Probability & Decision-Making: Risk, Payoffs & Optimal Choices

TAKEAWAYS:

• Learn how to build payoff tables from demand probabilities and cost/revenue data (e.g., milk salesman’s profit).
• Master expected value (EV) calculations to compare risky decisions (e.g., Daraz’s inventory ordering).
• Understand dominance rules in game theory to eliminate dominated strategies (e.g., Pathao’s surge pricing vs. competitor fares).
• Solve decision trees for sequential choices (e.g., Ncell’s network expansion plans).
• Apply minimax regret to minimize worst-case losses (e.g., NEPSE stock portfolio hedging).
• Connect theory to Nepal’s real economy: eSewa’s fraud risk, Khalti’s transaction fees, or NTC’s call-center queues.

1. Probability Basics for Decision Making

Probability quantifies uncertainty. In OR, we use it to model demand, risks, and outcomes before making choices.

Key Definitions

• Random Variable (X): A variable whose outcome depends on chance (e.g., daily milk demand).
• Probability Distribution: Shows all possible values of X and their likelihoods (e.g., demand for 1–5 litres with P(X=3)=0.3).
• Expected Value (EV): The long-run average outcome, calculated as: $$EV=\sum _i{x}_i\cdot P(x_i)$$ Example: If a milkman sells at Rs. 70/litre and buys at Rs. 60, his profit per litre is Rs. 10. For demand X, EV(profit) = 10 × P(X).

Worked Example: Milk Salesman’s Payoff

Given:

Cost to buy: Rs. 60/litre. Selling price: Rs. 70/litre. Step 1: Calculate profit per demand level.

• If demand=3 litres, profit = (3 × Rs. 70) – (3 × Rs. 60) = Rs. 30.
• Repeat for all demands:

Step 2: Compute EV(profit). $$EV=(10\times 0.1)+(20\times 0.15)+(30\times 0.3)+(40\times 0.25)+(50\times 0.2)=31\text{ Rs.}$$

Step 3: Compare EV to alternatives (e.g., buying 4 litres vs. 3 litres).

• If he buys 4 litres but demand=3, he loses Rs. 60 (unsold milk). EV drops to 29 Rs..
• Optimal choice: Buy 3 litres (higher EV).

(Shows real-world application of probability in perishable goods.)

2. Decision Making Under Uncertainty

When outcomes depend on states of nature (e.g., weather, competition), we use payoff tables to evaluate choices.

Types of Criteria

Worked Example: eSewa’s Transaction Fee Strategy

Scenario: eSewa charges:

• Rs. 1 for transactions <Rs. 1,000 (State A: Low competition).
• Rs. 2 for transactions ≥Rs. 1,000 (State B: High competition).

Payoff Table (Profit in Rs.): States’ Probabilities: P(A)=0.6, P(B)=0.4.

Step 1: Calculate EV for each strategy.

• Charge Rs1: $$EV=(500\times 0.6)+(300\times 0.4)=440\text{ Rs.}$$
• Charge Rs2: $$EV=(400\times 0.6)+(450\times 0.4)=430\text{ Rs.}$$ Optimal choice: Charge Rs1 (higher EV).

Step 2: Apply Minimax Regret.

• Regret Table:
• Minimax Regret: Choose Rs2 (minimizes worst regret of 100 Rs.).

Conclusion: eSewa should charge Rs. 2 to hedge against high competition.

(Links payoff tables to real payment apps.)

3. Game Theory Basics

Game theory models conflict and cooperation between decision-makers (players). Key terms:

• Players: Competitors (e.g., Pathao vs. Taxi Nepal).
• Strategies: Choices (e.g., surge pricing, flat fare).
• Payoffs: Outcomes (e.g., profit, market share).
• Nash Equilibrium: No player can benefit by unilaterally changing strategy.

Dominance Rule

A strategy is dominated if it always yields a worse payoff than another strategy, regardless of the opponent’s move.

Example: Pathao’s Fare Strategy Payoff Table (Profit in Rs.): Step 1: Check for dominated strategies.

• For Pathao:
  • Low fare vs. High fare when Taxi Nepal charges high: High fare gives Rs. 300 vs. Rs. 100 → Low fare is dominated.
• For Taxi Nepal:
  • Low fare vs. High fare when Pathao charges low: Low fare gives Rs. 150 vs. Rs. 100 → High fare is dominated.

Step 2: Eliminate dominated strategies.

• Pathao’s optimal: High fare.
• Taxi Nepal’s optimal: Low fare. Nash Equilibrium: (Pathao High, Taxi Nepal Low) with payoffs (150, 100).

Mermaid Diagram:

flowchart TD
    A["Pathao: High Fare"] -->|"Taxi Nepal Low"| B["Profit: Rs. 150"]
    A -->|"Taxi Nepal High"| C["Profit: Rs. 300"]
    D["Taxi Nepal: Low Fare"] -->|"Pathao Low"| E["Profit: Rs. 150"]
    D -->|"Pathao High"| F["Profit: Rs. 100"]

Caption: Nash Equilibrium in Pathao’s fare competition.

4. Decision Trees for Sequential Decisions

Used when choices unfold over time (e.g., Ncell’s 5G rollout).

Example: Ncell’s Network Expansion Problem: Expand to Kathmandu (cost: Rs. 50M) or Pokhara (cost: Rs. 30M). Demand depends on competition. Tree:

sequenceDiagram
    participant Ncell
    participant Nature
    Ncell->>Nature: Expand Kathmandu (Rs. 50M)
    Nature-->>Ncell: High Demand (P=0.7): Rs. 120M profit
    Nature-->>Ncell: Low Demand (P=0.3): Rs. 20M profit
    Ncell->>Nature: Expand Pokhara (Rs. 30M)
    Nature-->>Ncell: High Demand (P=0.6): Rs. 80M profit
    Nature-->>Ncell: Low Demand (P=0.4): Rs. 10M profit

EV Calculation:

• Kathmandu: $$(120\times 0.7)+(20\times 0.3)–50=53\text{ Rs. M}$$
• Pokhara: $$(80\times 0.6)+(10\times 0.4)–30=48\text{ Rs. M}$$ Optimal choice: Expand to Kathmandu.

(Shows real telecom strategy.)

5. Comparing Methods

Figure: Decision-Making Criteria Comparison

In the Real World

1. Khalti’s Transaction Fees:

   • Idea: Uses expected value to set fees (e.g., 2% for low-risk transactions, 3% for high-risk).
   • How: Probability of chargeback (P=0.01) vs. profit per transaction (Rs. 50) determines fee structure.
   • Worked Example: If Khalti processes 1,000 transactions/day:
     • Fee = Rs. 20 (2%) → EV loss from chargebacks = 1,000 × 0.01 × 20 = Rs. 200/day.
     • Optimal fee balances revenue and risk.

2. Daraz’s Inventory Ordering:

   • Idea: Decision trees model stockouts vs. overstock.
   • How: Probability of demand (P=0.8 for high season) vs. cost of stockout (Rs. 500) vs. holding cost (Rs. 50/litre) guides orders.
   • Worked Example: For 100 units:
     • Order 100 → EV profit = (100 × Rs. 200) – (100 × Rs. 100) – (0.2 × 100 × Rs. 500) = Rs. 14,000.
     • Order 80 → EV profit = (80 × Rs. 200) – (80 × Rs. 100) – (0.8 × 20 × Rs. 500) = Rs. 13,200.
     • Optimal: Order 100 units.

3. NTC’s Call-Center Queues:

   • Idea: Queuing theory (Unit 9) models wait times, but probability sets staffing needs.
   • How: Arrival rate (λ=96 patients/24h) vs. service rate (μ=1 patient/10min) determines staffing.
   • Worked Example: If NTC hires 4 agents (μ=4/10min=24/24h):
     • Utilization = λ/μ = 96/24 = 4 → Queue length grows indefinitely.
     • Solution: Hire 8 agents (μ=48/24h) to balance λ.

Figure: Daraz’s Inventory Decision Tree Caption: Daraz’s optimal order quantity is 100 units (EV=Rs. 14,400).

Exam Tip

1. Payoff Tables:

   • Always label rows as strategies and columns as states of nature.
   • For EV calculations, show each multiplication step (e.g., "30 × 0.3 = 9").
   • Common mistake: Forgetting to subtract costs (e.g., buying milk at Rs. 60).

2. Game Theory:

   • Dominance rule is worth 3–4 marks. Always:
     1. Identify dominated strategies (compare rows/columns).
     2. Eliminate them and re-evaluate.
     3. State the Nash Equilibrium clearly.
   • Example question: "A firm faces two competitors. Show the dominance rule for the payoff table below." → Show elimination steps.

3. Decision Trees:

   • Draw the tree step-by-step with probabilities at chance nodes.
   • Calculate backward (from end nodes to root).
   • Tip: Use foldback notation (e.g., "EV = 0.7×120 + 0.3×20 – 50").

4. Real-World Links:

   • Tie examples to Nepal’s economy:
     • Banks: Loan approvals (probability of default).
     • NEPSE: Stock portfolio diversification (minimax regret).
     • Pathao: Dynamic pricing (game theory).
   • Formula recall: Memorize EV = Σ(xi × Pi) and regret = max payoff – actual payoff.

5. Time Management:

   • Probability (30%): Spend 15 mins on payoff tables/EV.
   • Game Theory (20%): 10 mins for dominance rules.
   • Decision Trees (20%): 12 mins (draw carefully).
   • Review (30%): 10 mins for marginal analysis or arithmetic method.

Final Figure: Exam Checklist