🎱Game Theory Unit 5 – Mixed Strategies and Randomization

What's the Big Idea?

• Mixed strategies involve players randomizing their actions based on a probability distribution over their available strategies
• Randomization prevents opponents from exploiting a player's predictable behavior and can lead to better outcomes in certain strategic situations
• Mixed strategies are particularly relevant in games with no pure strategy Nash equilibria, where players can benefit from being unpredictable
• The concept of mixed strategies expands the scope of game theory beyond deterministic decision-making, allowing for more complex and realistic models of strategic interaction
• Understanding mixed strategies is crucial for analyzing a wide range of real-world scenarios, from competitive sports to military conflicts and business decisions
  • In poker, players often randomize their betting patterns to avoid being exploited by observant opponents
  • Companies may randomize their pricing strategies or product offerings to keep competitors guessing and maintain market share

Key Concepts

• Pure strategy: A deterministic choice of a single action or strategy
• Mixed strategy: A probability distribution over a player's available pure strategies
• Support of a mixed strategy: The set of pure strategies that are assigned positive probabilities in a mixed strategy
• Nash equilibrium: A state in which no player can improve their expected payoff by unilaterally changing their strategy
  • In a mixed strategy Nash equilibrium, players randomize their actions in a way that makes their opponents indifferent between their own strategies
• Indifference condition: The requirement that, in a mixed strategy Nash equilibrium, each player's expected payoff from playing any pure strategy in the support of their mixed strategy is equal
• Dominated strategies: Strategies that always yield lower payoffs than another strategy, regardless of the opponent's actions
  • Dominated strategies are never played in a Nash equilibrium, whether pure or mixed
• Payoff matrix: A table that summarizes the payoffs for each player based on their chosen strategies

Why Randomize?

• Randomization can help players avoid being exploited by opponents who might otherwise anticipate and counter their actions
• In some games, there may be no pure strategy Nash equilibria, meaning that players can always improve their payoffs by changing their deterministic strategy
  • In such cases, mixed strategies can lead to stable outcomes where no player has an incentive to deviate
• Randomization can be an effective way to keep opponents guessing and maintain an element of surprise
  • This is particularly relevant in competitive settings like sports, where predictable strategies can be easily countered
• Mixed strategies can help players hedge against uncertainty about their opponents' actions or preferences
• Randomization can also be useful in coordination games, where players need to avoid choosing the same action as their opponent
  • By randomizing, players can reduce the likelihood of coordination failures and improve their expected payoffs

Mixed Strategy Basics

• A mixed strategy is a probability distribution over a player's available pure strategies
• The probabilities assigned to each pure strategy must sum to 1, ensuring that the mixed strategy is a valid probability distribution
• Players choose their actions independently according to their mixed strategies, without knowing the specific actions chosen by their opponents
• The expected payoff of a mixed strategy is calculated by weighing the payoffs of each pure strategy by its assigned probability and summing the results
• In a mixed strategy Nash equilibrium, each player's mixed strategy must make their opponents indifferent between the pure strategies in the support of their own mixed strategy
  • This means that the expected payoff from playing any pure strategy in the support must be equal
• Dominated strategies are never played in a mixed strategy Nash equilibrium, as players can always improve their payoffs by shifting probability away from dominated strategies

Calculating Mixed Strategies

• To find a mixed strategy Nash equilibrium, players must solve a system of equations based on the indifference condition
• For each player, set up an equation equating the expected payoffs from playing each pure strategy in the support of their mixed strategy
  • The expected payoff is calculated by multiplying the probability of each opponent's action by the corresponding payoff and summing the results
• Use the fact that probabilities must sum to 1 to set up an additional equation for each player
• Solve the system of equations to find the probabilities that define each player's mixed strategy
  • This can be done using substitution, elimination, or matrix methods, depending on the complexity of the game
• Verify that the calculated probabilities indeed make each player indifferent between the pure strategies in their support and that no player can improve their payoff by deviating from the mixed strategy Nash equilibrium

Real-World Examples

• In soccer penalty kicks, goalkeepers often randomize their diving direction to avoid being predictable and exploited by kickers
  • Similarly, kickers may randomize their shot placement to keep goalkeepers guessing
• In military conflicts, armies may randomize their attack patterns or troop deployments to maintain an element of surprise and prevent enemies from anticipating their moves
• Businesses often use mixed strategies when setting prices or deciding on product offerings to keep competitors guessing and maintain market share
  • For example, a company may randomize between high and low prices to prevent competitors from consistently undercutting them
• In animal behavior, prey species may use randomized escape patterns to avoid being easily caught by predators
  • For instance, some lizards alternate between running and stopping in a seemingly random manner to confuse pursuing predators
• Randomized controlled trials in scientific research rely on the principles of mixed strategies to ensure unbiased treatment assignment and improve the validity of study results

Common Pitfalls

• Failing to recognize when a game has no pure strategy Nash equilibria and incorrectly assuming that players will choose deterministic actions
• Assigning probabilities to dominated strategies in a mixed strategy, which violates the rationality assumption of game theory
• Incorrectly calculating expected payoffs by failing to weigh each outcome by its respective probability
• Solving for mixed strategies without ensuring that the indifference condition holds for all players
• Interpreting mixed strategy probabilities as deterministic predictions of player behavior, rather than long-run frequencies of actions
• Neglecting to consider the strategic implications of mixed strategies and focusing solely on the mathematical calculations
• Overestimating the ability of players to implement complex mixed strategies in real-world settings, where cognitive limitations and external factors may influence decision-making

Putting It All Together

• Mixed strategies are a fundamental concept in game theory that allow for more realistic and complex models of strategic interaction
• By randomizing their actions according to a probability distribution, players can avoid being exploited by predictable behavior and achieve better outcomes in certain strategic situations
• Calculating mixed strategy Nash equilibria involves solving a system of equations based on the indifference condition, ensuring that players have no incentive to deviate from their mixed strategies
• Real-world examples of mixed strategies can be found in various domains, from competitive sports and military conflicts to business decisions and animal behavior
• When analyzing games using mixed strategies, it is essential to avoid common pitfalls such as assigning probabilities to dominated strategies or misinterpreting the meaning of mixed strategy probabilities
• A deep understanding of mixed strategies and their strategic implications is crucial for applying game theory to real-world problems and predicting the behavior of rational agents in complex environments
• By combining the concepts of randomization, expected payoffs, and Nash equilibria, mixed strategies provide a powerful tool for analyzing strategic interactions and guiding decision-making in a wide range of contexts