Game theory is a mathematical framework that is used to analyze and predict the behavior of individuals or groups in competitive situations. It provides a structured approach to decision-making that can be applied in numerous fields such as economics, political science, psychology, and biology. In this article, we will discuss the different methods used to solve game theory problems.

**1. Dominant Strategy**

The dominant strategy method is one of the most straightforward ways to solve a game theory problem. It involves identifying the strategy that provides the best outcome for a player regardless of what their opponent chooses. If one player has a dominant strategy, then they will always choose it, and the other player’s choice becomes irrelevant.

__Example:__

Consider a game where two players are offered two choices each: cooperate or defect. If both players cooperate, they will receive $2 each.

If one player cooperates while the other defects, the defector receives $3 while the cooperator gets nothing. Lastly, if both players defect, they each get $1.

Using this method, we can see that defecting is always the best choice for both players since it provides more payoff than cooperating regardless of what their counterpart chooses.

**2. Nash Equilibrium**

The Nash equilibrium method is perhaps one of the most popular ways to solve game theory problems because it considers all possible outcomes and identifies strategies that are mutually beneficial for both players. A Nash equilibrium occurs when no player can improve their payoff by changing their strategy while their opponent keeps theirs.

Taking our previous example of two players playing either cooperate or defect with different payoffs associated with each outcome, we can use this method to find out if there exists any strategy pair such that neither player wants to change his/her decision given his/her opponent’s decision.

We can construct a table as shown below:

Cooperate | Defect | |

Cooperate | (2, 2) | (0, 3) |

Defect | (3, 0) | (1, 1) |

In this case, we can see that there are two Nash equilibria, (Defect, Defect) and (Cooperate, Cooperate). In other words, both players choosing to cooperate or defect results in the best outcome for both players.

**3. Minimax Method**

The minimax method is a decision-making strategy that involves selecting the option with the least maximum loss. It assumes that each player aims to minimize their maximum possible loss while playing the game.

Consider a game of rock-paper-scissors between two players. Each player has three options: rock, paper or scissors.

If both players choose the same option, it is a tie. If they choose different options, rock beats scissors (rock wins), paper beats rock (paper wins) and scissors beat paper (scissors win).

Using the minimax method to solve this game means that each player will aim to minimize their maximum potential loss. This means that they will choose an option that gives them the best chance of winning while also considering what their opponent might choose.

**Conclusion**

Game theory provides a structured approach to decision-making in competitive situations. The dominant strategy method identifies the best strategy regardless of what their opponent chooses while Nash equilibrium identifies mutually beneficial strategies for both players.

Lastly, the minimax method involves selecting the option with the least maximum loss. By employing any of these methods, players can make more informed decisions while playing games and increase their chances of success.