Subgame perfect equilibrium

Okay, so let's imagine that you are playing a game with your friends, like hide-and-seek. But this is a special kind of game, because there are different parts or "subgames" within the main game. For example, before you start playing hide-and-seek, you might all decide who is going to be "it" first. That's a subgame.

Now let's say that you want to be really good at this game and win every time. How can you do that? Well, you need to make the best decisions in every part of the game, especially in the subgames. That means you need to think about what your friends might do, and what you should do in response.

But what if your friends are thinking the same thing? What if they are also trying to make the best decisions in every subgame? This is where things can get tricky. Each subgame has its own set of strategies and payoffs (like points or rewards). You need to figure out what the best strategy is for every subgame, taking into account what your friends might do.

A subgame perfect equilibrium (SPE) is a point in the game where every player is making the best possible decision in every subgame, based on what they know about the other players and the game as a whole. This means that no player can do better by changing their strategy or making a different move.

Think of it like a puzzle: you need to fit all the pieces together perfectly to get the best outcome. If any one player makes a mistake or doesn't play optimally, the whole puzzle falls apart and the equilibrium is broken. But if everyone plays perfectly, you reach an equilibrium where no one can do better, and that's the subgame perfect equilibrium.

So, to sum up: a subgame perfect equilibrium is a point in a game where every player is making the best possible decision in every subgame, and no one can do better by changing their strategy. It's like a perfect puzzle where all the pieces fit together perfectly.