Game theory is a branch of mathematics that deals with decision-making and strategic interactions between rational individuals. It is commonly used in economics, political science, psychology, and other fields. In this article, we will explore the various methods of game theory.

Normal-Form Games

Normal-form games are the most common type of game in game theory. In this type of game, each player chooses a strategy from a set of available options.

The outcome of the game depends on the combination of strategies chosen by all players. Normal-form games are represented using a matrix known as a payoff matrix.

Example: Consider a two-player normal-form game in which both players can either choose to cooperate (C) or defect (D). The payoff matrix for this game is shown below:

 Player 2 Player 1 (3,3) (0,5) (5,0) (1,1)

In this example, if both players cooperate (C,C), they will receive a payoff of 3 each. If both players defect (D,D), they will receive a payoff of 1 each. If one player cooperates and the other defects (C,D or D,C), the defector receives a higher payoff than the cooperator.

Extensive-Form Games

In extensive-form games, players make decisions sequentially rather than simultaneously. These types of games are represented using a tree-like structure called a game tree.

Example: Consider a two-player game in which Player 1 chooses whether to go left or right, and then Player 2 chooses whether to go up or down. The game tree for this game is shown below:

In this example, Player 1 chooses whether to go left or right. If they go left, Player 2 chooses whether to go up or down. If they go right, the game ends with a payoff of -1 for both players.

Cooperative Games

Cooperative games are games in which players can form coalitions and cooperate with each other to achieve a common goal. In this type of game, the payoff is divided among the members of the coalition.

Example: Consider a cooperative game in which there are three players who must divide \$100 among themselves. The players can form coalitions and split the money as they wish. The payoff for each coalition is shown below:

• [(50), (25,25), (0)]
• [(30), (20,10), (40)]
• [(10), (40,50), (0)]

In this example, the first coalition consists of one player who receives \$50 while the other two receive nothing. The second coalition consists of two players who split \$25 each while the third player receives nothing.

Bayesian Games

Bayesian games are games in which some aspects of the game are unknown to one or more players. Each player has a belief about what these unknown aspects might be.

Example: Consider a Bayesian game in which there are two players who must choose between two options: A or B. One player knows which option is better but does not know the other player’s preference.

The other player knows their own preference but does not know which option is better. The payoff matrix for this game is shown below:

 Player 2 Player 1 (5,3) (0,1) (3,0) (1,5)

In this example, if Player 1 chooses A and Player 2 chooses B, Player 1 receives a payoff of 3 and Player 2 receives a payoff of 5. If Player 1 chooses A and Player 2 also chooses A, Player 1 receives a payoff of 5 and Player 2 receives a payoff of 3.

### Conclusion

Game theory provides a useful framework for understanding strategic interactions between rational individuals. There are several types of games in game theory including normal-form games, extensive-form games, cooperative games, and Bayesian games.

Each type of game has its own unique characteristics and requires different methods of analysis. By understanding these methods, we can better understand the strategies that players use to make decisions in various situations.