Glossary

is a key concept in game theory, describing a stable state where no player can improve their outcome by changing strategy alone. It's crucial for analyzing strategic interactions in economics, providing insights into optimal behavior and market dynamics in competitive environments.

The concept involves strategic interdependence, rational decision-making, and mutual best responses. While not always Pareto optimal, Nash equilibrium helps predict outcomes in various economic scenarios, from oligopoly models to public goods provision and labor market negotiations.

Definition of Nash equilibrium

  • Fundamental concept in game theory describes a stable state where no player can unilaterally improve their outcome
  • Crucial for analyzing strategic interactions in economic scenarios involving multiple decision-makers
  • Provides insights into optimal behavior and market dynamics in competitive environments

Key components

Top images from around the web for Key components

  • Rational Decision Making vs. Other Types of Decision Making | Principles of Management View original

    Is this image relevant?

  • Nash equilibrium - Wikipedia View original

    Is this image relevant?

  • Rational Decision Making vs. Other Types of Decision Making | Principles of Management View original

    Is this image relevant?

1 of 3

Top images from around the web for Key components

  • Rational Decision Making vs. Other Types of Decision Making | Principles of Management View original

    Is this image relevant?

  • Nash equilibrium - Wikipedia View original

    Is this image relevant?

  • Rational Decision Making vs. Other Types of Decision Making | Principles of Management View original

    Is this image relevant?

1 of 3
  • Strategic interdependence among players' decisions
  • Rational decision-making based on available information
  • Mutual best responses to other players' strategies
  • Equilibrium state where no player has incentive to deviate

Conditions for existence

  • Finite number of players
  • Finite set of strategies for each player
  • Well-defined payoff functions for all strategy combinations
  • Compact and convex strategy spaces in continuous games
  • Mixed strategies allow for existence in broader range of games

Nash vs Pareto optimality

  • Nash equilibrium may not always be Pareto optimal
  • Prisoners' Dilemma illustrates divergence between Nash and Pareto outcomes
  • Coordination games can have multiple Nash equilibria with varying
  • Social welfare considerations often require mechanisms beyond Nash equilibrium

Mathematical representation

  • Utilizes game theory notation to formalize strategic interactions
  • Enables rigorous analysis of equilibrium properties and comparative statics
  • Facilitates application of mathematical tools from optimization and fixed-point theory

Pure strategy equilibria

  • Represented as a s∗=(s1∗,s2∗,...,sn∗)s^* = (s_1^*, s_2^*, ..., s_n^*)
  • Each player's strategy maximizes their payoff given others' strategies
  • Mathematically expressed as ui(si∗,s−i∗)≥ui(si,s−i∗)u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*) for all sis_i and all players ii
  • Can be found by identifying mutual best responses

Mixed strategy equilibria

  • Probability distributions over pure strategies σi∗=(p1,p2,...,pk)\sigma_i^* = (p_1, p_2, ..., p_k)
  • Extends existence of equilibria to broader class of games
  • Expected payoff maximization condition E[ui(σi∗,σ−i∗)]≥E[ui(σi,σ−i∗)]E[u_i(\sigma_i^*, \sigma_{-i}^*)] \geq E[u_i(\sigma_i, \sigma_{-i}^*)]
  • Indifference principle applies to strategies with positive probabilities

Best response functions

  • Maps opponents' strategies to player's optimal strategy
  • Denoted as BRi(s−i)=arg⁡max⁡siui(si,s−i)BR_i(s_{-i}) = \arg\max_{s_i} u_i(s_i, s_{-i})
  • Nash equilibrium occurs at intersection of all players' functions
  • Graphical representation useful for 2x2 games (best response curves)

Applications in economics

  • Provides framework for analyzing strategic behavior in various economic contexts
  • Helps predict market outcomes and inform policy decisions
  • Applicable to both micro and macroeconomic scenarios

Oligopoly models

  • Cournot competition quantity-setting firms simultaneously choose output levels
  • Bertrand competition price-setting firms compete on prices
  • Stackelberg model incorporates sequential decision-making (leader-follower dynamics)
  • Differentiated products oligopoly accounts for product heterogeneity

Public goods provision

  • Free-rider problem arises due to non-excludability of public goods
  • Voluntary contribution mechanisms often lead to suboptimal Nash equilibria
  • Lindahl equilibrium as a potential solution for efficient provision
  • Mechanism design approaches to incentivize optimal contributions

Labor market negotiations

  • Bargaining between unions and firms modeled as
  • Wage and employment levels determined by Nash equilibrium outcomes
  • Right-to-manage model vs efficient bargaining model
  • Strike threats and holdout strategies as equilibrium phenomena

Finding Nash equilibria

  • Multiple methods available depending on game structure and complexity
  • Analytical approaches for simple games, computational methods for complex ones
  • Understanding solution techniques crucial for applied game theory in economics

Dominant strategy method

  • Identify strategies that are optimal regardless of opponents' choices
  • Simplest case dominant strategy equilibrium exists for all players
  • Weakly dominant strategies may still lead to unique Nash equilibrium
  • Useful for games with clear strategic dominance (Prisoners' Dilemma)

Best response method

  • Iteratively determine each player's optimal response to others' strategies
  • Construct best response functions or tables for each player
  • Identify strategy profiles where all players are simultaneously best responding
  • Effective for games with continuous strategy spaces (Cournot duopoly)

Elimination of dominated strategies

  • Iteratively remove strategies that are always worse than alternatives
  • Reduces game complexity and may lead to unique Nash equilibrium
  • Strict domination vs weak domination considerations
  • Rationalizability concept related to iterated elimination process

Properties of Nash equilibrium

  • Characteristics that determine equilibrium behavior and implications
  • Important for understanding limitations and applicability of Nash concept
  • Guide refinements and extensions in more complex game-theoretic models

Stability and uniqueness

  • Stable equilibria resistant to small perturbations in strategies
  • Multiple equilibria possible in many games (coordination games)
  • Refinement concepts help select among multiple equilibria (risk dominance)
  • Trembling hand perfection ensures against small mistakes

Efficiency considerations

  • Nash equilibria may not maximize social welfare (Prisoners' Dilemma)
  • Pareto efficiency vs Nash efficiency in resource allocation
  • Mechanisms to align individual incentives with social optimality
  • Role of institutions and regulations in promoting efficient outcomes

Subgame perfection

  • Refinement for extensive form games ensures credible threats/promises
  • Eliminates non-credible Nash equilibria in multi-stage games
  • Backward induction method to find subgame perfect equilibria
  • Applications in sequential bargaining and dynamic oligopoly models

Extensions and variations

  • Adaptations of Nash equilibrium concept to more complex scenarios
  • Address limitations and expand applicability to diverse economic situations
  • Incorporate uncertainty, correlation, and evolutionary dynamics

Bayesian Nash equilibrium

  • Incorporates incomplete information about other players' types
  • Players have beliefs about opponents' types (probability distributions)
  • Strategies maximize expected payoffs given beliefs
  • Applications in auctions, signaling games, and mechanism design

Correlated equilibrium

  • Allows for correlation in players' strategies via a mediator
  • Generalizes Nash equilibrium can achieve higher payoffs
  • Implements coordinated behavior without explicit cooperation
  • Relevant for analyzing tacit collusion and focal points

Evolutionary stable strategies

  • Adapts Nash concept to population dynamics in evolutionary games
  • Strategy resistant to invasion by small proportion of mutants
  • Provides insight into long-term stability of behavioral traits
  • Applications in evolutionary economics and behavioral game theory

Limitations and criticisms

  • Understanding shortcomings crucial for appropriate application of Nash equilibrium
  • Motivates development of alternative solution concepts and modeling approaches
  • Informs interpretation of game-theoretic results in economic analysis

Rationality assumptions

  • Perfect rationality and common knowledge often unrealistic
  • Bounded rationality models incorporate cognitive limitations
  • Experimental evidence shows systematic deviations from Nash predictions
  • Behavioral game theory integrates psychological insights

Multiple equilibria problem

  • Many games have multiple Nash equilibria
  • Equilibrium selection becomes crucial issue
  • Refinement concepts (perfectness, stability) attempt to address this
  • Coordination problems and potential for inefficient outcomes

Real-world applicability

  • Simplified models may not capture full complexity of economic interactions
  • Difficulty in specifying accurate payoff functions and information structures
  • Learning and adaptation processes not fully captured by static equilibrium
  • Need for empirical validation and calibration of game-theoretic models

Nash equilibrium in game theory

  • Foundational concept applies to various game representations
  • Understanding different game forms crucial for modeling economic scenarios
  • Each representation captures different aspects of strategic interaction

Normal form games

  • Matrix representation of simultaneous-move games
  • Strategies and payoffs for all players explicitly listed
  • Nash equilibrium easily visualized for 2x2 games
  • Applications in static competition models (Cournot, Bertrand)

Extensive form games

  • Tree representation of sequential decision-making
  • Incorporates timing of moves and information sets
  • Subgame perfect Nash equilibrium refines solution concept
  • Relevant for dynamic economic interactions (bargaining, entry deterrence)

Repeated games

  • Multiple repetitions of a stage game
  • Allows for reputation building and punishment strategies
  • Folk theorem characterizes set of Nash equilibrium payoffs
  • Models long-term relationships in economics (tacit collusion, cooperation)

Computational approaches

  • Essential for analyzing complex games with many players or strategies
  • Enables application of game theory to large-scale economic models
  • Facilitates numerical experiments and sensitivity analysis

Algorithm for finding equilibria

  • Lemke-Howson algorithm for 2-player games
  • Simplicial subdivision methods for n-player games
  • Homotopy methods for computing all equilibria
  • Approximation algorithms for large-scale games

Software tools and simulations

  • Gambit open-source library for game-theoretic analysis
  • GAMUT suite for generating and studying game instances
  • Agent-based modeling platforms (NetLogo, MASON) for evolutionary games
  • Custom implementations in scientific computing environments (MATLAB, Python)

Complexity considerations

  • Computing Nash equilibria is PPAD-complete
  • Approximation algorithms for -Nash equilibria
  • Parameterized complexity analysis for specific game classes
  • Implications for practical solvability of large economic games

Key Terms to Review (18)

Auction theory: Auction theory is a branch of economics that studies how goods and services are allocated through bidding processes. It examines the strategies bidders use, the types of auctions, and how different auction formats can impact outcomes, like prices and efficiency. Understanding auction theory is crucial for analyzing competitive situations and predicting behavior in markets where buyers and sellers interact through bidding.
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.
Best response dynamics: Best response dynamics refers to a process in game theory where players adjust their strategies based on the strategies chosen by other players, aiming to maximize their own payoff. This iterative process continues until players reach a stable outcome, where no player can benefit from unilaterally changing their strategy. It connects closely with the concept of Nash equilibrium, as reaching an equilibrium can often involve players finding their best responses to one another's choices.
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.
Finite Games: Finite games are strategic interactions that involve a limited number of players, each with a defined set of strategies and outcomes. In these games, players have specific goals and the game concludes after a certain number of moves or a predetermined condition is met, making it essential for players to consider their opponents' potential actions to optimize their own outcomes. Finite games often exhibit clear rules and objectives, providing a structured environment for analyzing decision-making processes.
Fixed Point Theorem: A fixed point theorem states that under certain conditions, a function will have at least one point where the value of the function at that point is equal to the point itself. This concept is crucial in various areas of economics as it helps establish the existence and stability of equilibria, showing that certain solutions or outcomes are not only possible but also reliable under specific mathematical frameworks.
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.
Market Competition: Market competition refers to the rivalry among sellers in a market to attract customers and gain market share. This competition influences pricing, product quality, and innovation, pushing businesses to optimize their offerings. In a competitive market, no single entity can dominate prices or dictate terms, as multiple players vie for consumers' attention and resources.
Mixed Strategy Nash Equilibrium: A mixed strategy Nash equilibrium occurs in a game where players randomize over possible moves, ensuring that no player has an incentive to deviate from their strategy given the strategies of the others. This concept highlights situations where players may not choose a single action but instead select among multiple options with certain probabilities, leading to a stable state where each player's strategy is optimal when considering the choices of others.
Nash bargaining solution: The Nash bargaining solution is a concept in cooperative game theory that provides a solution for how two or more players can agree on a mutually beneficial outcome. It is derived from the idea of fairness and takes into account the players' preferences and their disagreement points, effectively balancing their interests to reach an optimal agreement. This solution extends the ideas of Nash equilibrium by considering cooperative interactions rather than just competitive scenarios.
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.
Nash Existence Theorem: The Nash Existence Theorem states that in a finite game with a finite number of players, where each player has a set of strategies, there exists at least one Nash equilibrium. This is a foundational result in game theory, establishing that even in complex strategic situations, players can reach a stable state where no one has an incentive to unilaterally change their strategy. This theorem is crucial for understanding how competitive interactions can lead to equilibrium outcomes.
Non-cooperative games: Non-cooperative games are strategic interactions where players make decisions independently, without the ability to enforce agreements or collaborate for mutual benefit. This framework emphasizes individual strategies and outcomes rather than cooperative efforts, focusing on how each player can maximize their own payoff given the choices of others. A key concept in this context is the Nash equilibrium, which represents a situation where no player can benefit by changing their strategy while the other players keep theirs unchanged.
Pareto Efficiency: Pareto efficiency is an economic state where resources are allocated in a way that no individual can be made better off without making someone else worse off. This concept is fundamental in understanding how markets operate and is closely related to various equilibrium analyses, demonstrating how optimal resource distribution can occur without wasting resources or creating inefficiencies.
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.
Stability: Stability refers to the property of a system to return to equilibrium after a disturbance. This concept is crucial in understanding how systems react to changes and whether they can maintain or regain their balance over time. Stability can apply to various contexts, including dynamic systems, economic models, and strategic interactions, helping to analyze the behavior of these systems when faced with external shocks or perturbations.
Strategy profile: A strategy profile is a complete description of the strategies chosen by all players in a game, representing their combined actions and decisions. It encapsulates the choices made by each participant, allowing for an analysis of potential outcomes and the interaction between different strategies. Understanding strategy profiles is essential for evaluating concepts like Nash equilibrium and dominant strategies, as they provide the framework for assessing how players' decisions impact each other.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide