Nash equilibrium is a crucial concept in game theory, representing a stable state where no player can benefit by changing their strategy alone. It applies to various strategic situations, from the Prisoner's Dilemma to market competition and political campaigns.

While powerful, Nash equilibrium has limitations. It assumes full rationality and common knowledge, which may not reflect real-world scenarios. Pure and mixed strategy equilibria offer different ways to model player behavior, each with unique applications and interpretations.

Nash Equilibrium Fundamentals

Characteristics of Nash equilibrium

• Represents a stable state where no player benefits from unilaterally changing their strategy given the strategies of the other players
• Consists of a set of strategies (one per player) where each strategy is the best response to the others' strategies
• Players cannot improve their payoff by changing their strategy alone
• Self-enforcing as players lack the incentive to deviate from their equilibrium strategy
• Involves simultaneous decision-making without knowing the choices of the other players

Nash Equilibrium in Practice

Nash equilibria in strategic situations

• Prisoner's Dilemma
  • Two suspects are questioned separately and must choose to either confess or stay silent
  • The Nash equilibrium is for both suspects to confess, despite a better outcome being possible if both stayed silent
  • Demonstrates how individual rationality can lead to a suboptimal collective result
• Battle of the Sexes
  • A couple is deciding on their evening plans, but they have different preferences (opera vs football game)
  • There are two pure strategy Nash equilibria: both attend the opera or both attend the football game
  • There is also a mixed strategy Nash equilibrium where they randomize their choice based on probabilities $p$ and $1-p$ such that each player is indifferent between the options given the other's mixed strategy

Real-world applications of Nash equilibrium

• Market competition
  • In an oligopoly market, the equilibrium may have firms setting prices or quantities above the competitive level, reducing consumer surplus
  • The Cournot model involves firms simultaneously choosing quantities, with the Nash equilibrium quantities depending on production costs and market demand
  • The Bertrand model involves firms simultaneously choosing prices, with the Nash equilibrium involving pricing at marginal cost if the goods are perfect substitutes
• Political campaigns
  • The median voter theorem suggests that candidates will converge on similar policies in equilibrium to avoid losing votes by deviating from the preferences of the median voter
  • The Hotelling model involves two ice cream stands on a beach choosing locations to maximize their market share, with the Nash equilibrium being both locating in the middle
• International conflicts
  • An arms race equilibrium involves countries heavily investing in their military despite the potential benefits of mutual disarmament
  • The Hawk-Dove game involves aggressive and passive strategies for resource disputes, with the Nash equilibrium depending on the value of the resource and the cost of fighting

Nash Equilibrium Refinements and Limitations

Limitations of Nash equilibrium

• Assumes full rationality, but people have bounded rationality and make decisions based on emotions, biases, and incomplete information
• Assumes common knowledge of the game structure and player rationality, which may not hold with imperfect or asymmetric information
• The existence of multiple equilibria can make predicting the outcome difficult without additional refinements or selection criteria
• Does not account for players learning, adapting, and changing strategies over time in repeated interactions

Pure vs mixed strategy equilibria

• Pure strategy equilibrium
  • Players choose a single strategy with 100% probability
  • Applies when players have a strictly dominant strategy or when a single strategy is the best response to others' strategies
  • Examples include the Prisoner's Dilemma, Cournot duopoly, and the median voter theorem
• Mixed strategy equilibrium
  • Players randomize their strategy choice based on probabilities
  • Applies when no pure strategy equilibrium exists or to make actions unpredictable
  • Requires player indifference between the randomized strategies given the others' mixed strategies
  • Examples include the Battle of the Sexes, Hawks-Dove game, and tennis serve directions
• Games can have both pure and mixed equilibria (e.g., Battle of the Sexes) or only one type (e.g., Matching Pennies only has mixed equilibria)