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 pure strategy Nash equilibrium 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

• Pure strategy represents a single action or move a player will follow in every possible attainment in a game
• Dominant strategy provides a player with the highest payoff available regardless of the strategies adopted by other players
• Strictly dominant strategy always results in a higher payoff for a player than any other strategy, no matter what strategies other players choose
• Weakly dominant strategy 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
• Iterated elimination of dominated strategies 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 Prisoner's Dilemma, 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 Cournot duopoly 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

• Multiple equilibria occur when there is more than one set of strategies that satisfy the conditions for a Nash Equilibrium in a game
• Pareto efficiency is achieved when no player can be made better off without making at least one player worse off
• Coordination games often have multiple equilibria, some of which may be more efficient or preferable than others (Stag Hunt, Battle of the Sexes)

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
• Focal points can help players coordinate on a particular equilibrium in games with multiple equilibria (Schelling's Meeting Place Problem)