Tri-Metric Decision Algorithm (TMDA)

Beyond Expected Value and Likely Profit: Adding Risk of Ruin

In a previous post, I introduced the Dual-Metric Decision Algorithm (DMDA), which combined Expected Value (EV) and Likely Profit (LP) to provide a more comprehensive framework for evaluating betting decisions. While this approach was an improvement over solely relying on EV, it still lacked a critical component: explicit risk management.

Today, I'm extending that framework with the Tri-Metric Decision Algorithm (TMDA), which adds a third dimension: Risk of Ruin (RoR). This addition addresses a fundamental question that every bettor must answer: "What is the probability that my bankroll will decline to an unacceptable level within a given time horizon?"

The Missing Piece: Risk of Ruin

Expected Value tells you if a bet is statistically profitable. Likely Profit tells you if it's sustainable in terms of geometric growth. But neither metric explicitly addresses the volatility risk — the chance that short-term fluctuations will devastate your bankroll before the long-term edge materializes.

Consider this: you might have a positive EV bet with decent LP, but if your stake size is too aggressive relative to the volatility of outcomes, you could hit your drawdown threshold long before experiencing the expected growth. This is where Risk of Ruin (RoR) becomes essential.

Key Insight: RoR quantifies the probability of your bankroll falling below a critical threshold (e.g., 50% drawdown) within a specified number of bets. It's not just about whether you'll win in the long run — it's about whether you'll survive to reach the long run.

Building the Foundation

Let's start with our familiar canonical example and build up to the full TMDA framework:

Parameter	Value
Initial Bankroll (B)	$10,000
Decimal Odds (O)	1.9091
Win Probability (P)	55.00%
Stake Fraction (F)	Variable
Drawdown Threshold	50%
Time Horizon	2,300 bets

Win-Balance and Loss-Balance Multipliers

As before, we calculate the bankroll state after wins and losses:

WB = 1 + (F × (O - 1))
LB = (1 - F)

Where WB represents the bankroll multiplier after a win, and LB represents the bankroll multiplier after a loss.

Expected Value and Likely Profit

The first two metrics remain unchanged from DMDA:

EV = (WB × P) + (LB × (1-P)) - 1
LP = (WB^P × LB^(1-P)) - 1

EV represents the arithmetic mean return per bet, while LP represents the geometric mean return, which accounts for compounding effects.

Log-Drift and Log-Volatility

To calculate Risk of Ruin, we need two additional statistics derived from the log-space representation of bankroll growth:

μ = P·ln(WB) + (1-P)·ln(LB)
σ² = P·(ln(WB)-μ)² + (1-P)·(ln(LB)-μ)²

Here, μ (mu) represents the expected log-growth per bet (drift), and σ (sigma) represents the standard deviation of log-returns (volatility). These metrics transform the problem into a continuous-time random walk, which allows us to apply diffusion approximations.

Technical Note: LP and μ are closely related but not identical. Since LP = exp(μ) − 1, they differ by higher-order terms. For small values (as in typical betting scenarios), LP ≈ μ to several decimal places, which is why the worked example shows them as equal when rounded to 0.000819.

Risk of Ruin Formula

Using the reflection principle from probability theory, we can approximate the probability of hitting a drawdown threshold d within n bets:

RoR ≈ Φ((-d - μ·n)/(σ·√n)) + exp((-2·μ·d)/σ²)·Φ((-d + μ·n)/(σ·√n))

Where Φ is the cumulative distribution function of the standard normal distribution, and d = -ln(drawdown_fraction). For a 50% drawdown threshold, d = 0.693.

Note: This formula assumes continuous betting and uses the normal approximation of the underlying diffusion process. For small sample sizes or extreme probabilities, the approximation may be less accurate, but it provides an excellent practical guideline for typical betting scenarios.

The Tri-Metric Decision Algorithm

With all three metrics in hand, we can now construct the TMDA decision framework:

if EV ≤ 0:
    decision = 'Avoid'  # Statistically unprofitable
elif LP ≤ 0:
    decision = 'Reduce stake'  # Geometric decay despite positive EV
elif RoR > tolerance:
    decision = 'Reduce stake'  # Risk exceeds acceptable threshold
else:
    decision = 'Accept'  # All metrics favorable

This hierarchical decision tree ensures that we only accept bets that satisfy all three conditions:

1. Positive EV — The bet is statistically profitable
2. Positive LP — The bet exhibits sustainable geometric growth
3. Acceptable RoR — The risk of significant drawdown is within tolerance

Worked Example

Let's examine our canonical example with a stake fraction of F = 2%:

Step 1: Calculate WB and LB

WB = 1 + (0.02 × (1.9091 - 1)) = 1.018182
LB = 1 - 0.02 = 0.98

Step 2: Calculate EV and LP

EV = (1.018182 × 0.55) + (0.98 × 0.45) - 1 = 0.001000
LP = (1.018182^0.55 × 0.98^0.45) - 1 = 0.000819

Step 3: Calculate μ and σ

μ = 0.55·ln(1.018182) + 0.45·ln(0.98) = 0.000819
σ² = 0.55·(ln(1.018182)-0.000819)² + 0.45·(ln(0.98)-0.000819)² = 0.000362
σ = 0.01901

Step 4: Calculate RoR

For a 50% drawdown over 2,300 bets:

d = -ln(0.5) = 0.693
RoR ≈ 4.15%

Step 5: Apply TMDA

• EV = 0.001000 ✓ Positive
• LP = 0.000819 ✓ Positive
• RoR = 4.15% ✓ Below 5% tolerance

Decision: Accept — The bet has positive EV, positive LP, and the risk of experiencing a 50% drawdown within 2,300 bets is within our 5% tolerance threshold.

Finding the Optimal Stake

One powerful application of TMDA is determining the maximum stake size that keeps RoR within acceptable bounds. Using binary search or numerical optimization, we can find the stake fraction F* that satisfies:

RoR(F*) = tolerance

The dashboard computes this optimal stake numerically. For any given parameters, the "Optimal Stake for Target RoR" section in the dashboard will display the maximum stake fraction that meets your risk tolerance, along with the corresponding dollar amount for your bankroll.

Interactive Dashboard

To explore TMDA across different parameters, we have created an interactive dashboard (coding by Copilot) where you can:

• Adjust odds, probabilities, and bankroll amounts
• Compare multiple stake fractions simultaneously
• Visualize the relationship between EV, LP, and RoR
• Find the optimal stake for your risk tolerance

(Opens in a new window; allow popups if prompted)

Conclusion

The Tri-Metric Decision Algorithm represents a significant evolution beyond traditional EV-only approaches. By incorporating Likely Profit, we account for geometric compounding effects. By adding Risk of Ruin, we explicitly manage volatility risk and ensure that our stake sizing aligns with our risk tolerance.

However, TMDA is not a silver bullet. It assumes:

• Independent, identically distributed bets
• Accurate probability estimates
• Continuous betting (for the RoR approximation)
• Fixed odds and probabilities across all bets

Real-world betting involves correlated outcomes, model uncertainty, and dynamic market conditions. TMDA should be viewed as a framework for thinking rather than a mechanical system. It provides a structured approach to balancing profitability and risk, but successful implementation requires judgment, experience, and continuous refinement.

Bottom Line: While EV tells you if a bet is profitable, and LP tells you if it's sustainable, RoR tells you if it's survivable. TMDA integrates all three perspectives to make more informed betting decisions.

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Sanity checked and dashboard implementation by Copilot.