6.2 Pure and mixed strategies
7 min read•august 21, 2024
Game theory provides a mathematical framework for analyzing strategic decision-making in competitive environments. It's crucial for understanding economic interactions, applying to scenarios like market competition, resource allocation, and policy decisions.
This section explores pure and mixed strategies in game theory. Pure strategies involve players choosing a single action with certainty, while mixed strategies involve randomizing choices according to probability distributions. Understanding both is essential for solving various economic games and predicting outcomes.
Fundamentals of game theory
- Game theory provides a mathematical framework for analyzing strategic decision-making in competitive environments, crucial for understanding economic interactions
- Applies to various economic scenarios including market competition, resource allocation, and policy decisions
- Helps predict outcomes and optimal strategies when multiple rational actors interact
Types of games
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- Simultaneous games involve players making decisions concurrently without knowledge of others' choices
- Sequential games feature players taking turns, with later players having information about earlier moves
- Cooperative games allow players to form binding agreements, while non-cooperative games do not
- Zero-sum games result in one player's gain exactly balancing another's loss (poker)
- Positive-sum games allow for mutual benefit (trade negotiations)
Players and payoffs
- Players represent decision-makers in the game, often firms, consumers, or governments in economic contexts
- Payoffs quantify the outcomes or utilities players receive based on the combination of strategies chosen
- Utility functions translate game outcomes into numerical values reflecting players' preferences
- Payoff matrices display all possible strategy combinations and resulting payoffs for each player
- Risk attitudes affect how players value uncertain payoffs (risk-averse, risk-neutral, risk-seeking)
Nash equilibrium concept
- Represents a stable state where no player can unilaterally improve their payoff by changing strategy
- Occurs when each player's strategy is the to others' strategies
- Multiple Nash equilibria can exist in a single game, leading to coordination problems
- Not always socially optimal, as seen in the where individual rationality leads to suboptimal collective outcomes
- Serves as a key solution concept in non-cooperative game theory, predicting likely outcomes of strategic interactions
Pure strategies
- Pure strategies involve players choosing a single action with certainty, fundamental to understanding basic game theory concepts
- Represent the simplest form of strategic decision-making in games, often used as a starting point for more complex analyses
- Critical for identifying dominant strategies and initial equilibrium points in economic models
Definition and characteristics
- Deterministic choices made by players, selecting one specific action from their available set
- Represented as distinct points in a player's strategy space
- Always yield the same payoff against a given opponent strategy
- Easier to analyze and interpret compared to mixed strategies
- Often insufficient to solve games with no
Dominant strategies
- Strategy that yields the highest payoff regardless of opponents' choices
- Strictly outperforms all other strategies against any opponent strategy
- Weakly dominant strategy performs at least as well as any other strategy, sometimes better
- Presence of dominant strategies simplifies game analysis and decision-making
- Rational players are expected to always choose dominant strategies when available
Best response functions
- Maps each possible strategy of opponents to the optimal strategy for a given player
- Crucial for finding Nash equilibria in games without dominant strategies
- Can be represented graphically for two-player games with continuous strategy spaces
- Intersection points of best response functions indicate pure strategy Nash equilibria
- Used in iterative processes to eliminate dominated strategies and solve complex games
Mixed strategies
- Involve players randomizing their choices according to probability distributions, expanding the strategy space
- Essential for solving games without pure strategy equilibria, common in many economic scenarios
- Provide a more nuanced approach to strategic decision-making, reflecting real-world uncertainty
Probability distributions over actions
- Assign probabilities to each pure strategy in a player's action set
- Sum of probabilities across all actions must equal 1
- Allow for a continuous range of strategies between pure strategy choices
- Represent strategic uncertainty or deliberate randomization by players
- Can be visualized as points in a simplex for games with finite strategy sets
Expected payoffs
- Calculate average payoff a player receives from a profile
- Computed by summing the products of pure strategy payoffs and their respective probabilities
- Used to compare different mixed strategies and determine optimal choices
- Affected by players' risk attitudes, particularly in games with uncertain outcomes
- Key to finding mixed strategy Nash equilibria through payoff equalization
Indifference principle
- States that in a mixed strategy equilibrium, a player must be indifferent between all pure strategies with non-zero probabilities
- Crucial for calculating mixed strategy equilibria in two-player games
- Leads to a system of equations that can be solved to find equilibrium probabilities
- Ensures that no player has an incentive to deviate from their mixed strategy
- Applies only to the strategies played with positive probability in the equilibrium
Pure vs mixed strategy equilibria
- Compares the two main types of Nash equilibria in game theory, essential for comprehensive economic analysis
- Understanding the differences helps in selecting appropriate solution concepts for various economic models
- Highlights the importance of considering both deterministic and probabilistic strategies in strategic interactions
Existence and uniqueness
- Pure strategy equilibria may not exist in all games, while mixed strategy equilibria always exist in finite games (Nash's theorem)
- Games can have multiple pure strategy equilibria, a single mixed strategy equilibrium, or a combination of both
- Coordination games often feature multiple pure strategy equilibria, requiring additional selection criteria
- Some games (rock-paper-scissors) have a unique mixed strategy equilibrium but no pure strategy equilibrium
- Existence of multiple equilibria can lead to strategic uncertainty and coordination problems in economic contexts
Computation methods
- Pure strategy equilibria found by checking best responses or eliminating dominated strategies
- Mixed strategy equilibria often require solving systems of equations based on the indifference principle
- Graphical methods useful for two-player games with continuous strategy spaces
- Computational algorithms (Lemke-Howson) employed for larger games with many strategies
- Evolutionary approaches simulate repeated play to approximate equilibrium strategies in complex games
Interpretation of probabilities
- In mixed strategy equilibria, probabilities represent long-run frequencies of action choices
- Can be viewed as beliefs held by opponents about a player's likely actions
- Do not necessarily imply conscious randomization by players in every instance
- Reflect strategic uncertainty in the game, even when players use pure strategies
- Important for predicting aggregate behavior in large populations (market share in oligopolies)
Applications in economics
- Game theory concepts apply to numerous economic scenarios, providing insights into strategic interactions
- Help explain and predict outcomes in markets, negotiations, and policy decisions
- Enable the development of more sophisticated economic models that account for strategic behavior
Oligopoly models
- Cournot competition models firms choosing output levels simultaneously
- Bertrand competition focuses on price-setting behavior in oligopolistic markets
- Stackelberg model introduces sequential decision-making with a first-mover advantage
- Capacity constraint games examine strategic capacity choices and their impact on market outcomes
- Collusion and cartel stability analyzed using repeated game frameworks
Bargaining scenarios
- Nash bargaining solution provides a framework for analyzing cooperative bargaining outcomes
- Rubinstein bargaining model examines sequential offers in non-cooperative settings
- Ultimatum game studies one-shot bargaining with complete information
- Asymmetric information models explore the impact of private information on bargaining power
- Applications include labor negotiations, international trade agreements, and merger discussions
Public goods provision
- Voluntary contribution mechanisms studied using game theoretic models
- Free-rider problem analyzed as a multi-player prisoner's dilemma
- Lindahl equilibrium concept for efficient provision of public goods
- Mechanism design approaches to incentivize truthful revelation of preferences
- Repeated interaction models examine how cooperation in public goods provision can be sustained
Limitations and extensions
- Recognizes the boundaries of basic game theory and introduces more advanced concepts
- Addresses real-world complexities that simple models may not capture adequately
- Provides direction for further study and application of game theory in economics
Information asymmetry
- Incorporates scenarios where players have different levels of information about the game
- Signaling games model situations where informed players can convey information through actions
- Screening games involve uninformed players designing mechanisms to elicit information
- Adverse selection and moral hazard analyzed using principal-agent models
- Applications in insurance markets, labor contracts, and financial markets
Repeated games
- Examines strategic interactions that occur multiple times over an extended period
- Folk theorem demonstrates how cooperation can emerge in indefinitely repeated games
- Tit-for-tat and other strategies studied for their effectiveness in sustaining cooperation
- Reputation effects modeled through beliefs about player types in repeated games
- Applications include tacit collusion in oligopolies and international relations
Evolutionary game theory
- Applies game theory concepts to populations of players using replicator dynamics
- Evolutionary stable strategies (ESS) represent equilibria robust to invasion by mutant strategies
- Models the evolution of behaviors and norms in large populations over time
- Provides insights into the emergence and stability of cooperative behaviors
- Applications in biology, sociology, and the study of cultural evolution in economics
Key Terms to Review (18)
Battle of the sexes: The battle of the sexes is a classic game in game theory that illustrates a coordination problem between two players with conflicting preferences. In this game, each player prefers to be with the other but has different favorite activities, leading to a scenario where both must decide on an action that ideally aligns their outcomes while compromising on personal preferences. This scenario can be analyzed using both pure and mixed strategies to find optimal solutions for each player's decision-making process.
Best response: A best response is a strategy that yields the highest payoff for a player, given the strategies chosen by other players in a game. Understanding this concept is crucial as it helps determine optimal decision-making in strategic situations, revealing how players can react to each other's choices. The best response forms the foundation for concepts like Nash equilibrium and is applicable to both pure and mixed strategies, highlighting the dynamics of competitive interactions.
Correlated Equilibrium: A correlated equilibrium is a solution concept in game theory where players coordinate their strategies based on signals received from an external source, leading to a situation where no player has an incentive to unilaterally deviate from the recommended strategy. This approach enhances the traditional Nash equilibrium by allowing players to condition their actions on observed signals, resulting in potentially more efficient outcomes. Players rely on these signals to make decisions that align with the group's overall strategy, which can include both pure and mixed strategies.
Dominance: Dominance refers to a strategy in game theory where one player's strategy is superior to another, regardless of what the other player does. In the context of decision-making, a dominant strategy yields a better outcome for a player than any other strategies available, leading them to choose it when pursuing the best possible result.
Dominant Strategy: A dominant strategy is a choice made by a player in a game that results in the highest payoff regardless of what the other players decide to do. This means that a dominant strategy is always the best option for a player, no matter the actions taken by opponents. Understanding dominant strategies is crucial for analyzing both pure and mixed strategies, as it helps determine which strategies players should adopt in various competitive scenarios.
Expected Payoff: Expected payoff refers to the anticipated return from a strategic decision, calculated by weighing all possible outcomes by their probabilities. This concept plays a critical role in assessing the effectiveness of pure and mixed strategies in decision-making, where individuals or entities make choices under uncertainty. Understanding expected payoff helps to inform strategies that maximize potential gains while minimizing risks.
John Nash: John Nash was a groundbreaking mathematician and economist known for his contributions to game theory, particularly the concept of Nash equilibrium. His work laid the foundation for understanding strategic interactions among rational decision-makers in competitive situations. Nash's ideas have not only influenced economics but also fields such as political science, biology, and computer science, showcasing the versatility and importance of his theories.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and polymath, known for his groundbreaking contributions to many fields, including game theory, economics, and computer science. His work laid the foundations for modern mathematical economics and provided essential tools for analyzing strategic interactions among rational agents.
Mixed strategy: A mixed strategy is a strategic decision-making approach where a player randomizes their choices among multiple actions, rather than consistently choosing a single action. This approach is crucial in situations where players aim to keep their opponents guessing, preventing predictability and allowing for potentially better outcomes in competitive environments. Mixed strategies are particularly relevant in game theory, where players may face uncertainty about their opponents' choices and must adapt accordingly.
Nash equilibrium: Nash equilibrium is a concept in game theory where no player can benefit by changing their strategy while the other players keep theirs unchanged. This idea highlights a state of mutual best responses, making it essential in analyzing strategic interactions among rational decision-makers. Understanding Nash equilibrium helps to explore various scenarios, including competitive markets, sequential games, and different strategic approaches, thus providing a foundation for equilibrium analysis and the existence of stable outcomes.
Payoff matrix: A payoff matrix is a table that represents the potential outcomes of a strategic interaction between players, showing the payoffs each player receives based on the combination of strategies they choose. This matrix is essential in analyzing competitive situations, helping to identify strategies that lead to equilibrium and informing decisions about whether to adopt pure or mixed strategies. It is also useful for determining dominant and dominated strategies among players.
Prisoner's dilemma: The prisoner's dilemma is a fundamental concept in game theory that illustrates how two rational individuals may not cooperate, even if it appears that it is in their best interest to do so. This scenario shows that when both players choose to betray each other, they end up worse off than if they had cooperated, highlighting the conflict between individual self-interest and mutual benefit. It connects to strategies where players can either choose pure strategies—consistently making one choice—or mixed strategies, where they randomize their decisions based on probabilities, as well as the identification of dominant strategies that could lead to suboptimal outcomes.
Probability Distribution: A probability distribution is a mathematical function that describes the likelihood of different outcomes in a random experiment. It provides a comprehensive summary of how probabilities are distributed over the possible values of a random variable, allowing for the analysis of both pure and mixed strategies in decision-making scenarios. By illustrating how outcomes are spread across a range, probability distributions play a critical role in understanding uncertainty and making informed choices.
Pure strategy: A pure strategy is a specific and consistent plan of action that a player employs in a game, whereby they choose one particular option or move in a given situation. In game theory, this contrasts with mixed strategies, where players randomize over different actions. A pure strategy leads to predictable behavior from the player, allowing them to fully commit to their choice without any variation.
Saddle Point: A saddle point is a point in a two-dimensional space where the slope is zero, meaning it can be a minimum in one direction and a maximum in another. In the context of game theory, it represents an optimal strategy for players when both pure and mixed strategies are used, balancing the interests of both competitors. Identifying saddle points helps determine the best possible outcome for each player in strategic interactions.
Subgame perfect equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium used in dynamic games, where players' strategies constitute a Nash equilibrium in every subgame of the original game. This concept ensures that players' strategies remain optimal not only for the overall game but also for every possible scenario that may arise as the game unfolds. By applying backward induction, players can determine their best responses at every stage of the game, leading to a more robust understanding of strategic interactions.
Utility Function: A utility function is a mathematical representation of a consumer's preferences, quantifying the satisfaction or happiness gained from consuming different goods and services. It connects to various concepts such as optimization, decision-making strategies, and welfare analysis by illustrating how individuals make choices to maximize their overall utility within constraints like budget or resources.
Zero-sum game: A zero-sum game is a situation in game theory where one player's gain is exactly balanced by another player's loss, resulting in a total change of zero. This concept highlights the competitive nature of certain strategic interactions, indicating that resources are fixed and each participant's success directly correlates with the other's failure. Understanding zero-sum games is crucial for analyzing strategies, as players must consider both pure and mixed strategies to optimize their outcomes and recognize the implications of dominant or dominated strategies.