# Game Theory Game theory studies strategic decision-making where outcomes depend on the actions of multiple rational players. It analyzes situations where each player's optimal choice depends on what they expect others to do. ### Information Structures in Games How information flows and what players know affects strategic outcomes: - **Perfect information:** All players know the complete history of the game - **Imperfect information:** Some information is private or unknown - **Complete information:** All players know the game structure and payoffs - **Incomplete information:** Uncertainty about game rules, payoffs, or player types Understanding information structures is crucial for predicting strategic behavior and designing mechanisms for desired outcomes. ### Common Knowledge vs. Mutual Knowledge #### Mutual Knowledge Information that all players know, but they may not know that others know it. #### Common Knowledge Information that all players know, all players know that all players know it, all players know that all players know that all players know it, and so on infinitely. **Why It Matters:** Common knowledge enables coordination that mutual knowledge cannot. Players can act on shared assumptions only when they're certain others share the same understanding. **Example:** In the consecutive numbers puzzle, when the clock strikes without anyone announcing, it transforms the mutual knowledge "neither player has number 1" into common knowledge, enabling further deductive reasoning. ### Signaling Through Inaction When choosing not to act conveys information to other players. Silence or inaction becomes a form of communication in strategic settings. **Mechanism:** Players observe what others don't do and infer information about their private knowledge or situation. **Example:** In consecutive numbers puzzle, each clock strike without an announcement signals "I don't have the lowest possible remaining number," progressively narrowing down possibilities.