Game theory is a mathematical study of decision making, conflict resolution, and strategy. It is widely used in various fields like economics, political science, biology, psychology, and computer science. In game theory, payoffs are the rewards or outcomes received by players based on their decisions or strategies.

## What are Payoffs?

Payoffs are the outcomes or rewards that players receive in a game. They can be in the form of money, points, satisfaction, etc.

Payoffs are determined by the decisions made by each player and the rules of the game. The goal of each player is to maximize their payoff while minimizing their opponent’s payoff.

## Types of Games

There are two types of games: zero-sum games and non-zero-sum games. In a zero-sum game, the total payoff for all players is zero. One player’s gain is another player’s loss.

Examples of zero-sum games include chess and poker. In a non-zero-sum game, there can be positive or negative payoffs for all players involved. Examples of non-zero-sum games include prisoner’s dilemma and chicken.

## Payoff Matrix

A payoff matrix is a table that shows all possible outcomes or payoffs for each player in a game based on their decisions or strategies. It is used to analyze and predict the behavior of players in different scenarios. Each cell in the matrix represents a specific outcome for both players.

### Example:

Consider a simple game between two players A and B where both can either choose to cooperate (C) or defect (D). The payoff matrix for this game can be represented as follows:

- If both A and B cooperate (C,C), they both receive 3 points.
- If A cooperates (C) but B defects (D), A receives 0 points while B receives 5 points.
- If A defects (D) but B cooperates (C), A receives 5 points while B receives 0 points.
- If both A and B defect (D,D), they both receive 1 point.

## Strategies

A strategy is a set of decisions or actions that a player follows to achieve their goal. In game theory, players can have different types of strategies like pure strategy, mixed strategy, and dominant strategy.

- A pure strategy is a single decision or action that a player follows in all situations. For example, always choosing to cooperate in the game mentioned above is a pure strategy.
- A mixed strategy is a combination of different decisions or actions that a player follows randomly.
For example, choosing to cooperate or defect with equal probability in the game mentioned above is a mixed strategy.

- A dominant strategy is a decision or action that always results in the highest payoff for a player regardless of their opponent’s decision. For example, always choosing to defect in the game mentioned above is a dominant strategy for both players.

## Nash Equilibrium

Nash equilibrium is a concept introduced by John Nash in which each player’s decision or action is optimal given their opponent’s decision or action. It is the point where neither player has an incentive to change their decision or action. In other words, it is the most stable outcome of the game.

### Example:

In the game between A and B mentioned above, (D,D) is the Nash equilibrium because neither player has an incentive to change their decision given their opponent’s decision.

## Conclusion

Payoffs are an essential aspect of game theory as they determine the success of players’ strategies. Understanding payoffs, strategies, and Nash equilibrium can help in predicting and analyzing the behavior of players in different games.