Game Theory is a branch of mathematics that deals with the study of decision-making among individuals or groups. It analyzes how these individuals or groups interact and make choices in different scenarios.

In Game Theory, strategies refer to the different options available to players to reach their objectives. Pure Strategies in Game Theory are an important concept as they help players make optimal decisions.

A Pure Strategy is a single choice made by a player that determines their actions throughout the game. It is a strategy that does not involve any randomization or uncertainty and is based on a fixed course of action. In other words, it is a strategy that is not dependent on chance or probability.

There are two types of Pure Strategies in Game Theory – Dominant and Nash Equilibrium.

Dominant Strategy:

A Dominant Strategy refers to the best option available to a player regardless of the choices made by other players. In other words, it is an optimal choice for a player irrespective of what other players do. This means that if all players choose their Dominant Strategies, it will lead to the best possible outcome for all players involved.

For example, imagine two companies competing against each other in the same market. They both have two options – either lower their prices or maintain them.

If both companies lower their prices, they both benefit as they attract more customers. However, if one company lowers its prices while the other maintains them, the former gains more customers while the latter loses out on revenue and market share. Therefore, lowering prices becomes the dominant strategy for both companies as it leads to the best possible outcome regardless of what their competitor does.

Nash Equilibrium:

Nash Equilibrium refers to a situation where each player chooses their best strategy given what others have chosen. It is named after John Nash who developed this concept in his famous paper ‘Non-Cooperative Games’ in 1950.

In Nash Equilibrium, no player can improve their outcome by changing their strategy as they are already making the best possible choice given what others have done. It is a stable state where each player is satisfied with their decision and no one has an incentive to change.

For example, imagine two players playing a game where they can choose between two options – either cooperate or defect. If both players cooperate, they both receive a reward of 3 points.

If one player defects while the other cooperates, the defector receives 5 points while the cooperator receives nothing. If both players defect, they both receive 1 point.

In this scenario, Nash Equilibrium is achieved when both players choose to defect as it is the best option for them given what the other player might do. Both players know that if they cooperate, there is a chance that the other player might defect and gain more points. Therefore, choosing to defect becomes their optimal choice in this situation.

In conclusion, Pure Strategies in Game Theory are an important concept that helps players make optimal decisions based on fixed courses of action. Dominant Strategies and Nash Equilibrium are two types of Pure Strategies that help players achieve their goals and reach better outcomes in different scenarios. Understanding these concepts can help individuals make better decisions not just in games but also in real-world situations involving competition and strategic thinking.