**What Is Subgame Perfect Equilibrium in Game Theory?**

Game theory is a mathematical framework used to study the behavior of rational decision-makers in interactive situations. It has applications in economics, political science, psychology, and many other fields.

In game theory, a subgame perfect equilibrium (SPE) is a solution concept that captures the idea of rationality and consistency throughout the game.

## What is a game?

A game in game theory refers to any situation where there are two or more players who interact with each other and have different goals or objectives. Examples of games include chess, poker, and even simple decisions such as where to go for dinner with friends.

## What is an equilibrium?

An equilibrium is a solution to a game where each player’s strategy is optimal given the strategies chosen by the other players.

## What is subgame perfect equilibrium?

In some games, there are smaller subgames within the larger game. A subgame perfect equilibrium is a solution concept that requires each player to choose a strategy that is optimal not only for the entire game but also for every subgame within it.

To understand this concept better, let’s take an example:

Consider a two-player game where each player can either cooperate or defect. If both cooperate, they each receive $3.

If one cooperates and the other defects, the defector receives $5 while the cooperator receives $1. Finally, if both defect, they each receive $2.

- If we look at this as just one big game, there are two equilibria: both players defecting or both cooperating.
- However, if we look at just one small part of this big game – the part where one player has already defected – there is only one optimal strategy for the other player: defect.
- This means that the equilibrium where both players cooperate is not subgame perfect because if one player defects, the other has no reason to continue cooperating.
- The only subgame perfect equilibrium in this game is for both players to defect, as this is the optimal strategy for each player in every subgame within the larger game.

## Conclusion

Subgame perfect equilibrium is a solution concept that captures the idea of rationality and consistency throughout a game. It requires each player to choose a strategy that is optimal not only for the entire game but also for every subgame within it.

In some games, there may be multiple equilibria, but only one of them may be subgame perfect. Understanding subgame perfect equilibrium can help us better predict how rational decision-makers will behave in interactive situations.