PS: Ask for a monthly pass, ask for a monthly pass.

"In life, we often encounter problems related to consensus. Let's take the following story, for example. A and B have something to meet in person, and they have to use text messages to agree on the time and place of meeting tomorrow. However, the time of both is very valuable, and they will only show up when they are confident that the other will be able to attend. A texted B and said, 'We'll see you tomorrow at 10:00 at Xizhimen subway station.'" However, it is not uncommon for text messages to be lost. To be confident that B was aware of this, A added, 'Please reply if you receive it'. After receiving it, B immediately replied: 'Received, see you at 10:00 tomorrow'. However, B also has his own concerns: Doesn't A only go after confirming that I am going? What if the other party does not receive my confirmation text message and does not show up at that time and asks me to wait for noon? After receiving the confirmation letter, A will naturally reply "Confirmation received". But A has a new concern: if B doesn't get a reply from me, he must be worried that I won't go because I didn't get a reply from him, so will he not go because of this? In order to make sure that B receives a reply, A also adds 'please reply received' at the end of the text message. The process continues, apparently endlessly. As a result, A and B have been confirming each other's information, but they have never been able to reach a consensus: 'We will all arrive at Xizhimen subway station at 10:00 tomorrow'. ”

Some smart people might say, that's not easy. Wouldn't it be enough for A to call B? In life, this is indeed the best solution to the above dilemma. An interesting question arises: What is the difference between making a phone call and texting that makes the two of them solve the problem at once? Calling is 'online' and texting is 'offline'. When talking on the phone, everyone can be sure that the other person is listening, and they can be sure that the other person is listening, and so on, so that anything they say will immediately become a common understanding: not only I know. And I know you know, and I know you know I know......"

"The archmage publicly declares 'there is at least one blue eye on the island' to let everyone know about it. And let everybody know everybody knows this, and nest infinitely like this. This is called a certain message and becomes the consensus of everyone. Let's take a look at what would have happened if this news hadn't become a consensus. ”

"For the sake of simplicity. Suppose there are only two blue eyes on the island. Both of them could see that each other had blue eyes. So they all know that 'there is at least one blue eye on the island'. However, since the mage did not appear, neither of them knew if the other knew about the blue eyes on the island. So, by the next day, the previous reasoning could not be continued, and everyone thought to themselves that it was entirely possible that the other party did not commit suicide because the other party did not know that there were blue eyes on the island. ”

"Similarly. If there are three blue eyes on the island, then unless they all know. Everybody knows that everybody knows about the blue eyes on the island, otherwise the reasoning on the fourth day is not valid, and on the third day, some people will think that the two men didn't commit suicide just because they didn't know that the other person also knew that there are blue eyes on the island. Continuing to expand to the case of 100 blue eyes, you will find that 'knowing each other' has to be nested in 100 layers for all reasoning to go smoothly. ”

"Actually, the conditions for my topic are also incomplete. All the people on the island know these terms and rules very well, and they should be changed to: These conditions and rules are the consensus of all the people on the island, or rather: all the people on the island know the above conditions and rules, and everyone knows that everyone knows them, and so on and so forth. If this condition is not true, the reasoning just now is not valid. For example, even though everyone is infinitely smart, if you don't know that everyone else is infinitely smart, or if you don't know that everyone else knows that others are infinitely smart, the reasoning will get stuck because you didn't kill himself last night just because he was too stupid to push it out. ”

Han Yan's eyes lit up after hearing this, and he seemed to understand.

Professor Liu Meng continued: "In fact, it is difficult for the human brain to imagine the assumption of this question, all people have the ability to reason, if it is implemented with a computer program, it is much clearer, which is equivalent to everyone's thinking in an infinite loop, and the sentence that the archmage said is to break the interruption of everyone's infinite loop, which will trigger a series of chain reactions. ”

Han Yan was thoughtful, Liu Meng was in a good mood, and said, "I won't tell you what the final result is, it's better to discuss game theory first." ”

"In 1950, Canadian mathematician Albert Tucker came up with the famous 'prisoner's dilemma'. Imagine that two members of a criminal gang are arrested and tried separately in two different rooms. The police said exactly the same thing to the two men: first, they confessed that they could only be sentenced to one year's imprisonment because of insufficient evidence, but that if one of them confessed and the other remained silent, the former would be acquitted and the latter would be sentenced to three years' imprisonment, and if both confessed, they would be sentenced to two years' imprisonment each. If both of them remain silent, they will only be locked up for a total of two years, which will be the best ending for them. But in reality, everyone will find that no matter what the decision is made, confessing truthfully will always save them a year. As a result, both of them will choose to confess in unison, so they are sentenced to two years each, which is actually the worst outcome for them. ”

"For the prisoner's dilemma to be true, one condition must be indispensable: the two will never see each other again. In this way, everyone can feel comfortable and boldly betray each other, without fear of retaliation. This is not the case if the decision is not a one-time decision, and the two parties to the decision will meet again and again in the future. Robert Axelrod's book "The Evolution of Cooperation" mentions that there was a very interesting phenomenon on the Western Front in World War I: after a period of "acquaintance" between British and German soldiers in trench warfare, a very subtle cooperation mechanism gradually emerged. Say. When one side drove into the war zone, the other side could have easily blown it up, but it didn't. Because they know the consequences of doing so - the other side will retaliate, which will leave both sides without food or drink. Over time, this cooperation will even develop to the point where the German soldiers walk back and forth within the range of the British army, and the British soldiers will not be moved!"

"It's a very complex society. Everyone wants to maximize their own interests, so there is cooperation where there should not be cooperation, and there is betrayal where there should not be betrayal. Mathematicians have built various models. to describe the way people make decisions driven by profits, and there is such a branch of mathematics - game theory. ”

"It's hard to talk about game theory, so I'll give you a few examples. You naturally understand what absolute rationality and infinite loops are. Professor Liu Meng said with a smile.

An airline lost two suitcases. The contents of these two suitcases are identical, but they belong to two different passengers, A and B. The airline sent a manager to negotiate compensation with the two passengers. The manager explained to the two passengers that the airline could not assess the value of the lost suitcase. Therefore, it is necessary for each traveler to write down a positive integer between 2 and 100 (including 2 and 100) independently. Indicates the valuation of the suitcase in yuan.

If the two passengers wrote exactly the same amount, the airline considered it to be the true value of the suitcase and paid both passengers according to that amount. However, if one of the passengers writes a lower number than the other, the airline will assume that the former's valuation is true. The airline will pay compensation to two passengers based on this estimate, but the passenger who quotes this price will receive an additional 2 yuan as a reward. Another traveler will receive $2 less as a penalty for overvaluation. For example, if A and B are valued at $50 and $40 respectively. A will receive $38 and B will receive $42.

If both travelers are absolutely rational and all the above conditions have become the consensus of these two travelers. So, what kind of numbers will these two travelers write down?

If this is the first time you've heard of this question, you won't believe the answer: the end result is that both people are only valued at $2. Why?

It is easy to think that for these two people, the best outcome is that both of them are valued at 100 yuan, so that both of them will get 100 yuan. However, one of them will definitely use his brain: "If the other party estimates 100 yuan and I estimate 99 yuan, then the airline will think that I am honest and I can get 101 yuan, and the other party can only get 97 yuan." The other person actually thought of this, so the two of them would write 99 yuan in unison, and the result was that each of them would get 99 yuan.

Interestingly, if both people think that the other person will write $99, then everyone will find that it is useless to raise their valuation to $100 again, but reducing their valuation to $98 will increase their income from $99 to $100. As a result, both would change the valuation to $98. In short, both people realized this: no matter how much money the other party quotes, it is always the best choice for me to report 1 yuan less than the other party. So, this vicious psychological warfare will continue until everyone pushes it out and should change the valuation from $3 to $2. At this point, the two finally stopped fighting, and they got the answer they just said.

If someone stands up at this time and says, "If you two continue to fight viciously like this, everyone will get the least money." This sentence seems to be nonsense, but it is not, which will make the two come to a consensus not to continue fighting.

Again, ask 10 people to play a game where they give each person 100 yuan, and then everyone can choose to donate a portion of the money, and the donations raised will be used for investment, and in the end, double the money will be recovered and divided equally among everyone, even if everyone doesn't contribute the same amount of money. The best outcome is, of course, that everyone comes up with 100 yuan, and everyone will get 200 yuan in the end.

However, rational decision-makers will think: "If I only give 99 yuan, then the fund used for investment is only 999 yuan, and in the end everyone will get a return of 1998 yuan, and everyone will get 199.8 yuan; but don't forget that I still have 1 yuan in my hand, so in the end, I will have 200.8 yuan? In fact, if I don't give a penny at all, I will be able to sit back and enjoy a return of 180 yuan, and I will have 280 yuan in my hand!" If everyone is absolutely rational, then everyone will find that they can always make more money by giving less than others. The end result turned out to be that everyone was not willing to give a penny!

In daily life, there are many such phenomena, such as the problem of primary and secondary school students making up classes. The best-case scenario should be that each school does not make up classes, which both ensures fairness and reduces the burden on children. However, every school thinks that if other schools don't make up classes, our school will only make up for an hour, and we will earn it. Of course, once all the schools realize this, each school will scramble to make up for an extra hour. As a result, every school is making up classes endlessly, and this is the tragic situation.

In game theory, if all the players make a good decision and make the decision public, each player finds that unilaterally changing their decision will not make them more profitable, and we call the decision of the crowd a "Nash equilibrium". This is named after the American mathematician John Nash, and you should be very familiar with this name after watching the movie "A Beautiful Mind", so it is easy to make yourself a psychotic when studying game theory, and it is easy to fall into an infinite cycle of thinking, and the smarter a person is to study game theory, the easier it is to be imprisoned in a mental hospital.

Liu Meng finally said: "Back to our blue-eyed problem, that is to say: after the archmage said that sentence, on the 101st day, all the blue-eyed people will find out that they are blue-eyed and commit mass suicide, and on the 102nd day, the remaining 900 brown-eyed people will also commit collective suicide because of the blue-eyed people, and they will know that the color of their eyes will also commit mass suicide, and the final result will be that the island will be completely extinct." ”

Han Yan knocked on his head, only to feel very swollen, and said embarrassedly: "Professor Liu Meng, my head hurts so much, I need to go back and rest." (To be continued......)