Game Theory

1.2 Key concepts and terminology in game theory

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Game theory is all about strategic decision-making. It looks at how players interact, make choices, and deal with outcomes in various situations. Understanding key concepts like players, strategies, and payoffs is crucial for grasping the fundamentals of game theory.

This intro to game theory covers essential ideas like equilibrium, rationality, and different game types. We'll explore how games are represented using matrices and trees, giving you a solid foundation for analyzing strategic interactions in various contexts.

Key Concepts in Game Theory

Players, Strategies, and Payoffs

Players are the decision-makers in a game
Each player has a set of strategies or actions they can choose from
A strategy is a complete plan of action specifying what a player will do in every possible situation throughout the game
Payoffs are the outcomes or rewards that players receive based on the strategies chosen by all players
- Payoffs can be represented numerically (profits, points) or ordinally (rankings)

Equilibrium and Nash Equilibrium

An equilibrium is a stable state where no player has an incentive to unilaterally change their strategy, given the strategies of the other players
Nash equilibrium is a key concept in game theory
- Defined as a set of strategies, one for each player, such that no player has an incentive to change their strategy given what the other players are doing
- In a Nash equilibrium, each player's strategy is a best response to the strategies of the other players
Nash equilibrium can be pure (players choose a single strategy) or mixed (players randomize over multiple strategies)
- Example: In the classic Prisoner's Dilemma game, the Nash equilibrium is for both players to defect, even though mutual cooperation would yield higher payoffs

Rationality in Game Theory

Assumptions of Rationality

Rationality in game theory assumes that players are self-interested and aim to maximize their own payoffs
Players are assumed to have complete information about the game structure, available strategies, and potential payoffs for all players
Rational players are expected to make decisions based on logical reasoning and strategic thinking, considering the actions of other players
Common knowledge of rationality assumes that all players are rational, and all players know that all players are rational, and so on
- This assumption allows players to anticipate and reason about the actions of others

Bounded Rationality

Bounded rationality recognizes the limitations of human decision-making
Suggests that players may not always make optimal choices due to cognitive constraints, limited information, or computational complexity
Players may use heuristics or rules of thumb to simplify decision-making in complex situations
Satisficing is a decision-making strategy where players seek a satisfactory solution rather than an optimal one
- Example: In a complex business environment, managers may make decisions based on limited information and time constraints, rather than exhaustively analyzing all possible options

Game Types: Simultaneous vs Sequential

Simultaneous Games

Simultaneous games, also known as static games, are those in which players make their decisions simultaneously or without knowing the actions chosen by other players
In simultaneous games, players cannot observe the actions of others before making their own decisions
The outcome is determined by the combination of all players' choices
Simultaneous games are often represented using the normal form or payoff matrix
- Example: The classic game of Rock-Paper-Scissors is a simultaneous game, where both players choose their action at the same time

Sequential Games

Sequential games, also known as dynamic games, are those in which players make their decisions in a specific order
Some players observe the actions of others before making their own choices
The order of play and the information available to players at each decision point are crucial factors in determining the game's outcome
Perfect information games are sequential games in which all players have complete knowledge of the previous actions taken by other players at each decision point
Sequential games are often represented using the extensive form or game tree
- Example: Chess is a sequential game with perfect information, where players alternate turns and can observe all previous moves

Game Representation: Matrices and Trees

Normal Form Representation (Payoff Matrix)

Normal form representation, also known as the strategic form, represents a game using a matrix that shows the players, their strategies, and the corresponding payoffs for each combination of strategies
In a normal form matrix, each row represents a strategy for one player, and each column represents a strategy for the other player
The payoffs for each player are listed in each cell of the matrix, typically with the row player's payoff listed first
Normal form is useful for analyzing simultaneous games and identifying Nash equilibria
- Example: The Prisoner's Dilemma can be represented in normal form, with each player having two strategies (Cooperate or Defect) and the resulting payoffs for each combination of strategies

Extensive Form Representation (Game Tree)

Extensive form representation, also known as the game tree, represents a game as a tree-like structure that shows the sequence of moves, decision points, and payoffs
In an extensive form game tree, nodes represent decision points for players, edges represent actions or moves, and terminal nodes represent the outcomes and payoffs for each combination of strategies
Game trees allow for the representation of sequential moves, imperfect information, and subgame perfect equilibria
- Subgame perfect equilibria are Nash equilibria in every subgame of the extensive form game
Information sets in game trees represent situations where players have the same information and available actions
- Example: The Ultimatum Game can be represented as an extensive form game tree, with the proposer choosing an offer and the responder deciding whether to accept or reject the offer at each decision node

Key Terms to Review (13)

Pareto efficiency: Pareto efficiency refers to a situation in which resources are allocated in such a way that no individual can be made better off without making someone else worse off. It is a key concept in understanding optimal resource allocation and plays a significant role in various strategic interactions, showing how individuals or groups can reach outcomes where any change would harm at least one party involved.

Subgame Perfect Equilibria: Subgame perfect equilibria is a refinement of Nash equilibrium used in extensive-form games, ensuring that players' strategies constitute a Nash equilibrium in every subgame of the original game. This concept highlights the importance of making credible threats and promises, as it requires players to act rationally not just in the game as a whole, but also in every possible situation that could arise within the game. By focusing on the entire structure of the game, subgame perfect equilibria helps identify stable outcomes where players' strategies are optimal at every decision point.

Minimax theorem: The minimax theorem is a fundamental principle in game theory that states that in a zero-sum game, the optimal strategy for a player is to minimize the possible loss for a worst-case scenario. This theorem asserts that players can determine their optimal strategies by evaluating the minimum gain of the opponent and choosing their moves to maximize their own minimum payoff. This concept is closely linked to key ideas such as Nash equilibria and mixed strategies, providing a crucial framework for understanding competitive interactions in strategic settings.

Normal Form Representation: Normal form representation is a way to describe a game using a matrix or a table format that outlines the players, their strategies, and the resulting payoffs for each combination of strategies. This format makes it easier to analyze strategic interactions, as it clearly lays out all possible actions and outcomes for the players involved, highlighting their preferences and choices. It's particularly useful in identifying equilibrium concepts and understanding the dynamics of strategic decision-making.

Perfect Information Game: A perfect information game is a type of game where all players have complete knowledge of all past actions taken by every player throughout the game. This means that there are no hidden moves or private information, allowing players to make fully informed decisions based on the game's history. The concept is crucial in understanding strategic interactions and analyzing optimal strategies in various settings.

Sequential Game: A sequential game is a type of game in game theory where players make decisions one after another, rather than simultaneously. This structure allows players to observe the previous actions of others before making their own choices, which can significantly influence the strategies they adopt. The strategic implications of these games are often analyzed through extensive form representations, such as game trees, and can lead to concepts like subgame perfect equilibrium where players optimize their strategies at every possible decision point.

Simultaneous game: A simultaneous game is a type of game in which all players make their decisions or choose their strategies at the same time, without knowledge of the other players' choices. This setup creates a situation where players must consider the potential decisions of their opponents when formulating their own strategies, leading to unique strategic interactions and outcomes. Simultaneous games are crucial for understanding various concepts in game theory, especially when calculating equilibria like the mixed strategy Nash equilibrium.

Utility Function: A utility function is a mathematical representation that assigns a real number to each possible outcome or choice, reflecting the level of satisfaction or preference that an individual derives from that outcome. This concept is crucial in understanding decision-making processes and is linked to how players evaluate their options in strategic interactions, providing insights into their preferences, strategies, and behaviors.

Payoff Matrix: A payoff matrix is a table that shows the possible outcomes of a strategic interaction between players in a game, detailing their payoffs based on different strategies they may choose. This matrix is fundamental in understanding how players make decisions, as it encapsulates all possible strategies and their respective outcomes, leading to insights on rational choice and strategic behavior.

Nash equilibrium: Nash equilibrium is a concept in game theory where no player can benefit from changing their strategy while the other players keep theirs unchanged. This situation arises when each player's strategy is optimal given the strategies of all other players, leading to a stable state in strategic interactions.

Equilibrium: Equilibrium in game theory refers to a state where players have chosen strategies such that no player can benefit by unilaterally changing their strategy. This concept is crucial as it highlights the stability of strategic interactions, where players' choices are mutually reinforcing. It connects to other important concepts, showing how individuals or groups balance their decisions in competitive and cooperative settings, leading to outcomes that can be analyzed through different lenses, including evolutionary strategies and population dynamics.

Bounded rationality: Bounded rationality refers to the idea that individuals, when making decisions, are limited by their cognitive abilities, available information, and time constraints. This concept highlights that humans often rely on simplifying strategies or heuristics rather than fully rational approaches, leading to decisions that may not always align with traditional economic models of rational choice.

Rational Player: A rational player is an individual or entity in game theory who makes decisions by logically evaluating their choices to maximize their utility or payoff. This concept assumes that the player has clear preferences and will consistently choose options that lead to the best possible outcome based on their knowledge and the available information, connecting deeply with the strategic interactions in competitive scenarios.

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