6.Mixed Strategies
Introduction to Mixed Strategies
In game theory, a mixed strategy refers to a strategy in which a player does not choose a single, fixed action but instead randomizes their choices, assigning probabilities to each possible action. This is in contrast to a pure strategy, where a player always chooses the same action in a given situation. Mixed strategies are essential when no pure strategy Nash Equilibrium exists or when a player’s best response requires unpredictability to avoid being exploited.
Mixed strategies allow players to unpredictably switch between available options, ensuring that they don’t become predictable or exploitable by their opponents. In situations where there’s no dominant strategy or when players are forced to make unpredictable choices to maximize their payoffs, mixed strategies offer an elegant solution.
Why Mixed Strategies Are Important
Mixed strategies are useful in competitive games where:
Predictability is dangerous: If a player consistently chooses the same strategy, they become easy to anticipate and can be outplayed by an opponent who adapts to their actions.
No pure strategy Nash Equilibrium exists: Some games don’t have a pure strategy equilibrium (like in Matching Pennies), so players must rely on mixed strategies to achieve a stable outcome.
Players must randomize to be competitive: In certain games, randomness is the key to ensuring that an opponent cannot predict your next move, as seen in games like Rock-Paper-Scissors.
Mixed vs. Pure Strategies
Pure Strategy: The player chooses one action and sticks to it in all situations. For example, if you’re playing Rock-Paper-Scissors, choosing Rock every time is a pure strategy.
Mixed Strategy: The player chooses a combination of actions, each with a certain probability. In Rock-Paper-Scissors, a mixed strategy might involve choosing Rock, Paper, or Scissors each with a probability of 1/3.
In many games, especially when there’s no clear advantage to choosing one strategy over another, players adopt mixed strategies to keep their actions unpredictable and to avoid being taken advantage of by the opponent.
Example 1: Rock-Paper-Scissors
Scenario: Two players, Player A and Player B, each choose one of three options: Rock, Paper, or Scissors. Rock beats Scissors, Scissors beat Paper, and Paper beats Rock.
The payoff matrix for this game is as follows:
If Player A and Player B both use pure strategies, one will always win and the other will always lose, resulting in a predictable pattern that can be exploited by the other player.
However, if both players randomize their choices (a mixed strategy), neither player can predict the other’s moves. In the mixed strategy Nash Equilibrium for Rock-Paper-Scissors, each player chooses Rock, Paper, and Scissors with equal probability, i.e., 1/3 for each option. This makes the game fair, as no player has an advantage.
Player A’s best strategy is to randomly choose Rock, Paper, or Scissors with a probability of 1/3 each.
Player B’s best strategy is the same: randomly choosing Rock, Paper, or Scissors with equal probability.
This is the Nash Equilibrium of the game, as neither player can improve their payoff by unilaterally changing their strategy.
Example 2: The Matching Pennies Game
Scenario: In this game, two players—Player A and Player B—simultaneously choose either Heads (H) or Tails (T). Player A wins if both players choose the same side, while Player B wins if they choose opposite sides.
The payoff matrix is as follows:
In this case, no pure strategy Nash Equilibrium exists. If Player A always plays Heads, Player B can simply play Tails to win every time. If Player A always plays Tails, Player B can play Heads to win. Therefore, both players need to randomize their choices to avoid being predictable.
To find the mixed strategy Nash Equilibrium, both players must choose Heads and Tails with equal probability (50%). Here’s why:
Player A’s expected payoff will be the same no matter whether they choose Heads or Tails, assuming Player B is also randomizing their choices.
Player B’s expected payoff will also be equal no matter whether they choose Heads or Tails, assuming Player A is randomizing.
Thus, the mixed strategy equilibrium is for both players to randomly choose Heads or Tails with equal probability (50%).
Finding Mixed Strategy Equilibria
To find mixed strategy Nash Equilibria, players must:
Set up the payoff matrix: List all possible strategies and outcomes for each player.
Define probabilities: Assign probabilities to each available action for the players. For example, let Player A choose Heads with probability ppp, and Tails with probability 1−p1 - p1−p.
Calculate expected payoffs: Calculate the expected payoff for each player based on the probabilities of the other player’s actions.
Solve for the equilibrium: Set up equations based on the condition that each player is indifferent between their strategies, meaning they get the same expected payoff no matter what action they take. This will give the probabilities that make each player’s mixed strategy optimal.
Real-World Applications of Mixed Strategies
Auctions: In an auction, bidders often use mixed strategies, especially when bidding for an item with uncertain value. By randomizing their bids, they can prevent other bidders from anticipating their next move and adjust to unpredictable auction environments.
Security: In cybersecurity, attackers may choose targets randomly to prevent defenders from predicting and strengthening defenses against certain types of attacks. This creates a mixed strategy scenario where both sides are constantly adapting.
Military Strategy: In warfare, unpredictability is crucial. A commander might use mixed strategies to keep the enemy from anticipating tactics. For example, varying attack times, locations, and troop movements ensures that the opponent cannot easily counter the strategy.
Marketing: Companies may use mixed strategies in their advertising or promotional efforts, randomly changing offers, timing, and channels to maintain competitive advantage without allowing rivals to predict their moves.
Questions for Reflection – Mixed Strategies
Why do mixed strategies help players avoid being exploited in competitive situations?
Answer: Mixed strategies prevent players from being predictable. By randomizing their choices, players ensure that their opponents can’t exploit patterns in their behavior, which is crucial in competitive games where predictability can lead to defeat.
In what types of games or situations might players be forced to use mixed strategies, even if a pure strategy seems more straightforward?
Answer: Players are forced to use mixed strategies when a pure strategy leads to a predictable pattern that can be exploited, such as in games like Matching Pennies or Rock-Paper-Scissors. Mixed strategies are also used when no pure strategy Nash Equilibrium exists, as seen in many competitive scenarios with multiple outcomes.
Can you think of a real-world example where a mixed strategy might lead to a better outcome than a pure strategy?
Answer: In business competition, companies may adopt mixed strategies in pricing to prevent competitors from undercutting them by constantly changing their prices in a predictable pattern. In sports, athletes may mix up their plays to make themselves unpredictable to opponents.
How does the use of mixed strategies impact decision-making in industries such as finance, advertising, or public policy?
Answer: In finance, mixed strategies are used in trading algorithms to avoid predictability. In advertising, companies randomize offers or campaigns to ensure they don’t become too predictable. In public policy, mixed strategies might be used in negotiations or trade agreements to maintain flexibility and respond to changing circumstances.
