Friday, August 01, 2025

DMDA Probability Thresholds

WCMI DMDA Probability Thresholds

A Weekend Warrior's approach to sustainable speculation using probability thresholds and illustrated with a Taleb's Barbell Strategy example

In our previous posts, we introduced the Dual-Metric Decision Algorithm (DMDA) as a framework combining Expected Value (EV) and Likely Profit (LP) for sports betting decisions. While EV measures average profit per bet, LP captures the expected geometric growth rate of our bankroll - the multiplicative reality facing bettors with limited capital.

This post extends that work by deriving the precise probability thresholds required for positive EV and LP, using the Rugby World Cup 2023 as a worked example of Taleb's Barbell Strategy in practice.

Mathematical Framework

For a bet with stake fraction F, decimal odds O, and bettor's estimated win probability P, the DMDA calculates win-balance and loss-balance multipliers:

  • Win-Balance Multiplier: W B = 1 + F ( O − 1 ) WB = 1 + F(O-1)
  • Loss-Balance Multiplier: L B = 1 − F LB = 1-F

From these, we derive:

  • Expected Value: E V = ( W B × P ) + ( L B × ( 1 − P ) ) − 1 EV = (WB \times P) + (LB \times (1-P)) - 1
  • Likely Profit: L P = W B P × L B ( 1 − P ) − 1 LP = WB^P \times LB^{(1-P)} - 1

Probability Thresholds

EV = 0 Threshold

Setting E V = 0 EV = 0 and solving for P gives:

P EV = 1 − L B W B − L B = F F × O = 1 O P_{\text{EV}} = \frac{1-LB}{WB-LB} = \frac{F}{F \times O} = \frac{1}{O}

Elegantly, a bet has non-negative expected value only if our probability estimate exceeds the market's implied probability 1 / O 1/O .

LP = 0 Threshold

Setting L P = 0 LP = 0 yields:

P LP = ln ⁡ ( 1 / L B ) ln ⁡ ( W B / L B ) P_{\text{LP}} = \frac{\ln(1/LB)}{\ln(WB/LB)}

This boundary is stricter than the EV threshold because geometric growth is more sensitive to downside volatility.

Canonical Example

We will focus initially on this canonical betting example:

Parameter Value
Total Bankroll (B) $10000
Markets (M) 1
Decimal Odds (O) 1.9091
Win Probability (P) 55.00%
Stake Fraction (F) 1.00%

Market Conditions:

  • Implied probability: 52.38% (decimal odds 1.9091)
  • Warrior assessment: >=55.00% (conservative minimum)

Threshold Calculations:

  • F = 0.01 F = 0.01
  • O = 1.9091 O = 1.9091
  • L B = 1 − 0.01 = 0.99 LB = 1 - 0.01 = 0.99
  • W B = 1 + 0.01 ( 1.9091 − 1 ) = 1.00909 WB = 1 + 0.01(1.9091-1) = 1.00909

Results:

  • P EV = 1 / 1.9091 = 0.5238 P_{\text{EV}} = 1/1.9091 = 0.5238 (52.38%)
  • P LP = ln ⁡ ( 1 / 0.99 ) ln ⁡ ( 1.00909 / 0.99 ) ≈ 0.52622 P_{\text{LP}} = \frac{\ln(1/0.99)}{\ln(1.00909/0.99)} \approx 0.52622 (52.62%)

Decision: Warrior assessment of 55% exceeds both thresholds and an effective lower bound s set at 53%.

Rugby World Cup 2023: Worked Example

Consider a Weekend Warrior's barbell allocation:

  • Total Bankroll: $10,000
  • Total Stake: $500 (5% of bankroll)
  • Safe Allocation: $450 (4.5% of bankroll, 90% of stake)
  • Risky Allocation: $50 (0.5% of bankroll, 10% of stake)

Safe Allocation (Favs)

Market Conditions:

  • Implied probability: 80% (decimal odds 1.25)
  • Warrior assessment: >=85% (conservative minimum)

Threshold Calculations:

  • F = 0.045 F = 0.045
  • O = 1.25 O = 1.25
  • L B = 1 − 0.045 = 0.955 LB = 1 - 0.045 = 0.955
  • W B = 1 + 0.045 ( 1.25 − 1 ) = 1.01125 WB = 1 + 0.045(1.25-1) = 1.01125

Results:

  • P EV = 1 / 1.25 = 0.80 P_{\text{EV}} = 1/1.25 = 0.80 (80%)
  • P LP = ln ⁡ ( 1 / 0.955 ) ln ⁡ ( 1.01125 / 0.955 ) ≈ 0.8045 P_{\text{LP}} = \frac{\ln(1/0.955)}{\ln(1.01125/0.955)} \approx 0.8045 (80.45%)

Decision: Warrior assessment of 85% exceeds both thresholds

Some Insights

Conservative Estimates as Lower Bounds

The warrior assessments function as conservative minimum thresholds rather than precise probability estimates. This approach provides safety margins: if actual probability assessments exceed these minimums (likely), returns will surpass calculated expectations.

LP Threshold Criticality

The LP threshold being consistently higher than the EV threshold demonstrates why stake sizing matters even for positive-EV bets. Excessive stakes can create negative geometric growth despite positive expected value - a critical insight for bankroll survival.

Real-World Application

South Africa's victory in the Rugby World Cup 2023 validated the safe allocation approach, while maintaining exposure to potential Black Swan events preserved optionality for extreme outsiders. The mathematical framework translated effectively to practical application.

Note: The final draft of this post was sanity checked by Claude.

# https://vendire-ludorum.blogspot.com/

<#
.SYNOPSIS
    DMDA Probability Thresholds Calculator
    
.DESCRIPTION
    Calculates the probability thresholds for positive Expected Value (EV) and 
    positive Likely Profit (LP) based on the Dual-Metric Decision Algorithm (DMDA).
    
.PARAMETER DecimalOdds
    The decimal odds offered by the market (e.g., 1.25, 50.0)
    
.PARAMETER StakeFraction
    The fraction of bankroll being staked (e.g., 0.045 for 4.5%)
    
.EXAMPLE
    .\DMDA-Thresholds.ps1 -DecimalOdds 1.25 -StakeFraction 0.045
    
.EXAMPLE
    .\DMDA-Thresholds.ps1 -DecimalOdds 50.0 -StakeFraction 0.005
#>

param(
    [Parameter(Mandatory=$true)]
    [ValidateRange(1.01, [double]::MaxValue)]
    [double]$DecimalOdds,
    
    [Parameter(Mandatory=$true)]
    [ValidateRange(0.001, 0.999)]
    [double]$StakeFraction
)

# Calculate multipliers based on DMDA framework
$LossBalanceMultiplier = 1 - $StakeFraction
$WinBalanceMultiplier = 1 + $StakeFraction * ($DecimalOdds - 1)

# Calculate probability thresholds
$P_EV = 1 / $DecimalOdds
$P_LP = [Math]::Log(1 / $LossBalanceMultiplier) / [Math]::Log($WinBalanceMultiplier / $LossBalanceMultiplier)

# Output results
Write-Host ""
Write-Host "DMDA Probability Thresholds Calculator" -ForegroundColor Cyan
write-Host "(https://vendire-ludorum.blogspot.com/)" -ForegroundColor Cyan
Write-Host "======================================" -ForegroundColor Cyan
Write-Host ""
Write-Host "Input Parameters:" -ForegroundColor Yellow
Write-Host "  Decimal Odds (O): $DecimalOdds"
Write-Host "  Stake Fraction (F): $StakeFraction ($($StakeFraction * 100)%)"
Write-Host ""
Write-Host "Calculated Multipliers:" -ForegroundColor Yellow
Write-Host "  Loss-Balance Multiplier (LB): $($LossBalanceMultiplier.ToString('F6'))"
Write-Host "  Win-Balance Multiplier (WB): $($WinBalanceMultiplier.ToString('F6'))"
Write-Host ""
Write-Host "Probability Thresholds:" -ForegroundColor Green
Write-Host "  P(EV) = $($P_EV.ToString('F6')) ($($($P_EV * 100).ToString('F4'))%)"
Write-Host "  P(LP) = $($P_LP.ToString('F6')) ($($($P_LP * 100).ToString('F4'))%)"
Write-Host ""
Write-Host "Market Implied Probability: $($($P_EV * 100).ToString('F2'))%" -ForegroundColor Cyan
Write-Host "Positive Bankroll Growth Probability: $($($P_LP * 100).ToString('F2'))%" -ForegroundColor Cyan
Write-Host ""

# Validation check
if ($P_LP -gt $P_EV) {
    Write-Host "LP threshold exceeds EV threshold" -ForegroundColor Green
} else {
    Write-Host "Warning: LP threshold does not exceed EV threshold" -ForegroundColor Red
}

Write-Host ""
Write-Host "For a bet to be considered sustainable (bankroll growth), "
Write-Host "our estimated win probability should be greater than both P(EV) and P(LP)."
Write-Host "As shown, the P(LP) threshold is always stricter than the P(EV) threshold." -ForegroundColor Cyan
Write-Host ""


# Example usage:

<# 
.\DMDA_Probability_Thresholds.ps1 -DecimalOdds 1.9091 -StakeFraction 0.01

DMDA Probability Thresholds Calculator
(https://vendire-ludorum.blogspot.com/)
======================================

Input Parameters:
  Decimal Odds (O): 1.9091
  Stake Fraction (F): 0.01 (1%)

Calculated Multipliers:
  Loss-Balance Multiplier (LB): 0.990000
  Win-Balance Multiplier (WB): 1.009091

Probability Thresholds:
  P(EV) = 0.523807 (52.3807%)
  P(LP) = 0.526188 (52.6188%)

Market Implied Probability: 52.38%
Positive Bankroll Growth Probability: 52.62%

LP threshold exceeds EV threshold

For a bet to be considered sustainable (bankroll growth),
our estimated win probability should be greater than both P(EV) and P(LP).
As shown, the P(LP) threshold is always stricter than the P(EV) threshold.

#>
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