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Probability and Optimization Probability and Optimization
Models for RacingModels for Racing
Victor S. Y. LoVictor S. Y. Lo
University of British ColumbiaUniversity of British Columbia
Fidelity InvestmentsFidelity Investments
Disclaimer: This presentation does not reflect the opinions of Fidelity Investments. The work here was completed at University of British Columbia and the University of Hong Kong.
22
OutlineOutline
Areas to Discuss in Racing:Areas to Discuss in Racing:
FavoriteFavorite--longshotlongshot Bias Bias
(Economics, Statistics)(Economics, Statistics)
Ordering Probabilities and Ordering Probabilities and
Optimal Investment (Probability, Optimal Investment (Probability,
Statistics, Finance)Statistics, Finance)
33
Part 1: FavoritePart 1: Favorite--LongshotLongshot BiasBias
Favorites are Favorites are underbetunderbet and and
longshotslongshots are are overbetoverbet –– Busche & Busche &
Hall (1988), Ali (1977)Hall (1988), Ali (1977)
A wellA well--known phenomenon in known phenomenon in
economic literature economic literature -- Ziemba Ziemba
(2004);(2004); Creating opportunities Creating opportunities ––Bolton & Chapman (1986), Bolton & Chapman (1986), BentorBentor
(1994)(1994)
Economic interpretation: RiskEconomic interpretation: Risk--
loving behaviorloving behavior
44
FavoriteFavorite--LongshotLongshot BiasBias
higharesifP
lowaresifP
iii
iii
'
'
ππ
ππ
>
<
Define: Pi = Bet fraction on horse i, i.e. consensus win probability, i = 1, …, n= (1- track take)/(1 + Oi), where Oi = Odds on i
πi = objective (true) win probability of i
55
Statistical ModelStatistical ModelMany techniques mentioned in the Many techniques mentioned in the
literature literature –– Ali (1977), Asch and Ali (1977), Asch and
QuandtQuandt (1984) (1984)
Propose to use a simple logit model, Propose to use a simple logit model,
BaconBacon--Shone, Lo, and Busche Shone, Lo, and Busche
(1992a):(1992a):1∑=njP
∑=
=n
j
j
ii
P
P
1
β
β
π
66
Statistical Model (continued)Statistical Model (continued)
ββ>1 >1 →→ riskrisk--preferprefer
ββ =1 =1 →→ riskrisk--neutralneutral
ββ<1 <1 →→ riskrisk--averseaverse
∑=
=n
j
j
ii
P
P
1
β
β
π
77
Universal ComparisonsUniversal ComparisonsUS racetracks consistently have a risk-prefer bias with β > 1
Racetrack # races Estimated β
p-value for
H1: β n.e. 1 Average pool size
US (Quandt's 83-84):
Atlantic City 712 1.10 0.08 unknownMeadowlands 705 1.12 0.02 $52K
US (Ali's 70-74):
Saratoga 9,072 1.16 ~0 $25K
Roosevlt 5,806 1.13 ~0 $218KYonkers 5,369 1.13 ~0 $228K
Japan (90) 1,607 1.07 0.01 $168K
Hong Kong (81-89):
Happy Valley 2,212 1.04 0.25 $1.1MShatin 1,943 0.94 0.04 $1.1M
China (23-35):Shanghai 730 1.03 0.38 unknown
88
Utility Function InterpretationUtility Function InterpretationExpected utility Expected utility maximizermaximizer is indifferent is indifferent
between betting on any horses in a race:between betting on any horses in a race:
It can be shown:It can be shown:
.,...,1)( niKUE i =∀=
ββ
β
β
)1()1()1(
1)1( ii
ij
j
i OOt
P
KOU +∝
++
+=+
∑≠
Power utility
lovers"-Risk" declinecapitalasriskmoretakeBettors
1 ifwealth,withincreasesand,0
)2(/)1()('
)(''
AversionRiskAbsoluteofMeasurePrattArrow Then,
⇒
><
−−=−=
−
β
β xxU
xU
See Ali (1977), Lo (1992)
99
Conclusion and Research Conclusion and Research
OpportunitiesOpportunitiesFavoriteFavorite--longshotlongshot bias exists in many US bias exists in many US racetracks (but not huge)…racetracks (but not huge)…
…but does not exist in some Asian …but does not exist in some Asian racetracks racetracks –– would it depend on the would it depend on the Pool Pool SizeSize??
Bias in other investment areas Bias in other investment areas –– see see ZiembaZiemba (2004)(2004)
Opportunity to understand bias or accuracy Opportunity to understand bias or accuracy in complicated bets, e.g. Lo and Busche in complicated bets, e.g. Lo and Busche (1994)(1994)
Opportunity to apply similar logit models in Opportunity to apply similar logit models in other applications and other sports, e.g. Lo other applications and other sports, e.g. Lo (1994a), Willoughby (2002)(1994a), Willoughby (2002)
1010
Part 2: Ordering ProbabilitiesPart 2: Ordering ProbabilitiesRunning time distribution (Running time distribution (TTii’s’s) is key to ) is key to
determine ordering probabilities:determine ordering probabilities:
)4()|()]|(1[)|(
}){(
:.','
.,(.)(.)
,)(
)3(,)|()]|(1[
}){(
0 ,
,
0
∫ ∏
∫∏
∞
≠
≠
∞
≠
≠
−=
<<=
=
−=
<=
jir
jjjrjij
rjir
jiij
ii
ii
ir
iiiri
rir
ii
dttftFtF
TMINTTP
computeThensforsolvesGiven
respcdfandpdfareFandfand
parameterlocationorTEwhere
dttftF
TMINTP
θθθ
π
θπ
θ
θθ
π
1111
Running Time DistributionRunning Time DistributionThe following types have been considered in The following types have been considered in
literature, all assuming literature, all assuming independentindependent running running
times:times:
parametershapetheisrwhere
rGammaTSternGamma
NTHeneryNormal
ttfHarvillelExponentia
ii
ii
ii
i
ii
),,(~:)1990(
)1,(~:)1981(
)/exp(1
)|(:)1973(
θ
θ
θθ
θ
−
−
−=−
1212
Exponential Running TimeExponential Running Time
Strictly speaking, we only need Strictly speaking, we only need g(Tg(T) ~ ) ~
Exponential, where g(.) is a Exponential, where g(.) is a
monotonically increasing functionmonotonically increasing function
)6()1)(1(
)3,2,1(
)5(,1
)21(
jii
kji
ijk
ii
i
ji
ij
rdfinisheskndfinishesjstfinishesiP
Pfractionbetbyestimatedbecanwhere
ndfinishesjandstfinishesiP
πππ
πππ
π
π
π
ππ
π
−−−=
=
−=
=
1313
Normal and Gamma Running TimeNormal and Gamma Running Time
The formulas are complex, as one The formulas are complex, as one has to solve (3), a system of integral has to solve (3), a system of integral equations, for equations, for θθii ’’s, and then s, and then compute (4)compute (4)
Henery(1981) proposed to use a Henery(1981) proposed to use a firstfirst--order Taylor series order Taylor series approximation under normal running approximation under normal running timetime
Lo and BaconLo and Bacon--Shone (2007) Shone (2007) proposed a simple approximationproposed a simple approximation……
1414
Simple ApproximationSimple Approximation
).6()5()7(
,1,
).2007(
,''
)7(
,
andtoreduces
timelExponentiaforthatNote
ShoneBaconandLoinvaluesparameterareand
sPfractionsbetbyestimatedbecanswhere ii
jit
t
k
is
s
j
iijk
==
−
=∑∑≠≠
τλ
τλ
π
π
π
π
πππ
τ
τ
λ
λ
Lo and Bacon-Shone (2007):
1515
Running Time Distribution Running Time Distribution CompetitonCompetitonSo, which distribution should be used?So, which distribution should be used?
Lo and BaconLo and Bacon--Shone (1994) found that Shone (1994) found that HarvilleHarvillemodel has a systematic bias in estimating ordering model has a systematic bias in estimating ordering probabilities based on Hong Kong data and probabilities based on Hong Kong data and HeneryHenerymodel is clearly superiormodel is clearly superior
BaconBacon--Shone, Lo, and Busche (1992b) had a similar Shone, Lo, and Busche (1992b) had a similar conclusion using Meadowlands data, however…conclusion using Meadowlands data, however…
… Lo (1994b) found that Stern model with r=4 is … Lo (1994b) found that Stern model with r=4 is better than both better than both HeneryHenery and and HarvilleHarville using Japan using Japan data!data!
1616
Correlated Running TimesCorrelated Running Times
Henery.toreducesit,0If
iance.higher var havewillhorsesweaker0,κifi.e.,
,)](exp[ :arianceconstant v-Non B)
.pairsstronger for higher nscorrelatio i.e.
,1
),()1
(logwhere
, :ncorrelatioconstant -Non A)
:cases dcomplicate more Henery;toreduces
,)Corr(i.e.n,correlatioConstant
==
>
−=
=−−−=−
≠∀=
≠∀=
∑
κγ
θθκσ
θθθθγδψ
ψ
ψψρ
ρ
ii
i
ii
i
i
jiij
ji
n
ji
ji,TT
1717
First order Taylor series approx employed for First order Taylor series approx employed for
complexitycomplexity
Empirical ResultsEmpirical Results
Non-constant correlation with slope γ only or non-constant variance shows some promise
Model Estimates
p-value of Lik
ratio test rel to
Henery
A) Non-constant
correlation (γ only) γ = 0.58 0.06
A) Non-constant
correlation (γ and δ) γ = 0.60, δ =0.05 0.18B) Non-constant
variance κ = 0.08 0.06
1818
Kelly Criterion for Optimal InvestmentKelly Criterion for Optimal InvestmentInstead of meanInstead of mean--variance criterion, we variance criterion, we
maximize expected log wealth maximize expected log wealth →→ growth rate growth rate
of capital:of capital:
Breiman(1960), Thorp(1971), Breiman(1960), Thorp(1971), AlgoetAlgoet & Cover(1988) & Cover(1988)
show longshow long--run asymptotic optimalityrun asymptotic optimality
Adopted by Hausch, Ziemba, & Rubinstein(1981) Adopted by Hausch, Ziemba, & Rubinstein(1981)
using exponential running times, and Lo, Baconusing exponential running times, and Lo, Bacon--
Shone, & Busche(1995) and Hausch, Lo, & Ziemba Shone, & Busche(1995) and Hausch, Lo, & Ziemba
(1994) using other running time distributions, all (1994) using other running time distributions, all
showed promisesshowed promises
ofendtheatwealthtotalwhere
|)(log{maxarg
(t,raceiniesopportunitallonWages
1,...1
=
≤=∑−
t
i
ttitXX
W
XWXWE
XX
tmt
t.raceofendtheatwealthtotalwhere
}0,|)(log{maxarg
),...,(t,raceiniesopportunitallonWages
1,...
**
1
1
=
∀≥≤= ∑ −
t
i
tittitXX
T
tmt
W
iXWXWE
XX
tmt
1919Conclusion and Research Conclusion and Research
OpportunitiesOpportunitiesKnowing the appropriate running time Knowing the appropriate running time distribution is key to determining ordering distribution is key to determining ordering probabilitiesprobabilities
There appears to be no universal best There appears to be no universal best distribution but distribution but HeneryHenery (Normal) and Stern (Normal) and Stern (Gamma) are competitive(Gamma) are competitive
Simple approximation is available for Simple approximation is available for HeneryHeneryand Sternand Stern
Correlated running time model is more Correlated running time model is more complex but may be bettercomplex but may be better
Other approximation methods may be Other approximation methods may be considered especially for more complicated considered especially for more complicated modelsmodels
(Fractional) Kelly is promising for optimal (Fractional) Kelly is promising for optimal bettingbetting
2020
References for FavoriteReferences for Favorite--LongshotLongshot BiasBias
Ali, M.M. (1977) “Probability and Utility Estimates for RacetracAli, M.M. (1977) “Probability and Utility Estimates for Racetrack Bettors,” k Bettors,” J. of J. of Political EconomyPolitical Economy, 84, p.803, 84, p.803--815.815.
Asch, P., Asch, P., MalkielMalkiel, B., and , B., and QuandtQuandt, R. (1984) “Market Efficiency in Racetrack Betting,” , R. (1984) “Market Efficiency in Racetrack Betting,” J. of BusinessJ. of Business 57, p.16557, p.165--174.174.
BaconBacon--Shone, J., Lo, V.S.Y., and Busche, K. (1992a) “Shone, J., Lo, V.S.Y., and Busche, K. (1992a) “ModellingModelling the Winning the Winning Probability,” Probability,” Research Report Research Report 10, Dept. of Statistics, the University of Hong Kong.10, Dept. of Statistics, the University of Hong Kong.
Benter, W. (1994) “Computer Based Horse Race Handicapping and WaBenter, W. (1994) “Computer Based Horse Race Handicapping and Wagering gering Systems: A Report,” in Hausch, D.B., Lo, V.S.Y., and Ziemba, W.TSystems: A Report,” in Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. ed. (1994) . ed. (1994) Efficiency of Racetrack Betting Markets, Efficiency of Racetrack Betting Markets, Academic Press, p.183Academic Press, p.183--198.198.
Bolton, R.N. and Chapman, R.G. (1986) “Searching for Positive ReBolton, R.N. and Chapman, R.G. (1986) “Searching for Positive Returns at the Track, turns at the Track, A Multinomial Logit Model for Handicapping Horse Races,” A Multinomial Logit Model for Handicapping Horse Races,” Management ScienceManagement Science, 32, , 32, p.1040p.1040--1059. Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. ed. (1994) 1059. Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. ed. (1994) Efficiency of Efficiency of Racetrack Betting Markets, Racetrack Betting Markets, Academic Press, p.237Academic Press, p.237--247.247.
Busche, K. and Hall, C.D. (1988) “An Exception to the Risk PrefeBusche, K. and Hall, C.D. (1988) “An Exception to the Risk Preference Anomaly,” rence Anomaly,” J. J. of Business, of Business, 61, p.33761, p.337--346.346.
Lo, V.S.Y. (1992) “Statistical Lo, V.S.Y. (1992) “Statistical ModellingModelling of Gambling Probabilities,” of Gambling Probabilities,” PhD ThesisPhD Thesis, Dept. , Dept. of Statistics, The University of Hong Kongof Statistics, The University of Hong Kong
Lo, V.S.Y. (1994a) “Application of Logit Models in Racetrack DatLo, V.S.Y. (1994a) “Application of Logit Models in Racetrack Data,” in Hausch, D.B., a,” in Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. ed. (1994) Lo, V.S.Y., and Ziemba, W.T. ed. (1994) Efficiency of Racetrack Betting Markets, Efficiency of Racetrack Betting Markets, Academic Press, p.307Academic Press, p.307--314.314.
Lo, V.S.Y. and Busche, K. (1994) “How Accurately do Betters Bet Lo, V.S.Y. and Busche, K. (1994) “How Accurately do Betters Bet in Doubles?,” in in Doubles?,” in Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. ed. (1994) Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. ed. (1994) Efficiency of Racetrack Efficiency of Racetrack Betting Markets, Betting Markets, Academic Press, p.465Academic Press, p.465--468.468.
Willoughby, K.A. (2002) “Winning Games in Canadian Football: A LWilloughby, K.A. (2002) “Winning Games in Canadian Football: A Logistic Regression ogistic Regression Analysis,” Analysis,” The College Mathematics JournalThe College Mathematics Journal, 33, No.3, p.215, 33, No.3, p.215--220.220.
Ziemba, W.T. (2004) “Behavioral Finance, Racetrack Betting and OZiemba, W.T. (2004) “Behavioral Finance, Racetrack Betting and Options and ptions and Futures Trading,” Futures Trading,” Mathematical Finance SeminarMathematical Finance Seminar, Stanford University., Stanford University.
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References for Ordering ProbabilitiesReferences for Ordering ProbabilitiesBaconBacon--Shone, J.H., Lo, V.S.Y., and Busche, K. (1992b) “Logistic AnalysShone, J.H., Lo, V.S.Y., and Busche, K. (1992b) “Logistic Analyses of es of
Complicated Bets,” Complicated Bets,” Research Report 11Research Report 11, Dept. of Statistics, the University of Hong , Dept. of Statistics, the University of Hong
Kong.Kong.
HarvilleHarville, D.A. (1973) “Assigning Probabilities to the Outcomes of Multi, D.A. (1973) “Assigning Probabilities to the Outcomes of Multi--Entry Entry
Competitions,” Competitions,” J. of American Statistical AssociationJ. of American Statistical Association, 68, p.312, 68, p.312--316.316.
Hausch, DB., Ziemba, W.T., and Rubinstein, M. (1981) “EfficiencyHausch, DB., Ziemba, W.T., and Rubinstein, M. (1981) “Efficiency of the Market for of the Market for
Racetrack Betting,” Racetrack Betting,” Management Science, Management Science, 27, p.143527, p.1435--1452.1452.
HeneryHenery, R.J. (1981) “Permutation Probabilities as Models for Horse Rac, R.J. (1981) “Permutation Probabilities as Models for Horse Races,” es,” J. of J. of
Royal Statistical Society BRoyal Statistical Society B, 43, p.86, 43, p.86--91.91.
HeneryHenery, R.J. (1985) “On the Average Probability of Losing Bets on Hors, R.J. (1985) “On the Average Probability of Losing Bets on Horses with es with
Given Starting Price Odds,” Given Starting Price Odds,” J. of Royal Statistical Society A, J. of Royal Statistical Society A, 148, p.342148, p.342--349.349.
Lo, V.S.Y. (1994b) “Application of Running Time Distribution ModLo, V.S.Y. (1994b) “Application of Running Time Distribution Models in Japan,” in els in Japan,” in
Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. ed. (1994) Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. ed. (1994) Efficiency of Racetrack Efficiency of Racetrack
Betting Markets, Betting Markets, Academic Press, p.237Academic Press, p.237--247.247.
Lo, V.S.Y. and BaconLo, V.S.Y. and Bacon--Shone, J. (1994) “A Comparison between Two Models for Shone, J. (1994) “A Comparison between Two Models for
Predicting Ordering Probabilities in MultiplePredicting Ordering Probabilities in Multiple--Entry Competitions,” Entry Competitions,” The StatisticianThe Statistician, ,
43, No.2, p.31743, No.2, p.317--327.327.
Lo, V.S.Y. and BaconLo, V.S.Y. and Bacon--Shone, J. (2007) “Approximating the Ordering Probabilities of Shone, J. (2007) “Approximating the Ordering Probabilities of
MultiMulti--Entry Competitions By a Simple Method,” To appear in: Hausch, D.Entry Competitions By a Simple Method,” To appear in: Hausch, D.B. and B. and
Ziemba, W.T. ed. (2007) Ziemba, W.T. ed. (2007) Handbook of Investments: Efficiency of Sports and Handbook of Investments: Efficiency of Sports and
Lottery Markets, Lottery Markets, Elsevier.Elsevier.
Stern, H. (1990) “Models for Distributions on Permutations,” Stern, H. (1990) “Models for Distributions on Permutations,” J. of American J. of American
Statistical Association, Statistical Association, 85, p.55885, p.558--564.564.
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References for Optimal Investment StrategyReferences for Optimal Investment StrategyAlgoetAlgoet, P.H. and Cover T.H. (1988) “Asymptotic Optimality and , P.H. and Cover T.H. (1988) “Asymptotic Optimality and Asymptotic Asymptotic EquipartitionEquipartition Properties of LogProperties of Log--optimum Investment,” optimum Investment,” The The Annals of Probability,Annals of Probability, 16, No.2, p.87616, No.2, p.876--898.898.
Benter, W. (1994) “Computer Based Horse Race Handicapping and Benter, W. (1994) “Computer Based Horse Race Handicapping and Wagering Systems: A Report,” in Hausch, D.B., Lo, V.S.Y., and ZiWagering Systems: A Report,” in Hausch, D.B., Lo, V.S.Y., and Ziemba, emba, W.T. ed. (1994) W.T. ed. (1994) Efficiency of Racetrack Betting Markets, Efficiency of Racetrack Betting Markets, Academic Press, Academic Press, p.183p.183--198.198.
BreimanBreiman, L. (1960) “Investment Policies for Expanding Businesses , L. (1960) “Investment Policies for Expanding Businesses Optimal in a LongOptimal in a Long--run Sense,” run Sense,” Naval Research Logistics QuarterlyNaval Research Logistics Quarterly, 7, , 7, p.647p.647--651.651.
HaighHaigh, J. (2000) “The Kelly Criterion and Bet Comparisons in Spread , J. (2000) “The Kelly Criterion and Bet Comparisons in Spread Betting,” Betting,” The StatisticianThe Statistician, 40, Part 4, p.531, 40, Part 4, p.531--539.539.
Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. (1994) “Pricing ExotiHausch, D.B., Lo, V.S.Y., and Ziemba, W.T. (1994) “Pricing Exotic c Racetrack Wagers,” in Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T.Racetrack Wagers,” in Hausch, D.B., Lo, V.S.Y., and Ziemba, W.T. ed. ed. (1994) (1994) Efficiency of Racetrack Betting Markets, Efficiency of Racetrack Betting Markets, Academic Press, p.469Academic Press, p.469--483.483.
Hausch, DB., Ziemba, W.T., and Rubinstein, M. (1981) “EfficiencyHausch, DB., Ziemba, W.T., and Rubinstein, M. (1981) “Efficiency of the of the Market for Racetrack Betting,” Market for Racetrack Betting,” Management Science, Management Science, 27, p.143527, p.1435--1452.1452.
Lo, V.S.Y., BaconLo, V.S.Y., Bacon--Shone, J., and Busche, K. (1995) “The Application of Shone, J., and Busche, K. (1995) “The Application of Ranking Probability Models to Racetrack Betting,” Ranking Probability Models to Racetrack Betting,” Management Science,Management Science,41, p.104841, p.1048--1059.1059.
Thorp E.O. (1971) “Portfolio Choice and the Kelly Criterion,” Thorp E.O. (1971) “Portfolio Choice and the Kelly Criterion,” Business and Business and Economics Statistics Section, Proceedings of the American StatisEconomics Statistics Section, Proceedings of the American Statistical tical Association.Association.
Ziemba, W.T. (2004) “Behavioral Finance, Racetrack Betting and OZiemba, W.T. (2004) “Behavioral Finance, Racetrack Betting and Options ptions and Futures Trading,” and Futures Trading,” Mathematical Finance SeminarMathematical Finance Seminar, Stanford University., Stanford University.
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