In game theory, pure and mixed strategies are concepts used to describe the decision$-$making approaches of players in a game.

Pure Strategy:

A pure strategy is a specific, definite plan or course of action that a player decides to follow in a game.

In a pure strategy, a player makes a single, unambiguous choice, committing to a particular action with certainty.

Pure strategies are easy to understand and analyze, as they involve clear and distinct choices.

Example:

Consider a simple game of Rock, Paper, Scissors. Choosing to always play "Rock" as your move is a pure strategy. You consistently select the same option without any randomness or variability.

Mixed Strategy:

A mixed strategy involves a player randomly choosing between different pure strategies according to a probability distribution.

In a mixed strategy, a player does not commit to a single action but instead assigns probabilities to various pure strategies, deciding on each move based on chance.

Mixed strategies introduce an element of unpredictability and variability into a player's decisions.

Example:

In the game of Poker, a player might adopt a mixed strategy by bluffing with a certain probability. Instead of always bluffing or never bluffing, the player might bluff $30\%$ of the time and play their hand truthfully $70\%$ of the time. This randomness makes it more challenging for opponents to predict the player's actions.

Main Difference:

The main difference between pure and mixed strategies lies in the certainty of actions. Pure strategies involve a clear, deterministic choice, while mixed strategies introduce randomness through a probabilistic approach. Pure strategies are straightforward and easy to analyze, while mixed strategies add complexity and unpredictability to a player's decision$-$making.

In summary, a pure strategy is a definite, specific plan of action, while a mixed strategy involves probabilistic choices among different pure strategies. The choice between using a pure or mixed strategy often depends on the goals, information, and characteristics of the game being played.