A Poisson Betting Model with a Kelly Criterion Element for ...
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1 A Poisson Betting Model with a Kelly Criterion Element for European Soccer Kushal Shah James Hyman Dominic Samangy [email protected] [email protected] [email protected] 1. Introduction Sports betting has experienced a rapid rise in popularity as accessibility and commercialization of daily fantasy and live betting has increased. As sports betting is legalized in different countries and states, we are presented with a new opportunity to create statistical models that we can utilize to predict outcomes of different sporting events that can then be used to find inefficiencies in sportsbooks. In this paper, we attempt to create a model for European Soccer and measure its performance against betting markets to understand if this model can be used to generate profits. The paper shows how we predicted the number of goals by each team in a game and utilized these predictions to create a Poisson distribution that determined the probability of each event (Win, Lose, Draw). To add another dimension to our model, we used an optimization technique known as Kelly Criterion to determine the optimal amount of money that should be bet on each match. This technique generates bet amounts while also creating a value (KCO Value) that acts as an accurate estimator of the risk associated with each match. By exploring the characteristics of this value, we were able to maximize the success of the model. After running the model for the 2018 and 2019 seasons across the five major European soccer leagues, we can safely say that our model was not only successful in predicting outcomes, but also in generating significant profit yields for a user. A profit percentage of 119.48% can be yielded using this model, which implies that a user would essentially double their money using this model. We also evaluate how the model performs in different leagues to understand which league characteristics benefit the model. The highest profit percentage was seen in the 2019 Bundesliga season with a profit percentage of 226.68%. The success of the model can not only help users generate significant profits, but it can also expose certain inefficiencies in the market. 2. Generating Values for Our Poisson Distributions The first step to use our Poisson Distribution is determining the to be used. In a Poisson Distribution, value represents the expected rate of occurrence within a given time interval. In this case, the value will represent the number of goals we expect a team to score in a game. To ensure we have the most accurate value of , we used four different models to create different values and tested our models with each value of . An important aspect of any betting model is accounting for the context surrounding the game. Specifically, if a team is playing on home or away and the strength of their opponent. Each model we ensure that our value we generate are adjusted for the strength of the opponent and if the team is home or away. In addition to this, we ensure that we
Transcript of A Poisson Betting Model with a Kelly Criterion Element for ...
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APoissonBettingModelwithaKellyCriterionElementforEuropeanSoccer
KushalShah JamesHyman [email protected] [email protected]@syr.edu
1. IntroductionSportsbettinghasexperiencedarapidriseinpopularityasaccessibilityandcommercializationofdailyfantasyandlivebettinghasincreased.Assportsbettingislegalizedindifferentcountriesandstates,wearepresentedwithanewopportunitytocreatestatisticalmodelsthatwecanutilizetopredictoutcomesofdifferentsportingeventsthatcanthenbeusedtofindinefficienciesinsportsbooks.Inthispaper,weattempttocreateamodelforEuropeanSoccerandmeasureitsperformanceagainstbettingmarketstounderstandifthismodelcanbeusedtogenerateprofits.ThepapershowshowwepredictedthenumberofgoalsbyeachteaminagameandutilizedthesepredictionstocreateaPoissondistributionthatdeterminedtheprobabilityofeachevent(Win,Lose,Draw).Toaddanotherdimensiontoourmodel,weusedanoptimizationtechniqueknownasKellyCriteriontodeterminetheoptimalamountofmoneythatshouldbebetoneachmatch.Thistechniquegeneratesbetamountswhilealsocreatingavalue(KCOValue)thatactsasanaccurateestimatoroftheriskassociatedwitheachmatch.Byexploringthecharacteristicsofthisvalue,wewereabletomaximizethesuccessofthemodel.Afterrunningthemodelforthe2018and2019seasonsacrossthefivemajorEuropeansoccerleagues,wecansafelysaythatourmodelwasnotonlysuccessfulinpredictingoutcomes,butalsoingeneratingsignificantprofityieldsforauser.Aprofitpercentageof119.48%canbeyieldedusingthismodel,whichimpliesthatauserwouldessentiallydoubletheirmoneyusingthismodel.Wealsoevaluatehowthemodelperformsindifferentleaguestounderstandwhichleaguecharacteristicsbenefitthemodel.Thehighestprofitpercentagewasseeninthe2019Bundesligaseasonwithaprofitpercentageof226.68%.Thesuccessofthemodelcannotonlyhelpusersgeneratesignificantprofits,butitcanalsoexposecertaininefficienciesinthemarket.
2. Generating𝝀ValuesforOurPoissonDistributionsThefirststeptouseourPoissonDistributionisdeterminingthe𝜆tobeused.InaPoissonDistribution,𝜆valuerepresentstheexpectedrateofoccurrencewithinagiventimeinterval.Inthiscase,the𝜆valuewillrepresentthenumberofgoalsweexpectateamtoscoreinagame.Toensurewehavethemostaccuratevalueof𝜆,weusedfourdifferentmodelstocreatedifferent𝜆valuesandtestedourmodelswitheachvalueof𝜆.Animportantaspectofanybettingmodelisaccountingforthecontextsurroundingthegame.Specifically,ifateamisplayingonhomeorawayandthestrengthoftheiropponent.Eachmodelweensurethatour𝜆valuewegenerateareadjustedforthestrengthoftheopponentandiftheteamishomeoraway.Inadditiontothis,weensurethatwe
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preventeddataleakagewithinourmodel.Dataleakageiswhenmodelsusedataforpredictionsthatatthetimeofthepredictionwouldnotbeavailable.The𝜆valueswecalculatedforeachmodelwerecalculatedastheseasonprogressed,forexamplewewouldusedataavailablethrough12weeksintotheseasontopredictoutcomesforthematchesingameweek13.Thisalsodenotesthatour𝜆valuesarebeingupdatedandimproveduponastheseasonprogresses.Tosummarize,foreachweensuredeachofour𝜆valuesfactorsin3majorcharacteristics:
1. The𝜆valueaccountsfortheopponent2. The𝜆valueaccountsforwhethertheteamisplayinghomeoraway3. The𝜆valueupdatesastheseasonprogresses,becomingmoreaccurate
2.1. 𝝀ValuesbasedonGoalsThefirstmodelwecreatedusedgoalsscoredandgoalsallowedtocreateattackinganddefendingstrengthsforeachteam(Smarkets,2020).Theformulaweuseinthemodeltodeveloptheaforementionedstrengthsare:
𝐴𝑡𝑡𝑎𝑐𝑘𝑖𝑛𝑔𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 𝐺𝑜𝑎𝑙𝑠𝑃𝑒𝑟𝐺𝑎𝑚𝑒𝐹𝑜𝑟𝑇ℎ𝑒𝑇𝑒𝑎𝑚
𝐺𝑜𝑎𝑙𝑠𝑃𝑒𝑟𝐺𝑎𝑚𝑒𝐴𝑐𝑟𝑜𝑠𝑠𝑇ℎ𝑒𝐿𝑒𝑎𝑔𝑢𝑒 (1)
𝐷𝑒𝑓𝑒𝑛𝑑𝑖𝑛𝑔𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 𝐺𝑜𝑎𝑙𝑠𝐴𝑙𝑙𝑜𝑤𝑒𝑑𝑃𝑒𝑟𝐺𝑎𝑚𝑒𝐹𝑜𝑟𝑇ℎ𝑒𝑇𝑒𝑎𝑚
𝐺𝑜𝑎𝑙𝑠𝐴𝑙𝑙𝑜𝑤𝑒𝑑𝑃𝑒𝑟𝐺𝑎𝑚𝑒𝐴𝑐𝑟𝑜𝑠𝑠𝑇ℎ𝑒𝐿𝑒𝑎𝑔𝑢𝑒 (2)
Thesestrengthswerecalculatedwithhomeandawaysplits.Thisresultedineveryteamhaving4differentstrengths:
1. HomeAttackingStrength2. HomeDefendingStrength3. AwayAttackingStrength4. AwayDefendingStrength
Weusethesestrengthstocalculateeachteam’s𝜆valueforagame.Theformulabelowisused:
𝐻𝑜𝑚𝑒𝐺𝑜𝑎𝑙𝑠𝜆 = 𝐻𝑜𝑚𝑒𝐴𝑡𝑡𝑎𝑐𝑘𝑖𝑛𝑔𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ × 𝐴𝑤𝑎𝑦𝐷𝑒𝑓𝑒𝑛𝑑𝑖𝑛𝑔𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ × 𝐴𝑣𝑒𝑟𝑎𝑔𝑒𝐿𝑒𝑎𝑔𝑢𝑒𝐻𝑜𝑚𝑒𝐺𝑜𝑎𝑙𝑠 (3)
𝐴𝑤𝑎𝑦𝐺𝑜𝑎𝑙𝑠𝜆 = 𝐴𝑤𝑎𝑦𝐴𝑡𝑡𝑎𝑐𝑘𝑖𝑛𝑔𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ × 𝐻𝑜𝑚𝑒𝐷𝑒𝑓𝑒𝑛𝑑𝑖𝑛𝑔𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ × 𝐴𝑣𝑒𝑟𝑎𝑔𝑒𝐿𝑒𝑎𝑔𝑢𝑒𝐴𝑤𝑎𝑦𝐺𝑜𝑎𝑙𝑠 (4)
Toprovidemoreclarity,wewillusetheCrystalPalace(H)vsWestBrom(A)gameasanexampleduringthelastgameweekofthe2018PremierLeagueseason.Belowisacalculationofbothteam’sattackinganddefendingstrengthsalongwithourfinal𝜆valueforeachteam:
CrystalPalaceAverageGoalsScoredandGoalsAllowedatHomeRespectively:1.5and1.5WestBromAverageGoalsScoredandGoalsAllowedwhenAwayRespectively:0.56and1.39AverageLeagueGoalsScoredandGoalsAllowedatHomeRespectively:1.52and1.15
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𝐶𝑅𝑌𝐻𝑜𝑚𝑒𝐴𝑡𝑡𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 1.51.52
= 0.99𝑊𝐵𝐴𝐴𝑤𝑎𝑦𝐴𝑡𝑡𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 0.561.15
= 0.48
𝐶𝑅𝑌𝐻𝑜𝑚𝑒𝐷𝑒𝑓𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 1.51.15
= 1.31𝑊𝐵𝐴𝐴𝑤𝑎𝑦𝐷𝑒𝑓𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 1.391.52
= 0.91
𝐶𝑅𝑌𝜆 = 0.99 × 0.91 × 1.52
𝑊𝐵𝐴𝜆 = 0.48 × 1.31 × 1.15
CrystalPalace𝜆Value≈1.37
WestBrom𝜆Value≈0.73
2.2. 𝝀ValuesbasedonExpectedGoals(xG)Thesecondmodeweusedtogenerate𝜆valuesisexactlylikethefirstmodel,howeverratherthangoalsweusedexpectedgoals(xG).Expectedgoalsisapopularmetricintheworldofsoccertodaythatrepresentsthenumberofgoalsthatareexpectedtobescoredbasedonthelocationandwayashotistaken.BysubstitutingxGinplaceofgoals,areformulasnowlooklikethis:
𝐴𝑡𝑡𝑎𝑐𝑘𝑖𝑛𝑔𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 𝑥𝐺𝑃𝑒𝑟𝐺𝑎𝑚𝑒𝐹𝑜𝑟𝑇ℎ𝑒𝑇𝑒𝑎𝑚
𝑥𝐺𝑃𝑒𝑟𝐺𝑎𝑚𝑒𝐴𝑐𝑟𝑜𝑠𝑠𝑇ℎ𝑒𝐿𝑒𝑎𝑔𝑢𝑒 (5)
𝐷𝑒𝑓𝑒𝑛𝑑𝑖𝑛𝑔𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 𝑥𝐺𝐴𝑙𝑙𝑜𝑤𝑒𝑑𝑃𝑒𝑟𝐺𝑎𝑚𝑒𝐹𝑜𝑟𝑇ℎ𝑒𝑇𝑒𝑎𝑚
𝑥𝐺𝐴𝑙𝑙𝑜𝑤𝑒𝑑𝑃𝑒𝑟𝐺𝑎𝑚𝑒𝐴𝑐𝑟𝑜𝑠𝑠𝑇ℎ𝑒𝐿𝑒𝑎𝑔𝑢𝑒 (6)
Thegameusedasanexampleearlier(CRYvsWBA)wouldhavethefollowingcalculationforthe𝜆valuesforthismodel:
CrystalPalaceAveragexGScoredandxGAllowedatHomeRespectively:1.73and1.2WestBromAveragexGScoredandxGAllowedwhenAwayRespectively:0.77and1.37AverageLeaguexGScoredandAllowedatHomeRespectively:1.39and1.08
𝐶𝑅𝑌𝐻𝑜𝑚𝑒𝐴𝑡𝑡𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 1.731.39
= 1.24𝑊𝐵𝐴𝐴𝑤𝑎𝑦𝐴𝑡𝑡𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 0.771.08
= 0.71
𝐶𝑅𝑌𝐻𝑜𝑚𝑒𝐷𝑒𝑓𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 1.21.39
= 1.11𝑊𝐵𝐴𝐴𝑤𝑎𝑦𝐷𝑒𝑓𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ = 1.371.08
= 0.98
𝐶𝑅𝑌𝜆 = 1.24 × 0.98 × 1.39
𝑊𝐵𝐴𝜆 = 0.71 × 1.11 × 1.08
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CrystalPalace𝜆Value≈1.70
WestBrom𝜆Value≈0.85
2.3.𝝀ValuesbasedonLinearRegression Thethirdmodelweusedtopredict𝜆waslinearregression.Weusedthecumulativestatisticsofateamandtheteam’sopponenttodevelopthemodel.Thelinearregressionpredictstheexpectednumberofhomeandawaygoalsusingthemetricsofbothteams.Thesemetricsinclude:
1. Goals2. Shots3. ShotsonTarget4. Fouls5. Corners6. YellowCards7. RedCards
The𝜆valuegeneratedfromthismodelfortheCrystalPalaceWestBromgameweusedasanexampleearlierwouldbe:
CrystalPalace𝜆Value≈1.54WestBrom𝜆Value≈0.42
2.4.𝝀ValuesbasedonaRandomForestRegressionThismodelpredicts𝜆valuesinthesamemannerasthethird(LinearRegression)model.Ratherthanalinearregression,thismodelmakesuseofaRandomForestregressioninstead.ThemetricsusedbythemodeltomakepredictionsarethesameusedbytheLinearRegressionmodel.
The𝜆valuegeneratedfromthismodelfortheCrystalPalaceWestBromgameweusedasanexampleearlierwouldbe:
CrystalPalace𝜆Value≈1.70WestBrom𝜆Value≈0.38
3. GeneratingProbabilitiesforEveryEvent
3.1.Using𝝀ValuestoGenerateProbabilitiesforaHomeWin,AwayWin,andDrawNowthatwehaveour𝜆values,wecanusethePoissonDistributiontocalculateourprobabilities.BelowistheformulathePoissondistributionusestopredict𝑥numberofevents:
𝑃(𝑥) =
𝑒!" × λ$
𝑥! (7)
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Forexample,ifwewouldwanttopredicttheprobabilityofateamscoring3goals,𝑥wouldtakethevalueof3.Essentially,tocalculateeachoutcome’stotalprobabilityweusedtheaboveformulatocalculateeverypossiblescorelinefrom0to0to5to5.Usingsimpleprobabilityrules,theprobabilityofeachscorelinecanbecalculatedinthefollowingmanner:
𝑃𝑟𝑜𝑏𝑜𝑓𝑆𝑐𝑜𝑟𝑒𝑙𝑖𝑛𝑒 = 𝑃(𝐻𝑜𝑚𝑒𝐺𝑜𝑎𝑙𝑠) × 𝑃(𝐴𝑤𝑎𝑦𝐺𝑜𝑎𝑙𝑠) (8)
So,theprobabilityofa3-2scorelinewouldbetheprobabilityofthehometeamscoringthreegoalstimestheprobabilityoftheawayteamscoringtwogoals.Wecanshowtheprobabilityofeachscorelineas“ScoreLineMatrix”asseenbelow:
TherowscorrespondtothenumberofgoalsscoredbyWestBrom(Away)andthecolumnscorrespondtothenumberofgoalsscoredbyCrustalPalace(Home).Thenumberpresentwithineachcellistheprobabilityofthatscorelinehappening.Asdiscussedearlierthelikelihoodofeachscorelineiscalculatedbymultiplyingtheprobabilities.Basedonthematrix,wecanseethatthemodelbelievesthemostlikelyscorelineis1–0whichhasa0.21238(0.31056×0.68386)chanceofoccurring.
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Thetotalprobabilityforeveryeventisthesumofallthescorelinesthatcorrelatewiththatoutcome.Belowaretheformulasweusedforthis:
𝐻𝑜𝑚𝑒𝑊𝑖𝑛 = S𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠𝑜𝑓𝑆𝑐𝑜𝑟𝑒𝑙𝑖𝑛𝑒𝑠𝑤𝑖𝑡ℎ𝐻𝑜𝑚𝑒𝑇𝑒𝑎𝑚𝑊𝑖𝑛𝑛𝑖𝑛𝑔 (9)
𝐴𝑤𝑎𝑦𝑊𝑖𝑛 = S𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠𝑜𝑓𝑆𝑐𝑜𝑟𝑒𝑙𝑖𝑛𝑒𝑠𝑤𝑖𝑡ℎ𝐴𝑤𝑎𝑦𝑇𝑒𝑎𝑚𝑊𝑖𝑛𝑛𝑖𝑛𝑔 (10)
𝐷𝑟𝑎𝑤 =S𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠𝑜𝑓𝑆𝑐𝑜𝑟𝑒𝑙𝑖𝑛𝑒𝑠𝑒𝑛𝑑𝑖𝑛𝑔𝑤𝑖𝑡ℎ𝑎𝐷𝑟𝑎𝑤 (11)
Foreachofourmodels,hereweretheprobabilitiesgeneratedfortheCrystalPalaceWestBromgame:
Weconsideredthescorelinesfrom0-0to5-5tocoverenoughoutcomestogenerateanaccurateprobabilityforeacheventinthegame.Belowisadistributionofthetotalprobabilityforeverygameforeachmodel.Ideally,wewanttheprobabilitytobeequalto1asthatwouldentailthatalloutcomesarecovered.
Model HomeWinProbability(CrystalPalaceWin)
AwayWinProbability(WestBromWin)
DrawProbability
Goals 0.52 0.198 0.297
xG 0.567 0.186 0.239
LinearRegression 0.649 0.098 0.248
RandomForest 0.696 0.077 0.22
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Ifwelookatthedistributionsofthetotalprobabilityforeachmodel,wecanseeanoverwhelmingmajorityhaveasumover0.9withmostbeingequaltoatleast0.99.Thesearealsofromgamesbeginninginmatchweek5,henceastheseasonprogresses,weanticipatethedistributiontomoveevencloserto1.Hence,thePoissonDistributionweusedwitheach𝜆valuewegeneratedaccuratelyaccountsforeachoutcomeinthegame.
4. ModelCalibration(BrierScore)WecalibratedthemodelusingaBrierScoretodeterminewhichmodelgeneratedthemostaccurate𝜆value.TheBrierScoreisascorefunctionthathelpsdeterminetheaccuracyofanyprobabilisticmodel.TheBrierscoreforpredictionsisgivenbytheformulabelow:
𝐵𝑆 =
1𝑛S(𝑝! − 𝑜!)"#
!$%
(12)
Essentially,wearemeasuringiftheoutcomeswepredictwithacertainprobabilityareoccurringinthatproportion.Thebestperformingmodelwouldhaveprobabilityvaluescorrespondingtotherespectiveinterval.Forexample,amongalltheeventsfortherestoftheseasonthatwepredictedtooccurwithaprobabilityof0.6and0.7,wewouldwanttobecorrectaround65%(0.65)ofthetime.Theoretically,calibrationresultsaftereachgameweekwouldimproveasthemodelswouldhavemoredata.Wewantedtoidentifywhichmodelandatwhichgameweekthemarginalimprovementisminimaltobeabletounderstandtheidealtimetobegintousethebestperformingmodel.Basedonourcalibrationresults,wedeterminedthatthebestmodeltoestimate𝜆wastheRandomForestModelaftergameweek13.Thisisthemodelweusewhenevaluatingourmodel’ssuccess.Thecalibrationresultswerecalculatedaftereachgameweek,withthebestcalibrationresultbeingshowninthegraphsbelow.Thegraphsshowcaseall4modelsandhowwelltheydoinpredictingthehometeamandtheawayteamtowin.Thesearegraphsforthecalibrationresultsaftergameweek13.Thedashedredlineiswhatperfectcalibrationwouldlooklikeandprovidesareferenceforcomparison.Themodelclosesttotheredlinewouldbethebestmodelintermsofpredictions.InthecaseofbothhomewinsandawaywinstheRandomForestisthebestmodel.
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5. BankrollManagementSystem(KellyCriterion)TheKellyCriterionmodelisaformofprobabilitytheoryoftenusedbyinvestorsthatweplantoincorporateintothismodel.Thegoalofthemodelistomaximizeprofitwhileaccountingfortheriskassociatedwithalostbet.Thisisdonebymaximizingthelogarithmofthepotentialendingbankrollsafterthebetisplaced.ManyonlinesourceschoosetosimplifythemathbehindtheKellyCriterionTheoryintoagenericformulathatlookslikethis:
(𝑂 × 𝑃&) − 𝑃'𝑂
= 𝐵
O = 𝑂𝑑𝑑𝑠, 𝑃& = 𝑃𝑟𝑜𝑏𝑜𝑓𝐵𝑒𝑡𝑊𝑖𝑛, 𝑃' = 𝑃𝑟𝑜𝑏𝑜𝑓𝐵𝑒𝑡𝐿𝑜𝑠𝑠, 𝑎𝑛𝑑𝐵 = 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝑜𝑓𝐵𝑎𝑛𝑘𝑟𝑜𝑙𝑙
(13)
WhilethisformuladoesinvolvesomeprinciplesoftheKellyCriterionTheory,itdoesnotreflectthetheoryitself.Also,asoccermatchhasthreeoutcomes,slightlycomplicatingthemathofthissimplifiedtwooutcomeformula.OurgoalistosimplymaximizethelogarithmofourpotentialbankrollaccordingtotheKellyCriterionTheory.BelowisouroptimizationfortheCrystalPalacevs.WestBromexamplediscussedearlier.Theinputsaretheoddsandprobabilitiesofeachpossibleresultaswellasthestartingbankroll.Usingthisinformation,wecancalculateourendingbankrollforeachpossibleeventandthentakethelogarithmofthat.Lastly,the“objective”(KCOValue)isaweightedaverageofthelogarithmsandtheirassociatedprobabilities.𝐾𝐶𝑂 = (𝐻𝑜𝑚𝑒𝑊𝑖𝑛𝑃𝑟𝑜𝑏 × 𝑙𝑜𝑔(𝐻𝑊)) +(𝐴𝑤𝑎𝑦𝑊𝑖𝑛𝑃𝑟𝑜𝑏 × 𝑙𝑜𝑔(𝐴𝑊)) +(𝐷𝑟𝑎𝑤𝑃𝑟𝑜𝑏 × 𝑙𝑜𝑔(𝐷𝑊))
HW=EndingBankrollHomeWin,AW=EndingBankrollAwayWin,DW=EndingBankrollDraw
(14)
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AboveisanimageconsistingoftheinputsandusingthisinformationandR,wemaximizetheobjectivecellbychangingthe“BetAmount”cells.Theimagebelowiswhatatypicaloutputwouldlooklike.Basedontheseoddsandprobabilities,themodelsuggestsabetof$22.34onCrystalPalacetowinandabetof$1.65onWestBromtowinaretheoptimalbetstoplacethatmaximizelongtermprofitability.Thesecondbetisaformofhedging(TheLines,2020).Theresultofthegamewas2–0infavorofCrystalPalace,hencewewouldhaveprofited$16.22forthisbet.Wecanseeherethatthemodelpredictsaformofhedging,tominimizelosstosomedegree
Tofurtherunderstandhowtheobjectivecellinteractswithdifferentbetamountswecanusethegraphbelow:
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Theabovegraphshowshowtheobjectivecellinteractswithdifferentbetamountsforasamplegame(blue)inwhichtheteambeingbetonhas+100bettingoddsanda60%chancetowinthegame.Itfollowssomewhatofaquadraticformthatvariesinshapedependingonthebettingoddsandprobabilityofwinning.Theleftsideofthecurveindicatesbetamountsthatareprofitable,butnotasprofitableashigherbetamounts.Therightsideofthecurveindicateshigherbetamountswouldbeconsideredhigherriskbetsthatwouldprohibitlongtermsuccess.Theorangelineindicatestheobjectivenumber(KCOValue)ifnobetisplaced.Inotherwords,betsinwhichthebluelineisbelowtheorangelineshouldnotbeplaced.Asmentionedbefore,ourmodelessentiallylookstofindthemaximumoftheblueline(HaghaniandDewey,2020).ThismethodofriskmanagementandprofitmaximizationhasbeenprovenveryeffectiveinastudyconductedbyVictorHaghaniandRichardDewey.Beforemovingontoourresults,itisimportanttopointouttwomoreaspectsofthismethodology.Thefirstishowtointerpretthe“objective”numberproducedbytheoptimization.Asmentionedearlier,theobjectivenumberistheproductofthelogarithmofallpossibleendingbankrolls.Sinceweassumedastartingbankrollof$100,theobjectivecellwhenplacingnobetis2sincethereisa100%chanceofendingwith$100andlog(2)=100.Afterwefindtheoptimalbet,theobjectivecellbecomesarepresentationofourconfidenceinprofitabilitywithaKCOvalueover2havingalotofconfidencerelativetoavaluebelow2.Lastly,therearesomegamesinwhichabetmightbeplacedonateamtowinaswellasasmallerbetononeoftheotheroutcomes.Thisoccursasaformofhedgingthatreducesthepotentiallossofourbetnothitting.6. ModelEvaluation6.1.InterpretingKCOValuesasRiskEstimatorsBeforeanalyzingourmodel’sresults,itisimportanttounderstandtheKCOvalueassociatedwitheachbet(GameWeek5onwards).BelowisadensityplotfortheKCOvaluesassociatedwitheachbet:
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Asmentionedearlier,aKCOvalueof2isassociatedwhennobetisplaced.Wecanseethatthepeakofthedistributionisatavalueextremelycloseto2.Thismakessenseasoddsmakersareextremelyeffectiveatsettingoddsandhenceamajorityofthemodel’ssuggestionsaretonotbetanything.Thevaluesontheleftsideofthedistributionarebetsthataredeemed“risky”withbetstotherightofdistributionbeingabetwithlessrisk.Wecanalsoseeaweakbutpositiverelationshipbetweenthesuccessofabetandthebet’sKCOvalue.Wecanseethisrelationshipbelow,withNetIncomebeingtheprofitorlossfromabet(GameWeek13onwards):
WecanseeastheKCOvaluesincreases,thenetincomeofabetalsotendstoincrease.WecanalsoseethattheproportionofbetsthatmakeaprofitaregreaterwhentheKCOvalueisgreaterthan2.ThisweakpositivecorrelationbetweentheNetIncomeandKCOValueshowsusthattheKCOValuecanbeusedasariskestimator.Thisalsoentails,thatifwefilterthemodelbasedontheKCOValue,wecanincreaseourprofitpercentage.Theoretically,ifweonlyplacebetswithaKCOvaluegreaterthan2or2.1,ourprofitpercentagewillimprove.Wefurtherexplorethishypothesisinthenextsection.
6.2.SuccessoftheModelintermsofProfitInthissection,wewillobservehowsuccessfulthemodelwasintermsofoverallprofit.ItisimportanttonotethattheKellyCriterionsuggestsdifferentamountstobetforeachgamebasedontherisk,hencewedecidedtouseprofitpercentage,asanindicatorforthemodel’ssuccessinadditiontosimplyprofit.Asmentionedearlier,weusedabankrollof$100foreachgamesothateachbetcouldbemadeindependentofthesuccessofanotherbet.ThefirstaspectwelookedatwastherelationshipbetweenprofitpercentageandtheKCOvalue.Ifwelookatthe2019LaLigaSeason,wecanseethatasweincreaseourfilterfortheKCOvaluewithincrementsof0.01,theprofitpercentagealsosteadilyincreases.Inadditiontothis,thenumberofbetssuggestedalsodecreasessteadilyinthiscase.Thisishighlightedbelow:
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Wethenapplytothistoeveryleague,lookingatprofitpercentagewith3categoriesforthebetsbasedontheirKCOValue.IftheKCOValueisgreaterthan2,greaterthan2.05,andgreaterthan2.1.
Wecanclearlyseeforeveryseason,theoverallprofitpercentagedrasticallyimprovesasweselectbetswithhigherKCOvalues.Everyorangebarishigherthantheleague’srespectivebluebarandeverypurplebarishigherthantheleague’srespectiveorangeandbluebaroutsidethePremierLeague.WhilethereisamassiveamountofsuccessbasedonprofitpercentageforbetswithaKCOvaluegreaterthan2.1,thenumberofbetsbeingplacedisminimal.TheBundesligain2019hadonly
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26betswithaKCOvaluegreaterthan2.1whichwasthemostwesawinaseason.Whiletheprofitpercentageishigher,theoverallprofitishigherwhenplacingbetswithaKCOvaluegreaterthan2or2.05.Hence,ifthegoalisprofitmaximizationausedcouldplacebetswithaKCOvaluegreaterthan2or2.05.Belowisagraphhighlightingtheamountsriskedandwonforthesebets:
TheKCOvalueprovidesauniqueaspectofthemodelasitcanbetailoredtoanindividual’spreferenceofhowriskytheywouldliketobewiththeirbets.Toprovideaholisticexplanation,belowisatablehighlightingtheresultsforthemodel:
LeagueKCOValueGreaterthan2 KCOValueGreaterthan2.1
AmountRisked AmountProfited AmountRisked AmountProfited
Bundesliga2018 $3251.95 $1273.59 $454.73 $616.80
Bundesliga2019 $4705.58 $7780.69 $1720.42 $5014.22
LaLiga2018 $4633.20 $1363.87 $1098.77 $703.33
LaLiga2019 $5517.66 $2847.05 $1215.85 $2555.30
Ligue12018 $4727.76 $1812.51 $894.89 $1117.37
Ligue12019 $4136.08 $1555.22 $674.37 $519.70
PremierLeague2018 $5261.52 $3483.21 $1148.28 $933.22
PremierLeague2019 $5464.42 $2060.52 $897.53 $176.17
SerieA2018 $4519.20 $796.81 $1178.63 $1200.40
SerieA2019 $4070.81 $804.77 $704.14 $625.49
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Ifweweretouseourmodelforthepast2years,placingbetsthathadaKCOvalueofgreaterthan2wewouldprofit$23,778.24frombettingon1164games.IfweonlyplacedbetswithaKCOvalueofgreaterthan2.05wewouldprofit$17,450.94frombettingon387games.Lastly,ifweonlyplacedbetswithaKCOvalueof2.1wewouldprofit$9,987.61frombettingon161games.InordertofullyunderstandtheimportanceoftheKCOvalue,wecanlookattheamountofmoneywewouldwinorlosewithoutimplementingtheKCOaspect.Ifweweretosimplybetontheoutcomewiththehighestprobabilityforeachgame,overthe10seasonswewouldprofit$62,340.Whilethisseemslikealargeamount,wewouldberiskingover$237,800.Thisyieldsaprofitpercentageof26.22%whichisnotverygoodcomparedtothemodelwiththeKCOelement.ThereisasignificantlylargerreturnonyourinvestmentwhenusingtheKCOaspectwithinmodelandhencewebelieveitisextremelyimportanttouse.Moreover,asdiscussedearlierthedifferentKCOintervalsallowforusertotailorthemodeltotheirpreferenceinhowriskytheywouldliketobe.ThisuniqueaspectisalsoabigadvantageprovidedbytheKCOaspectofthemodel.7. FutureResearch
7.1.BivariateandZeroInflatedPoissonDistributionsItwouldbeextremelyinterestingtorepeatthisprocesswithaBivariateorZeroInflatedPoissonDistribution.Abivariatedistributionprovidesprobabilitieswhenyouhavetwoindependentvariables,allowingforeachcombinationtobeaccountedfor.HencebyusingHomeGoalsandAwayGoalsastheindependentvariables,thedistributionwouldprovideprobabilitiesforeachcombinationofhomegoalsandawaygoal.Inasimilarmanneraswedointhispaper,wecansumtheprobabilitiesforeachcorrespondingevent.
AzeroinflatedPoissondistributionissimilartoaPoissonDistribution,howeveritisprimarilyusedwhenanexcessof0isexpectedwithinthedata.Consideringthatoutcomeswith0goalsbeingscoredbyateamtendtobehigherinnature,azeroinflatedPoissonDistributionmaybemoresuitedtopredictingprobabilities.
7.2.FactoringLineupSelectionsWithinOurPredictionsAmajorlimitationforourmodelisthatourmodeldoesnotaccountforinjuriesandlineupchanges.Ifwecanfactorinthestrengthoflineupsusingaholisticstatistic,ourpredictionscandrasticallyimproveintermsofitsaccuracy.AnexamplewouldbeusingtherecentlycreatedAdvancedRPMmetricbytheSyracuseUniversitySoccerAnalyticsClub.Byusingeachplayer’sAdvancedRPMvaluewithinourmodel,eachlineup’sstrengthisbeingaccountedfor.Thisfactorsinwhichplayersareplayingforeachteam,allowingustocreateevenmoreaccuratepredictions.
7.3.AdjustingforDifferentLeaguesWecanseeadecentvariationinsuccessbetweentheleagueswelookedat.Whileallofourmodelsarepositiveandyieldagreatamountofprofit,thereispossibleroomforimprovementbyadjustingforthetalentdistributionamongtheleagues.Forexample,thePremierLeagueisconsideredoneof
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themorecompetitiveleagues,ifwecanfactorthatwithinourmodel,thesuggestedbetsandoverallprofitfromthemodelcanbesignificantlyimproved.
8. ConclusionWhenappropriatelyusingpastdatatopredictgameoutcomeprobabilitiesandthenallocatingfundsefficientlytominimizeriskitisclearthatthereismoneytobemadebettinginEuropeansoccer.UtilizingourpredictivemodelandwiththeKellyCriterion,wefoundalargeamountofsuccess.ThefilteringofourKCOvaluesineachrespectiveleaguehelpusmaximizeourreturnoninvestmentwhenbetting.WefoundthemostsuccessandgreatestprofitpercentageforallgameswheretheKCOvaluewasgreaterthan2.1withaprofitpercentageincreasingupto291.45%intheBundesligain2019.Asdiscussedabove,therearealwaysimprovementsthatcanbemadewithourmodel.Wehopetoexpandourmodelandcontinuetogrowthoseprofitpercentages,findingthemostaccurateandprofitablewaytopredictEuropeansoccermatchoutcomes.
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References[1]HowtocalculatePoissondistributionforfootballbetting.(n.d.).Retrieved2020,fromhttps://help.smarkets.com/hc/en-gb/articles/115001457989-How-to-calculate-Poissondistribution-for-football-betting[2]ListofCompetitions.(n.d.).Retrieved2020,fromhttps://fbref.com/en/comps/[3]TherealKellyCriterion:Acriticalanalysisofthepopularstakingmethod.(2017).Retrieved2020,fromhttps://www.pinnacle.com/en/betting-articles/Betting-Strategy/the-real-kellycriterion/HZKJTFCB3KNYN9CJ[4]FootballResults,Statistics&SoccerBettingOddsData.(n.d.).Retrieved2020,fromhttps://www.football-data.co.uk/data.php[5]HowToHedgeABet:SportsBettingStrategyExplained.(n.d.).Retrieved2020,fromhttps://www.thelines.com/betting/hedge[6]Haghani,VictorandDewey,Richard,RationalDecision-MakingunderUncertainty:ObservedBettingPatternsonaBiasedCoin(October19,2016).Retrieved2020,fromhttps://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963
