📈business microeconomics review
9.2 Nash equilibrium and dominant strategies
Last Updated on July 30, 2024
Game theory helps us understand strategic decision-making. Nash equilibrium 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 John Nash, forming a cornerstone of non-cooperative game 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
- Simultaneous games 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 best response 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 dominant strategy 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
- Iterated elimination of dominated strategies (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