative for NBA basketball, and Avery and Chevalier (1999) show that they are informative for college football.

(4.) Indeed, the fact Gibbs (2007) finds probability discrepancies of 6.4% in the NBA, where salaries of players (as opposed to referees) are extremely high (the minimum salary for a benchwarmer with 2-yr experience exceeds $750,000), should also raise suspicions about the validity of the methodology: it is implausible that gamblers could provide enough compensation to induce professional players to jeopardize their livelihoods.

(5.) Michael Konik (2006) describes these markets based on his experiences as a beard for a large North American sports syndicate.

(6.) Bettors disproportionately bet on strong favorites, leading bookmakers to bias point spreads very slightly against strong favorites. "The public loves betting superior against bottom shelf squads, and this.., indisputable fact is accounted for in the line." Joe Duffy Online, 03/29/06, joeduffy.net.

(7.) The fact that strong favorites are more likely to win by blowout margins also indicates that a desire to avoid running up the score (e.g., by giving reserves playing time) cannot explain the patterns in winning margins all by itself.

(8.) More recently, players at Northwestern fixed games in which it was an underdog.

(9.) Arguing against this possibility, Table 1 reveals that if we regress winning margin on opening and closing line, the estimated coefficient on the closing line is highly significant and close to one, while the coefficient on the opening line is close to zero and statistically insignificant.

(10.) Lines also were not set for games played by colleges from Nevada and occasionally for games with unstable situations (e.g., an injured star player who may or may not play), where significant private information and hence adverse selection is likely.

(11.) Since teams play about 30 games each season, the contribution of a single game to power ratings is minimal.

(12.) Note that the adjusted R2 is higher than its counterpart using point spreads. This likely reflects that games without betting lines often feature overwhelming favorites (e.g., Duke vs. North Carolina A&T, Florida vs. Savannah State).

(13.) Because the estimated spread, S, is continuous, there are essentially no pushes.

(14.) In contrast, minimal incentives appear to exist between games (Ferrall and Smith 1999).

(15.) See Gibbs (2007) for play-by-play analysis documenting this end-gaming strategic behavior in the NBA.

(16.) Relative to a fitted normal, there is 4.96% too little probability mass on winning margins of 18-34 points, implying a total difference of 11.64%.

DAN BERNHARDT and STEVEN HESTON *

* D.B. acknowledges support from NSF grant SES0317700. We thank Judy Chevalier. Tony Smith, and Justin Wolfers for helpful comments.

Bernhardt: IBE Distinguished Professor of Economics and Finance, Department of Economics, University of Illinois, 1407 W. Gregory, Urbana, IL 61801. Phone 1-217-244-5708, Fax 1-217-244-6678, E-mail danber@illinois.edu

Heston: Associate Professor of Finance, Department of Finance, Robert H. Smith School of Business, University of Maryland, College Park, MD 20742-1815. Phone 1-301-405-9686, Fax 1-301-314-9120, E-mail sheston@rhsmith.umd.edu

TABLE 1
Efficiency of Opening and Closing Lines
 Both Open
 Open Only Close Only and Close (SE)

Constant -0.135 (0.064) 0.071 (0.063) 0.073 (0.064)
Opening line 1.003 (0.006) -- -0.018 (0.041)
Closing line -- 1.01 l (0.006) 1.028 (0.041)
SE 10.95 10.87 10.87
Adjusted [R.sup.2] .38 .39 .39

Notes: This table presents linear regression coefficients for
regressions of home team winning margins on opening and closing point
spreads. The sample consists of 39,713 NCAA Division I basketball games
with a home team between 1990 and 2006.

TABLE 2
Narrower Intervals around the Closing Line of S

Opening Line < Closing Line (%)

(0,S) (S,2S) Discrepancies
45.15 39.54 5.61 (1.06)
(S - 2,S) (S,S + 12) Discrepancies
37.54 32.44 5.10 (1.02)
(S - 6,S) (S,S + 6) Discrepancies
20.30 18.23 2.07 (0.85)

Opening Line [greater than or equal to] Difference in
 Closing Line (%) Discrepancies (SE) (%)

(0,S) (S,2S) Discrepancies 1.03 (1.45)
45.12 38.48 6.64 (0.99)
(S - 12,S) (S,S + 12) Discrepancies 1.39 (1.41)
39.12 32.65 6.50 (0.96)
(S - 6,S) (S,S + 6) Discrepancies 0.37 (1.16)
20.84 18.40 2.44% (0.80)

Notes: This table reports for strong favorites (teams favored by more
than 12 points) the percentage of games won by less than the point
spread, and the percentage of games won by more than the point spread,
but by less than twice the point spread. The probability discrepancy is
the difference in these two percentages. The left column considers the
4,350 games where the closing point spread exceeded 12 and exceeded the
opening point spread. The middle column considers the 4,956 games where
the closing point spread exceeded 12 but did not exceed the opening
point spread. The table also presents these statistics for narrower
intervals of 6 or 12 points around the point spread.

TABLE 3
Larger Moves in Line between Open and Close
toward Underdog

Change No. of (O,S) (S,2S) Discrepancy
in Line Observations (%) (%) (SE) (%)

>0 4,350 45.15 39.54 5.61 (1.06)
[less than
 or equal to] 0 4,956 45.12 38.48 6.64 (0.99)
[less than
 or equal to] -0.5 2,902 45.18 38.01 7.17 (1.29)
[less than
 or equal to] -1 2,088 45.74 37.31 8.43 (1.53)
[less than
 or equal to] -1.5 1,025 45.27 38.44 6.83 (2.18)

Notes: This table reports for strong Favorites (teams favored by more
than 12 points), the percentage of games won by less than the point
spread, and the percentage of games won by more than the point spread,
but by less than twice the point spread. The probability discrepancy is
the difference in these two percentages. The change in line is the
closing point spread minus the opening point spread.

TABLE 4
Moderate Favorites 5 < S [less than or equal to] 12, Larger Moves
in Line

Change No. of (O,S) (S,2S) Discrepancy
in Line Observations (%) (%) (SE) (%)

All 17,152 27.63 26.97 0.66 (0.48)
>0 6,952 28.04 26.73 1.31 (0.76)
[less than
 or equal to] 0 10,200 27.35 27.14 0.22 (0.62)
[less than
 or equal to] -0.5 6,052 27.96 26.77 1.19 (0.81)
[less than
 or equal to] -1 3,975 27.62 26.89 0.73 (1.00)
[less than
 or equal to] -1.5 1,884 28.03 25.96 2.07 (1.45)

Notes: This table reports for moderate favorite (teams favored by more
than 5 points but no more than 12 points) the percentage of games won
by less than the point spread, and the percentage of games won by more
than the point spread, but by less than twice the point spread. The
probability discrepancy is the difference in these two percentages. The
change in line is the closing point spread minus the opening point
spread.

TABLE 5
Efficiency of Power Ratings

 Estimated Adjusted
 Constant Winning Margin SE [R.sup.2]

 0.01 1.03 10.67 0.57
(SE) (0.28) (0.01)

Notes: This table presents linear regression coefficients
for regressions of home team winning margins on the estimated
winning margin. The estimated winning margin uses
Jeff Sagarin's estimates published in USA Today for years
1998-2006. The sample consists of 4,531 NCAA Division
I basketball games with a home team between 1998 and
2006, where Las Vegas point spreads were not offered.

TABLE 6
Games without Lines, Estimated Spread of S

(O,S) (S,2S) Discrepancy
48.55% 42.64% 5.91% (1.02%)
(S - 12,S) (S,S + 12) Discrepancy
40.76% 34.17% 6.58% (1.00%)
(S - 6,S) (S,S + 6) Discrepancy
23.35% 20.43% 2.92% (0.85%)

Notes: This table reports for teams with a predicted
margin of victory of more than 12 points the percentage
of games won by less than the predicted margin, and the
percentage of games won by more than the predicted margin,
but by less than twice the predicted margin. The probability
discrepancy is the difference in these two percentages. The
estimated winning margin uses Jeff Sagarin's estimates
published in USA Today for years 1998-2006. The sample
consists of 4,723 NCAA Division I basketball games with a
home team between 1998 and 2006, where Las Vegas point
spreads were not offered.