9.1 Basic concepts of game theory
4 min read•july 30, 2024
Game theory is all about strategic decision-making. It's like playing chess, but with real-life situations. You'll learn how people and businesses make choices when their actions affect each other's outcomes.
This topic covers the building blocks of game theory. You'll explore different types of games, how to represent them, and key concepts like . It's crucial for understanding how people interact strategically in various scenarios.
Game Theory Elements
Players, Strategies, and Payoffs
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- Players act as decision-makers in games with unique action sets and preferences
- Strategies outline complete action plans for players in all possible game situations
- Payoffs represent outcomes or utilities players receive based on chosen strategy combinations
- (normal form) displays strategy combinations and payoffs in a matrix
- illustrates game sequences and player information as a tree-like structure
- determine player knowledge about others' actions during decision-making
- consistently outperforms other strategies regardless of opponents' choices
- Example: In the , confessing dominates staying silent for both players
- Example: In a Cournot duopoly model, firms may have dominant strategies for production levels
Game Representations and Analysis
- Strategic form suits simultaneous decision games (rock-paper-scissors)
- Extensive form accommodates sequential decision games (chess)
- Perfect information games provide complete history to all players (chess)
- Imperfect information games involve uncertainty about game aspects (poker)
- Mixed strategies involve randomized choices based on probability distributions
- Example: In matching pennies, players might choose heads or tails with equal probability
- refines Nash equilibrium for sequential games
- solves sequential games by working from end to beginning
- Example: In a two-stage bargaining game, players use backward induction to determine optimal offers
Simultaneous vs Sequential Games
Simultaneous Games
- Players make decisions without knowledge of others' choices
- Often represented in strategic form as a
- Utilize mixed strategies for randomized decision-making
- Example: In rock-paper-scissors, players might randomize their choices to avoid being predictable
- Nash equilibrium serves as a key solution concept
- Example: In a Cournot duopoly, firms simultaneously choose production levels, reaching a Nash equilibrium
Sequential Games
- Players make decisions in a specific order with some information about earlier choices
- Typically represented in extensive form as a game tree
- Introduce subgame perfect equilibrium concept
- Employ backward induction for solution finding
- Example: In the centipede game, players use backward induction to determine optimal stopping points
- Perfect information games reveal complete history at each decision point (chess)
- Imperfect information games involve uncertainty about some game aspects (poker)
Cooperative vs Non-Cooperative Games
Cooperative Games
- Allow binding agreements and coalition formation among players
- Focus on payoff distribution among coalition members
- Utilize the as a solution concept for stable allocations
- Example: In a three-player voting game, the core represents allocations that no two-player coalition can improve upon
- Employ for fair surplus allocation
- Example: In a joint venture, Shapley value determines each partner's contribution to the overall profit
- Classify as or games
- Address bargaining problems modeling negotiations over surplus division
- Example: for two players dividing a fixed amount of money
Non-Cooperative Games
- Assume players cannot form enforceable agreements outside game model
- Emphasize strategic decision-making among players
- Rely on Nash equilibrium and its refinements as solution concepts
- Example: In the game, multiple Nash equilibria may exist
- Model competitive situations without explicit cooperation
- Example: Cournot and Bertrand models of oligopolistic competition
- Analyze strategic interactions in various economic and social contexts
- Example: Analyzing firms' decisions to enter or exit a market using game-theoretic models
Rationality in Game Theory
Rational Decision-Making
- Assumes players maximize payoffs or utilities based on beliefs about others' actions
- Incorporates common knowledge of rationality (CKR) concept
- Utilizes best response strategy optimizing outcomes given beliefs about others
- Example: In a Cournot duopoly, each firm's output decision is a best response to the other's expected output
- Forms consistent beliefs about other players' strategies
- Applies concept in games
- Example: In a first-price sealed-bid auction, bidders form beliefs about others' valuations
Bounded Rationality and Alternative Approaches
- Recognizes limitations in cognitive abilities and information availability
- Considers suboptimal decision-making due to constraints
- Example: In complex games, players might use heuristics or rules of thumb instead of full optimization
- Employs to test equilibria stability
- Example: In the centipede game, trembling hand perfection refines Nash equilibrium predictions
- Explores challenging traditional rationality assumptions
- Example: Analyzing the evolution of cooperation in repeated Prisoner's Dilemma games
- Models strategy adoption through natural selection or social learning
- Example: Studying the emergence of social norms using evolutionary game theory models
Key Terms to Review (30)
Backward induction: Backward induction is a method used in game theory to determine optimal strategies by analyzing a game from its end back to the beginning. This approach involves reasoning backwards through the potential decisions and outcomes, allowing players to make informed choices based on future consequences. It is especially important in games where players make moves sequentially, as it helps predict the actions of others based on their potential responses.
Battle of the sexes: The battle of the sexes is a classic game theory scenario that illustrates a conflict between two players who have different preferences over two outcomes. In this game, both players want to coordinate on an outcome but prefer different ones, creating a situation where compromise is needed. This scenario highlights the strategic decision-making involved when two parties have competing interests, making it a key concept in understanding cooperation and negotiation.
Bayesian Equilibrium: Bayesian equilibrium is a concept in game theory where players make decisions based on their beliefs about other players' types, which are private information. This equilibrium reflects the strategies that players adopt in response to their expectations of others' actions, allowing them to maximize their payoffs considering these beliefs. It combines elements of Bayesian probability with equilibrium strategies, highlighting how incomplete information affects decision-making in games.
Cooperative game: A cooperative game is a type of game in game theory where players can negotiate and form binding commitments to work together to achieve mutual benefits. This contrasts with non-cooperative games, where players make decisions independently. Cooperative games focus on the collective strategies of the players, emphasizing collaboration and shared payoffs rather than individual actions.
Core: In game theory, the core refers to a set of feasible allocations that cannot be improved upon by any subset of players through their collective action. It represents outcomes where no group of players can gain by breaking away from the proposed allocation, highlighting stability in cooperative settings. The core connects with concepts like fairness and efficiency, as it seeks to ensure that resources are distributed in a way that all participants feel satisfied and no one has an incentive to deviate.
Dominant Strategy: A dominant strategy is a decision-making rule in game theory where a player consistently chooses the same course of action regardless of what the other players decide. This strategy stands out because it provides a better outcome for the player than any other strategies available, no matter what others do. It connects to various concepts such as competition in markets, decision-making processes, and strategic interactions among firms.
Evolutionary game theory: Evolutionary game theory is a branch of game theory that studies strategic interactions where the success of strategies is determined by their evolutionary stability and adaptability over time. It combines principles of traditional game theory with concepts from evolutionary biology, focusing on how organisms evolve their behaviors and strategies based on interactions with others in a population.
Extensive form: Extensive form is a way of representing games that captures the sequence of players' moves, the choices available to them, and the potential outcomes of those choices. It uses a tree structure to illustrate how decisions unfold over time, making it particularly useful for analyzing situations where timing and order of moves matter, such as sequential games. This representation helps in understanding the strategic interactions between players by highlighting their decision points and possible paths.
Incomplete information: Incomplete information refers to a situation in game theory where players do not have perfect knowledge about the game environment, including the strategies, payoffs, or types of other players. This lack of information can lead to uncertainty in decision-making, as players must make assumptions or estimates based on their limited knowledge, influencing their strategies and the overall outcome of the game.
Information Sets: Information sets refer to the collection of all possible states of the game that a player cannot distinguish between at a given point in time. In games where players make decisions sequentially, these sets are crucial as they help determine what strategies players can employ based on their knowledge or lack thereof about the actions taken by others. Understanding information sets is key to analyzing strategic interactions, particularly in situations where one player's knowledge influences the choices available to others.
John Nash: John Nash was an American mathematician renowned for his contributions to game theory, particularly through the concept of Nash equilibrium. His work laid the foundation for understanding strategic interactions in competitive situations, influencing various fields such as economics, biology, and political science. Nash's insights into how individuals make decisions based on the actions of others have shaped our understanding of both cooperative and non-cooperative games.
John von Neumann: John von Neumann was a Hungarian-American mathematician and polymath who made significant contributions to various fields, including game theory, computer science, and economics. He is best known for his groundbreaking work in game theory, particularly for establishing the mathematical framework that underpins strategic interactions among rational decision-makers. His ideas paved the way for understanding concepts like Nash equilibrium and dominant strategies, making him a central figure in economic and strategic thought.
Market entry strategy: A market entry strategy is a plan of action that a company uses to start selling its products or services in a new market. This involves assessing the competitive landscape, understanding customer needs, and determining the best approach to enter the market effectively. It includes various methods such as exporting, franchising, joint ventures, or establishing wholly-owned subsidiaries, all of which can be analyzed using game theory principles to anticipate competitor reactions and optimize decision-making.
Mixed strategy: A mixed strategy is a decision-making approach in game theory where a player randomly chooses among available actions according to a specific probability distribution. This strategy is particularly useful when players want to keep their opponents uncertain about their choices, thereby preventing predictability and allowing for a more strategic advantage. In competitive scenarios, employing a mixed strategy can be essential for achieving optimal outcomes when pure strategies do not yield favorable results.
Nash Bargaining Solution: The Nash Bargaining Solution is a solution concept in game theory that describes how two parties can negotiate an agreement that maximizes their collective benefit while ensuring that each party receives at least a minimum payoff. It builds on the idea of cooperative game theory, focusing on how players can reach an efficient outcome by considering both their individual preferences and the potential gains from cooperation. This solution emphasizes fairness and balance in negotiations, making it crucial for understanding strategic interactions in various economic and social contexts.
Nash equilibrium: Nash equilibrium is a situation in a game where no player can benefit by changing their strategy while the other players keep theirs unchanged. This concept illustrates the balance of strategies among players, highlighting how individuals or firms make decisions based on the expected actions of others. The notion of Nash equilibrium connects to various strategic behaviors and interactions in competitive environments, especially in scenarios where players must consider the choices of others in their decision-making processes.
Non-Transferable Utility (NTU): Non-transferable utility refers to a situation in game theory where the benefits or payoffs from a particular strategy cannot be transferred between players. This concept is crucial in understanding cooperative games, as it implies that players cannot simply trade or share their utilities with one another, leading to unique dynamics in negotiations and collective decision-making processes. NTU challenges the traditional notions of utility in economic models, highlighting the complexities of individual preferences and the interactions among players.
Pareto Efficiency: Pareto efficiency refers to an economic state where resources are allocated in the most efficient manner, such that no individual's situation can be improved without making someone else's situation worse. This concept emphasizes the optimal distribution of resources, highlighting that once a Pareto-efficient outcome is reached, any further changes would require a trade-off that negatively impacts at least one party. Understanding Pareto efficiency is essential when analyzing market dynamics, pricing strategies, competitive interactions, and the management of public resources and externalities.
Payoff matrix: A payoff matrix is a table that outlines the potential outcomes for each participant in a game, depending on the strategies they choose. It visually represents how different strategies yield different payoffs for each player, helping to identify optimal choices and the interdependence of decisions. This tool is essential in understanding strategic interactions and evaluating the consequences of various actions within competitive settings.
Pricing Strategy: Pricing strategy refers to the method companies use to price their products or services in order to maximize profits, attract customers, and remain competitive in the market. This approach considers factors like demand elasticity, consumer income levels, and competitive pricing, which can significantly influence business decisions and overall market dynamics.
Prisoner's dilemma: The prisoner's dilemma is a fundamental concept in game theory that illustrates a situation where two individuals, acting in their own self-interest, may not cooperate even if it appears that it is in their best interest to do so. This scenario demonstrates how rational decision-making can lead to suboptimal outcomes when individuals are unable to communicate and trust each other, highlighting the tension between individual incentives and collective benefit.
Pure strategy: A pure strategy is a specific and unchanging plan of action that a player follows in a game, where they consistently choose the same option in a given situation. This approach contrasts with mixed strategies, where players randomize their choices. In game theory, employing a pure strategy allows players to have a predictable and defined response to various scenarios, thus influencing the outcomes based on their decisions.
Shapley Value: The Shapley Value is a concept in cooperative game theory that provides a way to fairly distribute the total gains or costs among participants based on their individual contributions. This value allocates a unique payoff to each player, considering the different ways they can collaborate with others and how their contributions change the outcome. It emphasizes fairness in resource allocation, making it crucial for understanding collective decision-making processes and strategies.
Signaling: Signaling refers to actions taken by informed parties to reveal private information to uninformed parties in order to reduce information asymmetry in various markets. This concept is crucial for understanding interactions where one party possesses more or better information than the other, influencing decisions and outcomes in situations like hiring, insurance, and product quality. Effective signaling can lead to better decisions, mitigate issues like adverse selection, and enhance overall market efficiency.
Strategic form: Strategic form is a representation of a game that captures the players, their available strategies, and the payoffs for each combination of strategies chosen. This format allows players to analyze their options and make decisions based on their expectations of other players' actions. It emphasizes the importance of strategy in determining outcomes and is fundamental in understanding interactions between rational decision-makers.
Subgame perfect equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable to dynamic games, where players' strategies are optimal at every point in the game. It ensures that players' decisions are rational not just at the start of the game but at every possible stage, taking into account all possible future actions. This concept is closely related to how strategic interactions unfold over time and plays a crucial role in analyzing decision-making processes in competitive environments.
Transferable Utility (TU): Transferable utility (TU) refers to a situation in cooperative game theory where the utility or payoff of one player can be transferred to another player without loss. This concept allows for the redistribution of resources or benefits among players, facilitating cooperation and negotiation to achieve mutually beneficial outcomes. TU is essential in understanding how players can form coalitions and share the total value created in a game, enabling more efficient solutions.
Trembling Hand Perfect Equilibrium: Trembling hand perfect equilibrium is a refinement of Nash equilibrium that considers the possibility of players making mistakes in their decision-making. This concept acknowledges that players may sometimes choose strategies they did not intend to, which creates a more robust equilibrium by eliminating strategies that would be irrational in light of potential errors. It connects to the broader understanding of game theory by addressing how players might behave in real-world situations where mistakes are possible.
Utility Function: A utility function is a mathematical representation that assigns a numerical value to the level of satisfaction or happiness a consumer derives from consuming goods and services. It captures preferences and helps explain how consumers make choices based on maximizing their utility. Understanding utility functions is essential for analyzing consumer behavior and the decision-making process in various economic scenarios.
Zero-sum game: A zero-sum game is a situation in game theory where one participant's gain or loss is exactly balanced by the losses or gains of other participants. This means that the total utility available in the game remains constant, and the net change in wealth or benefit is zero. In a zero-sum context, players are essentially in direct competition, as any advantage gained by one player is an equivalent disadvantage to another, highlighting the competitive nature of interactions in strategic scenarios.