4.3 Mixed strategy Nash equilibrium
3 min read•august 7, 2024
in mixed strategies adds unpredictability to games. Players choose strategies randomly based on probability distributions, making it harder for opponents to predict moves. This concept is crucial when pure strategy equilibria don't exist.
Mixed strategies involve calculating expected payoffs and applying the . Players must be equally likely to choose any strategy in their support. This approach helps solve games like and , where randomization is key.
Mixed Strategies and Probability Distributions
Defining Mixed Strategies
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- represents a over a player's set of pure strategies
- Involves choosing a pure strategy randomly according to the probability distribution
- Allows players to introduce an element of unpredictability into their decision-making process
- Useful when there is no pure strategy Nash equilibrium or when players want to avoid being exploited
Probability Distributions in Mixed Strategies
- Probability distribution assigns a probability to each pure strategy in a player's strategy set
- Probabilities must sum to 1, ensuring that the player chooses a strategy with certainty
- Mixed strategy can be represented as a vector of probabilities, with each element corresponding to a pure strategy
- Probability distributions capture the likelihood of a player choosing each pure strategy in their mixed strategy
Randomization and Support
- Randomization is a key aspect of mixed strategies, as players randomly select a pure strategy based on the probability distribution
- Randomization prevents opponents from predicting and exploiting a player's actions
- Support of a mixed strategy refers to the set of pure strategies that are assigned positive probabilities
- Pure strategies outside the support have zero probability of being played in the mixed strategy
- Support can be a subset of the player's entire strategy set, depending on the probability distribution
Expected Payoffs and Indifference
Calculating Expected Payoffs
- is the average payoff a player anticipates receiving when playing a mixed strategy against an opponent's mixed strategy
- Calculated by multiplying the probability of each outcome by its corresponding payoff and summing these products
- Represents the long-run average payoff a player would receive if the game were repeated many times
- Allows players to compare the effectiveness of different mixed strategies and make informed decisions
Indifference Principle in Mixed Strategies
- Indifference principle states that in a mixed strategy Nash equilibrium, each player must be indifferent between all pure strategies in their support
- Implies that the expected payoff of each pure strategy in the support must be equal
- If a player strictly prefers one pure strategy over another, they would always choose the preferred strategy, violating the concept of a mixed strategy
- Indifference principle ensures that players have no incentive to deviate from their mixed strategy, as all pure strategies in the support yield the same expected payoff
Applications of Mixed Strategy Nash Equilibrium
Matching Pennies Game
- Matching pennies is a two-player game where each player simultaneously chooses either "Heads" or "Tails"
- Player 1 wins if the choices match, while Player 2 wins if the choices differ
- No pure strategy Nash equilibrium exists, as players have an incentive to switch their choice if the opponent's choice is known
- Mixed strategy Nash equilibrium involves both players choosing "Heads" and "Tails" with equal probability (0.5 each)
- Randomization prevents players from being exploited and results in an expected payoff of zero for both players
Rock-Paper-Scissors Game
- Rock-paper-scissors is a classic game where players simultaneously choose either Rock, Paper, or Scissors
- Rock beats Scissors, Scissors beats Paper, and Paper beats Rock
- Similar to matching pennies, there is no pure strategy Nash equilibrium
- Mixed strategy Nash equilibrium involves players choosing each option with equal probability (1/3 for Rock, Paper, and Scissors)
- Randomization ensures that no player can gain an advantage by predicting their opponent's move
- Expected payoff for both players is zero in the mixed strategy Nash equilibrium
Key Terms to Review (16)
Dominant Strategy: A dominant strategy is a course of action that yields the highest payoff for a player, regardless of the strategies chosen by other players. This concept is key in understanding how individuals or firms make decisions in strategic situations where their outcomes depend on the choices of others.
Expected Payoff: Expected payoff refers to the average outcome that a player can anticipate when making a decision in a strategic setting, calculated by weighing the potential payoffs of all possible actions by their probabilities. This concept is crucial in determining optimal strategies within mixed strategy Nash equilibria, where players randomize their choices to keep opponents indifferent among their options. The expected payoff helps players evaluate their strategies against others and predict outcomes based on uncertain future events.
Indifference Principle: The indifference principle refers to the idea that a player in a game is indifferent between different strategies when they yield the same expected payoff. This concept is crucial in mixed strategy Nash equilibria, where players randomize their strategies to keep opponents uncertain about their actions, leading to a situation where all players are satisfied with their choices as no one has an incentive to deviate.
John Nash: John Nash was an influential mathematician and economist best known for his groundbreaking work in game theory, particularly the concept of Nash equilibrium. His theories have fundamentally shaped our understanding of strategic interactions among rational decision-makers, making them essential for analyzing competitive behaviors in various fields, including economics, political science, and biology.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and polymath, widely regarded as one of the founders of game theory. His groundbreaking work laid the foundation for analyzing strategic interactions among rational decision-makers, influencing fields such as economics, computer science, and social sciences.
Market Competition: Market competition refers to the dynamic environment in which businesses and firms compete for customers and market share. This competitive landscape influences pricing, product offerings, and overall business strategies as companies strive to attract consumers and outperform their rivals. In the context of game theory, market competition can lead to various strategic interactions, including mixed strategies where firms randomize their actions to maintain a competitive edge.
Matching Pennies: Matching pennies is a two-player zero-sum game where each player has two strategies: to choose heads or tails. The game is structured such that one player wins if both players choose the same side of the coin, while the other player wins if they choose opposite sides. This setup highlights the strategic complexities of mixed strategies, rationalizability, and normal form representation, serving as a foundational example in game theory.
Minimax Theorem: The minimax theorem is a fundamental principle in game theory that states that in zero-sum games, there exists a strategy for each player such that the maximum loss that a player can incur is minimized. This theorem asserts that each player's optimal strategy leads to a situation where neither player can benefit by changing their strategy unilaterally, linking the concept of optimal strategies with the existence of equilibrium in competitive situations.
Mixed strategy: A mixed strategy is a strategy in which a player randomizes over two or more available actions, assigning a probability to each action they might take. This concept is crucial because it allows players to keep their opponents guessing and can lead to outcomes where no pure strategy Nash equilibrium exists, thus providing insights into the strategic decisions players make in various scenarios.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where no player can benefit by unilaterally changing their strategy if the strategies of the other players remain unchanged. This means that each player's strategy is optimal given the strategies of all other players, resulting in a stable outcome where players have no incentive to deviate from their chosen strategies.
Negotiation Strategies: Negotiation strategies are the planned approaches individuals or groups use to reach an agreement or resolve a conflict. These strategies often involve tactics that account for the preferences and actions of the other party, which can lead to more favorable outcomes through cooperation or competition. Understanding negotiation strategies is essential for analyzing how players in a game context choose their actions based on expected moves from others, particularly when considering mixed strategies to achieve equilibrium.
Pareto Efficiency: Pareto efficiency is an economic state where resources are allocated in a way that no individual can be made better off without making someone else worse off. This concept emphasizes the idea of optimal distribution of resources among players in a game, relating closely to strategies, payoffs, and the rational behavior of individuals involved.
Probability Distribution: A probability distribution is a mathematical function that describes the likelihood of different outcomes in a random variable. It provides a complete description of the probabilities associated with each possible outcome, whether in discrete or continuous form. This concept is crucial for understanding decision-making under uncertainty, especially in games where players must consider mixed strategies, make decisions based on beliefs about other players' types, and handle situations involving sequential moves with incomplete information.
Rock-paper-scissors: Rock-paper-scissors is a simple hand game that serves as a classic example of game theory, particularly illustrating concepts such as mixed strategy Nash equilibrium and evolutionary stable strategies. In this game, each player simultaneously chooses one of three options: rock, paper, or scissors, where rock crushes scissors, scissors cut paper, and paper covers rock. This interaction showcases how players can use randomness to make their choices unpredictable, which connects to deeper strategic considerations in competitive scenarios.
Simultaneous Game: A simultaneous game is a type of strategic interaction where players make decisions at the same time, without knowing the choices of the other participants. This setup creates uncertainty as each player's strategy affects the outcome for all players involved. The inability to observe opponents' actions before making a choice makes simultaneous games distinct and often leads to mixed strategies, as players aim to optimize their payoffs based on anticipated responses from others.
Zero-sum game: A zero-sum game is a situation in game theory where one player's gain is exactly balanced by the losses of other players. In these games, the total utility or benefit available is fixed, meaning that any advantage gained by one participant comes at the expense of another. This concept is essential for understanding competitive scenarios and helps in analyzing strategic interactions across various fields.