Game Theory & Decision Making: Models, Games & Strategies

TAKEAWAYS:

• Game theory models strategic interactions (e.g., price wars between Daraz and Sastodeal) using payoff matrices and optimal strategies.
• Decision criteria (maximin, minimax regret, Laplace, Hurwitz) help managers pick the best action under uncertainty (e.g., Ncell’s network expansion).
• Zero-sum games (where one player’s gain = another’s loss) are solved via dominance rules or mixed strategies (e.g., eSewa vs. Khalti in digital payments).
• Real-world applications: Game theory explains pricing (Nepal Oil Corporation’s fuel price wars), queuing (Pathao’s surge pricing), and negotiations (Nabil Bank’s loan terms).
• Exam focus: Solve payoff tables, reduce games to 2×2 form, and justify decisions using criteria—always show calculations.
• Key tools: Payoff matrices, saddle points, expected value, and dominance rules are non-negotiable for full marks.

---

1. What is Game Theory?

Game theory studies strategic decision-making where outcomes depend on interdependent choices of two or more players. It’s used in:

• Competitive markets (e.g., Daraz vs. Sastodeal on discounts).
• Negotiations (e.g., NTC vs. private telecoms on spectrum auctions).
• Resource allocation (e.g., Kathmandu traffic police optimizing routes).

Core Concepts

A classic example of non-zero-sum games where cooperation fails. (Image: S4485581, CC BY-SA 4.0, via Wikimedia Commons)

---

2. Types of Games

Real-World Example:

• Nepal Oil Corporation (NOC) vs. Private Dealers:
  • Game: Fuel price adjustments.
  • Zero-Sum: If NOC cuts prices, private dealers lose sales (and vice versa).
  • Outcome: NOC uses dominance strategies (e.g., bulk discounts to retain customers).

---

3. Solving Two-Person Zero-Sum Games

Step 1: Payoff Matrix

For a game with players A and B, the payoff table shows A’s gains (B’s losses):

Step 2: Check for Saddle Points

• Saddle point: The minimum of the row maxima or the maximum of the column minima.
• If minimax = maximin, the game is strictly determinable (pure strategy solution exists).

Example: For the matrix above:

• Row maxima: max(3, -2) = 3, max(4, -3) = 4 → Minimax = 3.
• Column minima: min(3, 4) = 3, min(-2, -3) = -3 → Maximin = 3.
• Saddle point at (A1, B1) → Optimal pure strategy.

Step 3: Reduce to 2×2 by Dominance

Remove dominated strategies (always worse than another option).

Example: Original game:

• A3 is dominated by A2 (A2 always ≥ A3).
• B3 is dominated by B2 (B2 always ≥ B3 for A1/A2). Reduced game: | | B1 | B2 |

|-------|--------|--------| | A1| 3 | 2 | | A2| 4 | 4 |

---

4. Decision-Making Under Uncertainty

When probabilities are unknown, use decision criteria:

Worked Example (Minimax Regret): Payoff table:

1. Regret Matrix:
   • For A: max(4,0,-5)=4 → Regrets: 0, 6, -3
   • For B: max(0,6,2)=6 → Regrets: 6, 0, 3
   • For C: max(-5,9,7)=9 → Regrets: 14, 0, 7
2. Max Regrets: [6, 3, 14] → Minimax Regret = S2.

---

5. Real-World Applications in Nepal

Case 1: eSewa vs. Khalti (Digital Payments)

• Game Type: Zero-sum (market share).
• Strategies:
  • eSewa: Lower transaction fees, cashback.
  • Khalti: Partnerships with merchants.
• Outcome: Mixed strategies (e.g., Khalti offers discounts during festivals to attract users).

Case 2: Pathao’s Surge Pricing

• Game Theory: Dynamic pricing during peak hours (e.g., Dashain).
• Model: Non-zero-sum (drivers and riders both benefit from efficient matching).
• Strategy: Increase fares when demand > supply (like Uber’s surge pricing).

Case 3: NTC’s Network Expansion

• Decision Criterion: Hurwitz (α=0.6 for optimism).
• Options:
  • Expand in Kathmandu (high cost, high demand).
  • Expand in rural areas (low cost, low demand).
• Calculation:
  • Kathmandu: (0.6×10M + 0.4×(-2M)) = 4.4M profit.
  • Rural: (0.6×3M + 0.4×1M) = 2.2M profit.
• Choice: Kathmandu (higher expected value).

---

6. Game Theory in Global Companies

---

Exam Tip: How to Score Full Marks

1. For payoff matrices:

   • Always label rows/columns clearly (Player A/B, Strategies 1/2).
   • Show step-by-step dominance reduction (circle dominated strategies).
   • For mixed strategies, solve using linear equations (e.g., $3p+4q=1$).

2. For decision criteria:

   • Write the formula (e.g., Hurwitz: $\alpha \times \text{max}+(1-\alpha )\times \text{min}$).
   • Calculate all options before choosing the best.
   • Justify (e.g., “Minimax regret is chosen to avoid high opportunity loss”).

3. For case studies:

   • Map the game type (zero-sum/non-zero-sum).
   • Identify strategies and payoffs (even if hypothetical).
   • Link to real data (e.g., “NTC’s revenue dropped 10% when private dealers undercut prices”).

4. Common Pitfalls:

   • ❌ Forgetting to reduce the game to 2×2 before solving.
   • ❌ Mislabeling rows/columns (Player A’s payoffs go first).
   • ❌ Skipping the “fairness” check for strictly determinable games.

---

Practice Questions (Exam-Style)

1. Reduce and solve: Player A | B1 (2,4) | B2 (3,1) | B3 (1,5) Use dominance to find optimal strategies.

2. Decision criteria: For the payoff table below, find the best decision using: i) Laplace criterion ii) Hurwitz (α=0.3)

   	S1	S2	S3
   A	5	2	8
   B	3	7	4

3. Case Study: Nepal Telecom (NTC) is deciding whether to launch a new 5G plan in Pokhara or Chitwan. The payoffs (in millions) are:

   • Pokhara: High demand (15), Low demand (5).
   • Chitwan: High demand (8), Low demand (10).
   • Probability of high demand: 60%. Which location should NTC choose using expected value?

---

Final Note: Game theory is not just math—it’s about real-world strategy. Whether it’s negotiating with suppliers (like Daraz) or optimizing routes (like Pathao), these tools help managers make data-driven decisions. Practice payoff matrices daily, and you’ll ace the exam!