In game theory, pure strategies involve always choosing one action, while mixed strategies use probabilities to randomize choices. Mixed strategies offer more flexibility and can prevent opponents from exploiting predictable patterns. They're crucial when there's no pure strategy Nash equilibrium.

Mixed strategies shine in competitive scenarios where unpredictability is key, like sports or military conflicts. They help players secure the best expected payoff in uncertain situations. However, mixed strategies can lead to more complex outcomes and potentially lower payoffs compared to pure strategies.

Pure vs Mixed Strategies

Defining Pure and Mixed Strategies

• Pure strategies are actions that players choose with a probability of 1 meaning they will always choose that particular action in a given situation
• Mixed strategies are probability distributions over the set of available actions where players randomly choose actions based on predetermined probabilities
  • In a mixed strategy, the probability of choosing each action must sum up to 1 ensuring that the player selects an action with certainty

Relationship between Pure and Mixed Strategies

• Pure strategies can be considered a special case of mixed strategies where the probability of choosing one action is 1 and the probability of choosing other actions is 0
• Mixed strategies allow for a wider range of strategic choices compared to pure strategies by incorporating randomization
• The use of mixed strategies can prevent opponents from exploiting predictable patterns in a player's decision-making which is a limitation of pure strategies

Mixed Strategies: Optimal Scenarios

Games with No Pure Strategy Nash Equilibrium

• Mixed strategies are optimal when there is no pure strategy Nash equilibrium in a game meaning that no player can improve their payoff by unilaterally changing their strategy
• In games with a mixed strategy Nash equilibrium, players can exploit their opponent's predictability if they use pure strategies making mixed strategies the best response
  • For example, in the game of "Rock-Paper-Scissors," always choosing "Rock" (a pure strategy) can be easily exploited by an opponent who adapts and consistently selects "Paper"

Competitive Scenarios Favoring Unpredictability

• Mixed strategies are often employed in competitive scenarios where players aim to be unpredictable such as in sports, military conflicts, or business competition
  • In tennis, players may randomize between serving to the opponent's forehand or backhand to keep them guessing and off-balance
  • In military conflicts, commanders may employ a mix of offensive and defensive strategies to avoid being predictable and vulnerable to enemy tactics
• Games with symmetric payoff structures, such as "Matching Pennies," typically have mixed strategy equilibria as players aim to avoid being predictable

Rationale for Mixed Strategies

Introducing Unpredictability

• Mixed strategies introduce an element of unpredictability making it difficult for opponents to anticipate and counter a player's actions
  • In poker, players may randomize their betting patterns (bluffing) to prevent opponents from accurately gauging the strength of their hand
• By randomizing their actions according to a specific probability distribution, players can prevent their opponents from exploiting any discernible patterns in their decision-making

Securing the Best Expected Payoff

• Mixed strategies can help players secure the best possible expected payoff in situations where no pure strategy guarantees a better outcome
  • In a game of "Chicken," where two players drive towards each other, a mixed strategy of swerving or not swerving can lead to a better expected payoff than the pure strategies of always swerving or never swerving
• Employing mixed strategies allows players to balance risk and reward by varying their actions based on the likelihood of different outcomes

Mixed Strategies: Impact on Outcomes

Complexity and Dynamism in Game Outcomes

• Mixed strategies can lead to more complex and dynamic game outcomes compared to pure strategies as players continually adapt to their opponents' randomized actions
  • In a repeated "Prisoner's Dilemma," players using mixed strategies may engage in a series of cooperations and defections based on their chosen probabilities leading to a more intricate pattern of outcomes than pure strategies
• The use of mixed strategies can result in a mixed strategy Nash equilibrium where no player has an incentive to deviate from their chosen probability distribution over actions

Potential Drawbacks and Challenges

• In games with mixed strategy equilibria, the expected payoffs for players are typically lower than in games with pure strategy equilibria as the randomization of actions reduces the likelihood of consistently achieving the best possible outcome
• The effectiveness of mixed strategies depends on players correctly identifying the optimal probability distribution for their actions which may require extensive analysis of the game's payoff structure and their opponents' tendencies
  • Miscalculating the optimal mixed strategy can lead to suboptimal outcomes and vulnerability to exploitation by opponents
• The use of mixed strategies can prolong games and make it more difficult to reach a definitive resolution as players continue to randomize their actions in an attempt to outmaneuver their opponents
  • In a protracted conflict, the use of mixed strategies by both sides may lead to a stalemate or a war of attrition rather than a clear victory for either party