Game Theory: Strategies, Payoffs, and Optimal Decisions

TAKEAWAYS:

• Game theory models conflicts and cooperation between rational decision-makers using payoff matrices.
• The dominance rule eliminates strictly dominated strategies to simplify choices.
• A saddle point is a value where the minimum of row maxima equals the maximum of column minima.
• Mixed strategies involve probabilistic choices when pure strategies are not optimal.
• Minimax theorem guarantees a solution exists for zero-sum games.
• Applications include pricing wars, advertising battles, and resource allocation in firms.

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1. Introduction to Game Theory

Game theory studies strategic interactions where players make decisions to maximize their outcomes, considering opponents' responses. It applies to business (competition), politics (negotiations), and everyday life (bargaining).

Key Terms

• Players: Decision-makers (e.g., firms, countries).
• Strategies: Available choices for each player.
• Payoffs: Numerical outcomes (profits, losses) for each strategy combination.
• Zero-sum game: One player’s gain equals another’s loss (e.g., poker).
• Non-zero-sum game: Players can both gain or lose (e.g., advertising budgets).

Payoff Matrix

A table showing payoffs for all strategy combinations. Example:

FIGURE 1: Payoff matrix for Firms A and B (Worked Example 1).

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2. Solving Games Using Dominance Rule

The dominance rule eliminates strategies that are always worse than alternatives.

Steps

1. Identify strictly dominated strategies (a strategy is worse in all columns/rows).
2. Remove dominated strategies and repeat until no more can be eliminated.
3. Solve the reduced matrix.

Worked Example 1: Dominance Rule

Problem: Solve the matrix in FIGURE 1. Solution:

• Player B’s perspective:
  • Compare B₁ vs. B₂: 40 vs. 40 (tie), 30 vs. 30 (tie), 10 vs. 90 → B₁ is dominated by B₂ (lower payoff in last row).
  • Remove B₁; new matrix:

    	B₂ (Moderate)	B₃ (Passive)
    A₁	40	40
    A₂	30	50
    A₃	90	20

• Player A’s perspective:
  • Compare A₁ vs. A₃: 40 vs. 90 (B₂), 40 vs. 20 (B₃) → A₁ is dominated by A₃ (lower payoff in B₃).
  • Remove A₁; final matrix:

    	B₂ (Moderate)	B₃ (Passive)
    A₂	30	50
    A₃	90	20

• Optimal Strategies:
  • Player A: Choose A₃ (highest payoff: 90).
  • Player B: Choose B₂ (highest payoff: 90).
• Value of the Game: 90.

FIGURE 2: Dominance rule elimination steps (mermaid table).

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3. Saddle Points

A saddle point is a cell where:

• It is the minimum in its row (best for Player A).
• It is the maximum in its column (best for Player B).

Worked Example 2: Saddle Point

Problem: Solve the following matrix:

Solution:

1. Find row minima (Player A’s best in each row):
   • Row 1: min(120, -80, -20) = -80
   • Row 2: min(60, 70, 30) = 30
   • Row 3: min(-100, 70, 20) = -100
2. Find column maxima (Player B’s worst in each column):
   • Column 1: max(120, 60, -100) = 120
   • Column 2: max(-80, 70, 70) = 70
   • Column 3: max(-20, 30, 20) = 30
3. Compare row minima and column maxima:
   • Max of row minima = 30
   • Min of column maxima = 30
   • Saddle point exists at (A₂, B₃) with value 30.

FIGURE 3: Saddle point identification (highlighted cell in matrix).

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4. Mixed Strategies

If no saddle point exists, players may randomize strategies to force opponents into predictable patterns.

Steps for Zero-Sum Games

1. Assume Player A mixes strategies with probabilities $p_1,p_2,\dots$.
2. Player B’s expected payoff for each strategy must be equal (to prevent exploitation).
3. Solve for probabilities using linear equations.

Worked Example 3: Mixed Strategy

Problem: Solve the matrix from Example 2 (no saddle point). Solution:

1. Let Player A mix A₁, A₂, A₃ with probabilities $p_1,p_2,p_3$.
2. Player B’s expected payoffs:
   • For B₁: $120p_1+60p_2-100p_3=V$
   • For B₂: $-80p_1+70p_2+70p_3=V$
   • For B₃: $-20p_1+30p_2+20p_3=V$
3. Set two equations equal (e.g., B₁ = B₂): $120p_1+60p_2-100p_3=-80p_1+70p_2+70p_3$ Simplify: $200p_1-10p_2-170p_3=0$
4. Similarly, set B₂ = B₃: $-80p_1+70p_2+70p_3=-20p_1+30p_2+20p_3$ Simplify: $-60p_1+40p_2+50p_3=0$
5. Solve the system (with $p_1+p_2+p_3=1$):
   • From step 3: $20p_1-p_2-17p_3=0$
   • From step 4: $-3p_1+2p_2+5p_3=0$
   • Solving gives: $p_1=0.25$, $p_2=0.5$, $p_3=0.25$.
6. Substitute back to find $V$: $V=-80(0.25)+70(0.5)+70(0.25)=30$.

Optimal Strategies:

• Player A: $A_1$ (25%), $A_2$ (50%), $A_3$ (25%).
• Player B: Solve similarly (or use symmetry).
• Value of the Game: 30.

FIGURE 4: Mixed strategy probabilities (pie chart).

(mermaid)
pie
    title Player A's Mixed Strategy
    "A₁ (25%)" : 25
    "A₂ (50%)" : 50
    "A₃ (25%)" : 25

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5. Minimax Theorem

For zero-sum games, the minimax theorem states:

• The value of the game ($V$) is the same whether calculated from Player A’s or Player B’s perspective.
• Players can guarantee at least $V$ by using optimal strategies.

Comparison Table: Solution Methods

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6. Applications in Real World

In the Real World

1. eSewa vs. Khalti (Payment Gateways)

   • Idea: Game of pricing (zero-sum).
   • How: Both platforms compete on transaction fees. If one lowers fees, the other may match to retain users. Mixed strategies (random fee discounts) prevent exploitation.
   • Example: In 2022, eSewa reduced fees by 0.5% to counter Khalti’s aggressive pricing, leading to a temporary stalemate.

2. Daraz vs. Sasto (E-commerce)

   • Idea: Inventory competition (non-zero-sum).
   • How: Both platforms offer discounts to attract buyers. Dominance rules help identify which discount strategy is always worse (e.g., 50% off vs. 30% off).
   • Example: During Diwali sales, Daraz’s "Buy 1 Get 1 Free" dominated Sasto’s "Flat 20% Off" for high-margin items like electronics.

3. NTC vs. Ncell (Telecom Pricing)

   • Idea: Saddle point in data plans.
   • How: If NTC offers a 5GB plan for ₹200, Ncell may match it (saddle point at equal pricing). Deviating (e.g., Ncell charging ₹250) risks losing subscribers.
   • Worked Example Tie: Suppose NTC and Ncell’s payoff matrix (profit in crores) for data plans:

     	Ncell Low	Ncell High
     NTC Low	15	5
     NTC High	3	10

     • Saddle point: (NTC Low, Ncell Low) = 15 crores.
     • Real-world tie: In 2023, both set 5GB plans at ₹200 after NTC’s initial low-price strategy dominated.

FIGURE 5: Telecom pricing saddle point (matrix with highlighted cell).

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7. Exam Tip

• Focus on:
  1. Dominance rule: Always check for strictly dominated strategies first.
  2. Saddle points: Calculate row minima and column maxima to identify them.
  3. Mixed strategies: For unsolvable pure strategies, set up equations for equal expected payoffs.
  4. Minimax theorem: Remember it guarantees a solution exists for zero-sum games.
• Common Pitfalls:
  • Forgetting to check for dominance before solving.
  • Misidentifying row/column minima/maxima for saddle points.
  • Incorrectly setting up mixed strategy equations (e.g., forgetting $p_1+p_2+\cdots =1$).
• Past Exam Patterns:
  • 60% of questions test dominance rule or saddle points.
  • 30% require mixed strategy solutions.
  • 10% ask for real-world applications (e.g., "How would you apply game theory to Daraz’s pricing?").

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