Game Theory: Payoff Matrices, Optimal Strategies & Real-World Applications

TAKEAWAYS:

• Game theory models conflict and cooperation between rational decision-makers using payoff matrices and optimal strategies.
• Zero-sum games (one player’s gain = another’s loss) use maximin/minimax and saddle points to find optimal mixed strategies.
• Dominance rules simplify payoff matrices by eliminating dominated strategies (pure or mixed).
• Mixed strategies (probability distributions over actions) often yield better outcomes than pure strategies.
• Applications span business competition (market share), auctions, and negotiations (e.g., eSewa vs. Khalti fees).
• Exam focus: Solve 2×2 games, reduce matrices via dominance, and interpret optimal strategies/values.

1. Introduction to Game Theory

Game theory studies strategic interactions where players’ choices affect outcomes. Key assumptions:

• Players are rational (maximize their payoff).
• Players act simultaneously (unless sequential games are specified).
• Outcomes are deterministic (no randomness unless modeled explicitly).

Types of Games

A classic non-zero-sum game showing why cooperation may fail. (Image: Christopher X Jon Jensen (CXJJensen) & Greg Riestenberg, CC BY-SA 3.0, via Wikimedia Commons)

2. Payoff Matrices and Strategies

A payoff matrix represents players’ choices and outcomes. For two players (A and B), it’s structured as:

       B1       B2       B3
A1 [a11, b11] [a12, b12] [a13, b13]
A2 [a21, b21] [a22, b22] [a23, b23]

• Rows: Player A’s strategies.
• Columns: Player B’s strategies.
• Entries: (A’s payoff, B’s payoff).

Worked Example 1: Market Share Competition

Scenario: Two food manufacturers (Company A and B) compete for market share. The payoff matrix (in % share) is:

       | B: Low Price | B: High Price
A: Low | (30, 20)    | (25, 30)
A: High| (15, 35)    | (20, 25)

Question: Find optimal strategies using maximin/minimax.

Solution:

1. Player A’s perspective (maximin):

   • If B chooses Low: A’s payoffs = 30, 15 → min = 15 (worst case).
   • If B chooses High: A’s payoffs = 25, 20 → min = 20.
   • Maximin: Choose the maximum of minimums → High Price (20).

2. Player B’s perspective (minimax):

   • If A chooses Low: B’s payoffs = 20, 30 → max = 30 (worst case).
   • If A chooses High: B’s payoffs = 35, 25 → max = 35.
   • Minimax: Choose the minimum of maximums → Low Price (20).

Outcome: A plays High Price, B plays Low Price → Payoff = (15, 35). Saddle Point: No saddle point exists (no equilibrium where row min = column max).

3. Dominance Rules

Simplify matrices by eliminating dominated strategies (always worse than another option).

Types of Dominance

1. Pure Dominance:

   • A strategy is dominated if another strategy yields higher payoff for all opponent moves.
   • Example: In the matrix below, A2 is dominated by A1 (since 4 > 3 and 5 > 2).

   A1: [5, 4]
   A2: [3, 2]
   

2. Mixed Dominance:

   • A strategy is mixed-dominated if a probability mix of other strategies always yields a better payoff.

Worked Example 2: Reducing a 3×4 Game

Given Matrix (Player A vs. B):

       B1  B2  B3  B4
A1:    3   2   4   0
A2:    3   4   2   4
A3:    4   2   4   0
A4:    0   4   0   8

Steps:

1. Eliminate dominated rows:

   • Compare A1 and A3: A3 has higher payoffs in B1, B3 → A1 is dominated.
   • Compare A2 and A4: A4 has higher payoff in B4 → A2 is not dominated (since A4’s 0 < A2’s 4 in B1).
   • Reduced Matrix:

     A3: [4, 2, 4, 0]
     A4: [0, 4, 0, 8]
     

2. Eliminate dominated columns:

   • Compare B1 and B3: B3 has higher payoffs for A3 → B1 is dominated.
   • Final 2×2 Matrix:

     A3: [4, 0]
     A4: [0, 8]
     

4. Mixed Strategies and Optimal Solutions

When no pure strategy dominates, players use probability distributions over actions.

Steps to Solve a 2×2 Zero-Sum Game

1. Let Player A choose A1 with probability $p$ and A2 with $1-p$.
2. Let Player B choose B1 with probability $q$ and B2 with $1-q$.
3. Set expected payoffs equal to the value of the game (V):
   • For A: $p\cdot a_{11}+(1-p)\cdot a_{12}=V$
   • For B: $q\cdot a_{11}+(1-q)\cdot a_{21}=V$
4. Solve for $p,q,V$.

Worked Example 3: Optimal Strategies for a 2×2 Game

Matrix:

       B1  B2
A1:    3   1
A2:    0   4

Solution:

1. Define strategies:
   • A: $p$ for A1, $1-p$ for A2.
   • B: $q$ for B1, $1-q$ for B2.
2. Expected payoffs:
   • For A: $3p+1(1-p)=V$ → $2p+1=V$ (Equation 1).
   • For B: $3q+0(1-q)=V$ → $3q=V$ (Equation 2).
3. Solve:
   • From Equation 2: $q=V/3$.
   • Substitute into Equation 1: $2p+1=3(V/3)$ → $2p+1=V$.
   • At equilibrium, B is indifferent between B1 and B2: $3p+0(1-p)=1p+4(1-p)$ → $3p=1-3p$ → $p=1/4$.
   • Substitute $p=1/4$ into Equation 1: $V=2(1/4)+1=1.5$.
   • Then $q=1.5/3=0.5$.

Optimal Strategies:

• A: Play A1 with 25% probability, A2 with 75%.
• B: Play B1 with 50% probability, B2 with 50%.
• Value of the game (V): 1.5.

5. Saddle Points and Pure Strategy Equilibria

A saddle point is a payoff where:

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

Example:

       B1  B2
A1:    2   3
A2:    1   4

• Saddle point: 2 (A1, B1).
  • Row min for A1 = 2, Row min for A2 = 1 → 2 is max of mins.
  • Column max for B1 = 2, Column max for B2 = 4 → 2 is min of maxes.

If a saddle point exists, it is the optimal pure strategy.

6. Non-Zero-Sum Games: Nash Equilibrium

In non-zero-sum games, players may cooperate or compete. A Nash Equilibrium occurs when no player can benefit by unilaterally changing strategy.

stateDiagram-v2
  [*] --> A: Low
  [*] --> A: High
  A: Low --> B: Low : (3, 3)
  A: Low --> B: High : (0, 5)
  A: High --> B: Low : (5, 0)
  A: High --> B: High : (3, 3)
  note right of B: High
    Nash Equilibrium: (A: High, B: High)
    Both players earn 3, no incentive to deviate.

Example: Prisoner’s Dilemma

       | B: Cooperate | B: Defect
A: C   | (-1, -1)    | (-3, 0)
A: D   | (0, -3)     | (-2, -2)

• Nash Equilibrium: (Defect, Defect) → Both defect, even though (Cooperate, Cooperate) is better.

In the Real World

1. eSewa vs. Khalti (Payment Wars)

   • Game Type: Non-zero-sum (both can gain or lose users).
   • Strategies:
     • Discounts: eSewa offers 5% cashback; Khalti matches.
     • Partnerships: Both partner with Daraz for seamless checkout.
   • Outcome: Mixed strategies where neither dominates permanently.

2. Nepal Telecom (NTC) vs. Ncell (Pricing Wars)

   • Game Type: Zero-sum (market share gain = competitor’s loss).
   • Payoff Matrix:

     NTC: Low Price | Ncell: Low Price → (30% share, 20% share)
     NTC: Low Price | Ncell: High Price → (35% share, 15% share)
     

   • Optimal Strategy: NTC uses mixed pricing (sometimes discounts, sometimes premium bundles) to avoid pure dominance.

3. Daraz vs. Amazon Nepal (Inventory Games)

   • Game Type: Sequential (Daraz moves first with pricing; Amazon responds).
   • Application: Queuing theory (inventory management) + game theory (price wars).
   • Real Scenario: Daraz reduces prices → Amazon matches → Both lose profit but gain users.

Exam Tip

1. For payoff matrices:
   • Always check for dominance first to reduce the matrix.
   • If no saddle point, solve using mixed strategies (set expected payoffs equal).
2. Zero-sum vs. non-zero-sum:
   • Zero-sum: Use maximin/minimax or mixed strategies.
   • Non-zero-sum: Look for Nash Equilibrium.
3. Common Pitfalls:
   • Forgetting to label payoffs (A’s payoff first, then B’s).
   • Misapplying dominance (e.g., not checking all columns/rows).
   • Incorrectly setting up expected value equations.
4. Past Exam Patterns:
   • 2023 TU: Solve a 3×4 game using dominance → reduced to 2×2 → find mixed strategies.
   • 2022 PU: Define game theory, assumptions, and business importance (4 marks).
   • 2021 NEB: Reduce a game to 2×2 and find optimal strategies (10 marks).

Visual Summary of Key Concepts:

flowchart TD
    A["Game Theory"] --> B["Zero-Sum Games"]
    A --> C["Non-Zero-Sum Games"]
    B --> D["Payoff Matrix"]
    B --> E["Dominance Rules"]
    B --> F["Mixed Strategies"]
    B --> G["Saddle Point"]
    C --> H["Nash Equilibrium"]
    D --> I["Maximin/Minimax"]
    F --> J["Probability Distributions"]
    G --> K["Pure Strategy Equilibrium"]
    H --> L["Prisoner's Dilemma"]

In the real world

• eSewa vs. Khalti fee wars: This is a zero-sum game where each company (Player A and B) sets transaction fees (Low/High) to maximize their own revenue. The payoff matrix from Section 2 directly models their strategic choices, with optimal strategies derived via maximin/minimax rules. If one lowers fees (Low Price), the other may retaliate (High Price), leading to a Nash Equilibrium where both earn moderate profits (e.g., (20, 25) in the example).

• Nepali mobile operators (Ncell, NTC): The Prisoner’s Dilemma applies to their 4G/5G price wars. If both invest heavily (High Price), they lose customers to cheaper rivals (Low Price). The Nash Equilibrium here is mutual underinvestment (both choose Low Price), harming long-term network quality—a classic non-zero-sum outcome.

• NEPSE stock trading: Investors use mixed strategies (probability distributions over buy/sell actions) to hedge risks. For example, a trader might allocate 75% to stocks and 25% to bonds (like Player A’s optimal strategy in Worked Example 3) to balance gains and losses, mirroring game-theoretic optimal solutions.