Decision Analysis: Trees, Payoffs, and Optimal Choices

TAKEAWAYS:

• Decision trees and payoff matrices visually model choices, probabilities, and outcomes to identify optimal strategies.
• Expected value (EV) quantifies risk by weighting outcomes by their likelihood, while maximin/minimax rules handle extreme uncertainty.
• Sensitivity analysis reveals how changes in probabilities or payoffs affect the best decision.
• Real-world tools like Khalti’s fraud detection (decision trees) and Pathao’s route optimization (decision analysis) rely on these techniques.
• Simulation models complement decision analysis by testing strategies against probabilistic scenarios.

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Decision-Making Under Uncertainty: The Core Problem

Every business decision involves trade-offs. Should Daraz expand to a new district? Should Ncell invest in 5G infrastructure? Should a bank approve a loan? These questions share a common structure:

1. Alternatives (Choices): Actions you can take (e.g., "Expand," "Maintain," "Withdraw").
2. States of Nature: Uncontrollable events (e.g., "High demand," "Low demand," "Competitor enters").
3. Payoffs: Outcomes (profits, losses, costs) tied to each choice-state combination.

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1. Payoff Tables: Structuring the Problem

A payoff table organizes choices, states, and outcomes in a matrix. For example, consider NTC’s decision to launch a new broadband service in Pokhara:

Key Terms:

• Payoff: Net outcome (profit/loss) for each choice-state pair.
• States of Nature: Assumed independent (e.g., demand and competition are not directly linked here).

Worked Example: NTC’s Broadband Decision Assume NTC estimates:

• Probability of high demand = 0.4
• Probability of low demand = 0.3
• Probability of competitor entering = 0.3

Step 1: Calculate Expected Value (EV) for each choice EV = Σ (Payoff × Probability)

• Launch: (₹50 × 0.4) + (-₹20 × 0.3) + (-₹30 × 0.3) = ₹20 - ₹6 - ₹9 = ₹5 lakh
• Delay Launch: (₹30 × 0.4) + (₹10 × 0.3) + (₹25 × 0.3) = ₹12 + ₹3 + ₹7.5 = ₹22.5 lakh
• Withdraw: ₹0

Optimal Choice: Delay Launch (highest EV of ₹22.5 lakh).

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2. Decision Trees: Visualizing Sequential Decisions

Decision trees extend payoff tables to multi-stage decisions, where later choices depend on earlier outcomes. Example: Khalti’s fraud detection system uses decision trees to flag suspicious transactions.

Mermaid Diagram: Decision Tree for Khalti Fraud Detection

How It Works:

1. Root Node: Initial decision (e.g., "Transaction amount?").
2. Branches: Probabilities or conditions (e.g., "User history clean?").
3. Leaf Nodes: Final outcomes (e.g., "Block" or "Approve").

Worked Example: Pathao’s Route Optimization Pathao uses decision trees to decide whether to:

1. Accept a ride request (if demand is high and driver is nearby).
2. Reject or delay (if traffic is heavy or driver is far).

Assumptions:

• Probability of high demand = 0.6 → EV(accept) = ₹80
• Probability of low demand = 0.4 → EV(accept) = -₹20
• EV(reject) = ₹0

Decision Tree Calculation:

EV(accept) = (₹80 × 0.6) + (-₹20 × 0.4) = ₹48 - ₹8 = **₹40**
EV(reject) = ₹0

Optimal Choice: Accept (higher EV).

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3. Decision Criteria: Choosing Without Probabilities

When probabilities are unknown, use non-probabilistic criteria:

Worked Example: NEPSE’s Investment in New Stocks NEPSE must decide whether to invest in Stock A or Stock B under uncertain market conditions:

Apply Criteria:

1. Maximax: Choose Stock A (max payoff = ₹10 lakh).
2. Maximin: Choose Stock B (min payoff = -₹2 lakh).
3. Minimax Regret:
   • Regret for Stock A: max(0, 10-8) = ₹2 (bull), max(0, 5-2) = ₹3 (bear).
   • Regret for Stock B: max(0, 8-10) = ₹2 (bull), max(0, 2-5) = ₹3 (bear).
   • Both have equal regret → Choose either (or use other criteria).

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4. Sensitivity Analysis: Testing Robustness

Sensitivity analysis checks how changes in probabilities or payoffs affect the optimal decision. Example: How does Ncell’s 5G investment hold up if demand probabilities shift?

Graph: EV vs. Probability of High Demand

EV(Launch) = 50P - 20(0.3) - 30(0.7 - P)
EV(Delay) = 30P + 10(0.3) + 25(0.7 - P)

Plot EV for both choices as P (probability of high demand) varies from 0 to 1.

Key Insight:

• If P > 0.45, Launch becomes optimal.
• If P < 0.45, Delay remains better.
• Break-even point: P = 0.45 (where EV(Launch) = EV(Delay)).

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5. Simulation Models: Extending Decision Analysis

When probabilities are complex, Monte Carlo simulation models thousands of scenarios. Example: Daraz’s warehouse location decision simulates demand fluctuations.

Mermaid Diagram: Simulation Pipeline

Worked Example: Bank Loan Approval A bank uses simulation to decide whether to approve a loan:

• Input: Customer credit score (normal distribution, μ=650, σ=100).
• Output: Probability of default (if score < 600, default = 0.3; else 0.1).
• Decision Rule: Approve if EV(approve) > EV(reject).

Simulation Steps:

1. Generate 10,000 random credit scores.
2. For each, calculate EV(approve) and EV(reject).
3. Approve if 70% of simulations favor approval.

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In the Real World

1. Khalti’s Fraud Detection
   • Idea Used: Decision trees classify transactions as fraudulent or legitimate.
   • How: Splits on features like amount, time, and user history to minimize false positives.
   • Real Output:

2. Pathao’s Driver Assignment

   • Idea Used: Decision analysis balances ride demand vs. driver availability.
   • How: Uses EV to decide whether to assign a driver to a request.
   • Example: If 60% chance of high demand, accept the ride (EV = ₹40 > ₹0).

3. NTC’s Network Expansion

   • Idea Used: Payoff tables and sensitivity analysis for infrastructure projects.
   • How: Models payoffs for "Expand," "Maintain," or "Withdraw" under demand uncertainty.
   • Real Data: If demand grows >5%/year, expand; else maintain.

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Comparing Decision Analysis Tools

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Exam Tip

1. Always draw diagrams: Decision trees and payoff tables are worth 50% of marks. Label branches clearly with probabilities and payoffs.
2. Show calculations: For EV, write every step (e.g., "EV(Launch) = (₹50 × 0.4) + ..."). Partial credit is given for correct formulas.
3. Link to real-world: Examiners love examples from Nepali businesses (e.g., "How would Khalti use decision trees?").
4. Sensitivity analysis is key: If asked to "test robustness," plot a graph or state the break-even point.
5. Common mistakes to avoid:
   • Forgetting to multiply payoffs by probabilities in EV.
   • Misapplying maximin/maximax (e.g., choosing maximin when the question asks for maximax).
   • Ignoring the "states of nature" in payoff tables.

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Final Note: Decision analysis is about turning uncertainty into actionable insights. Whether it’s Ncell’s network upgrades or Daraz’s logistics, the same principles apply. Practice with real data (e.g., use NEPSE stock prices or Khalti transaction logs) to master this unit.