11

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

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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.

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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.

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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.