11.2 Nash equilibrium and dominant strategies
3 min read•august 16, 2024
Game theory explores strategic decision-making in competitive situations. and dominant strategies are key concepts that help predict outcomes when players interact. These tools analyze how rational players choose strategies to maximize their payoffs.
Nash equilibrium occurs when no player can improve their outcome by changing strategy alone. Dominant strategies always yield the best payoff regardless of others' choices. Understanding these concepts helps predict behavior in various real-world scenarios, from business competition to international relations.
Nash Equilibria in Games
Defining Nash Equilibrium
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- Nash equilibrium occurs when no player can unilaterally improve their payoff by changing strategy
- Each player's strategy optimizes their payoff given other players' strategies
- Applies to various game types (simultaneous-move, sequential, repeated)
- Can exist in pure strategies (single action chosen with certainty) or mixed strategies (randomized choices)
- Multiple Nash equilibria may exist in a single game
- Some games lack Nash equilibria in pure strategies
Identifying Nash Equilibria
- Analyze each player's to other players' strategies
- Find strategy profiles where all players simultaneously play best responses
- Existence and uniqueness depend on game structure and payoffs
- Use best response analysis to determine equilibria
- Apply iterated elimination of dominated strategies to simplify complex games
- Construct best response curves for graphical solutions (two-player games with continuous strategies)
- Evaluate expected payoffs for mixed strategy equilibria
Dominant Strategies and Game Outcomes
Types of Dominant Strategies
- provides highest payoff regardless of other players' choices
- Strictly dominant strategy yields strictly higher payoff than any other ()
- yields payoff at least as high as others, with strict inequality for at least one opponent strategy (Stag Hunt game)
- Rational players always choose dominant strategy if one exists
- Simplifies Nash equilibria identification process
Impact on Game Outcomes
- Games where all players have dominant strategies always result in Nash equilibrium
- Can lead to suboptimal outcomes for all players (Prisoner's Dilemma)
- Dominance concept eliminates dominated strategies
- Simplifies analysis of complex games through strategy elimination
- Influences equilibrium selection in games with multiple equilibria
- Affects strategic thinking and decision-making in real-world scenarios (oligopoly pricing)
Finding Nash Equilibria: Pure vs Mixed Strategies
Pure Strategy Nash Equilibria
- Identify each player's best response to every possible opponent strategy
- Find strategy profiles where all players play best responses simultaneously
- Use iterated elimination of dominated strategies to simplify games
- Apply backward induction for sequential games (Subgame Perfect Nash Equilibrium)
- Check for stability by confirming no player can improve payoff by deviating
- Consider all possible strategy combinations in normal form games
- Evaluate payoff matrices to determine equilibria (Battle of the Sexes)
Mixed Strategy Nash Equilibria
- Exist in games without pure strategy Nash equilibria
- Players randomize choices according to specific probabilities
- Set up equations to make each player indifferent between pure strategies
- Apply indifference principle: player must be indifferent between all pure strategies played with positive probability
- Calculate expected payoffs for mixed strategies
- Solve systems of equations to determine equilibrium probabilities
- Use graphical methods (best response curves) for two-player games with continuous strategy spaces (Cournot duopoly)
Key Terms to Review (16)
Bertrand Competition: Bertrand competition is a market structure in which firms compete by setting prices rather than quantities, leading to outcomes where prices can drop to marginal cost levels. In this model, each firm assumes that its competitors' prices remain constant when deciding its own price, which can result in a Nash equilibrium where no firm has an incentive to change its price unilaterally. This scenario illustrates how firms may engage in strategic decision-making regarding pricing in order to maximize profits.
Best response: A best response is the strategy that yields the highest payoff for a player, given the strategies chosen by other players in a game. It highlights how an individual's choice is influenced by the actions of others, showing that optimal decision-making relies on anticipating opponents' moves. Understanding best responses is essential in analyzing strategic interactions, as they play a critical role in determining Nash equilibria and dominant strategies.
Correlated Equilibrium: Correlated equilibrium is a solution concept in game theory where players can coordinate their strategies through signals received from a correlation device, which helps them achieve better outcomes than in a Nash equilibrium. It extends the idea of Nash equilibrium by allowing players to base their actions on shared information, which can lead to more efficient results in terms of collective payoffs. This concept is particularly useful in games where cooperation and communication between players can enhance the overall welfare.
Cournot competition: Cournot competition is a model in which firms compete on the quantity of output they produce, and each firm decides its output level based on the anticipated output levels of its competitors. This interdependence means that each firm's decision affects the market price and the profits of all firms involved, leading to a Nash equilibrium where no firm has an incentive to change its output given the output of others.
Deviation: Deviation refers to the act of straying from an established strategy or equilibrium in a game-theoretic context. In the realm of Nash equilibrium and dominant strategies, deviation highlights how players may change their actions to potentially achieve a better outcome for themselves, even when it might disrupt the balance of strategies that others are employing. Understanding deviation helps explain the dynamics of strategic interactions, as it can indicate whether players are satisfied with their current strategies or whether incentives exist for them to alter their choices.
Dominant Strategy: A dominant strategy is a course of action that is the best choice for a player to make, regardless of what the other players choose. It implies that the chosen strategy yields a higher payoff than any other strategy, no matter what strategies the other players adopt. This concept is crucial in analyzing strategic interactions among players, helping to understand how decisions are made in competitive situations.
John Nash: John Nash was an influential mathematician and economist best known for his contributions to game theory, particularly the concept of Nash equilibrium. His work revolutionized the understanding of strategic interactions among rational decision-makers, impacting various fields including economics, political science, and biology. Nash's theories provide a framework for analyzing how individuals or firms can optimize their decisions in competitive environments, such as cartels and bargaining scenarios.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and polymath who made significant contributions to various fields, including game theory, which is essential in understanding Nash equilibrium and dominant strategies. His work laid the groundwork for modern economic theory by formalizing strategic interactions among rational agents. Von Neumann's principles of rational behavior in games have influenced the study of competitive situations where individuals make decisions that depend on the choices of others.
Mixed strategy Nash equilibrium: A mixed strategy Nash equilibrium occurs when players in a game randomize their strategies in such a way that no player can gain an advantage by unilaterally changing their strategy. This concept emerges when there is no dominant strategy for players, meaning they must make decisions that involve probabilities over their available strategies to keep their opponents indifferent. In this scenario, each player's strategy is a best response to the mixed strategies of the others, creating a stable outcome where everyone is optimizing their payoff given the strategies employed by others.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where players in a strategic interaction choose their optimal strategy, given the strategies of others, resulting in no player having an incentive to deviate from their chosen strategy. This concept is crucial in understanding how firms operate in competitive markets, particularly where their decisions are interdependent.
Pareto Efficiency: Pareto efficiency refers to a situation in which it is impossible to make any individual better off without making someone else worse off. This concept is central to understanding resource allocation and welfare economics, as it helps to identify optimal distribution of resources in various economic settings. In the context of competition and market dynamics, Pareto efficiency highlights the conditions under which markets can operate effectively and allocate resources in a way that maximizes overall utility without harming others.
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, end up with a worse outcome than if they had cooperated. This scenario often highlights the conflict between individual incentives and collective benefits, revealing how rational decision-making can lead to suboptimal results in competitive situations. It connects deeply with concepts of strategic interactions, where players must consider the choices of others to determine their own best action.
Pure strategy: A pure strategy is a predetermined and consistent plan of action that a player follows in a strategic game, where they choose a specific action with certainty, as opposed to randomizing their actions. This concept is crucial for understanding how players make decisions in both static and dynamic games, as it reveals the thought process behind achieving optimal outcomes based on the strategies of other players. In contexts like Nash equilibrium and dominant strategies, pure strategies highlight the best choices players can make when faced with predictable actions from their opponents.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable in dynamic games, where players' strategies form a Nash equilibrium in every subgame of the original game. This concept ensures that players make optimal decisions at every point in the game, anticipating future actions and outcomes. By focusing on sequential rationality, it highlights how players can strategically navigate situations where they have to make decisions at different stages.
Weakly dominant strategy: A weakly dominant strategy is a choice in game theory that provides a player with outcomes that are at least as good as any other strategy, regardless of what the opponents choose, and sometimes even better. This concept is crucial in understanding decision-making within Nash equilibria, where players choose strategies that lead to stable outcomes. It highlights situations where one strategy may not always outperform others but does not perform worse in any scenario.
Zero-sum game: A zero-sum game is a type of situation in game theory where one participant's gain is exactly balanced by the losses of another participant. This concept implies that the total benefit or loss among all players in the game remains constant, meaning that any advantage one player gains results in a corresponding disadvantage for another player. This situation is often analyzed using strategies like Nash equilibrium and dominant strategies, as players must choose actions that maximize their own payoff while minimizing their opponent's payoff.