Mixed strategy Nash equilibria are a key concept in game theory, where players randomize their strategies to keep opponents guessing. This topic dives into how to calculate these equilibria, exploring the math behind finding optimal probabilities for each strategy.

Understanding mixed strategies is crucial for analyzing complex games without clear dominant strategies. We'll learn how to compute equilibria, interpret their meaning, and explore their existence in various game types. This knowledge is essential for predicting player behavior in strategic interactions.

Mixed Strategy Nash Equilibria

Computing Mixed Strategy Nash Equilibria

• Mixed strategy Nash equilibria occur when players randomize their strategies according to specific probabilities, and no player can improve their expected payoff by deviating from their mixed strategy
• In a mixed strategy Nash equilibrium, each player's strategy must make the other player indifferent between their available pure strategies
• To compute mixed strategy Nash equilibria, players assign probabilities to their pure strategies such that the expected payoffs for each pure strategy are equal
• The probabilities assigned to each pure strategy in a mixed strategy Nash equilibrium must sum to 1 for each player
• In a 2x2 game, the mixed strategy Nash equilibrium can be found by solving a system of linear equations derived from the players' expected payoffs (Matching Pennies)
• In larger games, finding mixed strategy Nash equilibria may require more advanced techniques, such as the Lemke-Howson algorithm or the use of linear programming (Rock-Paper-Scissors)

Properties of Mixed Strategy Nash Equilibria

• Mixed strategy Nash equilibria represent situations where players randomize their strategies to keep their opponents indifferent between their available pure strategies
• The expected payoffs in a mixed strategy Nash equilibrium are the same for all pure strategies played with positive probability, meaning that players are indifferent between these strategies
• Mixed strategy Nash equilibria can be seen as a way for players to protect themselves against exploitation by their opponents, as randomization prevents opponents from taking advantage of predictable patterns
• The probabilities assigned to each pure strategy in a mixed strategy Nash equilibrium indicate the frequency with which each strategy should be played in the long run (Battle of the Sexes)

Interpreting Mixed Strategies

Strategic Uncertainty

• Mixed strategy Nash equilibria can be interpreted as a state of strategic uncertainty, where players are unable to predict their opponent's specific action in any given round
• In a mixed strategy equilibrium, players are indifferent between their available pure strategies, making it difficult for opponents to anticipate their actions (Prisoner's Dilemma with mixed strategies)
• The randomization of strategies in a mixed strategy equilibrium creates an unpredictable environment, where players cannot rely on their opponents to choose a specific action consistently

Long-Run Frequencies and Expected Payoffs

• The probabilities assigned to each pure strategy in a mixed strategy Nash equilibrium indicate the frequency with which each strategy should be played in the long run
• Over many rounds of play, the mixed strategy probabilities represent the optimal distribution of strategies to maximize a player's expected payoff (Hawk-Dove game)
• The expected payoffs in a mixed strategy Nash equilibrium are calculated by multiplying the probabilities of each strategy combination by their respective payoffs and summing the results
• In a mixed strategy equilibrium, players are indifferent between their available pure strategies because the expected payoffs for each strategy are equal (Matching Pennies)

Existence of Mixed Equilibria

Existence Theorems

• Every finite strategic-form game has at least one Nash equilibrium, which can be either a pure strategy or a mixed strategy equilibrium (Nash's Theorem)
• If a game does not have a pure strategy Nash equilibrium, it must have at least one mixed strategy Nash equilibrium
• Games with a unique Nash equilibrium in pure strategies do not have mixed strategy Nash equilibria (Prisoner's Dilemma)
• In 2x2 games, a mixed strategy Nash equilibrium exists if neither player has a strictly dominant strategy (Battle of the Sexes)

Determining Existence

• The existence of mixed strategy Nash equilibria can be determined by analyzing the players' best response functions and looking for intersections or fixed points
• In games with continuous strategy spaces, the existence of mixed strategy Nash equilibria can be proven using fixed-point theorems (Brouwer's Fixed-Point Theorem)
• The existence of mixed strategy Nash equilibria can also be demonstrated through the use of graphical methods, such as best response curves or polytopes (Rock-Paper-Scissors)
• In some cases, the existence of mixed strategy Nash equilibria can be inferred from the structure of the game, such as the absence of pure strategy equilibria or the presence of cyclic best responses (Matching Pennies)

Finding Mixed Equilibria in Complex Games

Computational Methods

• In games larger than 2x2, finding mixed strategy Nash equilibria may require solving systems of nonlinear equations or using linear programming techniques (3x3 Rock-Paper-Scissors)
• The Lemke-Howson algorithm is a computational method for finding mixed strategy Nash equilibria in two-player games by identifying complementary pivoting solutions
• In games with more than two players, finding mixed strategy Nash equilibria becomes more complex and may require the use of specialized software or algorithms, such as the Govindan-Wilson algorithm (Three-player Matching Pennies)

Graphical and Evolutionary Methods

• Graphical methods, such as best response curves or polytopes, can be used to visualize and analyze mixed strategy Nash equilibria in games with continuous strategy spaces (Cournot duopoly)
• Evolutionary game theory techniques, such as replicator dynamics or adaptive learning, can be used to model the emergence of mixed strategy Nash equilibria in large populations of players (Hawk-Dove population dynamics)
• These methods simulate the process of strategy adaptation and convergence to equilibrium over time, providing insights into the stability and attractiveness of mixed strategy Nash equilibria (Evolutionary stable strategies)
• Evolutionary game theory can also be used to study the evolution of cooperation and the emergence of social norms through the analysis of mixed strategy equilibria in repeated games (Iterated Prisoner's Dilemma)