Normal-Form Games

🎯 Learning Objectives

• Represent any static game of complete information using a normal-form payoff matrix
• Identify players, strategies, and payoffs from a game description
• Apply common knowledge of rationality assumptions
• Analyze classic games: Prisoner's Dilemma, Battle of the Sexes, Coordination, Chicken
• Distinguish between strict and weak dominance

📋 Prerequisites

• Basic algebra and set notation from high school mathematics
• From BSCS3003-AI Search: Minimax algorithm for zero-sum games provides intuition for strategic thinking

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1.1 What Is a Game?

1. Two or more players (decision-makers)
2. Each player chooses from a set of strategies $S_i$
3. The payoff for each player depends on the combination of strategies chosen

🔑 Key Insight: Your payoff depends not only on your own choice, but on others' choices too.

Formal Definition

• $N = \{1, 2, \dots, n\}$ — set of players
• $S_i$ — strategy set for player $i$
• $u_i: S_1 \times \cdots \times S_n \to \mathbb{R}$ — payoff function For two-player games, we use a payoff matrix:

$$\begin{array}{c|cc} & L & R \\ \hline U & (a_{11}, b_{11}) & (a_{12}, b_{12}) \\ D & (a_{21}, b_{21}) & (a_{22}, b_{22}) \end{array}$$

1.2 Rationality Assumptions

1.3 Classic Games

Prisoner's Dilemma

$$\begin{array}{c|cc} & \text{Silent} & \text{Confess} \\ \hline \text{Silent} & (-1, -1) & (-10, 0) \\ \text{Confess} & (0, -10) & (-5, -5) \end{array}$$

• Confess strictly dominates Silent for both
• Outcome $(\text{Confess}, \text{Confess})$ is worse for both than $(\text{Silent}, \text{Silent})$
• Paradox: Individual rationality → collective irrationality

Battle of the Sexes

$$\begin{array}{c|cc} & \text{Ballet} & \text{Football} \\ \hline \text{Ballet} & (2, 1) & (0, 0) \\ \text{Football} & (0, 0) & (1, 2) \end{array}$$

• Two Nash equilibria: (Ballet, Ballet) and (Football, Football)
• Coordination problem with distributional conflict

Chicken (Hawk-Dove)

$$\begin{array}{c|cc} & \text{Swerve} & \text{Stay} \\ \hline \text{Swerve} & (0, 0) & (-1, 1) \\ \text{Stay} & (1, -1) & (-5, -5) \end{array}$$

• Two asymmetric equilibria: (Swerve, Stay) and (Stay, Swerve)
• Models brinkmanship

1.4 Dominance

Strict Dominance

$$u_i(s_i, s_{-i}) > u_i(s_i', s_{-i}) \quad \forall s_{-i} \in S_{-i}$$

Weak Dominance

$$u_i(s_i, s_{-i}) \geq u_i(s_i', s_{-i}) \quad \forall s_{-i}$$

Iterated Elimination of Strictly Dominated Strategies (IESDS)

1.5 Best Response

$$u_i(s_i, s_{-i}) \geq u_i(s_i', s_{-i}) \quad \forall s_i' \in S_i$$

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📊 Formula Summary

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✅ Practice Questions

$$\begin{array}{c|cc} & H & L \\ \hline H & (4, 4) & (1, 6) \\ L & (6, 1) & (2, 2) \end{array}$$

Solution

For Firm 1: If P2 plays $H$, $u_1(L)=6 > u_1(H)=4$. If P2 plays $L$, $u_1(L)=2 > u_1(H)=1$. $L$ strictly dominates $H$. By symmetry, $L$ also dominates for Firm 2. This is a Prisoner's Dilemma. Q2: Find weakly dominated strategies:

$$\begin{array}{c|ccc} & A & B & C \\ \hline X & (5, 3) & (4, 2) & (3, 1) \\ Y & (3, 1) & (4, 4) & (5, 2) \\ Z & (2, 0) & (1, 5) & (4, 3) \end{array}$$

Solution

$Z$ is strictly dominated by $Y$ ($3>2, 4>1, 5>4$). $C$ is strictly dominated by $B$ ($2>1, 4>2, 5>3$). After eliminating $Z$ and $C$, no further dominated strategies remain. Q3: Find all best responses:

$$\begin{array}{c|cc} & L & R \\ \hline U & (10, 2) & (3, 4) \\ D & (5, 6) & (8, 1) \end{array}$$ </details> [Join Discord](https://discord.gg/gE2m4Qrdqv) [Next**Strategic Thinking**](/notes/04-degree-electives-bsms4023-game-theory-week01-01b-strategic-thinking)