# Mixed strategies

## Overview of Mixed Strategies

• Mixed strategies are a vital component of game theory where a player opts to randomise their actions, choosing different strategies with certain probabilities to keep their opponents guessing.
• Unlike pure strategies, where a player consistently plays a single best move, mixed strategies add an element of unpredictability to a game.
• Adopting a mixed strategy approach doesn’t necessarily mean each option is equally likely; different moves could be assigned varying probabilities depending on the player’s estimation of their success.

## Understanding Probability in Mixed Strategies

• The key to mastering mixed strategies involves comprehending how probability affects outcomes. Probabilities assigned to a player’s actions must sum to 1, representing a 100% certainty that one of the strategies will be played.
• The use of probabilities in this aspect of game theory widens the scope of possible outcomes beyond those stipulated in original payoff matrices, creating a more dynamic and less deterministic play environment.

## Finding an Optimal Mixed Strategy

• Determining an optimal mixed strategy often involves using mathematical techniques, typically based on linear programming or calculus, to ascertain the probabilities that maximise the player’s expected payoff.
• By varying these probabilities and calculating the resultant expected payoff, a player can identify the set of probabilities (or mixed strategy) that maximises their expected gain.

## Purpose and Application of Mixed Strategies

• Application of mixed strategies is commonly found in situations where repeat play is involved, and predictability needs to be minimised. Examples include poker, sports tactics, and financial market investments.
• In particular, mixed strategies are often the key to finding Nash equilibrium in many games which do not have one under pure strategies.
• Some games may use mixed strategies as part of their dominant strategy, especially when no single best response exists.

## Challenges with Mixed Strategies and Expected Payoff

• A significant challenge with mixed strategies is to accurately predict the opponent’s responses, especially if they are also playing a mixed strategy.
• Care should be taken to note that the optimum expected payoff doesn’t guarantee the optimum outcome in every iteration of the game; it refers to the average payoff over many repeated games.

Repititon of games involving mixed-strategy situations will enhance your understanding and intuition to handle these dynamic strategic decisions.