9.2 Nash equilibrium and dominant strategies
3 min read•july 30, 2024
Game theory helps us understand strategic decision-making. and dominant strategies are key concepts that show how players make choices based on what others might do. These ideas explain why certain outcomes happen in competitive situations.
In this part of the chapter, we'll look at how Nash equilibrium works in simple games. We'll also explore dominant and dominated strategies, which can help predict what rational players will do. These concepts are super useful for understanding real-world strategic interactions.
Nash equilibrium in game theory
Concept and definition
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- Nash equilibrium represents a state where no player can unilaterally improve their outcome by changing their strategy, given the strategies of other players
- Each player's strategy proves optimal given the strategies of all other players
- Developed by mathematician , forming a cornerstone of theory
- Represents strategic stability where players lack incentives to deviate from chosen strategies
- Multiple Nash equilibria can exist in a single game
- Not all games have a Nash equilibrium in pure strategies
Applications and significance
- Applies to various fields (economics, political science, biology) to analyze strategic decision-making
- Predicts stable outcomes in competitive situations
- Informs decision-making in strategic environments
- Provides insights into complex interactions between rational decision-makers
- Helps understand market dynamics, political negotiations, and evolutionary processes
Nash equilibrium in simple games
Simultaneous games and payoff matrices
- involve players making decisions without knowledge of other players' choices
- Payoff matrices represent outcomes of different strategy combinations
- Cells in payoff matrices where each player's strategy proves a to others' strategies represent Nash equilibrium
- Common examples include Prisoner's Dilemma, Battle of the Sexes, and Matching Pennies
Finding Nash equilibrium
- Identify each player's best response to every possible strategy of their opponents
- In 2x2 games, check each cell to see if it satisfies the equilibrium condition
- Mixed strategy Nash equilibria exist when players randomize choices according to specific probabilities
- Process may reveal multiple Nash equilibria or no pure strategy equilibrium
Dominant vs Dominated strategies
Dominant strategies
- Provide a player with the best outcome regardless of strategies chosen by other players
- Strictly dominant strategies yield strictly better payoffs than any other strategy
- Weakly dominant strategies yield payoffs at least as good as any other strategy, strictly better against at least one opponent strategy
- Example: In Prisoner's Dilemma, confessing proves a for both prisoners
Dominated strategies
- Provide a player with a worse outcome than some other strategy, regardless of strategies chosen by other players
- Strictly dominated strategies yield strictly worse payoffs than some other strategy
- Weakly dominated strategies yield payoffs no better than some other strategy, strictly worse against at least one opponent strategy
- Example: In a product pricing game, setting an extremely high price might be a dominated strategy
Importance in game analysis
- Identifying dominant and dominated strategies simplifies game analysis
- Helps predict rational player behavior
- Provides insights into optimal decision-making in strategic situations
- Serves as a starting point for more complex game-theoretic analyses
Iterated elimination of dominated strategies
Process and assumptions
- (IEDS) simplifies games and potentially identifies Nash equilibria
- Involves repeatedly removing dominated strategies from the game
- Assumes all players prove rational and this rationality proves common knowledge among players
- Order of elimination does not affect the final outcome
- May impact the number of steps required to reach that outcome
Outcomes and applications
- Can lead to a unique solution in some games, guaranteed to be a Nash equilibrium
- May reduce the game to a smaller set of strategies without yielding a unique solution
- Process terminates when no further dominated strategies can be eliminated
- Remaining strategies after IEDS are considered rationalizable
- Helps analyze complex games by systematically eliminating non-optimal choices
- Provides insights into strategic thinking and decision-making processes
Key Terms to Review (22)
Best response: A best response refers to the strategy that yields the highest payoff for a player, given the strategies chosen by other players in a game. It is a fundamental concept in game theory, as it helps determine how players will react to one another's choices, leading to important outcomes like Nash equilibrium and identifying dominant strategies. The best response showcases the optimal choice of action in situations where players are interdependent, making it essential for analyzing strategic interactions.
Best Response Function: A best response function is a strategy that a player in a game chooses that yields the highest payoff given the strategies chosen by other players. It reflects how one player's optimal choice is influenced by the actions of others, highlighting the interdependence of decisions in strategic situations. Understanding best response functions is crucial for identifying Nash equilibria and analyzing dominant strategies, as they show how players adapt their strategies based on their expectations of other players' actions.
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.
Dominance solvability: Dominance solvability refers to a situation in game theory where a game can be solved by iteratively removing dominated strategies until only undominated strategies remain. This process leads players to a clear and unambiguous outcome, often facilitating the identification of Nash equilibria. It showcases the power of rational decision-making by allowing players to disregard options that are inferior regardless of what others do.
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.
Equilibrium Strategies: Equilibrium strategies are strategic choices made by players in a game where no player has an incentive to deviate from their chosen strategy, given the strategies chosen by other players. This concept is crucial as it leads to outcomes where all participants are optimizing their decisions, leading to a state of balance in competitive situations, often analyzed through Nash equilibrium and dominant strategies.
Externalities: Externalities are costs or benefits of a market activity that affect third parties who did not choose to be involved in that activity. They can be positive or negative and play a significant role in influencing market efficiency, resource allocation, and social welfare.
Iterated elimination of dominated strategies: Iterated elimination of dominated strategies is a method used in game theory to simplify decision-making by repeatedly removing strategies that are inferior to others, regardless of the opponents' actions. This process helps players identify optimal strategies by narrowing down their choices to those that could potentially be best responses, ultimately aiding in finding Nash equilibria and understanding dominant strategies.
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 competition: Market competition refers to the rivalry among businesses in the same industry to attract customers and increase market share. This competition drives innovation, influences pricing strategies, and determines the availability of goods and services, fostering an environment where firms must continually adapt to meet consumer demands. Understanding how firms interact within competitive markets is crucial for analyzing strategic decision-making, particularly regarding the dynamics of Nash equilibrium and dominant strategies.
Market failure: Market failure occurs when the allocation of goods and services by a free market is not efficient, leading to a net social welfare loss. This often happens due to various reasons such as externalities, public goods, or monopolies that prevent the market from reaching an optimal equilibrium where supply equals demand.
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-cooperative game: A non-cooperative game is a type of game in which players make decisions independently, without collaboration or agreements with other players. Each participant aims to maximize their own payoff based on their strategies and the anticipated actions of others. In this setting, the outcome depends not only on individual strategies but also on the strategic interactions among players, leading to concepts like Nash equilibrium and dominant strategies.
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 strategies: Pricing strategies refer to the methods businesses use to determine the best price for their products or services, considering factors like costs, competition, and market demand. These strategies can influence consumer behavior, profitability, and market positioning, making them crucial in a competitive landscape. Different pricing strategies are often employed based on the market structure in which a business operates, the nature of the product or service, and the intended consumer segment.
Simultaneous games: Simultaneous games are a type of strategic interaction where players make decisions at the same time without knowing the choices of the other players. This scenario creates a situation where each player must consider the potential actions of their opponents when formulating their own strategies. The outcome often hinges on predicting how others will respond, making these games crucial for understanding concepts like Nash equilibrium and dominant strategies.
Strictly Dominant Strategy: A strictly dominant strategy is a choice that yields a higher payoff for a player regardless of what the other players choose. This concept is important in game theory as it helps to identify optimal strategies in competitive situations. When a player has a strictly dominant strategy, they will always choose that option, as it maximizes their benefits no matter the actions taken by others.
Strictly Dominated Strategy: A strictly dominated strategy is a choice in a game that is always worse than another strategy, regardless of what the other players do. If a player has a strictly dominated strategy, they can always achieve a better outcome by choosing an alternative strategy, making it rational to abandon the dominated option. This concept is closely tied to Nash equilibrium and dominant strategies, as identifying strictly dominated strategies helps simplify decision-making in strategic interactions.
Weakly dominant strategy: A weakly dominant strategy is an action that results in a player achieving at least the same payoff as any other strategy, regardless of what the other players choose, and sometimes provides a better payoff. This concept is important because it helps to identify strategies that are preferable for a player in a game, contributing to their decision-making process. Understanding weakly dominant strategies can also highlight the existence of Nash equilibria in games where players choose strategies based on their payoffs.
Weakly dominated strategy: A weakly dominated strategy is one where a player has another strategy that always gives them at least the same payoff and sometimes a better payoff, regardless of what the other players do. This concept helps in understanding strategic choices in games, especially when analyzing Nash equilibria and dominant strategies, as it allows players to eliminate suboptimal strategies and focus on more advantageous options.
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.