Common Knowledge

Game Theory · Axiom Academy

Understanding the infinite regress of mutual awareness in strategic reasoning Everyone knows that everyone knows p Everyone knows that everyone knows that everyone knows p Common knowledge is more than just mutual knowledge (where everyone happens to know something). It requires that this mutual awareness extends to all levels - an infinite regress of "I know that you know that I know..." 2. Why Common Knowledge Matters Common knowledge is essential for coordination and strategic reasoning. Consider this scenario: In game theory, common knowledge of rationality, the game rules, and payoffs allows us to solve games using concepts like Nash equilibrium. Without it, players cannot reliably predict each other's behavior. 3. The Infinite Regress of Knowledge Let's formalize this. We use K i to denote "player i knows." For common knowledge, we need: Level 2: Both players know that both players know p Level 3: Both players know that both players know that both players know p And this pattern continues infinitely... 4. Famous Example: The Blue Eyes Puzzle Remarkably, even though everyone already knew there were people with blue eyes, this public announcement changes everything. After 100 days, all 100 blue-eyed people leave simultaneously!

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