Understanding Price Movements in Point Spread Betting Markets: Evidence from NCAA Basketball

Transcription

1 Understanding Price Movements in Point Spread Betting Markets: Evidence from NCAA Basketball Brad R. Humphreys Rodney J. Paul Andrew Weinbach Department of Economics School of Business Department of Economics University of Alberta * Saint Bonaventure University Coastal Carolina University This Draft: June 2009 Abstract We analyze point spread changes and betting volume for NCAA Division IA men s basketball games using survival analysis techniques. Estimates based on point spread changes from 2,548 games at the Hilton sports book and 1,681 games at the Mirage sports book in the 2007 season indicate that observed changes in point spreads are not related to imbalances in bet volumes. Sports books do not appear to change point spreads to induce equal volumes of bets on either side of propositions; the balanced book model of sports book behavior does not describe observed point spread changes in this market. JEL Codes: L83, G12, C14 Key words: sports gambling, hazard model, balanced book model Introduction We analyze the relationship between changes in point spreads on NCAA Division IA men s basketball games and the volume of bets placed on either side. While some previous research has examined changes in point spreads, and some studies have used survival analysis to analyze changes in prices set in financial markets, no previous research has used survival analysis techniques to analyze changes in point spreads. Survival analysis techniques are well suited to the analysis of price changes because they exploit both the direction and timing of price changes, both of which are important in financial markets, including betting markets where the outcome of the wager has a well defined end point and relatively little time elapses between the opening of trading in the * Corresponding author. Department of Economics, University of Alberta, 8-14 HM Tory, Edmonton, Alberta T6G 2H4 Canada. brad.humphreys@ualberta.ca; phone: ; fax:

2 market and the revelation of the outcome. In addition, survival analysis can analyze multiple price changes, providing an understanding of the dynamics of point spread movements. Understanding the relationship between movements in point spreads and betting volume is important because it has the possibility to extend our understanding of the behavior of sports books, the suppliers of bets in sports betting markets. In addition, new insight into the relationship between changes in point spreads and bet volumes can be generalized to other related financial markets, like equities, furthering our understanding of the functioning of these important markets. We find that observed changes in point spreads set by the Mirage and Hilton sports books in Las Vegas on several thousand regular season NCAA men s basketball games played in the season to be unrelated to the volume of bets placed on either side of the proposition in these games. In particular, point spreads on games with unbalanced betting on either side of the do not appear to change more often than point spreads on games with balanced betting on either side of the proposition. The most commonly discussed model of sports book behavior in the literature, the balanced book model, assumes that sports books set point spreads in order to balance the volume of bets on either side of propositions in sports betting markets. According to the balance book model, changes in point spreads should be more likely to occur in games where the early betting is unbalanced, as the sports book attempts to induce later bettors to bet on the other side of the proposition by adjusting the point spread. The evidence developed here is not consistent with this prediction; observed point spread changes do not appear to be related to imbalances in betting volumes on college basketball games. Instead, sports books are more likely to change the point spread on games with relatively large opening point spreads. Games with large point spreads are played between teams with relatively unequal abilities. The outcome, in terms of the score difference, in games played between teams of unequal abilities may be more difficult to forecast than the outcome in games played between teams with relatively equal abilities. In this case, observed changes in point spreads can be interpreted as attempts to improve the forecast of the point difference in these games, and not to alter the volume of bets made on each side of the proposition. 2

3 Prices in financial markets change frequently. Because of the frequent nature of price changes in financial markets, survival analysis has been used to analyze changes in stock prices and the time between trades (Dufour and Engle 2000) in order to better understand the determinants of these changes. In the sports betting literature, a number of previous studies have analyzed the difference between the opening line and the closing line in betting markets for National Football League games (Gandar, Zuber, O Brien and Russo 1988, Avery and Chevalier 1999, ) the National Basketball Association (Gandar, Dare, Brown and Zuber 1998, Gandar, Zuber and Dare 2000, Colquitt, Godwin and Shortridge 2007), and NCAA football (Dare, Gandar, Zuber and Pavlik 2005, Durham, Hertzel, and Martin 2005, Durham and Perry 2007). In general the analysis of changes in point spreads assess opening and closing lines as forecasts for actual game outcomes, and estimates reduced form regression models explaining variation in the change in the point spread from the opening to closing of the market. In both approaches, the empirical analysis proceeds from the assumption that observed changes in point spreads reflect attempts by sports books to balance betting volume on either side of the proposition. The literature emphasizes the idea that either private information on the part of bettors, or bettor biases in the form of belief in hot hand effects or other misperceptions lead to imbalances in betting volume on specific games. In general, closing lines forecast game outcomes better than opening lines, and changes in point spreads from the opening line to the closing line are not well explained by public information like poll standings, expert s opinions about game outcomes, or recent performance. These results lead researchers to conclude that significant private information exists in sports betting markets, or that bettors are noise traders who exploit short term volatility in the market. Every paper in this literature explicitly assumes that the balanced book model provides the underlying rationale for observed point spread changes. 3

4 The Economic Basis for Sports Book Decision Making Sports books are profit maximizing firms that have clearly defined outputs and prices and make production decisions in an environment of uncertainty factors that would appear to generate theoretical interest. Yet there has been surprisingly little theoretical research on sports book behavior. The most commonly discussed basis for sports book behavior in the literature is what we call the balanced book model. A formal version of this model has never been written down, perhaps because it is so simple. In the balanced book model, firms set the price of a bet so that an equal amount is wagered on either side of the bet. Because sports books charge a commission on losing bets, the balanced book model generates a sure profit for the sports book no matter what the outcome of the game. It is important to emphasize that although sports wagering appears similar to horse or dog racing, there are distinct differences. Horse and dog racing offers pari-mutuel odds, where the actual odds on a horse or dog are not known until betting is closed and the race has started. In pari-mutuel wagering, payoffs are determined by the actions of all bettors, with the pool of money being proportionately distributed to winners after subtracting the take of the track. Therefore, at the time of the wager, the horse or dog bettor does not know the actual odds on their wager. In sports betting, the exact opposite occurs. When a wager is placed, an explicit contract at a specific price is generated. This contract is fixed, as the bettor will win, lose, or push on his wager based upon the score differential of the game compared to the point spread at the time of his contract (same for totals or odds wagers the total or odds are fixed once the contract is initiated). Subsequent price movements do not affect the contract (betting ticket) of the individual. Therefore, all prices and odds are known at the time of the sports wager. Suppose that the total dollars bet on a game is H, the fraction of the total dollars bet on the favorite is f, and the fraction of the total dollars bet on the underdog is (1-f). Let v be the commission or vig charged by the bookmaker. In the point spread betting market, commission is only collected on losing bets. A basic point spread bet is a wager of $1(1+v) to win $1.The balanced book model assumes that sports books set the point spread to induce an equal amount of wagers on the favorite and underdog. In this case 4

5 f=0.5; no matter which team wins the sports book collects $H(1+v)f from the losers and owes only $Hf to the winners. Levitt (2004) developed a useful expression for understanding the behavior of book makers based on this concept. This expression showed that the expected value of a sports book s profit per unit bet, R, is a function of the probability p that a bet on the favored team wins, the commission charged by the sports book and the fraction of the total dollars bet on the favorite E(R) = [(1-p)f + p(1-f)](1+v) [(1-p)(1-f)pf]. (1) The first term on the left hand side represents the amount of money kept by the sports book on losing bets and the second term represents the amount of money returned to the winners. Levitt (2004) treats p as a choice variable of the sports book, so p must be a function of the point spread. In particular, if p is the probability that the favored team wins, then the lower the point spread, the larger is p other things equal. If team A is stronger than team B, then the probability that a bet on team A wins will be larger if team A is favored by 10 points than if team A is favored by 5 points. Equation (1) can be simplified to E(R) = (2+v)(f+p-2pf) 1 (2) which shows that expected profits are an increasing function of the commission. The balanced book model is a special case of equation (2), where either the probability of the favored team winning is 0.5 or the fraction of the total dollars bet on each team is equal (f=0.5). In either case, the expected profits are v/2 and the book maker is indifferent to the outcome of the game, since profits are identical no matter who wins. Note that we do not attribute the balanced book model to Levitt (2004), but simply discuss the balanced book model in the context of Levitt s expression for the expected profits earned per unit bet by a sports book; the balanced book concept was discussed as far back as Lawrence (1950). The balanced book model predicts that sports books attempt to balance the betting volume evenly on each side of a bet, and set p optimally to achieve this outcome. In 5

6 order to explore the implications of the balanced book model, Levitt (2004) assumed that the fraction of money bet on the favorite is a function of the probability that a bet on the favorite wins f=f(p), f >0, where the fraction of money bet on the favorite increases with the probability that a bet on the favorite wins. Again, in this context, the probability that a bet on the favorite wins is in turn a function of the point spread set by the sports book. Given this relationship between f and p, the balanced book hypothesis requires that bettors preferences are such that half the bets will be placed on the favorite when the probability that a bet on the favorite wins is 50% [f(p=0.5)=0.5]. In this case, the sports book sets the point spread to make p=0.5, the betting is balanced on each side, and the expected profit earned by the sports book is the commission. Levitt (2004) also discusses the case where bettors preferences result in more than half the money being bet on the favorite when the probability that a bet on that team will win is 50% [f(p=0.5)>0.5]; in other words, the situation when bettors prefer to bet on favorites at even odds. In this case, the sports book can earn higher profits by setting the point spread in a way to attract more than half of the money to be bet on the favorite. We refer to this case as the unbalanced book model. Under the unbalanced book model, sports books take positions on bets in that they allow the dollars bet to be unequally distributed across the two teams. The unbalanced book model is gaining considerable empirical support in the literature. Levitt s (2004) experiment in betting on NFL games showed strong evidence of unbalanced book outcomes. Paul and Weinbach (2007, 2008) present evidence of unbalanced betting on NFL and NBA games. Two distinct behavioral explanations for the unbalanced book model have been identified in the literature. In one case, sports books with unbalanced betting exploit known bettor preferences to shade the pointspread in the direction of the bias. In this case, the sports book uses its knowledge of the market to exploit bettors by setting inflated prices in the direction of bettor bias. This strategy results in the less popular side of the proposition (typically big underdogs, home underdogs, and unders in games with the highest totals) winning more often than implied by efficiency, possibly generating profits. This 6

7 outcome was demonstrated to exist in point spread betting on NFL games by Levitt (2004), where bets on home underdogs earned positive returns. The alternative explanation is that point spreads and totals set by sports books are forecasts of the outcome of the game, and not designed to balance the volume of bets. In this case, the sports book accepts an unbalanced book without attempting to exploit known bettor preferences for favorites and overs. Accurate forecasts will result in expected win percentages of 50% for all basic wagering strategies, and the sports book will earn the commission on losing bets in the long-run. Given the repeated nature of gambling (due to consumption benefits or bettor addiction), the sportsbook will not necessarily be concerned with game-to-game or day-to-day returns, but focuses on the long-run earnings from commissions. This setting of an accurate forecast by the book eliminates incentives for informed bettors to enter into this market to exploit a pricing bias (i.e. setting the pointspread too high on favorites). Pricing as a forecast for the majority of games, coupled with shading of the pointspread or total in specific situations with known large bettor biases (big favorites, road favorites, overs on high totals) explains the results found in market efficiency studies of sports gambling. Furthermore, pricing as a forecast in the unbalanced book model is not inconsistent with the well-established efficiency of sports betting markets. Tests of the efficiency of sports betting markets are based on the hypothesis that point spreads are unbiased predictors of actual game outcomes (Sauer, 1998). The unbalanced book model allows the possibility that bettors prefer to bet on favorites even when the point spread is an unbiased predictor of the actual outcome of the game. In general, in the overall market for nearly all sports bets, market efficiency cannot be rejected. In several common subsets of the data, however, market efficiency has consistently been rejected 1. Bettors tend to prefer the best teams, resulting in the overbetting of the biggest favorites 1 Evidence of inflated betting lines for big favorites has been identified across betting markets for virtually all college and professional team sports (some examples include Major League Baseball (Woodland and Woodland, 1994; Gandar, Zuber, Johnson, and Dare, 2002), professional hockey (Woodland and Woodland, 2001; Gandar, Zuber, and Johnson, 2004), professional football (Dare and MacDonald, 2004; Levitt, 2004), college football (Paul, Weinbach, and Weinbach, 2003), professional basketball (Paul and Weinbach, 2005), and college basketball (Paul and Weinbach, 2005)). 7

8 and road favorites. In addition, in totals markets, bettors prefer to wager on the over in games with relatively high totals. In these situations, the sportsbook allows discretion in their pricing, setting the price too high, resulting in underdogs and unders winning more than 50% of the time. In some cases, this results in simple strategies earning returns which overcome the sportsbook commission. To prevent informed bettor activity in these situations, sports books set binding limits on specific bettors who are suspected to be informed bettors through the practice known as booking to face. This strategy allows the sportsbook to exercise market power in a subset of games (where biases are largest) by limiting the actions of certain bettors who could reduce the sports book s profits by taking a position opposite to the general betting public. The idea behind booking to face is to limit the actions of actual or suspected informed bettors ( wiseguys ). Instead of allowing these informed bettors to earn large profits at the expense of the betting public, sports books limit the amount that can be wagered by these bettors and keep the profits generated by bettor preferences for themselves. This practice of booking to face is accomplished by setting binding limits on informed bettors (or agents of informed bettors often called runners ). General limits on bets are set by the sportsbook, but, in practice, are often at the discretion of the sportsbook manager. Suspected wiseguys will only infrequently be extended the courtesy of an above-the-limit bet. In addition, limits may be lowered on these players, the sportsbook could offer part of the desired wager at the current price (pointspread, total, odds) and the rest at a less opportune price for the wiseguy, or in extreme cases, betting action can be outright refused from these suspected informed bettors. 2 Outside of the subset of games where bettor preferences are important enough to generate large profit opportunities, the sports book does not appear to attempt to balance the volume of bets. Instead, prices are set as a forecast of the outcome of the game. In this paper, we examine observed changes in point spreads on college basketball games in the context of these competing models of sports book behavior. Sports books 2 For a detailed description of this practice, refer to chapter 2 of The Odds: One Season, Three Gamblers, and the Death of their Las Vegas Millman (2001). 8

9 frequently change the line on games. Based on the balanced book model, changes in point spreads should be related to imbalances in the betting on the game; if too much money is bet on the favored team, the balanced book model predicts that a sports book will try to induce more bettors to place bets on the other side. One way to do this is to change the point spread. The unbalanced book model predicts that sports books maximize profits by taking a position on bets; if this model describes sports books behavior, then observed changes in point spreads should be explained by factors other than the amount of money bet on each side of a game. Data Description The data come from two sources. The first data set contains point spreads set by the Mirage and Hilton sports books in Las Vegas for regular season and conference tournament NCAA men s college basketball games in the season. Data on changes in point spreads and overs were collected for 2,672 games from G & J Update Sports Information Services, which monitors and records instant updates on posted point spreads and over/under lines from a number of Las Vegas and offshore sportsbooks. G & J posts all changes in pointspreads and totals with individual time stamps, which was collected daily to construct the dataset. The initial sample contained 25,803 records. Each record represented a specific change in the point spread or over/under for a game made by the Mirage and Hilton sports book. A number of these records were deleted because of data errors. These errors included duplicate records (about 2% of the initial sample), typos that generated implausibly large changes in the point spread (about 20% of the initial sample), and a handful of games for which we lack point scores. We also dropped 2,579 records from NCAA tournament games from the data set. The final sample contains 16,165 records; about 65% of the records are from the Hilton and about 35% are from the Mirage. The sample is not evenly divided between the sports books because the Hilton sets point spreads and over/under lines for more games and there were relatively more data read errors for the Mirage. 9

10 The second data set contains information on individual game outcomes and betting percentages for each proposition. These data were purchased from Sports Insights, an online sports betting information service. Sports Insights has agreements to obtain betting volume data from BetUS, Carib Sports, Sportbet, and Sportsbook.com, four large on-line sports books. The percentage of bets reported on each side of a proposition represents an average across the four participating sportsbooks. The betting volume is not available for all sportsbooks and is not available for each game in the dataset. However, as a benchmark, for 35 college basketball games with posted pointspreads played on a Wednesday night in February, BetUS, Carib Sports, and Skybook reported more than 247,000 wagers, with an average of roughly 7,200 wagers per game. In general, most wagers reflect pointspread wagers, followed by wagers on totals, followed by moneyline and parlay bets. We matched the point spread and over/under movement data obtained from G & J Update Sports Information Services with the betting volume data obtained from Sports Insights to construct the final analysis data set. By combining these two data sources, we explicitly assume that the bet volumes at BetUS, Carib Sports, Sportbet, and Sportsbook.com are similar enough to the bet volumes at the Hilton and Mirage sports books. The opening and closing lines reported by G & J Update Sports Information Services for the four on-line sports books are identical to the opening and closing lines for the Hilton. The opening and closing lines for the Mirage are slightly lower than the opening and closing lines reported by G & J Update Sports Information Services, and the closing line reported by the Mirage is slightly lower than the closing line reported by the Hilton. However, in all cases, these differences are extremely small. The difference between the closing line for the four on-line sports books and the Hilton is 0.02 points. The difference in opening lines is 0.01 points. These tiny differences can be due to the fact that G & J Update Sports Information Services continuously monitors the web pages of the Hilton and Mirage, while the opening and closing lines reported by Sports Insights are sometimes the early and late lines, missing initial or last minute line moves. The fact that the reported opening and closing lines are very similar indicates that our 10

11 assumption that the bet volumes are very similar is a good assumption. We cannot explicitly test the assumption, because we lack bet volume data from he Vegas sports books. Analysis Time Each record in the data shows the current point spread or over/under at a point in time. Survival analysis assigns meaning to the passage of time, because time is a proxy for changes in other unobservable factors that are correlated with the passage of time. This makes it important to appropriately define analysis time, because the measurement of time needs to be an appropriate proxy. Defining analysis time has two aspects 1. Ensuring that whenever two subjects have equal t values, the risk they face would be the same if they also have the same covariates. 2. Deciding which particular value of t should be labeled t=0, marking the onset of risk. When is the onset of risk? In this case, it seems clear that the onset of risk begins when the line is posted, so we measure time from the instant that each point spread for each game was posted. When do two games have the same risk? In this application, the question is when do the lines on two different games have the same forces acting on them? The underlying forces include bets made on each side and the objective of the sports books. If bettors are relatively homogenous and both sports books have the same objectives, then this would seem to be true in our sample. We assume that the two sports books move their lines for some reason, and both attract similar bettors, implying that analysis time begins when each sports book posts the first point spread on each game and that the risks are identical across games and across the two sports books. Data Analysis We apply standard survival time analysis techniques to changes in point spreads on NCAA college basketball games. Because these techniques are not commonly used to analyze sports betting, we begin with a brief introduction to survival time analysis. Suppose that T is a non-negative random variable that represents the amount of time that 11

12 elapses between the time a sports book first sets a point spread and the time that the sports book changes the point spread. F(t) is T s probability density function and F(t) = Pr(T t) is T s cumulative distribution function. Survival time analysis is based on T s survivor function S(t) S(t) = 1 F(t) = Pr(T > t) (3) which is simply the probability that the point spread has not been changed by time t. The survivor function has a value of 1 at t=0 and decreases toward zero as t increases. Another important concept in survival time analysis is the hazard function h(t), which is also called the conditional failure rate. The hazard function is the instantaneous rate of point spread changes; in other words the probability that a sports book changes the point spread over a given interval, conditional on the point spread not being changed prior to that interval. The hazard function is h(t) = f(t)/s(t). (4) Hazard functions take values between zero (no chance of the point spread being changed during the interval) and infinity (the point spread will be changed with certainty during the interval). The hazard function can have many shapes; it can change over time. The key is that the underlying process of sports books taking bets from bettors on each bet, the effects of these cumulative bets on the sports book s position, and the objective of the sports book all influence the shape of the hazard function. In the jargon of survival time analysis, the effect of these factors is called risk. While we cannot hope to quantify the underlying risk, or the underlying process that generates this risk, changes in point spreads over time can be observed, and these changes can be used to learn something about the underlying process. The survivor function and the hazard function lie at the heart of survival time analysis. The hazard function provides a natural way to interpret the process that generates observed changes in point spreads. The survivor function is a simple measure of the probability of no change in the point spread. 12

13 Finally, to complete the link between the hazard function and the probability density function, define the cumulative hazard function H(t) = h(u) du = -ln[s(t)]. (5) The cumulative hazard function reflects the total amount of risk the cumulative factors that lead sports books to change their point spreads that have accumulated up until time t. The cumulative hazard function also has a natural interpretation beyond a summation. For any given hazard function, the cumulative hazard function tells us the number of times we would expect a sports book to change the point spread over a given period of time. Given these three functions, the relationship between the survivor function, the hazard function, and the probability density function for t is straightforward S(t) = exp{-h(t)} F(t) = 1 - exp{-h(t)} (6) F(t) = h(t) exp{-h(t)}. Nonparametric Analysis of Point Spread Changes Preliminary analysis of survival time data cannot be done using standard univariate data analysis tools like means, medians, and variances because of the amount of censoring and truncation present in survival time data. Descriptive statistics in survival time analysis are based on nonparametric estimates of the survivor function and hazard function. Table I shows some basic information about the sample. The Hilton posts sides and totals for more games than the Mirage. The sample contains more observations than point spread changes because it also contains changes in the over/under line and the typical practice is to first post the point spread and then post the over/under line later in the day. Table I: Basic Duration Statistics Book Observations Line Changes Games Hilton 10,432 3,466 2,546 Mirage 5,733 3,242 1,675 Point Spread Changes per Game Time at Risk (hours) Mean Median Min Max Mean Median Min Max Hilton Mirage

14 Although the Hilton posts point spreads and over/under lines for more games, the Mirage changes its posted point spread more often than the Hilton. In order to explain why the data set contains so many more observations than games or spread moves, consider a small part of the data set for the game played between Cal and Washington State in Pullman Washington on 31 January The relevant records and fields are shown below. team_v team_h date time book ou spread CALIFORNIA WASHINGTON STATE 31jan :12:46 Hilton. -9 CALIFORNIA WASHINGTON STATE 31jan :34:37 Hilton CALIFORNIA WASHINGTON STATE 31jan :37:19 Hilton CALIFORNIA WASHINGTON STATE 31jan :38:17 Hilton CALIFORNIA WASHINGTON STATE 31jan :01:04 Mirage CALIFORNIA WASHINGTON STATE 31jan :37:18 Mirage The Hilton posted its first point spread, Washington State as a 9 point favorite, at 8:12 am on the 31 st. 22 minutes later, at 8:34 am, the Hilton increased the point spread to 9.5 points. A few minutes later, the Hilton posted its first over/under line for the game, 133 total points. The Mirage posted its first point spread (Washington State as a 9.5 point favorite) and over under line (133.5 points) at 11:01 am. That afternoon, at about 2:38 pm, both the Mirage and the Hilton increased the point spread to 10.5 points. For the Hilton, analysis time begins at 8:12 am; for the Mirage, analysis time begins at 11:01 am. There are two point spread changes for the Hilton, from 9 points to 9.5 points and from 9.5 points to 10 points; there is only one point spread change. The third observation does not contribute to the time at risk in this instance, because the point spread did not change. Incidentally, Cal won this game 69-64, so all bets on the underdog and bets on the under placed after 2:38 pm paid off. Over/under bets placed before 2:38 pm were a push. The time at risk for this game is the time that elapsed between 8:12 am and tip-off for the Hilton and from 11:01 am and tip-off for the Mirage. One commonly used diagnostic for survival time data is the nonparametric Kaplan-Meier estimator for the survival function. For a data set that contains observed times of point spread changes t 1, t 2, t k where k is the number of point spread changes observed, the Kaplan-Meier estimate of the survivor function up until time t is S km = j (n j d j )/ n j (7) 14

15 where n j is the number of games that could have been bet on at time t j, and d j is the number of point spread changes at time t j. The Kaplan-Meier estimator summarizes what happened at each point in time in the data set, and shows the probability that a given point spread will be changed after point t j. Figure 1 shows the Kaplan-Meier estimates of the survivor function for the point spreads set by the two sports books in the sample for all games with a point spread. The graph shows the probability that each book has not changed the point spread on a given game up until point t. The survivor function for the Mirage lies below the survivor function for the Hilton at all points in analysis time. This would be expected based on information on Table I, which shows that the Mirage changes its point spreads more than the Hilton (a lower survivor function implies a greater chance of failure a change in the point spread at any point in time. Also note that the survivor functions are steep at the left of the graph, indicating that the probability of an observed change in the point spread increases in the first few hours after a point spread is posted, and then gets much flatter as analysis time increases. Finally, from Table I, the Hilton posts its point spreads earlier than the Mirage, which is why the estimate of the survivor function has a longer right tail for the Hilton. 15

16 Figure 1: Survivor Function Estimates Kaplan-Meier survival estimates analysis time Hilton Mirage Another commonly used diagnostic for survival time data is the Nelson-Aalen estimator for the cumulative hazard function, equation (5). Recall that the cumulative hazard function can be interpreted as the total number of point spread changes we would expect to see over the period (0,t). Although the Kaplan-Meier estimator could be used to obtain an estimate of the cumulative hazard function from equation (6), the Nelson-Aalen estimator H na (t) = j d j /n j (8) has better small sample properties. Figure 2 shows the Nelson-Aalen estimates of the cumulative hazard function for point spreads set by the two sports books in the sample. Note that, for any game, we would expect to observe a change in the point spread set by the Mirage in the first five hours or so, but would only expect to see a change in the point spread set by the Hilton in the first ten hours. The cumulative hazard function estimates also show a rapid increase in the expected number of observed changes in the point spread in the first few hours after the point spread is posted and a decrease as time 16

17 increases. This is consistent with anecdotal evidence that informed bettors, or wiseguys, bet early in the market and the public bets late. Wiseguys may bet early in this market as they can lock their bet at a price that is expected to be favorable for them (unlike pari-mutuel wagering where the prices and odds are not known until after betting is closed). Many informed bettors generate their own pointspread, totals, and odds through computer models. Betting early in the week or day, soon after the market opens, allows informed bettors to wager on games where the sportsbook prices differ from prices generated by their own models. Most public betting action is similar to illegal betting action, where the majority of wagers are placed within two hours of the start of the game 3. The outcome above is also consistent with the theory of booking to face, where only wiseguy action is important to the sportsbook and its manager. The general picture that emerges from the nonparametric analysis of the timing of price changes is that sports books change the posted point spread for most games and these changes typically happen in the first five to ten hours after the initial point spread is posted. Unfortunately, no information on the timing of bets exists. We can t determine what underlying factors lead sports books to change point spreads. However, we do have access to information on the characteristics of games, and the total volume of bets that were placed on each side in these games. In the next section, we turn to a semiparametric analysis of these data that takes into account observable characteristics of games and teams when explaining observed changes in point spreads in betting markets. 3 In chapter 2 of The Odds: One Season, Three Gamblers, and the Death of their Las Vegas, Millman (2001), referring to wagering on college basketball, Millman states This two hours before gametime is the most crucial part of the day for the bookmakers. Lines of people stretch from all thirteen betting counters to the back of the book, with hundreds of thousands of dollars coming in on nearly 100 games Squares, John Q. Public, can bet as much as they want whenever they want. Those guys are just gambling, says Lupo (sportsbook manager at Stardust in Las Vegas). We don t respect their plays. We ll take their money anytime. 17

18 Figure 2: Cumulative Hazard Function Estimates Nelson-Aalen cumulative hazard estimates analysis time Hilton Mirage Semiparametric Analysis of Point Spread Changes The semiparimetric analysis of survival time data is analogous to regression analysis. Semiparametric analysis examines the relationship between a vector of covariates, either time varying or time invariant but varying across games, and observed changes in point spreads. The most common semiparametric empirical approach for analyzing the effect of observable covariates and unobservable factors on observed changes in survival time data is the Cox proportional hazards model. This regression model relates the hazard rate for the j th subject to some baseline hazard function, h 0 (t), where all the factors that affect the hazard rate are equal to zero, and a vector of covariates. Formally, the Cox proportional hazards regression model is h(t x j ) = h 0 (t)exp(x j $ x ) (9) 18

19 where h(t x j ) is the hazard rate for college basketball game j, x j is a vector of explanatory variables that affect the hazard rate, and $ x is a vector of unknown parameters to be estimated. The Cox model is frequently used because it is computationally feasible, and because it does not require any parameterization of the baseline hazard rate, h 0 (t). Because the baseline hazard rate is not parameterized, the estimator is semiparametric. The vector of unknown regression parameters, $ x, are estimated using maximum likelihood. We estimate a random effects Cox model (called a shared frailty in the survival time literature. Frailty models assume that there is an unobservable positive factor, α i, that affects the hazard rate for different groups in the sample. In this case, groups are defined as individual games; frailty models account for within-group correlation in hazard rates from unobservable sources. These game-specific unobservable factors include any information about games that is available to sports books but not observable by econometricians, like injuries or past betting patterns. Formally, the model estimated is h(t x j ) = h 0 (t) α i exp(x j $ x ) (10) where α i,is a random variable with mean 1 and variance θ that captures the random effect. Note that the covariates in equation (10) only vary by game. We do not have access to variables that vary hour by hour like the observed point spread changes. The game characteristic variables come from the Sports Insights web site and are described above. Table II contains summary statistics for the game level variables used as covariates in the Cox proportional hazard model. Table II: Summary Statistics, Game Variables Variable Mean Std. Dev. Min Max Difference in Bets abs(opening Line) % Bet on Home Team % Bet on Favorite % Bet on Home Favorite % Bet on Road Favorite % Home Favorites % Road Favorites

20 The average opening line was just under 8 points. The difference in bets variable is the absolute value of the difference in the fraction of bets placed on each game in the sample. The mean value of 33 for this variable implies that the average book on any game was unbalanced with about 66% of the bets placed on one team and 33% of the bets placed on the other team. From the betting % variables, home teams, favored teams, and especially teams favored on the road attracted a majority of the bets placed in games. Although most home teams are favored, the home team was the underdog in 23.8 percent of the games, and the visiting team was bet heavily in these games. Table III reports estimated hazard ratios and P-values for the null hypothesis that the estimated parameter is equal to 1, and other summary statistics for several different specifications of equation (10). The reported hazard ratios are the ratio of the baseline hazard, h 0 (t), to the hazard associated with a one unit change in the covariate. Hazard ratios greater than one indicate an increased likelihood that the point spread changes associated with a change in that covariate; hazard ratios less than one indicate a reduced likelihood that the point spread changes associated with a change in that covariate. We are looking for evidence that observed changes in point spreads are consistent with either the balanced book model or the unbalanced book model of sports book behavior. Model 1 on Table III is a baseline model for generating such evidence. The baseline model contains a dummy variable for point spreads offered by the Mirage sports book because the nonparametric analysis indicates that the Mirage changes its point spread more than the Hilton. The baseline model also contains the absolute value of the opening line the first point spread posted by each sports book and this value squared. These variables capture any tendency to change large point spreads. The larger the absolute value of the point spread, the greater the disparity between the perceived strength of the two teams. We expect that the relationship between the size of the opening line and changes in the point spread might be non-linear, because sports books might have less of an incentive to change smaller lines. Given a higher likelihood for the possibilities of 20

21 being middled (where the score differential of a game falls between posted lines leaving the potential for bettors on both sides to win, opening the sportsbook to considerable risk) or sided (where the score differential of a game falls on one of the posted line leaving one group of bettors to win and the others to push) with small pointspreads (particularly around 2 or 3), sportsbooks may be less inclined to change these pointspreads. Table III: Estimated Hazard Ratios and P-Values Cox Proportional Hazards Model with Random Effects Variable Model 1 Model 2 Model 3 Model 4 Mirage Difference in Bet Volume abs(opening Line) abs(opening Line) % Bet on Home Team % Bet on Favorite % Bet on Home Favorite % Bet on Road Favorite Wald Statistic log likelihood Games 4,217 4,217 4,217 4,217 Line Changes 6,707 6,707 6,707 6,707 (P-values of 0.00 mean the null is rejected at better than 0.1%) In Sports Book Management: A Guide for the Legal Bookmaker, Roxborough and Rhoden (1998) note that in games with large pointspreads, full point movements are the norm in the sports gambling industry. Roxborough and Rhoden (1998) also note that basketball is similar to football in this respect (p. 33). With respect to football, Roxborough and Rhoden (1998) suggest When football pointspreads are in double digits, the house may wish to move the line by a full point on each limit bet. When the 21

22 pointspread is that large, the game is unlikely to fall on the numbers. Thus, movement of large spreads by only a half point will not effectively correct unbalanced action. 4 This practice also leads to the result of higher pointspreads having larger overall movements associated with these games. The key variable in the baseline model is the difference in bets placed on the two teams. This variable reflects how unbalanced the book is on each game. The balance book model predicts that sports books try to attract an equal amount of betting on each side of a bet; the larger the difference in betting volume, the larger the observed difference in bets. The balance book model predicts that observed moves in point spreads should be related to unbalanced books, as the point spread an important instrument available to sports books to influence the amount of money bet on each team. The estimated parameter on the difference in bets variable is our primary test for evidence supporting the balanced book model in our data. Column 1 on Table III contains the results for the baseline model. The estimated hazard ratio on the Mirage dummy variable is greater than one and significant, indicating that the Mirage is more likely to change its point spread than the Hilton, conditional on the other observed variables in the regression model. The parameters on the absolute value of the opening line variables are both statistically different from one. The estimated parameter on the absolute value of the opening line is less than one and the estimated parameter on the squared term is greater than one. These estimated parameters imply that sports books are less likely to change small point spreads and more likely to change large point spreads, other things equal. As noted above, this may be due to the risk involved in middles and sides and the practice of sportsbooks of moving large pointspreads by a full point, rather than a half point. Based on these estimated parameters, the overall hazard ratio is equal to one at an opening line of 12.5 points, which is above the average opening line of 7.5 in the sample. 4 Roxborough and Rhoden (1998) also note at various points in their book that balanced betting action may be the ideal, as this situation is riskless for the sportsbook, it is not the normal situation as the majority of games are typically unbalanced. 22

23 The estimated parameter on the difference in bets variable is quite interesting. This parameter estimate is equal to one, indicating that the difference in bets has no effect on the likelihood that a sports book changes the point spread, other things equal. Sports books are no more likely to have changed the point spread on a game where 75% of the dollars bet were on one team and 25% of the dollars bet were on the other than they were on a game where half of the bets were placed on each team. The balanced book model predicts that sports books want to attract an equal amount of dollars bet on each team. The observed changes in point spreads in our data are not consistent with this idea; points spread changes are not associated with unbalanced betting on college basketball games. The alternative to the balanced book model is the unbalanced book model, where sports books take positions on games when the betting is not equal on either side of a bet. Model specifications 2-4 on Table III explore the idea that observed point spread changes are consistent with the unbalanced book model. Model 2 includes two additional explanatory variables, the fraction of bets placed on the home team and the fraction of bets placed on the favored team. Since Model 2 still contains the difference in bets variable, this specification holds the effects of an unbalanced book on the likelihood of a change in the point spread constant, so these alternative specifications nest the balanced book model. The parameter estimates from model specification 2 indicate that observed changes in point spreads are not related to the fraction of the bets placed on the favorite team in that game, but the point spreads are less likely to change when the fraction of bets on the home team increases. Model specifications 3 and 4 explore the possibility that heterogeneity in the type of favorite drives the insignificant estimated parameter on the fraction of bets placed on the favorite variable in Model 2. Model 3 includes the fraction of bets on home favorites and Model 4 includes the fraction of bets on road favorites ( home dogs ) as explanatory variables. The results from these two model specifications indicate that observed point spreads are less likely to change as the fraction of bets on favored home teams increases, and more likely to move as the fraction of bets against home underdogs increases. The estimated parameter on the home favorite variable in Model 3 is consistent with the 23

24 estimated parameter on the home team variable in Model 2. Given that road favorites, in this sample and over longer timeframes, have been shown to win more than 50% of the time, these price movements toward road favorites (against home underdogs) may reflect sports book reaction to the actions of expected informed bettors wagering on road favorites. The results for Model 4 on Table III imply that sports books do not change the point spread on games with heavy betting on the home team. As noted above, the imbalanced book model is consistent with the idea that sports books shade their posted lines in order to exploit known patterns in bettors preferences. For example, Strumpf (2002) found evidence that illegal book makers in New York City offered worse odds to customers known to always bet on the home team, the New York Yankees. The observation that sports books do not change their point spreads when the number of bets placed on the home team increases could be consistent with this result, if there is evidence of point spread shading in the game results. Table IV presents the results of a number of alternative betting simulations we undertook to determine if shading of the point spread occurs in this sample. If sportsbooks shade their pointspreads toward home teams or favorites, simple betting strategies, like wagering on the road team or the underdog, should win more often than implied by efficiency. In these simulations, we calculate the percentage of the specific bets that would have won for a number of specific simple betting strategies like always bet on the favorite and test weather this strategy resulted in a fair bet (probability of a win is 50%), and weather this bet would have earned a profit (following this strategy led to wins more than 52.4% of the time). Results of all simulations are compared to the null hypothesis of a fair bet (win percentage of 50%) and, where necessary, tested against the null of no profitability (win percentage of 52.4% - percentage needed to break even given the sports book commission) using a log-likelihood ratio test. The test statistic a χ 2 distribution with one degree of freedom. Critical values for this test are for the 10% significance level, for the 5% level and for the 1% level. Details of this test can be found in Even and Noble (1992). 24