BIASES IN ASSESSMENTS OF PROBABILITIES: NEW EVIDENCE FROM GREYHOUND RACES. Dek Terrell* August Abstract

Transcription

1 BIASES IN ASSESSMENTS OF PROBABILITIES: NEW EVIDENCE FROM GREYHOUND RACES Dek Terrell* August 1997 Abstract This paper investigates biases in the perceptions of probabilities using data from the 1989 and 1994 seasons at the Woodlands greyhound park in Kansas City, Kansas. Results reveal consistent evidence that the gambler s fallacy exists. The results also reveal that gamblers overestimate the probability of a win by the favorite and the dog in the lucky seven position. However, the comparison also suggests some learning by bettors between the first season of operation in 1989 and the 1994 season. JEL classifications: L83, D81, G14 Keywords: pari-mutual game, uncertainty, gambler s fallacy *Department of Economics, Louisiana State University, Baton Rouge, LA I would like to thank Derick Rienicher at the Woodlands for providing data and other information about the Woodlands. I am grateful to Danielle Lewis and Geoffrey Turnbull for helpful comments.

2 1. Introduction The assumption that economic agents can correctly process information to calculate probabilities of events plays a crucial role in countless studies examining the behavior of agents under uncertainty. However, experiments and empirical evidence suggest several common biases in the assessments of probabilities. This paper examines the perceptions of probabilities by bettors at the Woodlands greyhound park in Kansas City, Kansas over two years, 1989 and Terrell and Farmer (1996) used 1989 data from the Woodlands to support a theoretical model by analyzing average returns to different strategies. Their results indicated evidence of three biases among bettors. This paper explores bettors assessments of probabilities at the Woodlands using Ali s (1977,1979) method of transforming odds into subjective probabilities. Applying Ali s methodology to a large set of dog races serves as a test of whether the biases found by Ali also exist in other samples. This approach also allows an assessment of whether Terrell and Farmer s conclusions are sensitive to methodology and permits us to evaluate whether biases persist over time. Because 1989 was the first year of operation, one might expect gamblers assessments of probabilities would improve between 1989 and 1994 due to learning. This paper begins with a description of three common misperceptions of probabilities which Terrell and Farmer documented among bettors at the Woodlands in Section 3 describes the methodology, including a new method of allowing for rounding in odds when calculating subjective probabilities. Section 4 describes the data, variable construction, and the rules governing races at the Woodlands. Section 5 contains the empirical results examining biases among bettors. 1

3 2. Common errors in assessments of probabilities by bettors The strategies and biases of gamblers are best described using several gambling terms. The odds on an event are the odds against the event occurring and define the return to a winning wager on an event. For example, if the odds are four to one (4:1), a wager of $1 yields $4 if the event occurs. The event with the lowest odds is the favorite, while the event with the highest odds is the longshot. The pool is the total amount wagered on an event. Pari-mutual games split the pool (less a portion reserved for the track) between all bets on the winner. Suppose $4 is wagered on event one and $2 on event two. If the track takes out 1/6, $5 is left to be divided among the winners. A winning wager of $1 on event one earns $5/4=$1.25, while a winning wager of $1 on event two earns $5/2=$2.50. The odds of 1.25:1 on event one and 2.5:1 on event two reflect bettors beliefs concerning the likelihood of events and thus can be used to examine how agents assess probabilities. 1 The literature suggests several common errors in those assessments. Kahneman and Taversky s (1979,1984) experiments consistently found that subjects understate the probability of high probability events and overstate the probability of low probability events. Because the odds on longshots at racetracks often translate into probabilities near those used in Kahneman and Taversky s (1979,1984) experiments, one might expect bettors to overestimate the probability of wins by longshots. The odds on favorites do not imply probabilities as extreme as those considered by Kahneman and Taversky, but earlier empirical results suggest that bettors underestimate the probability of a win by the favorite. The proposition that bettors underestimate the probability of a win by the favorite and overestimate the probability of a win by the favorite is called the longshot bias. 1 The method of determining the odds and how odds are translated into subjective probabilities is discussed in section 4. 2

4 Studies finding evidence of the longshot bias date back to Griffith (1949). Perhaps the strongest evidence of the bias comes from Ali s (1977) study of 20,247 harness races at three tracks over five years. His study compares the subjective probabilities implied by the odds to win percentages. The results indicate that bettors in his sample understate the probability of wins by favorites and overstate the probability of wins by longshots. In fact, Ali finds that bettors understate the probability of wins by both the favorite and the dog with the third lowest odds. Similarly, bettors overstate the probabilities of wins by other dogs. Moreover, the ratio of subjective probability to objective probability rises almost monotonically as the odds rise. Terrell and Farmer (1996) also find evidence of the longshot bias using data from the 1989 season at the Woodlands. Although expected return from betting the favorite far exceeded the expected return from betting the longshot, the overall pattern was weaker than Ali s results. For example, the strategy of betting the dog with the second highest odds earned a higher expected return than betting the favorite. Taversky and Kahneman (1974) summarize another common error in experimental subjects views of chance by the statement: Chance is commonly viewed as a self-correcting process in which a deviation in one direction induces a deviation to restore equilibrium. This incorrect view of probability leads to the gambler s fallacy which is an incorrect belief that the recent occurrence of an event reduces the probability of the event. 2 Empirical evidence confirms the presence of the gambler s fallacy among bettors. Clotfelter and Cook (1993) and Terrell (1994) find that the number of wagers on winning lottery numbers falls dramatically after the 2 Clotfelter and Cook (1993) define the gambler s fallacy as the incorrect belief that the recent occurance of an event lowers the probability of an independent event. We prefer a more general definition which labels the fact that bettors tend to underestimate the probability of the events that recently occurred as the gambler s fallacy. 3

5 number wins, even though the outcome of drawings is independent across days. 3 Likewise, Metzger (1985) finds a bias against betting on repeat winners by post position in horse races and Terrell and Farmer (1996) document a similar bias in the 1989 Woodlands data. The combination of these experimental and empirical studies suggests that the gambler s fallacy also clouds assessments of probabilities. Superstition may also affect assessments of probabilities. Clotfelter and Cook (1991) note that the numbers 333 and 777 were bet nine times the average rate in the Maryland three digit numbers game. These popular numbers were bet 35 times for every bet on numbers 086 and 092. In pari-mutual games, such overbetting translates into low returns. Terrell and Farmer (1996) find that the worst strategy during the 1989 Woodlands season was a wager on lucky dog number 7 which earned far lower expected returns than any other strategy except betting the longshot. In addition to evidence on these three biases, many other studies show more generally that the wagers of gamblers lead to odds that fail to reflect the true probabilities of outcomes. Most studies focus on returns and search for strategies that earn positive expected returns using publicly available information. The existence of such strategies implies that bettors fail to use all information available to correctly assess probabilities. Asch and Quandt (1987), Golec and Tamarkin (1991), Goodwin (1996), and Hausch, Ziemba, and Rubinstein (1981) find strategies earning positive expected profits. 4 We choose to focus on subjective probabilities directly linked to the biases discussed above. 3 Johnson and Klotz s (1993) tests for uniformity of winning Lotto America numbers revealed that some numbers appeared slightly more likely to be drawn than others. Because these numbers are more likely to be drawn as winners, a recent winning number is more likely to win than another number if events are not independent. However, the difference in probabilities was so small that the draws are probably still best viewed as independent. 4 Dolbear (1993) questioned Asch and Quandt s (1991) results due to a potential bias in the calculation of subjective probabilities for exacta bets. 4

6 3. Methodology Following Ali (1977,1979), we explore biases among bettors by translating odds in each race into the subjective probabilities of bettors. To understand the transformation, we must first describe how odds are determined in the pari-mutual game. Let B i denote the amount wagered on a particular dog at the Woodlands. The gross pool in the race is simply the sum of the wagers on all eight dogs, 8 j= 1 B j. The track takes a portion, α, of the pool called the breakage. In a pari-mutual game, the remainder of the pool is divided among winning bets. The odds supply the payout to a winning one dollar, and are calculated as: 5 odds i = a = i ( 1 α ) B B i 8 j= 1 j (1) Following Ali (1977), subjective probability of dog i is defined as the proportion of the win pool bet on dog i. 6 Transforming equation (1), the subjective probability of a win by dog i is: p i Bi ( 1 α ) ( 1 α ) = = = 8 oddsi ai B j= 1 j (2) 5 The odds in our data set are defined slightly different from Ali s (1977) odds. Ali s formula assumes that the track returns the bettors original wager plus the amount dictated by the odds. At the Woodlands, the return of the original wager is included in the odds. The result is that Ali s odds equal our odds minus one. This difference exists throughout the paper and leads to slight differences between Ali s calculations and ours. 6 The practice of defining subjective probabilities in this way dates back to Griffin (1949) and McGlothlin (1956). 5

7 Combining equations (1) and (2) it is possible to show that the subjective probabilities should sum to one. However, in practice odds are rounded. At the Woodlands and many other tracks, the odds are rounded down to the nearest multiple of twenty cents. The Woodlands breakage (α) is.18, and Goodwin (1996) notes that rounding at most increases the breakage to We address rounding in two ways. The first, used by Ali(1977), adjusts the breakage to reflect rounding. To determine the adjustment, we note that the probabilities in each race should sum to one. Due to the rounding of odds, the probabilities calculated using equation (2) often sum to a value slightly larger than one. Therefore, we calculate initial subjective probabilities using equation (2), and then divide each probability by the sum of the initial probabilities to adjust for rounding. The method is essentially equivalent to increasing the breakage because it scales all probabilities by the same factor. An alternative method is to directly incorporate rounding into equation (2). Let u i denote the amount of downward rounding which randomly varies between 0 and.20. Assuming the digits in the odds are random, u i followings a uniform distribution with supports of 0 and After rounding the odds are a i - u i, rather than a i as stated in equation (1). Thus, after accounting for rounding the subjective probabilities are: p i Bi ( 1 α ) = = 8 ai + ui B j= 1 j (3) 7 Our results also indicate that rounding increases the breakage by less than More specifically, the range is [0,.20), and doesn t include the endpoint.20. 6

8 Because u i is not observed, the best measure of the subjective probability is the expected value of p i. Integrating over u, we find: E[ p ] = i. 20 ( ) 1 α 1 du a + u i α = i i ( ln( a. 20) ln( a )) i (4) It is important to note the relationship between this rounding adjustment and odds. Intuitively, the impact is obvious. A.10 reduction in 40 to one odds has little effect, while a.1 reduction in even money (one to one) odds is a major change. The same intuition applies in the calculation of subjective probabilities. For favorites with low odds, adjusting for rounding decreases the subjective probability substantially, and the effect of the rounding adjustment falls as odds increase. Table 1 contains subjective probabilities calculated with and without this adjustment for rounding. There are arguments for both methods above. Because the first method translates odds into subjective probabilities more directly, it leads to subjective probabilities more closely linked to returns. However, the second method of adjusting for rounding leads to subjective probabilities closer to the proportion of wagers on the event. In greyhound races where low odds are unusual, both methods generally lead to very similar estimates of subjective probabilities. We present subjective probabilities calculated both ways for comparison. Using the frequentist definition, the objective probability of a win is the proportion of wins in an infinite number of races. Let π i denote the objective probability of a win by dog i. Letting e i represent the error in assessment of probabilities by bettors: p = π + e i i i (5) 7

9 If bettors accurately assess probabilities E[e i ]=0. The subjective probability p i can be calculated for each race, but π i is unknown. Thus, to examine the ability of bettors to accurately assess probabilities, we use averages over many similar wagers. The average subjective probability over n wagers can be computed directly from the subjective probabilities for each race: p = 1 n n j= 1 p j (6) Likewise the average objective probability (denoted π ) is just the ratio of the number of wins to the total number of wagers. Taking the averages of both sides of equation (5) implies the bias in assessments of probabilities is consistently estimated as: e = p π (7) Assuming the averages are calculated over a large number of wagers, the central limit implies that this estimate of the bias ( e ) is normally distributed. The methodology can easily be used to test for the biases discussed earlier. For example, to assess the longshot bias we might focus on wagers on the favorite. The next step is to calculate the average objective and subjective probabilities across all wagers on the favorite. The difference between these averages supplies a consistent estimate of the bias in the subjective probability of a win by the favorite among bettors. Finally, hypothesis tests can be constructed based on the normal distribution to determine whether the bias can reasonably be attributed to sampling error. 8

10 4. The Data The Woodlands Greyhound Park in Kansas City, Kansas opened in Over its first four month season, the Woodlands attracted 590,003 bettors who wagered over 65 million dollars. By 1994 the Woodlands ran a nine month season and ranked among the largest parks in the United States in dollars wagered. During that season 836,915 people went to the track and wagered $104,360,787 on live greyhounds. 9 The track offers a number of wagers and the pool is widely divided among those wagers. For example, only $6,516,629 of the total pool in 1994 was wagered on the win bets considered in this paper. 10 At the track, bettors may purchase a program containing detailed records of each dog and wins by starting position in the current and past seasons races. Bettors can also get information from local handicappers in the newspaper. Personnel at the Woodlands observe a significant difference in the effort devoted to picked winners at the track. While most bettors limit their effort to analysis of the statistics in the program or handicappers picks, a few bettors view films of past races or use computers to project the winners of the races. Terrell and Farmer (1996) draw a sharp distinction between these informed bettors who sometimes attempt to earn a living at the track and the other "pleasure bettors. The Woodlands holds afternoon and evening racing sessions, which consist of either thirteen or fifteen races. Races generally include eight dogs randomly placed in one of eight starting or post positions. The dogs wear numbers based on the assigned post position. Dog 1 starts closest to the rail with dog 2 immediately to the right. The sequence continues until reaching 9 The track also allows gamblers to wager on simulcasts of races at other tracks. This figure excludes simulcast wagers. 9

11 dog 8 on the outside. Some positions offer advantages over others. The dog in position 1, nearest the rail, has less distance to cover than the other dogs. Likewise dog 1 and dog 8 are less likely to be bumped and stumble than the dogs 4, 5, or 6 which are in the center positions. However, wins by post position in the current and past season are recorded in the program; thus, the advantages should be known by all bettors. Our data set includes the post position, odds, and final finish for 4,121 races in 1989 and 5,931 races in After eliminating races involving fewer than eight dogs and a few races that ended in ties, the data set is reduced to 3,795 races in 1989 and 5,639 races in Although the number of variables is somewhat limited, the data permit us to evaluate all three potential sources of bias in assessments of probabilities discussed earlier. We compute subjective probabilities of bettors based on eighteen groupings of wagers which we call strategies. The first eight strategies are based on odds. Strategies one consists of betting the favorite each race, strategy two requires betting the dog with the second lowest odds,, and strategy eight requires betting the longshot in each race. Unfortunately, the odds increment by twenty-cent units and thus the odds on two dogs are the same in some races. In the event of a tie in odds, the strategy consists of betting both dogs. For example, if two dogs tie with lowest odds the strategy of betting the favorite requires betting both dogs. Because of ties, the number of wagers varies across strategies. The next strategy considered requires betting the dog number winning the last race. For example, the strategy requires betting dog 2 in the third race if dog 2 won the second race. Of course this strategy cannot be implemented for the first race each day. Likewise the strategy does not include wagers in any races following a scratch (a race with fewer than eight dogs). Thus, this 10 A win bet is the simplest wager. The gambler wins if their dog finishes first and looses otherwise. The Woodlands also offers bets on dogs to place or show and exacta, quinnella, trifecta, super trifecta 10

12 strategy includes fewer wagers than the number of races included in the data set. The next eight strategies require betting the dog in a single post position in every race. For example, betting dog 1 (the dog in position 1) in every race is the first strategy considered. Unlike the earlier strategies, these strategies generate a bet in every race. 4. New Empirical evidence from Greyhound Racing Table 2 contains the number of wins and number of wagers for seventeen strategies calculated using the 1989 data. The table also contains subjective probabilities, objective probabilities, and biases calculated as described in section 2. Two sets of subjective probabilities are presented in the table. The subjective probabilities without brackets are calculated using equation (2) and then scaled so that probabilities sum to one following Ali (1977). The probabilities in brackets are calculated as described in equation (4). The subjective probabilities are similar for all strategies. Thus, the bias is calculated only for the subjective probabilities calculated using equation (2) and the text refers to those numbers unless otherwise stated. Objective probabilities of a win differ substantially across strategies As expected, the post position affects the probability of winning. Dogs in position 1 won 16.5% of 1989 races, while dogs in position 7 won only 10.6%. Not surprisingly, subjective probabilities generally follow objective probabilities. Favorites win more than longshots, and the subjective probability of a win by dog 1 exceeds bettors assessments of the probability of a win by dog 7. However, the subjective and objective probabilities do differ in some cases. Although these differences or biases appear small, they are estimated quite precisely given the large sample size. Small biases also may translate into big differences in expected returns. For example, the which require picking the top two or three dogs, sometimes in several races. The largest pool in 1994 was 11

13 results in table 2 indicate that bettors understated the probability of a win by a longshot by.013 in 1989 while assessments of the favorite are essentially unbiased. This bias translates into an expected payout of 67.5 cents per dollar bet on the longshot, compared to an 80.2 cent expected payout to a one dollar bet on the favorite. 11 Begin with the analysis of the bias in assessments of the favorite. Ali (1977) finds that subjective assessments by bettors understate the probability of a favorite s winning by The estimate is large and statistically significant. Our results in table 2 fail to find this bias. The point estimate of the bias indicates that bettors overstate the probability of a win by the favorite by However, the standard error suggests that this bias may be attributable to sampling error. In addition, when subjective probabilities are calculated using equation (4) to adjust for rounding, this bias falls to zero. Several other biases do appear present in the odds-based strategies, although there is no evidence of a general rise in the ratio of subjective to objective probabilities with increases in odds similar to that Ali (1977) found. The longshot exhibits the largest bias. Subjective probabilities overstated the probability of a longshot win by.0135, a bias that cannot be attributed to sampling error. Ali s (1977) results indicated a smaller bias of.007, although the direction of the bias was the same. The results in table 2 also contain evidence of other biases not examined by Ali. Consistent with the gambler s fallacy, the point estimate suggests that subjective assessments by bettors understate the probability of a repeat winner by post position. The subjective probability of a win for the strategy of betting the dog winning the last race was.1297, compared to an objective probability of Likewise, the results suggest that bettor s subjective probabilities understate $35,147, 494 on the trifecta which requires picking the dogs finishing first, second and third. 11 Terrell and Farmer s (1996) analysis of returns reports this expected return, which we replicated using the current data set. 12

14 the probability of a win by dogs 1 and 4, but overstate the probability of a win by the dog in 7 position. Table 3 contains similar results computed from the 1994 statistics at the Woodlands. Some biases from 1989 disappear and other new biases appear. Table 4 summarizes the biases in both years to ease comparisons. Overall, several biases appear in both years. First, bettors consistently overestimate the probability of a win by the longshot. The bias is smaller in 1994 season, but is large and statistically significant in both years. The results also show that bettors underestimate the probability of a win by the dog in the post position of the dog winning the last race. This bias is large in both years, although the smaller number of wagers in 1989 leads to a failure to reject the hypothesis of zero bias in that year. These biases serve as additional evidence that the bettors suffer from the gambler s fallacy and limited form of the longshot bias. The results also show evidence of consistent biases by post position. The largest bias occurs for the strategy of betting dog 7. Bettors consistently overstate the probability of a win by the dog in position 7. The bias is large and statistically significant in both years and translates into low expected returns in both years. The expected return to a one dollar wager on dog 7 was approximately -.32 in both years, far below an expected The results also show weaker evidence that bettors underestimate the probability of wins by the dogs in position 1 and 4. The comparison of returns also reveals some evidence of smaller biases in 1994 than in For example, the longshot bias in 1994 fell to half the value of the bias in To measure the overall size of the bias, the last two lines in table 4 report the mean square error of all odd-based strategies and post-position strategies. These calculations suggest that subjective probabilities matched objective probabilities slightly better in the 1994 season than in the

15 season. Because 1989 was the first year of operation at the Woodlands, this result might be attributable to learning. However, Terrell and Farmer s (1996) model of gambling with informed bettors supplies another explanation. The increase in the pool between 1989 and 1994 could have attracted more informed bettors, which tends to drive subjective probabilities closer to objective probabilities. 5. Conclusion This paper analyzes biases in assessments of probabilities based on over 9,000 races at the Woodlands greyhound park in Kansas City, Kansas. Results indicate that odds do not accurately reflect the probabilities of outcomes in races. Combining these results with previous research reveals overwhelming evidence of certain predictable biases in bettors assessments of probabilities. First, bettors subjective probabilities overstate the objective probability of a win by the longshot in our sample. Our results on this bias closely mirror Ali s (1977) results based on over 20,000 harness races. The combination of both studies supplies strong evidence that the general population of bettors consistently overestimates the probability of wins by longshots. Kahneman and Taversky s (1979,1984) experiments suggest that this bias may simply reflect the fact that people in general overestimate the probability of low probability events. On the other hand, we find no evidence that bettors underestimate the probability of wins by favorites as Ali (1977) found. Results also reveal strong evidence of the gambler s fallacy. Bettors underestimate the probability of a repeat win by the dog in the post position of the last winner. Combined with studies by Clotfelter and Cook (1991, 1993), Terrell (1994), and Terrell and Farmer (1996), there 12 The return of -.18 reflects the fact that 18 cents of every dollar, the breakage of the track, is removed 14

16 appears to be overwhelming evidence that the gambler s fallacy exists among bettors. This bias also may simply reflect misperceptions of the laws of probability. Taversky and Kahneman (1974) document a similar bias among subjects in experiments. This study also reveals evidence that bettors overestimated the probability of a win by the dog in position 7 in both 1989 and We attribute this result to the psychological perception of seven as a lucky number. This result suggests that superstitions affect betting patterns, corroborating Clotfelter and Cook s (1993) conjecture from analysis of betting in lotteries. While a clear picture of the biases is emerging, interpretation of the results is more difficult. The simplest assumption is that gamblers are a representative sample from the entire population; thus, the biases in the odds at racetracks reveal general biases of all economic agents. Laboratory evidence reveals that subjects in experiments suffer from biases similar to those observed at racetracks, and provides evidence that these biases exist in other settings. In this case, the wagers of bettors supply a unique opportunity to observe the perceptions of probabilities by bettors wagering their own money. If information differs across gamblers, the biases of the general population may be distorted in the odds. In Terrell and Farmer s (1996) model of informed bettors, the odds are a mix of probability assessments by pleasure bettors and assessments of informed bettors. The biases appear initially because pleasure bettors exert little effort to determine the correct probabilities of wins. However, any bettor may purchase the true probabilities for a cost. Because of the breakage of the track and the small pool available to gamblers, equilibrium entry fails to drive the odds to a point of equal returns for all bets. The final odds still contain evidence of the pleasure bettors biases in the assessments of probabilities, but also reflect by the wagers of informed bettors. from the pool. 15

17 Several features of the model are worth noting. First, the biases of subjective probabilities in our study may in general understate the initial biases of pleasure bettors. This occurs because wagers of informed bettors drive odds closer to those implied by true probabilities. Second, some biases may be created or amplified by the wagers of informed bettors. For example, if longshots and favorites earn equal returns in the absence of informed bettors, entry of informed bettors will lead to odds that imply larger expected returns to wagers on the favorite than to wagers on the longshot. Finally, the breakage of the track and size of the pool influence odds. If the breakage is small and pool is large, the gambling market closely resembles a large financial market. In this case, the wagers of informed bettors drive the odds to the point where all bets earn an expected return of zero. At this point, odds imply subjective probabilities equal to objective probabilities. Under the second interpretation, the biases might be best explained in terms of an agent capable of learning. If little effort is devoted to obtaining and processing information, the agent s perception of probabilities is influenced by superstition and inaccurate perceptions about the laws of probability. In circumstances where the benefit of learning more about the true probabilities of events exceeds the cost of gathering and processing the information, the biases disappear as the agent learns. This implies that when the benefits of learning true probabilities of events are large, such as in analysis of stock markets, agents perceptions of probabilities may be unbiased. 13 However, economists should be aware that when the reward for accurate assessment of probabilities is smaller, agents perceptions of probabilities will likely suffer from the biases documented in this study. 13 Harrison (1989) makes a similar point that subjects in experiments may appear irrational when the reward to rational behavior is small. His observation led to an increased emphasis on evaluating the rewards for alternative courses of action when evaluating the results of experiments. 16

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19 References Ali, Mukhtar M. (1977), Probability and utility estimates for racetrack bettors, Journal of Political Economy, vol. 85(4), pp Ali, Mukhtar M. (1979), Some evidence on efficiency of a speculative market, Econometrica, vol. 47(2),pp Asch, Peter and Richard E. Quandt (1987), Racetrack betting and profitability in exotic bets, Economica, vol. 54, pp Clotfelter, Charles T. and Phillip J. Cook (1991), Lotteries in the Real World, Journal of Risk and Uncertainty, vol. 4, pp Clotfelter, Charles T. and Phillip J. Cook (1993), The gambler s fallacy in lottery play, Management Science, vol. 39, pp Dolbear, F. Trenery (1993), Is racetrack betting on exactas efficient? Economica, vol. 60, pp Golec, Joseph and Maurry Tamarkin (1991), The degree of inefficiency in the football betting market, Journal of Financial Economics, vol. 30, pp Goodwin, Barry (1996), Semi-parametric testing of speculative efficiency in a parimutuel gambling market, Journal of Business and Economic Statistics, vol. 14 (4), pp Griffith, R. M. (1949), Odds adjustment by American horse-race bettors, American Journal of Psychology, vol. 62, pp Haush, Donald B., William T. Ziemba, and Mark Rubinstein (1981), Efficiency of the market for racetrack betting, Management Science, vol. 27 (12), pp Harrison, Glenn W. (1989), Theory and Misbehavior of First Price Auctions, American Economic Review, vol. 79, pp Johnson, Richard and Jerome Klotz (1993), Estimating hot numbers and testing for uniformity for the lottery, Journal of the American Statistical Association, vol. 88, pp

20 Kahneman, Daniel and Amos Taversky (1979), Prospect theory: An analysis of decision under risk, Econometrica, vol. 47, pp Kahneman, Daniel and Amos Taversky (1984), Choices, Values, and Frames, American Psychologist, vol. 39, pp Metzger, Rebecca S. (1985), Biases in betting: an application of laboratory findings, Psychological Reports, vol. 56 (3), pp McGlotlin, W. H. (1956), Stability of Choices under Uncertain Alternatives, American Journal of Psychology, vol. 69, pp Quiggin, John (1991), On the optimal design of lotteries, Economica, vol. 58, pp Taversky, Amos and Daniel Kahneman (1974), Judgement under Uncertainty: Heuristics and Biases, Science, vol. 185, Terrell, Dek (1994), A test of the gambler s fallacy: evidence from parimutuel games, Journal of Risk and Uncertainty, vol. 8, pp Terrell, Dek and Amy Farmer (1996), Optimal betting and efficiency in parimutuel betting markets with information costs, Economic Journal, vol. 106, pp

21 Table 1: The effects of the rounding adjustment on subjective probabilities (1) Odds (2) Subjective Probability: No Adjustment (3) Subjective Probability: Adjusted for Rounding Notes: Column (2) of the table translates the odds into subjective probabilities using equation (2) in the paper. Column (3) converts odds into subjective probabilities accounting for rounding using equation (4).

22 Table 4: A Summary of Results Strategy 1989: Baseline 1994: Baseline Bet favorite Bet second lowest odds ** Bet third lowest odds Bet fourth lowest odds Bet fifth lowest odds Bet sixth lowest odds 0.008** Bet seventh lowest odds Bet longshot 0.013** 0.006** Bet number of dog winning the last race ** Bet dog ** Bet dog ** Bet dog Bet dog * ** Bet dog Bet dog * Bet dog ** 0.014** Bet dog Mean Square Error over Odds-based strategies x10, Mean Square Error over Post position strategies x 10, Notes: This table presents the bias, calculated as subjective probability minus objective probability. A positive value indicates that bettors overestimate the probability of a win using the strategy while a negative value implies that the odds underestimate the objective probability of a win. * denotes significant at the 10% level. ** denotes significant at the 5% level.

23 Table 2: Subjective and objective probabilities for the 1989 season Strategy Number of Wagers Number of Wins π se( π ) p se( p) e = p π se( e) z-value Bet favorite [.2937] Bet second lowest odds [.1831] Bet third lowest odds [.1392] Bet fourth lowest odds [.1102] Bet fifth lowest odds [.0897] Bet sixth lowest odds [.0737] Bet seventh lowest odds [.0608] Bet longshot [.0497] Bet number of dog winning the last race [.1268] Bet dog [.1486] Bet dog [.1276] Bet dog [.1198] Bet dog [.1182] Bet dog [.1150] Bet dog [.1146] Bet dog [.1184] Bet dog [.1377] Notes: The column labeled π contains the objective probability of a win using each strategy and the column labeled p contains the subjective probability calculated two ways. The number without brackets is calculated using equation (2) with the breakage scaled up to reflect rounding of odds. The number in brackets is the subjective probability calculated using equation (4). The bias e is the subjective probability minus the objective probability and the z-value is the test statistic for the null hypothesis of no bias ( e = 0 ).

24 Table 3: Subjective and objective probabilities for the 1994 season Strategy Number of Wagers Number of Wins π se( π ) p se( p) e = p π se( e) z-value Bet favorite [.2840] Bet second lowest odds [.1840] Bet third lowest odds [.1412] Bet fourth lowest odds [.1136] Bet fifth lowest odds [.0930] Bet sixth lowest odds [.0760] Bet seventh lowest odds [.0612] Bet longshot [.0469] Bet number of dog winning the last race [.1277] Bet dog [.1596] Bet dog [.1348] Bet dog [.1224] Bet dog [.1193] Bet dog [.1147] Bet dog Bet dog Bet dog Notes: The column labeled π contains the objective probability of a win using each strategy and the column labeled p contains the subjective probability calculated two ways. The number without brackets is calculated using equation (2) with the breakage scaled up to reflect rounding of odds. The number in brackets is the subjective probability calculated using equation (4). The bias e is the subjective probability minus the objective probability and the z-value is the test statistic for the null hypothesis of no bias ( e = 0 ). 1

25 Table 2a: Subjective and objective probabilities for the 1989 season: Rounding Adjustment II Strategy Number of Wagers Number of Wins π se( π ) p se( p) e = p π se( e) z-value Bet favorite Bet second lowest odds Bet third lowest odds Bet fourth lowest odds Bet fifth lowest odds Bet sixth lowest odds Bet seventh lowest odds Bet longshot Bet number of dog winning the last race Bet dog Bet dog Bet dog Bet dog Bet dog Bet dog Bet dog Bet dog

26 Table 3a: Subjective and objective probabilities for the 1989 season: Rounding adjustment II Strategy Number of Wagers Number of Wins π se( π ) p se( p) e = p π se( e) z-value Bet favorite Bet second lowest odds Bet third lowest odds Bet fourth lowest odds Bet fifth lowest odds Bet sixth lowest odds Bet seventh lowest odds Bet longshot Bet number of dog winning the last race Bet dog Bet dog Bet dog Bet dog Bet dog Bet dog Bet dog Bet dog

27 4