6.3 Dominant and dominated strategies

Game theory is a powerful tool in mathematical economics, helping us understand strategic decision-making. Dominant and dominated strategies are key concepts that simplify complex interactions, guiding players towards optimal choices regardless of their opponents' actions.

These strategies play a crucial role in predicting behavior and finding equilibria in various economic scenarios. By identifying dominant strategies, we can analyze competitive markets, policy decisions, and social dilemmas, providing valuable insights for businesses and policymakers alike.

Concept of strategic dominance

• Fundamental principle in game theory and economic decision-making guides players to choose strategies that yield better outcomes regardless of opponents' actions
• Crucial concept in mathematical economics helps analyze and predict behavior in competitive situations, market interactions, and policy decisions

Definition of strategy dominance

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• Occurs when one strategy consistently outperforms another strategy for a player, regardless of the strategies chosen by other players
• Provides a rational basis for decision-making in competitive environments (auctions, pricing strategies)
• Evaluated by comparing payoffs across all possible combinations of opponents' strategies

Types of strategic dominance

• Includes strict dominance and weak dominance as two main categories
• Strict dominance requires a strategy to always yield strictly higher payoffs than alternatives
• Weak dominance allows for equal payoffs in some scenarios but never worse outcomes
• Determined by analyzing payoff matrices and comparing outcomes across different strategy combinations

Dominant strategies

• Central concept in game theory and mathematical economics shapes understanding of rational decision-making in strategic interactions
• Provides insights into equilibrium outcomes and helps predict player behavior in various economic scenarios

Characteristics of dominant strategies

• Always yield the best outcome for a player regardless of opponents' choices
• Simplify decision-making process by providing a clear optimal course of action
• Remain unchanged even if opponents' strategies become known
• Often lead to Nash equilibrium in games with multiple players

Identifying dominant strategies

• Analyze payoff matrix to compare outcomes across all possible opponent strategies
• Check if a strategy consistently yields higher (or equal) payoffs than alternatives
• Utilize decision trees or extensive form games for sequential decision-making scenarios
• Apply mathematical techniques (linear programming) for complex games with multiple strategies

Examples in economic scenarios

• Pricing strategies in oligopolistic markets (undercutting competitors)
• Investment decisions in research and development (continuous innovation)
• Environmental policies (adopting clean technologies regardless of other countries' actions)
• Advertising strategies in competitive markets (maintaining a consistent brand presence)

Strictly dominant strategies

Definition and properties

• Strategy that always yields strictly better payoffs than any other strategy, regardless of opponents' choices
• Provides unambiguous optimal choice for a player in all scenarios
• Eliminates need to consider opponents' strategies when making decisions
• Often leads to unique Nash equilibrium in games with multiple strictly dominant strategies

Comparison with weakly dominant

• Strictly dominant strategies always yield strictly higher payoffs, while weakly dominant may allow for equal payoffs in some cases
• Strictly dominant strategies provide stronger predictions about player behavior
• Weakly dominant strategies may still allow for multiple equilibria, unlike strictly dominant ones
• Elimination of strictly dominated strategies is a more powerful tool for game simplification than weakly dominated strategies

Weakly dominant strategies

Definition and properties

• Strategy that yields payoffs at least as good as any other strategy, with some scenarios resulting in strictly better outcomes
• Provides a rational choice for players but may not always lead to unique equilibria
• Allows for indifference between strategies in some cases, complicating decision-making
• Often used in analyzing more complex economic scenarios (auctions, bargaining situations)

Comparison with strictly dominant

• Weakly dominant strategies may yield equal payoffs in some scenarios, while strictly dominant always yield better outcomes
• Weakly dominant strategies can lead to multiple equilibria, unlike strictly dominant ones
• Elimination of weakly dominated strategies requires more careful analysis than strictly dominated ones
• Weakly dominant strategies provide less definitive predictions about player behavior compared to strictly dominant strategies

Dominated strategies

Strictly dominated strategies

• Always yield lower payoffs than at least one other strategy, regardless of opponents' choices
• Rational players should never choose strictly dominated strategies
• Can be safely eliminated from consideration in game analysis
• Help simplify complex games by reducing the strategy space

Weakly dominated strategies

• Yield payoffs no better than another strategy, with some scenarios resulting in strictly worse outcomes
• May still be chosen by players in certain situations (trembling hand perfection)
• Require more careful analysis before elimination from game consideration
• Often used in refining Nash equilibria and analyzing evolutionary stable strategies

Elimination of dominated strategies

• Iterative process of removing dominated strategies to simplify game analysis
• Helps identify Nash equilibria in complex games with multiple strategies
• Can lead to unique solution in some games (dominance solvable games)
• May not always result in a single equilibrium, especially with weakly dominated strategies

Nash equilibrium vs dominance

Relationship between concepts

• Dominant strategies often lead to Nash equilibria, but not all Nash equilibria involve dominant strategies
• Elimination of dominated strategies can help identify Nash equilibria in complex games
• Nash equilibrium provides a broader solution concept for games without dominant strategies
• Dominance analysis often serves as a preliminary step in finding Nash equilibria

Differences in application

• Dominance focuses on individual player's optimal strategies, while Nash equilibrium considers all players simultaneously
• Dominance analysis can be applied to single-player decision problems, unlike Nash equilibrium
• Nash equilibrium can exist in games without dominant strategies (mixed strategy equilibria)
• Dominance provides stronger predictions about player behavior when applicable

Dominance in game theory

Prisoner's dilemma example

• Classic game theory scenario illustrates concept of dominant strategies
• Both players have a strictly dominant strategy to confess, leading to a suboptimal outcome
• Demonstrates how individual rationality can lead to collectively inferior results
• Serves as a model for various economic and social situations (environmental agreements, arms races)

Applications in oligopoly models

• Analyze firms' pricing and output decisions in markets with few competitors
• Cournot model uses best response functions to find equilibrium output levels
• Bertrand model demonstrates price competition can lead to marginal cost pricing
• Stackelberg model incorporates sequential decision-making and first-mover advantage

Limitations of dominance analysis

Incomplete information scenarios

• Dominance may not apply when players lack full knowledge of payoffs or opponents' strategies
• Bayesian games incorporate probabilistic beliefs about unknown information
• Requires more complex solution concepts (Bayesian Nash equilibrium)
• Introduces uncertainty into decision-making process, complicating strategy selection

Mixed strategy considerations

• Pure strategy dominance may not exist in games with mixed strategy equilibria
• Requires analysis of expected payoffs rather than deterministic outcomes
• Introduces probabilistic decision-making into game theory models
• Complicates identification and elimination of dominated strategies

Practical applications

Business strategy decisions

• Pricing strategies in competitive markets (penetration pricing, premium pricing)
• Product differentiation to create unique market positions
• Investment in research and development to maintain technological advantage
• Market entry decisions based on potential competitor responses

Public policy implications

• Design of incentive structures to encourage desired behaviors (tax policies, subsidies)
• Regulation of industries to prevent anticompetitive practices
• Environmental policies to address global challenges (carbon pricing, emissions trading)
• International trade agreements and negotiations based on game-theoretic principles

Mathematical representation

Payoff matrices for dominance

• Two-dimensional arrays represent players' payoffs for different strategy combinations
• Rows and columns correspond to players' strategies
• Entries contain payoff values for each player given strategy choices
• Facilitate visual analysis of dominance relationships and equilibria

Formal notation and proofs

• Utilize set theory and functions to define strategies and payoffs
• $s_i \in S_i$ represents player i's strategy from their strategy set
• $u_i(s_i, s_{-i})$ denotes player i's utility function given all players' strategies
• Formal proofs of dominance involve showing $u_i(s_i^*, s_{-i}) \geq u_i(s_i, s_{-i})$ for all $s_{-i}$ and $s_i \neq s_i^*$