Mathematics Game Theory is a mathematical framework for analyzing strategic interactions among rational decision-makers. It involves the study of various games and scenarios where individuals choose actions that maximize their payoffs, often in the presence of competing players. Game Theory encompasses concepts such as Nash equilibrium, dominant strategies, and zero-sum games, applying to diverse fields such as economics, political science, psychology, and computer science. By modeling conflicts and cooperation, Game Theory provides insights into optimal decision-making in competitive environments.
Basic Concepts of Game Theory Strategic Games Definition of Strategic Games Explanation of what constitutes a strategic game Differences between games played once versus iterated strategic games Players Role and definition of players in a game Number of players in strategic games Nature and attributes affecting players' decisions Strategies Pure Strategies Definition and examples Implications of choosing a pure strategy Mixed Strategies Definition and scenarios for employing mixed strategies Benefits and potential outcomes from mixed strategies Dominant and Dominated Strategies Definition and identification of dominant strategies Concept of domination in strategic games Impact of dominant strategies on game outcomes Payoffs Definition of payoffs in the context of strategic games Utility representation of payoffs Payoff structure and its influence on player strategies Examples of typical payoff matrices Extensive Games Game Trees Definition and components of a game tree How game trees represent sequential decision-making Interpretation of nodes, edges, and terminal nodes Histories Definition and role of histories in extensive games Tracking of sequences of actions Use of histories to determine available strategies Information Sets Definition of information sets and their significance Representation of players' knowledge in the game Differences between perfect and imperfect information sets Preferences and Utility Definition of preferences in the context of game theory Representation of utility functions Concepts of ordinal versus cardinal utility How preferences influence strategy selection Utility maximization as a goal of rational players Rationality Definition of rationality in game theory Rationality assumptions for players Consequences of rational behavior in games Common Knowledge of Rationality Explanation of common knowledge and its relevance in games Role of mutual beliefs and rationality in strategic thinking Impact of common knowledge on equilibrium and outcomes