Game Theory & Decision Making: Strategies, Payoffs & Nash Equilibrium

TAKEAWAYS:

• Model conflicts as payoff matrices to analyze strategic interactions between rational players.
• Compute optimal strategies using mixed strategies and expected payoffs for zero-sum and non-zero-sum games.
• Identify Nash equilibrium—stable outcomes where no player can benefit by unilaterally changing strategy.
• Apply minimax theorem to guarantee optimal play in zero-sum games under perfect information.
• Use decision trees to visualize sequential decisions and calculate expected values.
• Solve real-world problems in pricing wars (e.g., Daraz vs. Hamrobazaar), auctions (e.g., NEPSE IPOs), and negotiations (e.g., labor strikes).

1. Introduction to Game Theory

Game theory studies strategic interactions where the outcome for one player depends on the actions of others. It is used to model conflicts, cooperation, and decision-making in economics, politics, and computer science.

Key Definitions

• Player: A decision-maker (e.g., two food manufacturers competing for market share).
• Strategy: A complete plan of action (e.g., "Manufacturer A increases advertising").
• Payoff: The benefit/reward a player receives (e.g., market share gain).
• Game: A structured interaction with rules, strategies, and payoffs (e.g., Prisoner’s Dilemma).

Types of Games

A labelled payoff matrix for the Prisoner’s Dilemma with payoffs in years of imprisonment. (Image: נעמ״ה, CC0, via Wikimedia Commons)

2. Payoff Matrices and Strategies

A payoff matrix represents all possible outcomes of a game. Each cell shows the payoffs for both players based on their strategies.

Example: Food Manufacturer Dilemma

Two manufacturers, A and B, compete for market share. Their strategies and payoffs (in % market share gain) are:

Question: What is the Nash Equilibrium? Solution:

1. A’s best response to B’s "Increase Ads" → A prefers "Increase Ads" (3 > 1).
2. A’s best response to B’s "Keep Ads Same" → A prefers "Increase Ads" (5 > 4).
3. B’s best response to A’s "Increase Ads" → B prefers "Increase Ads" (3 > 1).
4. B’s best response to A’s "Keep Ads Same" → B prefers "Increase Ads" (5 > 4).
5. Nash Equilibrium: Both increase ads → (Increase Ads, Increase Ads) with payoff (3, 3).

Why? Neither can improve their payoff by unilaterally changing strategy.

3. Mixed Strategies and Expected Payoffs

When pure strategies lead to unstable outcomes, players randomize their choices using probabilities.

Example: Matching Pennies (Zero-Sum Game)

Two players simultaneously choose Heads (H) or Tails (T). Payoffs:

• If both choose the same → Player 1 loses 1, Player 2 gains 1.
• If different → Player 1 gains 1, Player 2 loses 1.

Payoff Matrix:

Solution:

1. Let A choose H with probability p and T with 1-p.
2. Let B choose H with probability q and T with 1-q.
3. A’s expected payoff (E_A):
   • If B plays H: $E_A=-p+(1-p)=1-2p$
   • If B plays T: $E_A=p-(1-p)=2p-1$
4. For B’s optimal strategy, set $E_A=0$: $(1-2p)q+(2p-1)(1-q)=0$ Solving gives $p=0.5$, $q=0.5$.

Optimal Strategy: Both players randomize 50-50 between H and T.

4. Minimax Theorem (Zero-Sum Games)

In zero-sum games, the minimax theorem states that:

There exists a mixed strategy for each player that guarantees at least the value of the game, regardless of the opponent’s strategy.

Example: Rock-Paper-Scissors

Payoff matrix (Player 1’s payoff):

Solution:

1. A’s optimal strategy: Play each move with probability $\frac{1}{3}$.
2. Expected payoff: 0 (fair game).
3. Minimax value: 0 (no player can guarantee a positive payoff).

A 3x3 matrix with payoffs and optimal mixed strategy probabilities. (Image: HowieKor, CC BY-SA 3.0, via Wikimedia Commons)

5. Decision Trees for Sequential Games

When players move sequentially (e.g., auctions, negotiations), use decision trees to model choices.

Example: Auction with Two Bidders

Two bidders, A and B, value an item at $100 and $80, respectively. They bid sequentially.

Decision Tree:

graph TD
    A["A bids"] -->|"90"| B1["B sees $90"]
    A -->|"100"| B2["B sees $100"]
    B1 -->|"Bid 85"| Outcome1["A wins, pays $90"]
    B1 -->|"Fold"| Outcome2["A wins, pays $90"]
    B2 -->|"Bid 95"| Outcome3["B wins, pays $100"]
    B2 -->|"Fold"| Outcome4["A wins, pays $100"]

Optimal Strategy:

1. A bids $90 (maximizes expected payoff).
2. B folds if A bids ≤ $80, else bids $85.
3. Expected payoff for A: $0.5\times 10+0.5\times 90=50$.

6. Real-World Applications

A. Pricing Wars (Daraz vs. Hamrobazaar)

• Game: Two e-commerce platforms compete on prices.
• Payoff Matrix:

  	Hamrobazaar: Low Price	Hamrobazaar: High Price
  Daraz: Low Price	(50, 50)	(70, 30)
  Daraz: High Price	(30, 70)	(60, 60)

• Nash Equilibrium: Both set low prices (50, 50).

B. Labor Strikes (NTC vs. Unions)

• Game: NTC (employer) vs. unions (employees).
• Payoffs:
  • If both cooperate → stable wages.
  • If either strikes → disruptions and losses.
• Outcome: Often leads to mixed strategies (partial strikes, negotiations).

C. NEPSE IPO Allocations

• Game: Investors bid for IPO shares.
• Strategy: Use sealed-bid auctions (like Vickrey auctions) to maximize revenue.
• Payoff: Higher bids may win, but overbidding risks losing money.

7. Exam Tips

1. Memorize Nash Equilibrium Definition:

   "A strategy profile where no player can improve their payoff by unilaterally changing strategy."

2. Practice Payoff Matrix Problems:
   • Always check best responses for each player.
   • For mixed strategies, set expected payoffs to zero.
3. Decision Trees:
   • Draw backwards (from end to start).
   • Calculate expected values at each node.
4. Minimax Theorem:
   • Applies only to zero-sum games.
   • Guarantees a saddle point in pure strategies.
5. Real-World Linkages:
   • Relate to auctions (NEPSE), pricing wars (Daraz), and negotiations (labor strikes).
6. Common Mistakes:
   • Ignoring mixed strategies when pure strategies are unstable.
   • Misidentifying dominant strategies (check all scenarios).

Final Note: Game theory is about predicting rational behavior. Master payoff matrices, Nash equilibrium, and decision trees to ace this unit! 🚀