Mixed strategy Nash equilibria (MSNE) offer a way to analyze games where players randomize their strategies. By assigning probabilities to each pure strategy, players can create unpredictable outcomes that keep opponents guessing.

Calculating MSNE involves setting up equations based on expected payoffs and solving for strategy probabilities. This approach can be extended to larger games, providing insights into complex strategic interactions and helping players make optimal decisions.

Calculating Mixed Strategy Nash Equilibria

Calculation of mixed strategy equilibria

• Mixed strategy Nash equilibrium (MSNE) is a solution concept where players randomize their strategies
  • Each player assigns a probability to each pure strategy they can play
  • Players choose their strategies independently of one another
• To find MSNE in a 2x2 game:
  • Let $p$ be the probability of Player 1 playing Strategy 1 (e.g., Cooperate)
  • Let $q$ be the probability of Player 2 playing Strategy 1 (e.g., Defect)
  • Set up equations based on the expected payoffs for each player
    • Player 1's expected payoff from playing Strategy 1 equals Player 1's expected payoff from playing Strategy 2
    • Player 2's expected payoff from playing Strategy 1 equals Player 2's expected payoff from playing Strategy 2
  • Solve the system of equations simultaneously to find the values of $p$ and $q$ that satisfy both equations

Interpretation of mixed strategy results

• The values of $p$ and $q$ represent the probabilities with which each player should play their respective strategies in the MSNE
  • Player 1 should play Strategy 1 with probability $p$ and Strategy 2 with probability $1-p$ (e.g., Cooperate 60% of the time, Defect 40% of the time)
  • Player 2 should play Strategy 1 with probability $q$ and Strategy 2 with probability $1-q$ (e.g., Cooperate 30% of the time, Defect 70% of the time)
• In MSNE, players are indifferent between their strategies given the other player's mixed strategy
  • A player has no incentive to deviate from their own mixed strategy as long as the other player sticks to theirs
• MSNE can be interpreted as a stable state of the game where no player can improve their expected payoff by unilaterally changing their strategy

Extension to larger game matrices

• The process of finding MSNE can be extended to games with more than two strategies per player (e.g., Rock-Paper-Scissors)
• In an $m \times n$ game:
  • Let $p_1, p_2, \ldots, p_m$ be the probabilities of Player 1 playing each of their $m$ strategies
  • Let $q_1, q_2, \ldots, q_n$ be the probabilities of Player 2 playing each of their $n$ strategies
  • Set up equations based on the expected payoffs for each player
    1. For Player 1, the expected payoff from playing each strategy should be equal
    2. For Player 2, the expected payoff from playing each strategy should be equal
  • Solve the system of equations to find the values of $p_1, p_2, \ldots, p_m$ and $q_1, q_2, \ldots, q_n$
• The probabilities must sum to 1 for each player to ensure a valid probability distribution
  • $\sum_{i=1}^{m} p_i = 1$ for Player 1 (e.g., $p_1 + p_2 + p_3 = 1$ in a 3x3 game)
  • $\sum_{j=1}^{n} q_j = 1$ for Player 2 (e.g., $q_1 + q_2 + q_3 = 1$ in a 3x3 game)

Conditions for mixed strategy existence

• MSNE exist in games where no player has a pure strategy that strictly dominates all other strategies
• For a 2x2 game, MSNE exist if:
  • Neither player has a strictly dominant strategy (e.g., Prisoner's Dilemma)
  • The game does not have two pure strategy Nash equilibria (e.g., Battle of the Sexes)
• In larger games, MSNE may exist even if pure strategy Nash equilibria are present (e.g., Rock-Paper-Scissors has a unique MSNE despite having no pure strategy equilibria)
• The existence of MSNE is guaranteed in finite games by Nash's Existence Theorem
  • The theorem states that every finite game (a game with a finite number of players and strategies) has at least one Nash equilibrium, which can be either pure or mixed