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# ECONOMICS AND SPORTS ECON 324

Clemson

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Date Created: 09/26/15

Sauer Economics ofWagem39ng Markets 2035 COMPARISON OF SUBJECTIVE PROBABILITIES AND ACTUAL WINNING FREQUENCIES Y ODDS BANK OF HORSE Odds Rank of Horse No of No of Entries Races 1 2 3 4 5 6 7 8 9 10 11 12 5 69 Subj prob 42 25 17 11 06 Obs freq 41 30 20 07 03 6 181 Subj prob 36 23 17 12 08 04 Obs freq 43 21 20 11 03 02 7 312 Subj prob 33 22 16 12 09 06 03 Obs freq 34 21 16 12 08 08 02 8 352 Subj prob 31 20 15 12 09 06 04 03 Obs freq 33 25 13 09 07 06 04 02 9 283 Subj prob 30 20 05 11 09 06 05 03 02 Obs freq 35 15 17 13 08 06 02 01 02 10 241 Subj prob 29 19 14 11 08 06 05 03 02 02 Obs freq 31 17 16 10 07 07 06 04 02 01 11 154 Subj prob 27 18 14 11 08 07 05 04 03 02 01 Obs freq 27 18 19 08 05 05 05 05 04 04 01 12 233 Subj prob 26 17 13 10 08 07 05 04 03 02 02 01 Obs freq 28 14 17 12 10 06 02 05 03 03 01 00 Source Hoerl and Fallin 1974 data are from all 1825 races run at Aqueduct and Belmont Park NY in 1970 Observed frequency MEAN ORDER OF FINISH BY ODDS BANK OF HORSE Odds Rank of Horse No of No of Entries Races 1 2 3 4 5 6 7 8 9 10 11 12 69 21 24 29 34 41 6 181 22 29 32 36 42 49 7 312 28 32 37 40 43 46 54 8 352 28 32 39 42 47 51 57 64 9 283 31 36 41 46 51 53 60 64 71 10 241 31 40 43 51 53 56 62 65 70 79 11 154 38 40 47 52 57 58 63 69 72 78 85 12 233 39 46 51 54 60 62 67 72 76 77 87 91 Source Hoerl and Fallin 1974 data are from all 1825 races run at Aqueduct and Belmont Park NY in 1970 studies including Griffith McGlothlin Snyder aggregated the rates of return and Weitzrnan encompassing 50000 for various odds categories with the races in North America each docu takeout added back as in McGlothlin rnenting the favoritelong shot bias The pretakeout rate of return varies 2036 journal ofEconomic Literature Vol XXXVI December 1998 7 E 20 r 3 E I53 40 7 760 0 l l l l l l l l 075 125 250 500 750 1000 1500 3300 Midpoint of Odds Range Figure 1 The FavoriterLongshot Bias in the US Parirmutuel Snyder and UK Bookmaking Dowie Markets Note Snyder s are parirmutuel returns with the takeout added back hence the norm is zero Dowie s are prertax returns at SF odds Hence the norm for Dowie is less than zero to account for the costs ofbookmaking from 91 percent for oddson horses 0 lt 1 to 7237 percent for horses with the highest odds 3371 and up Jack Dowie 1976 found the same pattern of returns in the British book making market hence this finding is not unique to parimutuel markets Indeed the bias is more pronounced in Dowie s data which encompass all 2777 races run in Britain during 1973 Figure 1 displays the rates of return from Snyder 1978 and a similar construction using Dowie s data The returns to extreme long shots are markedly lower in the UK market In addition the returns to low odds horses present a puzzle Breaking down the data more finely Dowie s figures indicate that book makers lost money when taking bets on extreme favorites horses with 01305 There were 107 such horses with a be foretax rate of return of 085 at final odds which is roughly the bookmaker s loss for these bets26 Ali 1977 analyzed betting in a sam ple of 20247 harness races in a fash ion similar to Hoerl and Fallin In con trast to Hoerl and Fallin s data Ali s data were strongly inconsistent with the null hypothesis that subjective and objective probabilities were equal Table 4 contains estimates of objective 25An explanation consistent with evidence in Section 44 is 39 ing in improved but imperfect estimates 0 the probability of wmning 2037 Sauer Economics of Wagering Markets THE DIFFERENCE BETWEEN SUBJECTIVE AND OBJEcrIVE PROBABILITIES IN THREE SAMPLES Ali 1977 Asch Malkiel 8r Quandt 1982 Busche 8 Hall 1988 Rank fik Wk 7 t 13k Wk 7 t 13k Wk 7 t 1 0358 70035 0361 40036 0276 0008 2 0205 0003 0218 40013 0190 40003 3 0153 70001 0170 40025 0151 40009 4 0105 0007 0115 40011 0099 0012 5 0076 0006 0071 0001 0084 0003 6 0055 0005 0050 40002 0063 0003 7 0034 0008 0030 0004 0048 0001 8 0021 0007 0017 0008 0047 40010 9 0006 0012 0034 40006 10 0021 0000 11 0023 40005 12 0014 40001 13 0010 0001 14 0013 40005 Notes The probabilities need not sum to one because different numbers of horses participated in each race pk is the estimated objective probability for horses in each group and Wk 7 pk is the difference between the subjective and objective probabilities A positive value for Wk 7 pk indicates that horses in this group were overbet relative to a smndard in which expected returns were equal across all classes of horses The original sources reported tests of smtistical significance for the variate wk 7 pk For the samples of Ali and Asch Mallciel and Quandt favorites rank 1 were significantly underbet and extreme longshots were significantly underbet there is no such pattern in the Busche and Hall data Alis sample is from 20247 harness races run at 3 New York tracks from 1970774 Asch Mallciel and Quandt s sample is from 729 races run at Atlantic City during the 1978 season Busche and Halls sample is from 2653 races in Hong Kong from 1981 to 1986 probabilities and the differences be tween subjective and estimated objec tive probabilities wk k for three studies Ali 1977 Peter Asch Burton Malkiel and Richard Quandt 1982 and Kelly Busche and Christopher Hall 1988 Ali s findings are in the first two columns of Table 4 where the bias is clearly evident27 Asch Malkiel and Quandt 1982 found the same pattern in betting data from Atlantic City s race course The data from Hong Kong are differ ent Busche and Hall 1988 analyzed a sample of 2653 races run in Hong Kong 27The tiratios reported by Ali testing the h 7 othesis that wkpk are 7103 for the favorite insignificant for the second and third ranked horses and above 30 for the 49 78 h ranked rses between 1981 and 1986 They found no evidence of biased returns in the Hong Kong parimutuel market wig x is negative 7035 and 7036 for the top ranked horse in the North American studies indicating the favorite was rela tively underbet28 In Hong Kong this difference is positive Whereas the sub jective probability exceeds the objective by about 01 for North American long shots there is no pattern in the Hong Kong long shots A Psychological Explanation There are several potential reasons for the ex istence of the favoritelong shot bias An obvious explanation is that bettors may underestimate the chances of 28 It was also statistically significant about 10 times its standard error mean in Ali and twice as large in Asch Malkiel and Quandt Sauer Economics of Wagering Markets TAB LE 6A RATIO OF FINAL ODDS AND MARGINAL ODDS TO MORNING LINE ODDS BY FINISH POSITION FOR 729 RACES RUN AT ATLANTIC CITY NJ IN 1978 Horses Finishing OFINAL OML OLATE OML First 096 082 Second 116 106 Third 122 117 Also Rans 159 163 Source Asch Mallciel and Quandt 1982 p 193 Notes Figures in the columns are ratios of average odds OFINAL are the final odds OML are the morning line odds projected by the tracks handicapper and OLATE are the marginal odds produced by bettors in the last third of the betting period of a technical analystithat something is up Betting late into a larger pool thus reduces the likelihood of creating bandwagon effects in the od s Asch et a1 use data from the 765 races run at Atlantic City Race Course in 1978 They find that the winning horse is bet down that is its final odds OFINAL are lower than the morn ing line estimate of the odds produced by the tracks expert handicapper OML These results are presented in Table 6A OFINALOML is 096 for horse the ultimately wins the race The ratio for all other horses including the second and third place finishers ranges from 106 to 163 Furthermore late money appears to be more informed than early money Asch Malkiel and Quandt calculated the marginal odds based exclusively on wagers made in the last eight minutes of the betting Using marginal odds OFINALOML is 082 for the winner and remains above 1 for horses that don t win As Asch Malkiel and Quandt 1982 p 306 put it winning horses are especially preferred by the late bettors 2043 N F R Crafts 1985 studied the bookmaking market in the UK along these lines In this market the odds analogous to the North American morn ing line are issued as the FF or forecast price by Sporting Life a daily trade publication Representatives of Sport ing Life observe the betting noting in particular the very large bets and the odds at which they are transacted A de scription of the betting is then printed in a subsequent edition of the paper At the end of each betting period these representatives determine the starting prices for the horses SP which are the odds offered in the oncourse bookmak ing market at the end of the betting pe riod Crafts notes that the practice of paying off at odds offered at the time of each transaction enhances the value of inside information since bandwagon ef fects from betting large sums do not af fect the payoff40 Craft s sample covers 16769 horses that ran between September 1982 and January 1983 Denote the subjective probabilities at FF and SP odds as SFP and 33p Horses for which either 15 S SSPSFP lt 2 0139 SSPSFP 2 2 are considered to have been heavily backed ie the wagers in the market 401t is important to understand how bookmaki ers change odds during the betting period Stan 2028 journal ofEconomic Literature Vol XXXVI December 1998 INFORMED AGENTS OPTIMAL WAGER SZE AND EXPECTED RETURNS Number of Inform ed Opiimal Wager Tomi Informed Expected Return Probability Subjective M x Mx ka R13 0 i 00 2000 0333 1 333 333 1333 0500 2 289 577 1155 0577 3 233 699 1097 0608 4 192 768 1070 0623 5 162 811 1055 0632 6 140 841 1045 0638 7 123 863 1038 0642 8 110 879 1033 0645 9 99 892 1029 0648 10 90 903 1026 0650 15 62 934 1017 0655 20 48 951 1013 0658 25 38 960 1010 0660 50 20 980 1005 0663 Notes The calculaiions assume Pk 23 and 5k 13 N is set to 100 so that the wagers listed can be interpreted as percenmges relaiive to the size of the uninformed betiing pool be rejected when private information is important and limited to small groups Horse race betting is just such a case Many results in this literature which de part from the constant expected returns standard are related to these factors Finally it is well known but worth re iterating that efficiency is not a stand alone concept Efficient prices embody properties that are implied by a given model and are therefore dependent on the behavioral assertions constraints and information structure that charac terize the model Hence the source of error when efficiency is rejected is by no means immediately obvious My own View is that the generic efficient mar kets hypothesis is a very useful bench mark Its generality is at once a great strength since it can be widely applied and a great weakness since it will be rejected in settings where idiosyncratic conditions are important But rejections of efficiency don t just highlight limita tions of the basic model they must be studied carefully for it is these cases which add the most to our under standing of the forces that create mar ket prices 3 Models of Gambling Behavior and Gambling Markets 31 Utility of Wealth Models of Gambling Models of gambling based pected utility date back to Bernoulli s famous solution to the St Petersburg paradox Bernoulli posited that individuals value a gamble usin a probabilityweighted utility function instead of the standard mathematical expectation see Kenneth Arrow 1952 pp 420721 and Mark Machina 1987 p 12272313 Since the solution requires on ex Daniel 13Arrow 1952 provides a particularly comprei 39 39 ssion of axiomatic foundations for and methodological ob39ections to expected utility theory Machina 1987 provides a elp intro duction to noniexpected utility models in whic r m 1 E lt m m o s 2044 journal ofEconomz c Literature Vol XXXVI December 1998 E6B RATES OF RETURN AT FP AND SP ODDS FOR HORSES CHARACTERIZED BY ODDS MOVEMENTS FOR 16769 HORSES IN THE UK 19883 Odds Movement Number of Horses Rate of Return at FP Rate of Return at SP Heavily Backed psPprZZ 397 141 7009 15 S PSP pr lt 2 712 064 7001 Drifting Out 15 S PFPPSP lt 2 858 7063 7038 pm psp 2 2 317 7064 7013 Source Crafts 1985 p 298 Notes 135 is the implied subjective probability at SP odds ppp the same at FP odds Horses whose odds decline in the betting will have psP pr ratios which exceed 10 vice versa for horses whose odds increase during the betting Rates ofretuin do not include the 4 10 Ex on course off course in effect at the time have pushed the odds down well below the forecast level Table 6B presents Crafts results There are 712 horses satisfying the former and 397 horses satisfying the latter more pronounced condition Bets on these horses at SP odds are not profitable earning pretax returns of 701 and 709 respectively Bets made at FP odds would have yielded returns of 64 and 141 the lat ter being phenomenally profitable On the flip side the 1175 horses whose odds drifted out were very poor bets horses with stsspz 2 yielded returns of 764 at FF and 713 at SP odds horses with 1533Fpssplt2 yielded returns of 7063 at FF and 7038 at SP odds Using parimutuel data from races run in Chicago Robert Losey and John Talbott Jr 1980 obtained a similar re sult Losey and Talbott s simulation placed 579 bets on all horses for which the Daily Racing Form s expert handi capper placed a morning line of 371 or less but whose final parimutuel odds exceeded this estimate The returns were 7284 percent which exceeds by a large margin the 17 percent loss take out rate breakage expected if the rates of return were equal across horses What does one make of these rates of return It is clear that trading in these markets creates measures of the prob ability of winning which are signifi cantly better than measures produced by an individual or group of experts This suggests that the betting market aggregates disparate sources of informa tion into a superior probability estimate of the races outcome Second the odds adjustment stops short of achieving con stant returns to equalize returns at SP odds between horses whose odds have fallen and those that have risen would require additional reductions and addi tional increases for each group41 Third and this is Crafts main point these re turns clearly point to the existence of an informed class of bettors The pub lished descriptions of the betting are helpful in this regard for they establish that bets were made at odds substan tially greater than SP odds for many of these winners Consider these descrip tions first for a winner that had never 41 Incomplete adjustment is related to the favor iteilong shot bias since orses w ose odds shorten are more 139 e y to be favorites and ose w o lengthen long shots This feature is closely related to Hurley and McDonoughs 1995 model of the bias 2050 journal ofEconomic Literature Vol XXXVI December 1998 SCORE DIFFERENCES AND POINT SPREADS FOR GAMES A Sample Frequencies Differencing Method Sample Partition Cam es Beb Wins Ties WinsBets A1 Hom eiAWay All Games 5636 5510 2789 126 506 Home Favorites 4341 4243 2148 98 506 Home Underdogs 1209 1181 600 28 508 Pick em Games 86 86 41 0 477 A2 FavoriteiUnderdog 5550 5424 2729 126 503 B Sample Means and Smndard Deviations Differencing Method Sample Partition DP PS DPiPS testat B1 HomeiAWay All Games 4621242 438559 0241115 162 Home Favorites 6871182 681362 0061107 037 Home Underdogs 3091174 A05230 0961145 291 Pick em Games 40911058 000000 70911058 i079 B2 FavoriteiUnderdog 6050183 621356 70160116 106 Notes Sample Characteristics The sample encompasses all regular season NBA games played in the six seasons from 19883 through 1987488 Score differences were obmined from the annual edition of the Sporting News NBA Guide Point spreads were obtained from The Basketball Scoreboard Book These point spreads are those prevailing in the Las Vegas market about25 hours prior to the start of play 5 PM Eastern time on a typical night No point spread is reported for 22 games during this period which reduces the sample from 5658 all games played to 5636 all games with point spreads Panel A This panel lists the number of games beb the number of games in which DP i PS which are ties and the number of bets Won by Wagering on the team in the first position of the score difference WinsBeb is the sample estimate of p the proportion of such bets Won Since this proportion always lies inside the bounds given by 2 no test statistic is required to evaluate this implication of efficient pricing Panel B Smndard deviations are given in parentheses The testatistic tesb the null hypothesis that the mean forecast error DP 7 PS is zero Although the null is rejected in the case of home underdogs the failure to reject efficient pricing in panel A for this partition indicates that the rejection in B is caused by a departure from the symmetry assumption all relevant information Denoting the set of all relevant information by 9 this requires that EDP PS lQ0 7 which says that the forecast error is un related to relevant information because this is already present in equal amounts in both DP and PS Table 7 presents data from the bet ting market on NBA games This table can be used to assess the implications listed above using both the home team away team and favoriteunderdog order ing of score differences Also included are all partitions of these orderings in cluding games in which there was no fa vorite PS 0 Panel A examines equa tion 5 In no partition is the ratio of Winning to total bets outside the bounds implied by 5 Sample means and standard devia tions of DP PS and DPPS are pre sented in panel B of Table 7 Note the 2052 journal ofEconomic Literature Vol XXXVI December 1998 THE PROFITABILITY OF SIMPLE BETTING RULES A Vergin and Scriabins Sample 196974 N p pivalue p 05 pivalue p lt 05238 Bet on big underdogs 674 546 017 249 Bet against big winner 78 538 499 795 Bet on turnaround team 59 627 051 112 Bet on strongest team 57 667 012 012 B Tryfos et alfs Sample 197571981 N p pivalue p 05 pivalue p lt 05238 Bet on big underdogs 735 535 060 277 Bet against big winner NA NA NA NA Bet on turnaround team 76 447 359 i Bet on strongest team 71 459 486 i Notes 15 is the proportion of winning beb from N tries The pivalues in the final two columns test the hypothesis that the true probability of winning is 5 and lt 5238 respectively Big underdogs are predicted to lose by more than 5 poinb by the point spread The big winner is the team with the largest margin of victory in the prior week The turnaround team is the team that beat the spread by the largest amount over the prior 4 weeks The strongest team is the team with the largest victory margin over the prior 4 weeks Tryfos et a1 did not reexamine the big winner strategy 52 The Simple Linear Prediction Model Equation 6 has been repeatedly examined in the context of a linear prediction model The basic form of this model is DP0LHBPS8 8 where H is a vector of ones 0i and B are regression coefficients and 8 is an error term Market efficiency is examined by testing the joint null hypothesis that 0i 0 and B 1 ie that PS is an unbiased linear predictor of DP With scores and spreads ordered on a home team minus away team basis it is clear that the inter cept term 0i reflects advantages of the home team that are not priced in the betting market T e first versions of equation 54 were estimated by Ben AmoakoAdu Harry Marmer and Joseph Yagil 1985 and Richard Zuber John Gandar and Benny Bowers 1985 and were pre sented as evidence that point spreads were very poor and probably inefficient predictors of score differences of pro fessional football games Zuber et al estimated separate regressions for each of the 16 weeks of the 1983 NFL regu lar season and failed to reject the noninformative null hypothesis that DP is unrelated to PS ie that 0i B 0 in 15 of the 16 weeks50 They conclude p 802 that the noninformative null hypothesis is as consistent with the sample data as is the efficiency hy pothesis AmoakoAdu et al reversed the de pendent and independent variables in the regression Their estimated equa tion is PS 447004 DP with an R2 of 04 from a sample of 233 games51 50 Note that even in the absence of point spread effects let B 0 Zuber et al s noniinformative nu hypothesis could have been rejected if 0L were welliknown positive effect of playing at home us ing a single weeks samp e o ames 1In contrast to most stu 39es the ordering of point differences in AmoakoiAdu et al is as 2058 BLE 9 THE INFORMATION CONTENT OF THE IDIOSYNCRATIC COMPONENT IN NBA POINT SPREADS Number 0 MFE Smndard ll Observations PS HAT Error 00 432 7035 1014 05 744 092 1119 10 624 088 1017 15 535 126 1137 20 398 107 1106 25 307 261 1177 30 211 275 1134 35 162 144 1053 40 90 298 1227 45 57 297 960 50 35 569 1235 55 21 588 884 60 14 229 1265 65 7 7074 705 70 5 800 485 75 8 450 1369 280 4 7350 690 Source Brown and Sauer 1993a The data on scores and spreads are the same as in Table 51 11 PS 7 PSHAT is the idiosyncratic component of point spreads based on out of sample forecasts using pa rameters obtained from estimating equation 13 Note that negative values of PS 7 PSHAT and the corre7 sponding forecast errors have been multiplied by 71 This conserves mble space and smtistical power For example suppose that MFEPS HAT 43 for observa7 tions where 11 74 The adjushn ent in the spread of 74 poinb accounted for by the unobserved component is thus 1 point too large Converting these numbers to 3 and 4 poinm respectively yields the same interpretai lion and allows the positive and negative observations to be pooled of MFEltPSHAT in Table 9 indicates that 111 does not represent meaningless noise if it were these values would be centered on 0 A fundamentalsbased null hypothesis is that 111 DP PSHAT 111 0 which requires that MFEltPSHAT increase with the value of 111 Brown and Sauer fail to reject this journal ofEconomz39c Literature Vol XXXVI December 1998 hypothesis Brown and Sauer interpret these results in light of Roll s inability to explain stock price changes with a small scale model Since the forecast errors in Table 9 tend to increase with 111 this indicates that 111 represents unobserved fundamentals which are excluded from the team strength model Brown and Sauer also identify cases in which the estimates of team strength change from season to season and show that the changes in these estimates are essential to unbiased prediction Brown and Sauer do not model the adjustment in team ability estimates however Dana and Knetter 1995 model this process explicitly using NFL data Dana and Knetter examine how the market incorporates information from score dif ferences in revising its estimates of team abilities They estimate a version of equation 12 in which the team strength estimates follow a random walk Efficient updating of these esti mates is complicated by the low signal to noise ratio in score differences Past score differences are used to estimate the team strength parameters with an endogenously estimated threshold level beyond which increases in the score dif ference are discounted rather than at tributed to relative ability Other indi cators of noiseinet turnovers an penaltiesiare used as regressors in the model in an attempt to clean the ability estimates from these factors Dana and Knetter calculate a dis count factor of 25 and a threshold of 83 points implying that score differ ences beyond 8 points add about 14 of the information on relative ability com pared to score differences within 8 points The key question is whether market participants efficiently discount the noise in large score differences Dana and Knetter conduct betting simula tions both in and out of sample to exam ine this question These simulations fail

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