Normal Form Games

Game Theory · Axiom Academy

Unit 1 - Introduction to Game Theory 1. Formal Definition of a Normal Form Game A normal form game is a mathematical structure that captures strategic interactions between decision-makers. It consists of three essential components that completely specify the game. N = Set of players (e.g., 1, 2, ..., n ) S = Strategy sets S₁, S₂, ..., Sₙ (one for each player) u = Payoff functions u₁, u₂, ..., uₙ (one for each player) The payoff function uᵢ maps each strategy profile (s₁, s₂, ..., sₙ) to a real number representing player i's payoff. 2. Understanding the Components Let's break down each component with a concrete example. Consider a game with two players choosing between two strategies each. Players (N): Player 1 and Player 2 S₁ = Top, Bottom - Player 1's available strategies S₂ = Left, Right - Player 2's available strategies Payoff Functions: u₁ and u₂ assign payoffs to each of the 4 possible outcomes (Top,Left), (Top,Right), (Bottom,Left), (Bottom,Right). For 2-player games, we represent the game as a matrix where: Rows represent Player 1's strategies Columns represent Player 2's strategies Each cell contains a payoff pair (u₁, u₂) Watch as we construct the matrix step by step, placing each strategy and its corresponding payoffs. Understanding the convention for reading payoff matrices is crucial: The first number in each cell is the row player's (Player 1's) payoff The second number is the column player's (Player 2's) payoff

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