An understanding of common knowledge in Game Theory

After reading a piece of "the essence of game theory knowledge", I suddenly realized something. In fact, what attracted me initially was the case that was already well known to many people.

It is the famous "dirty face problem", also known as "red and white hats". In order to respect the original work, I will describe it according to the dirty face problem. There are three people whose faces are dirty, but they do not know. They can only see whether the faces of others are dirty or clean. At this time, if they were asked to judge whether their faces were clean or dirty, it would be impossible for three people to say anything. As an outsider, I told them, "at least one of you has a black face !", In fact, this is obviously a "nonsense" (this is the term in the book ...), Because everyone can see that the faces of the other two are dirty, but the game can proceed because this sentence seems useless. At this time, I asked the first person whether his face was dirty or clean. He still couldn't answer the question and asked the second person, but when I asked the third person, if he is smart enough, he should be sure to answer the question. My face is dirty! The reasoning process is also very simple. If the first person cannot answer this question, it means 2. The first person may know that his or her face is dirty ), the second person knows the reasoning of the first person. If he sees the third person's face clean, he can quickly judge whether his face is dirty. The third person cannot tell the second person, your face must be dirty.

This process can be easily understood by many people. The key is, why does a seemingly useless sentence make the results different? In other words, if you don't say "at least one of you has a dirty face", everyone knows about it, and everyone else knows about it. The problem is that before we say this, everyone does not know that others know about this ~.~. This is a bit difficult, so there is no specific explanation in the book. I will use the reasoning process to clarify the difference.

To illustrate the differences between the two situations, we only need to infer a situation in which the proposition that at least one person's face is dirty is not true (because if we say that, at least one person's face is dirty and becomes "common knowledge", which will be applied in any situation ). When I didn't say that, the three people A, B, and C first knew that at least one person's face was dirty. For a, a will think that B must also know that "at least one person's face is dirty", because a can see that C's face is dirty, so this is certain. For a, because in a's opinion, B may only see one dirty face c, because a knows that B does not know whether B's face is dirty, so let's think about it again, A think of B and think that C may see clean faces. After thinking about the third layer, there will be a situation that is not in line with "common knowledge", and the proposition is proved. Therefore, in the absence of "common knowledge", if the above reasoning is still carried out, the third party cannot know the reasoning of the second party, so he cannot judge.

After talking about the word "common knowledge", a very important concept in game theory is actually not only the knowledge that everyone knows, it must also be known from public channels (such as popular media, or the president's speech) so that everyone can be certain that others know, I know others know... In this way, no matter how many layers, they are all "known. Many things in game theory play a guiding role in real life and enterprise competition. In the future, we will have the opportunity to study them in depth.