🆚Game Theory and Economic Behavior Unit 4 – Nash Equilibrium: Pure & Mixed Strategies

Key Concepts

• Nash equilibrium represents a stable state in a game where no player has an incentive to unilaterally change their strategy
• Pure strategies involve players choosing a single action with certainty
• Mixed strategies allow players to randomize over their available actions according to a probability distribution
• Best response refers to a player's optimal strategy given the strategies of the other players
• Payoff matrix summarizes the outcomes for each combination of strategies chosen by the players
• Dominant strategy provides a player with the highest payoff regardless of the other players' actions
• Zero-sum games have the property that one player's gain is always equal to the other player's loss

Nash Equilibrium Basics

• Nash equilibrium is a fundamental concept in game theory that describes a stable state in a game
• In a Nash equilibrium, each player's strategy is a best response to the strategies of the other players
• No player can improve their payoff by unilaterally deviating from their equilibrium strategy
• Nash equilibrium can be reached through pure strategies or mixed strategies
• The existence of a Nash equilibrium does not guarantee that it is unique or Pareto optimal
• Nash equilibrium assumes that players are rational, have complete information, and act simultaneously
• The concept of Nash equilibrium has wide-ranging applications in economics, political science, and other social sciences

Pure Strategy Nash Equilibrium

• Pure strategy Nash equilibrium occurs when each player chooses a single action with certainty
• In a pure strategy equilibrium, no player has an incentive to unilaterally deviate from their chosen strategy
• Pure strategy equilibria can be identified by finding the best response for each player given the other players' strategies
• Dominant strategy equilibrium is a special case of pure strategy equilibrium where each player has a dominant strategy
• Coordination games (Stag Hunt, Battle of the Sexes) often have multiple pure strategy equilibria
• Prisoner's Dilemma is a classic example of a game with a unique pure strategy equilibrium that is not Pareto optimal
  • In Prisoner's Dilemma, both players confessing is the unique Nash equilibrium, despite mutual cooperation being Pareto superior

Mixed Strategy Nash Equilibrium

• Mixed strategy Nash equilibrium involves players randomizing over their available actions according to a probability distribution
• Players choose a probability distribution over their actions rather than a single action with certainty
• In a mixed strategy equilibrium, each player's mixed strategy is a best response to the other players' mixed strategies
• Players are indifferent between the actions they play with positive probability in a mixed strategy equilibrium
• The expected payoff for each action played with positive probability must be equal in a mixed strategy equilibrium
• Mixed strategy equilibria can exist even when no pure strategy equilibria exist (Matching Pennies)
• Computing mixed strategy equilibria involves solving a system of equations based on the players' indifference conditions

Finding Nash Equilibria

• Finding Nash equilibria involves identifying the best response strategies for each player
• In games with finite strategy spaces, Nash equilibria can be found by examining the payoff matrix
• Identify the best response for each player given the other players' strategies
• Look for strategy profiles where each player's strategy is a best response to the others' strategies
• In games with continuous strategy spaces, finding Nash equilibria may require solving a system of equations or inequalities
• Iterated elimination of strictly dominated strategies can simplify the process of finding Nash equilibria
  • Strictly dominated strategies are never played in a Nash equilibrium and can be safely removed from consideration
• Brouwer's Fixed Point Theorem guarantees the existence of a Nash equilibrium in finite games with mixed strategies

Applications in Economics

• Nash equilibrium is widely used in economics to model strategic interactions between firms, consumers, and other economic agents
• Oligopoly models (Cournot, Bertrand) use Nash equilibrium to analyze firm behavior and market outcomes
• In Cournot competition, firms simultaneously choose their output levels, and the Nash equilibrium determines the market price and quantities
• In Bertrand competition, firms simultaneously choose their prices, and the Nash equilibrium determines the market shares and profits
• Nash bargaining solution is used to model bargaining outcomes between two parties
• Public goods provision and externalities can be analyzed using Nash equilibrium concepts
• Auction theory relies on Nash equilibrium to characterize bidding strategies and revenue outcomes

Common Mistakes and Pitfalls

• Confusing Nash equilibrium with Pareto optimality or social welfare maximization
  • Nash equilibrium is a stability concept and does not necessarily lead to socially optimal outcomes
• Failing to consider all possible deviations when checking for Nash equilibria
  • A strategy profile is a Nash equilibrium only if no player can unilaterally improve their payoff by deviating
• Incorrectly computing expected payoffs in mixed strategy equilibria
  • The expected payoff for each action played with positive probability must be equal in a mixed strategy equilibrium
• Ignoring the possibility of multiple Nash equilibria in a game
  • Some games may have multiple pure strategy, mixed strategy, or a combination of both types of equilibria
• Assuming that players can coordinate on a particular equilibrium without explicit communication
  • Nash equilibrium does not address the issue of equilibrium selection when multiple equilibria exist
• Misinterpreting mixed strategies as players actively randomizing their actions
  • Mixed strategies represent a player's beliefs about the other players' actions and can arise from population distributions or learning processes

Practice Problems

• Find all pure strategy Nash equilibria in a 2x2 game with the following payoff matrix:
  • $\begin{pmatrix} (2, 1) & (0, 0) \\ (0, 0) & (1, 2) \end{pmatrix}$
• Determine whether the following strategy profile is a Nash equilibrium: Player 1 plays A with probability 1/3 and B with probability 2/3, while Player 2 plays C with probability 1/4 and D with probability 3/4.
• Consider a Cournot duopoly where two firms simultaneously choose their output levels. The market demand is given by $P = 100 - Q$, where $P$ is the price and $Q$ is the total output. Each firm has a constant marginal cost of $20. Find the Nash equilibrium output levels and profits for each firm.
• In a first-price sealed-bid auction with two risk-neutral bidders and private values uniformly distributed between 0 and 100, find the symmetric Nash equilibrium bidding strategy.
• Analyze the following extensive-form game and find all pure strategy and mixed strategy Nash equilibria:

       Player 1
      /     \
     A       B
    /\      /\
  

Player 2 Player 2 /\ /
C D E F (2,1) (0,0) (1,2) (3,1)