Trader Probabilities Derivation And K-L Divergence (Part 3)

Returning one last time to the trader win probabilities derivation (TWPD) question Trader Probabilities Derivation And K-L Divergence (Part 1) and Trader Probabilities Derivation And K-L Divergence (Part 2):

How do we create a coherent set of trader win probabilities that does not stray too far from the implied win market probabilities while taking into account our own limited and possibly vague insights?

On this occasion, we will focus on soccer and add a late-breaking injury report from social media to our own 'gut instinct' about the likely outcome.

As before, we are using our 'no-frills' python script to access APMonitor-GEKKO:

import json
import os
import shutil

from gekko import GEKKO

d_path = os.path.dirname(os.path.realpath(__file__))
file = open(d_path + '/apm')
model = file.read()
file.close()

m = GEKKO(remote=False)

m.Raw(model)

m.solve(disp=False)

print('Objective: ', m.options.OBJFCNVAL)

shutil.copy(m.path+'/Results.json', d_path + '/json')

with open(m.path+'/Results.json') as f:
    results = json.load(f)
print(results)

with the input - apm:

Model 20210223_Follis_LeagueMatch

  Variables

    ! --- Public market information ---

    ! Trader win probabilities.
    ! Initial values set to market price implied probabilities.
    ! Lower-bound set to 17% - half the natural-odds implied 
    ! probability (1/3).
    ! Upper-bound set to 100%.
    home = 1.00/4.20, >=0.17, <=1.00
    away = 1.00/3.60, >=0.17, <=1.00
    draw = 1.00/1.98, >=0.17, <=1.00

    kld

  End Variables

  Equations
  
    ! Total trader win probabilities must sum to one.
    home + &
    away + &
    draw = 1.00
    

    ! --- Private trader opinions ---

    ! Gut instinct.
    draw >= 0.50
    draw <= 0.60

    ! Late Twitter away-team injury report.
    home >= 0.25
    home <= 0.30


    ! --- Combination of public and private information ---

    ! Minimize K-LD.
    kld=(1.00/4.20)*log(((1.00/4.20)/home)) + &
        (1.00/3.60)*log(((1.00/3.60)/away)) + &
        (1.00/1.98)*log(((1.00/1.98)/draw))

  End Equations

End Model

and the generated output - json:

{
  "home" : [   2.7046072411E-01],
  "away" : [   2.0172750176E-01],
  "draw" : [   5.2781177413E-01],

  "kld" : [   3.6252143687E-02]
}

Once again, as a sanity check. we can test the derived set of trader win probabilities against our constraints:

• min(home, away, draw) >= 17% [0.20173];
• 0.25 <= home <= 0.30 [0.27046];
• 0.50 <= draw <= 0.60 [0.52781].

In sum, we have expertly combined the wisdom of the crowds with our own limited insights to derive a coherent and valid set of trader win probabilities (TWPD) Historically, professional traders have 'melded' (e.g. Bill Benter) their own odds-line with the win-market prices to gain a more informed opinion. We have revised that process by baselining on the win-market prices and adding a few constraints where we believe we have additional information (e.g. late-breaking injury or weather reports from social media) while relying on our automatic method to generate the updated trader win probabilities and, of course, indirectly our own odds-line:

Outcome  Private   Public
Home:    3.70      4.20
Away:    4.95      3.60
Draw:    1.89      1.98

For completion here is an updated horse-racing model template file:

Model 20210223_14-30_RacingPark

  Variables

    ! --- Public market information ---

    ! Trader win probabilities.
    ! Initial values set to market price implied 
    ! probabilities.
    ! Lower-bound set to 6% - half the natural-odds 
    ! implied probability (1/8).
    alpha   = 1/25.00, >=0.06, <=1.00
    bravo   = 1/34.00, >=0.06, <=1.00
    charlie = 1/13.50, >=0.06, <=1.00
    delta   =  1/2.78, >=0.06, <=1.00
    echo    =  1/3.60, >=0.06, <=1.00
    foxtrot = 1/17.00, >=0.06, <=1.00
    golf    =  1/8.80, >=0.06, <=1.00
    hotel   = 1/17.50, >=0.06, <=1.00

    kld

  End Variables

  Equations
  
    ! Total trader win probabilities must sum to one.
    alpha + &
    bravo + &
    charlie + &
    delta + &
    echo + &
    foxtrot + &
    golf + &
    hotel = 1.00
    

    ! --- Private trader opinions ---

    ! Half the field have a better chance.
    (alpha + echo + foxtrot + hotel) > (bravo + charlie + delta + golf)

    ! Two horses have between 30% and 40% combined win probability.
    echo + foxtrot >= 0.30
    echo + foxtrot <= 0.40
    
    ! Between 3% and 7% edge on one horse.
    (17.00 * foxtrot) - 1.00 >= 0.03
    (17.00 * foxtrot) - 1.00 <= 0.07


    ! --- Combination of public and private information ---

    ! Minimize K-LD.
    kld=(1/25.00)*log(((1/25.00)/alpha)) + &
        (1/34.00)*log(((1/34.00)/bravo)) + &
        (1/13.50)*log(((1/13.50)/charlie)) + &
        (1/2.78)*log(((1/2.78)/delta)) + &
        (1/3.60)*log(((1/3.60)/echo)) + &
        (1/17.00)*log(((1/17.00)/foxtrot)) + &
        (1/8.80)*log(((1/8.80)/golf)) + &
        (1/17.50)*log(((1/17.50)/hotel))

  End Equations

End Model