Mixed strategy

What Is Mixed Strategy?

A mixed strategy is a probabilistic approach to decision-making within game theory, where a player chooses among their available pure strategies with specific probabilities rather than always picking a single action. This element of randomization makes a player's actions less predictable to opponents, which can be crucial in competitive scenarios. It represents a probability distribution over the set of possible actions. When a player employs a mixed strategy, their expected payoff is calculated based on these probabilities and the potential outcomes of each chosen action. The concept is central to understanding strategic interaction where deterministic choices might lead to predictable or exploitable patterns.

History and Origin

The foundational ideas behind mixed strategies emerged in the early 20th century. French mathematician Émile Borel is often credited with introducing the notion of a randomized strategy around 1920 when studying elementary duels. However, the rigorous mathematical framework and proof of their existence in a broader context came from Hungarian-American mathematician John von Neumann. In his 1928 paper "Zur Theorie der Gesellschaftsspiele" ("On the Theory of Board Games"), von Neumann proved the minimax theorem, demonstrating that every two-person zero-sum game has optimal mixed strategies.³⁰, ³¹

This work culminated in the seminal 1944 book Theory of Games and Economic Behavior, co-authored by von Neumann and economist Oskar Morgenstern, which is widely considered the birth of modern game theory.²⁷, ²⁸, ²⁹ Their groundbreaking text established a mathematical theory for economic and social organization based on games of strategy. While von Neumann's early work focused primarily on two-person zero-sum games, the concept of mixed strategies was later extended by John Nash in 1950, who proved that every finite n-player, non-zero-sum game has at least one Nash equilibrium in mixed strategies. This expansion solidified the significance of mixed strategies across a wider array of strategic environments.²⁵, ²⁶

Key Takeaways

• A mixed strategy involves selecting actions with specific probabilities, introducing an element of randomness.
• It is particularly useful in games where no single pure strategy consistently offers the best outcome.
• Mixed strategies are essential for the existence of Nash equilibrium in many games, especially those without a pure strategy Nash equilibrium.
• The concept helps players avoid predictable patterns that could be exploited by rational opponents.
• Calculating the expected payoff is crucial when employing a mixed strategy to assess potential outcomes.

Formula and Calculation

The calculation of a mixed strategy involves determining the probabilities with which a player should choose each of their pure strategies to make their opponent indifferent between their own pure strategies. This indifference is key to a mixed strategy Nash equilibrium. The expected payoff for a player using a mixed strategy is the sum of the payoffs from each pure strategy, weighted by the probability of choosing that strategy.

For a player I with two pure strategies, (S_{I1}) and (S_{I2}), and corresponding probabilities (p) and (1-p), and an opponent J with strategies (S_{J1}) and (S_{J2}), the expected payoff for Player I when Player J chooses (S_{J1}) is:

$E_I(S_{J1}) = p \cdot Payoff(S_{I1}, S_{J1}) + (1-p) \cdot Payoff(S_{I2}, S_{J1})$

Similarly, for Player J choosing (S_{J2}):

$E_I(S_{J2}) = p \cdot Payoff(S_{I1}, S_{J2}) + (1-p) \cdot Payoff(S_{I2}, S_{J2})$

In a mixed strategy Nash equilibrium, player I chooses (p) such that (E_I(S_{J1}) = E_I(S_{J2})) for player J. This makes player J indifferent between their strategies, preventing them from exploiting player I's choices. The same logic applies to player J choosing their probabilities to make player I indifferent.

Interpreting Mixed Strategy

Interpreting a mixed strategy goes beyond simply randomizing actions. While one interpretation suggests players literally randomize their choices (e.g., using a coin toss), other views are more common in economic analysis. A prominent interpretation is that a mixed strategy represents an opponent's belief about what a player might do, or the proportions of a large population of players choosing each pure strategy. For instance, if a player's mixed strategy assigns a 50% probability to action A and 50% to action B, it could mean that in a population, half the players choose A and half choose B.

In game theory, a mixed strategy is typically chosen because it is a "best response" to the other players' strategies, meaning no player can improve their expected payoff by unilaterally changing their own strategy.²⁴ This makes mixed strategies a vital part of understanding Nash equilibrium in situations where no pure strategy equilibrium exists, ensuring stability in strategic interaction. The optimal use of a mixed strategy aims to maximize a player's long-term average outcome by making their behavior unpredictable.

Hypothetical Example

Consider a classic example in game theory known as "Matching Pennies," a zero-sum game. Two players, Player A and Player B, each place a penny on a table simultaneously, either heads (H) or tails (T) up. Player A wins if the pennies match (both H or both T), and Player B wins if they don't match (one H, one T). Let's assign payoffs: Player A gets +1 and Player B gets -1 if they match; Player A gets -1 and Player B gets +1 if they don't match.

The payoff matrix for Player A (rows) and Player B (columns) looks like this:

In this game, there is no pure strategy Nash equilibrium. If Player A always plays H, Player B will always play T to win. But if Player B always plays T, Player A will switch to T to win, and so on. This creates an endless cycle of predictable, exploitable moves.

A mixed strategy resolves this. Player A chooses H with probability (p_A) and T with probability (1-p_A). Player B chooses H with probability (p_B) and T with probability (1-p_B). To achieve a mixed strategy Nash equilibrium, each player must make the other player indifferent between their pure strategies.

For Player A to make Player B indifferent between playing H and T:
Expected Payoff for B if B plays H = ((-1)p_A + (1)(1-p_A))
Expected Payoff for B if B plays T = ((1)p_A + (-1)(1-p_A))

Setting these equal:
(-p_A + 1 - p_A = p_A - 1 + p_A)
(1 - 2p_A = 2p_A - 1)
(2 = 4p_A \implies p_A = 0.5)

So, Player A plays H with 50% probability and T with 50% probability. Similarly, to make Player A indifferent, Player B must also play H with 50% probability and T with 50% probability. This probability distribution makes the game unpredictable for both players, leading to an expected payoff of zero for both players over many rounds.

Practical Applications

Mixed strategies, as a concept within game theory, have various practical applications in economics and competitive environments. While direct numerical calculation of mixed strategies might be complex in highly dynamic situations, the underlying principle of unpredictability and optimal randomization informs decision-making.

In business, firms operating in competitive markets, particularly oligopolies, may implicitly use mixed strategies in their pricing or marketing strategic interaction. For example, airlines may adjust ticket prices with a degree of randomness to avoid predictable patterns that competitors could exploit, reflecting a form of mixed strategy in action within market dynamics.²³ Similarly, companies might randomize the timing or nature of product launches or promotions to maintain a competitive advantage.²²

Beyond business, mixed strategies appear in areas like sports (e.g., a soccer goalie randomly choosing which way to dive on a penalty kick, or a tennis player varying serve direction) and auditing strategies (e.g., tax authorities randomizing audits to deter evasion).²⁰, ²¹ The design of auctions also heavily relies on game theory principles, including optimal bidding strategies that can involve mixed strategies to maximize seller revenue or efficient resource allocation.¹⁷, ¹⁸, ¹⁹

Limitations and Criticisms

Despite their theoretical elegance and importance in establishing Nash equilibrium in a wide range of games, mixed strategies face several limitations and criticisms, particularly when applied to real-world human decision-making.

One significant criticism is the assumption of perfect rational choice. Game theory often posits that players are perfectly rational agents who can calculate complex probability distributions and act accordingly.¹⁵, ¹⁶ In reality, individuals may struggle with true randomization and often exhibit cognitive biases or bounded rationality, making it challenging to implement precise mixed strategies.¹³, ¹⁴ Behavioral economists, who study the psychological influences on economic decision-making, note that real players often deviate from purely rational game theory predictions.⁹, ¹⁰, ¹¹, ¹²

Another critique revolves around the interpretation of mixed strategies. While mathematically sound, the idea of an individual literally randomizing their actions (e.g., using a random number generator for every decision) can seem unrealistic for many scenarios.⁸ Some scholars argue that mixed strategies are better understood as reflecting an observer's uncertainty about a player's choice or as aggregate behavior within a large population, rather than a conscious randomization by a single player.⁷

Furthermore, identifying the optimal mixed strategy can be complex, especially in games with many players or intricate payoff matrix structures. Games with mixed strategies can also have multiple equilibria, making it difficult to predict which outcome will occur without further refinement concepts.⁵, ⁶ The assumption that players are indifferent to pure strategies within a mixed strategy equilibrium also raises questions about their motivation to stick to a probabilistic choice.⁴

Mixed Strategy vs. Pure Strategy

The distinction between a mixed strategy and a pure strategy is fundamental in game theory.

A pure strategy involves a player choosing a single, specific action with certainty in a given situation. For example, if a company decides to always lower its prices in response to a competitor's price cut, that's a pure strategy. The player's choice is deterministic and entirely predictable if their strategy is known.

In contrast, a mixed strategy involves a player choosing among their available pure strategies with predetermined probabilities. Instead of always lowering prices, the company might lower prices 60% of the time and hold prices steady 40% of the time, based on a calculated optimal mixed strategy. The introduction of probability distribution makes the player's action unpredictable in any single instance, though the long-run frequency of their choices adheres to the specified probabilities.

The confusion between the two often arises because a pure strategy can be seen as a special case of a mixed strategy where one action is chosen with 100% probability and all others with 0% probability. However, the core difference lies in the element of randomization: pure strategies are fixed choices, while mixed strategies incorporate deliberate unpredictability to maximize expected payoff in situations where a fixed choice could be exploited.

FAQs

What is the main purpose of using a mixed strategy?

The main purpose of using a mixed strategy is to make a player's actions unpredictable to opponents, preventing them from exploiting a predictable pattern. This can lead to a more stable outcome, often a Nash equilibrium, in games where no single fixed choice would be optimal.³

Are mixed strategies always rational in real-world scenarios?

While theoretically rational within the framework of game theory assuming perfect rationality, the direct application of mixed strategies in real-world scenarios is debated. Human decision-making is influenced by psychological factors and may not always adhere to strict probabilistic choices, leading to deviations from theoretical predictions.²

Can a mixed strategy exist if there's a pure strategy Nash equilibrium?

Yes, a game can have both pure strategy Nash equilibria and mixed strategy Nash equilibria. However, mixed strategies become particularly important in games where no pure strategy Nash equilibrium exists, providing a way to find a stable outcome.

How do players determine the probabilities in a mixed strategy?

Players determine the probabilities in a mixed strategy such that their opponent is indifferent between their own pure strategies. This means that no matter which action the opponent chooses, their expected payoff remains the same. This indifference point helps create the equilibrium where neither player has an incentive to unilaterally change their strategy.¹