4.2 Pure strategy Nash equilibrium
2 min read•august 7, 2024
Nash equilibrium in pure strategies is a key concept in game theory. It occurs when no player can benefit by changing their strategy unilaterally. This solution concept helps predict outcomes in strategic interactions.
Understanding is crucial for analyzing various economic scenarios. It provides insights into decision-making processes and helps identify stable outcomes in competitive situations, forming the foundation for more complex equilibrium concepts.
Dominance and Equilibrium Concepts
Pure Strategies and Dominance
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- Pure strategy represents a single action or move a player will follow in every possible attainment in a game
- provides a player with the highest payoff available regardless of the strategies adopted by other players
- always results in a higher payoff for a player than any other strategy, no matter what strategies other players choose
- provides payoffs at least as high as any other strategy, regardless of other players' actions, but can result in equivalent payoffs for some strategy combinations
- involves removing strictly dominated strategies for each player in succession, reducing the game to a smaller set of remaining strategy profiles
Applications and Examples
- In the , confessing is a strictly dominant strategy for both players as it always results in a reduced sentence regardless of the other player's choice (Nash Equilibrium)
- In a model, each firm has a weakly dominant strategy to produce the quantity that maximizes their profit given the quantity produced by the other firm
- Iterated elimination can simplify complex games by progressively removing dominated strategies until only dominant or equilibrium strategies remain (Centipede Game)
Equilibrium Properties and Types
Multiple Equilibria and Efficiency
- occur when there is more than one set of strategies that satisfy the conditions for a Nash Equilibrium in a game
- is achieved when no player can be made better off without making at least one player worse off
- often have multiple equilibria, some of which may be more efficient or preferable than others (, )
Coordination and Examples
- In the Stag Hunt game, both players hunting the stag is Pareto optimal, but hunting rabbits individually is also a Nash Equilibrium
- The Battle of the Sexes game has two pure strategy Nash Equilibria, one favoring each player, but no equilibrium is more efficient than the other
- can help players coordinate on a particular equilibrium in games with multiple equilibria ()
Key Terms to Review (14)
Battle of the Sexes: The Battle of the Sexes is a classic game in game theory that illustrates a coordination problem between two players who prefer different outcomes but must reach an agreement on one. The game typically represents a situation where two players (often depicted as a male and female) want to go out together but have different preferences on where to go, such as one preferring to attend a football game while the other prefers going to the ballet. This scenario highlights the challenges of achieving a mutually beneficial outcome while dealing with conflicting interests, and it connects with various concepts in game theory such as normal and extensive form representations, dominant strategies, Nash equilibria, and rationalizability.
Coordination Games: Coordination games are a type of game in which players benefit from making the same choices or decisions, leading to mutual gains. These games highlight the importance of players aligning their strategies to achieve the best possible outcomes, often resulting in multiple equilibria where players can coordinate on different strategies. Such games are crucial for understanding various economic scenarios and behaviors, especially in contexts where cooperation is needed to avoid suboptimal results.
Cournot Duopoly: A Cournot duopoly is a market structure where two firms compete by choosing the quantity of output they will produce independently and simultaneously, assuming their rival's output remains constant. In this setup, each firm aims to maximize its profit given the quantity produced by the other, leading to a specific equilibrium known as the Nash equilibrium. The Cournot model illustrates how firms can engage in strategic decision-making under conditions of limited competition.
Dominant Strategy: A dominant strategy is a course of action that yields the highest payoff for a player, regardless of the strategies chosen by other players. This concept is key in understanding how individuals or firms make decisions in strategic situations where their outcomes depend on the choices of others.
Focal Points: Focal points are solutions that people tend to choose in strategic situations when there is ambiguity or uncertainty, often due to social conventions or cultural norms. These points help players coordinate their strategies, especially when they cannot communicate directly. The significance of focal points lies in their ability to guide players toward a mutual decision, making them crucial in analyzing situations like pure strategy Nash equilibria and scenarios involving collusion and tacit cooperation.
Iterated Elimination of Dominated Strategies: Iterated elimination of dominated strategies is a process used in game theory to simplify games by systematically removing strategies that are inferior to others, thus narrowing down the available choices for players. This method helps players focus on more viable strategies, ultimately leading to a clearer understanding of potential outcomes and facilitating the identification of pure strategy Nash equilibria. By eliminating dominated strategies iteratively, players can streamline decision-making and enhance strategic interactions in competitive scenarios.
Multiple equilibria: Multiple equilibria refer to a situation in a game where there are two or more distinct outcomes that can be stable under the conditions defined by the players' strategies. Each of these outcomes is a Nash equilibrium, meaning that no player can benefit by unilaterally changing their strategy while the others keep theirs unchanged. This concept highlights the potential for different stable states within a strategic interaction, often influenced by factors such as initial conditions or player preferences.
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 emphasizes the idea of optimal distribution of resources among players in a game, relating closely to strategies, payoffs, and the rational behavior of individuals involved.
Prisoner's dilemma: The prisoner's dilemma is a standard example of a game in which two players must choose between cooperation and betrayal, with the outcome for each dependent not only on their own choice but also on the choice made by the other player. This scenario highlights the conflict between individual rationality and collective benefit, demonstrating how two rational individuals may not cooperate even if it appears that it is in their best interest.
Pure Strategy Nash Equilibrium: A pure strategy Nash equilibrium occurs when each player in a game chooses one specific strategy and no player can benefit by unilaterally changing their strategy, given the strategies of all other players. This concept highlights the stability of strategy choices in a competitive environment, where each player's decision is optimal considering the choices made by others. It serves as a foundational element in understanding strategic interactions and the outcomes that result from rational decision-making.
Schelling's Meeting Place Problem: Schelling's Meeting Place Problem is a concept in game theory that illustrates how individuals can coordinate their actions even when they have no communication, relying on common knowledge or assumptions. This problem emphasizes the challenges of coordination when participants have to decide on a meeting point without prior discussion, leading to multiple potential solutions based on shared expectations. The essence of this problem lies in understanding how players converge on a solution that may not be the most optimal but is acceptable to all involved.
Stag Hunt: The stag hunt is a game theory scenario that illustrates the conflict between safety and social cooperation. In this setting, two players can either collaborate to catch a stag, which requires both to cooperate, or they can choose to hunt a hare, which can be done alone but provides a lower payoff. This scenario connects deeply to concepts of rationalizability and pure strategy Nash equilibrium by highlighting how the decisions of individuals can lead to different outcomes based on their expectations of others' behavior.
Strictly dominant strategy: A strictly dominant strategy is an action that results in a better outcome for a player, regardless of what the other players do. This means that when a player has a strictly dominant strategy, they will always prefer that strategy over any other, no matter the choices made by others. Recognizing strictly dominant strategies can simplify decision-making processes, leading to clearer predictions about player behavior in strategic situations and providing insights into the existence of equilibria.
Weakly Dominant Strategy: A weakly dominant strategy is a strategy that results in at least as good an outcome for a player as any other strategy, regardless of what the other players do, and sometimes even a better outcome. This concept is crucial because it helps identify choices that a player can rely on without fear of being worse off compared to alternative strategies. Recognizing weakly dominant strategies aids in understanding broader concepts like Nash equilibrium, where players make optimal decisions based on the strategies of others.