1.2 Key concepts: players, strategies, payoffs, and rationality

Game theory explores how players make strategic decisions. It looks at the choices available, the outcomes they lead to, and how players try to maximize their gains. Understanding these key concepts is crucial for analyzing real-world situations.

Players, strategies, payoffs, and rationality form the foundation of game theory. By examining how these elements interact, we can predict behavior, identify optimal choices, and understand the dynamics of competition and cooperation in various scenarios.

Game Participants

Players and Strategies

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• Players are the decision-makers in a game who choose from a set of available actions or strategies
• Strategies represent the complete plan of action for a player, specifying what action to take in every possible situation throughout the game
• Players can have pure strategies, which involve choosing a single action with certainty, or mixed strategies, which assign probabilities to each available action
• The set of all possible strategies for a player is called their strategy space

Decision-Making Process

• Players make decisions based on their preferences, beliefs, and available information
• The decision-making process involves evaluating the potential outcomes and payoffs associated with each strategy
• Players may engage in strategic thinking, considering the possible actions of other players and how they might respond to their own choices
• In simultaneous games, players make decisions independently without knowing the choices of others, while in sequential games, players make decisions in a specific order, potentially observing the actions of previous players

Outcomes and Incentives

Payoffs and Utility

• Payoffs represent the outcomes or rewards that players receive based on the combination of strategies chosen by all players in the game
• Payoffs can be monetary, such as profits or losses, or non-monetary, such as satisfaction or prestige
• Utility is a measure of a player's preferences over different outcomes, assigning a numerical value to each possible payoff
• Players aim to maximize their expected utility, which is the average utility weighted by the probabilities of each outcome occurring

Optimal Choices and Equilibrium

• An optimal choice for a player is the strategy that maximizes their expected utility given the strategies of other players
• Nash equilibrium is a key concept in game theory, representing a situation where each player's strategy is optimal given the strategies of the other players
• In a Nash equilibrium, no player has an incentive to unilaterally change their strategy, as doing so would not improve their payoff
• Dominant strategies are those that provide the highest payoff for a player regardless of the strategies chosen by other players (Prisoner's Dilemma)

Assumptions

Rationality and Decision-Making

• Game theory typically assumes that players are rational, meaning they make decisions based on maximizing their own utility or payoffs
• Rational players are assumed to have consistent preferences, use all available information, and make logical choices to achieve their goals
• Bounded rationality acknowledges that players may have cognitive limitations and may not always make perfectly rational decisions
• Common knowledge of rationality assumes that all players know that the other players are rational and that this knowledge is shared among all players

Information and Uncertainty

• The information set refers to the knowledge and beliefs that players have about the game, including the strategies available to themselves and other players, payoffs, and any prior actions taken
• Games can have perfect information, where players have complete knowledge of all previous actions and the payoff structure, or imperfect information, where some aspects are unknown or uncertain
• Incomplete information arises when players are uncertain about some characteristics of the other players or the game itself, such as their preferences or payoffs (Auction theory)
• Asymmetric information occurs when some players have more or better information than others, leading to strategic advantages or disadvantages (Principal-agent problem)