ps: Please give me a monthly ticket, please give me a monthly ticket.

"In life, we often encounter problems related to consensus. For example, the following story. A and B have something to interview, and they have to use text messages to agree on the meeting time and place tomorrow. However, both of them are very precious, and they will only be present when they are sure that the other party can attend. A sent a text message to B and said, "Let's meet at Xizhimen Metro Station at 10:00 tomorrow." However, it is common for the text message to be lost. In order to be sure that B learned about this news, a added, "Please reply when you receive it." After b received it, he immediately replied: "I have received it, see you tomorrow at 10:00 tomorrow." However, b also has his own concerns: A will only go after confirming that I am going?

What should I do if the other party did not receive my confirmation text message, and I was not present at that time and asked me to wait for noon in vain? Therefore, b also attached a sentence: 'Please reply to receive this confirmation message'. After a confirmation message, a will naturally reply 'Received Confirmation message'. But a has a new concern: If b does not receive my reply, he will definitely worry that I will not go because he did not receive his reply. Will he stop going? To ensure that b receives the reply, a also added 'Please reply to receive a reply' at the end of the text message. The process continued, obviously endlessly. As a result, a and b have been confirming each other's information, but they have never reached such a consensus: 'We will all arrive at Xizhimen Metro Station at 10:00 tomorrow'."

"Some smart people may say, that's not simple. Isn't it enough to call a b? In life, this is indeed the best solution to the above difficulties. Interesting problems arise: What is the difference between making a phone call and sending a text message, which makes the two solve the problem in one go? The main reason may be. Calling is 'online', while sending a text message is 'offline'. When making a phone call, everyone can be sure that the other party is listening, and can be sure that the other party is listening, etc. Therefore, any words the two say will immediately become a consensus: not only I know, but I know you know, but I know you know I know..."

"The ** teacher announced in public that 'there is at least one blue eye on the island' is to let everyone know this. And let everyone know this and nest it infinitely like this. This is called a message becoming a consensus among everyone. Let's take a look at what happens if this message does not become a consensus."

"For simplicity, assume that there are only two blue eyes on the island. Both of them can see that the other party has blue eyes. Therefore, they all know that 'there are at least one blue eyes on the island'. However, since the mage did not show up, neither of them knows whether the other party knows whether the other party has blue eyes on the island. So, by the next day, the previous reasoning could not continue. Everyone would think in their hearts that the other party did not commit suicide was entirely because the other party did not know that 'there are blue eyes on the island'."

"Similarly. If there are three blue eyes on the island, then unless they all know. Everyone knows that everyone knows that 'there are blue eyes on the island', otherwise the reasoning on the fourth day is not true. On the third day, someone will think that the two people did not commit suicide just because they didn't know that the other party also knew that 'there are blue eyes on the island'. Continue to expand to 100 blue eyes, you will find that 'know each other' must be nested 100 layers in order for all the reasoning to proceed smoothly."

"In fact, my question conditions are also incomplete. Everyone on the island knows the above conditions and rules very clearly, and should be changed to: the above conditions and rules are the consensus of everyone on the island, or: everyone on the island knows the above conditions and rules, and everyone knows that everyone knows, etc. Without this condition, the reasoning just now is not valid. For example, although everyone is infinitely smart, if everyone doesn't know that others are infinitely smart, or if everyone doesn't know that others are also infinitely smart, the reasoning will be stuck because of ideas such as 'He didn't commit suicide last night just because he was too stupid and didn't come out'."

Han Yan's eyes lit up after hearing this, as if he understood.

Professor Liu Meng continued: "In fact, it is difficult for the human brain to imagine the assumption of this question. Everyone has the ability to reason. If it is implemented using a computer program, it will be much clearer. It is equivalent to everyone's thinking in an infinite loop. The words said by Master ** are to break the interruption of everyone's infinite loop and trigger a series of chain reactions."

Han Yan thought about it, Liu Meng was in a good mood and said, "I won't tell you what the final result is, so why not discuss game theory first."

"In 1950, Canadian mathematician Alberttucker proposed the famous 'prisoner's dilemma'. Imagine two members of a criminal gang were arrested and they were locked up in two different rooms for trial. The police said exactly the same thing to the two: First, they admitted that due to insufficient evidence, they could only sentence two people to one year in prison; but, as long as one of them confesses and the other remained silent, the former would be acquitted, and the latter would be sentenced to three years in prison; and, if both of them confessed, they would be sentenced to two years in prison. If both of them remained silent, they would be only two years in total, which would be the best end for them. But in fact? Everyone would find that no matter what the decision made by the other party was, truthful confession would always make them less imprisoned. The result was that both of them would choose to confess, so they would be sentenced to two years each, which was actually the worst end for them."

"If the prisoner's dilemma is established, there is a condition that is indispensable: the two will never meet in the future. In this way, everyone can be content with confidence and boldness, and not worry about revenge. If the decision is not one-time, the decision-making parties will meet again and again in the future, and the situation will be different. In the book "Evolution of Cooperation" by robertaxelrod mentioned that a very interesting phenomenon occurred on the Western Front battlefield of World War I: English and German in trench warfare

After the soldiers "know each other" for a period of time, a very subtle cooperation mechanism will gradually develop. For example, after one party's food supply vehicle entered the war zone, the other party could have easily blown it up, but did not do it. Because they knew the consequences of doing so - the other party would take revenge, which would make both sides eat and drink. Over time, this cooperation will even develop to the point where German soldiers walk back and forth within the range of the British army, and the British soldiers were indifferent!”

"This is a very complex society. Everyone wants to maximize their own interests, so cooperation appears in places where there should not be cooperation, and betrayal appears in places where there should not be betrayal. Mathematicians have established various models to describe the way people make decisions driven by interests, so there is such a branch of mathematics - game theory."

"It may be difficult to understand game theory in boring. Let me tell you a few examples below. You will naturally understand what absolute rationality and infinite dead cycles are." Professor Liu Meng said with a smile.

An airline lost two suitcases. The items in these two suitcases were exactly the same, but they belonged 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 estimate the price of the lost suitcase. Therefore, the two passengers needed to write a positive integer between 2 and 100 (including 2 and 100) respectively. It means that the valuation of the suitcase is RMB.

If the numbers written by the two passengers are exactly the same, the airline believes that this is the real value of the suitcase and pays the two passengers according to this number. However, if one of the passengers writes lower than the other passenger, the airline will believe that the former valuation is real. The airline will compensate the two passengers according to this estimate, but the passengers who quoted this price will receive 2 yuan as a reward. The other passenger will receive 2 yuan less, as a penalty for excessive valuation. For example: if a and b respectively estimate 50 yuan and 40 yuan, a will receive 38 yuan, and b will receive 42 yuan.

If both passengers are absolutely rational and all the above conditions have become the consensus of these two passengers, then what numbers will these two passengers write?

If you first heard of this question, you would definitely not believe the answer to this question: the end result is that both of you are only estimated at 2 yuan. Why?

It is easy to think that for these two people, the best ending is that both of them are estimated to be 100 yuan, so that both of them will get 100 yuan. However, one of them will definitely have a bad idea: "If the other party estimates 100 yuan and I estimates 99 yuan, then the airline will think that I am honest and I can get 101 yuan, while the other party can only get 97 yuan." The other person actually thought of this, so the two people will write 99 yuan at the same time, and the result is that the two people each get 99 yuan.

Interestingly, if both of them think that the other party will write down 99 yuan, then everyone will find that it is useless to increase their valuation to 100 yuan, but reducing their valuation to 98 yuan will increase their income from 99 yuan to 100 yuan. As a result, both of them will change their valuation to 98 yuan. In short, both of them realize this: no matter how much the other party claims, it is always the best choice for me to report 1 yuan less than the other party. Therefore, this vicious psychological warfare will continue until everyone launches it and they should change their valuation from 3 yuan to 2 yuan. At this point, the two finally no longer have any fighting, so they get the answer they just mentioned.

If someone stands up at this time and says, "If you two continue to fight like this, everyone will get the least money." This sentence seems like nonsense, but it is not the case. This will lead the two to reach a consensus and no longer continue to fight.

Furthermore, let 10 people play a game like this: give everyone 100 yuan, and then everyone can choose to donate a part of the money; the donations raised will be used for investment, and the money will be recovered in the end and distributed to everyone, even if everyone pays different amounts of money. The best outcome is, of course, everyone will get 200 yuan in the end.

However, rational decision makers will think like this: "If I only pay 99 yuan, then the fund used for investment will only be 999 yuan, and everyone will get a return of 1,998 yuan in the end; but don't forget that I still have 1 yuan in my hand, so I will add it to the same amount. In fact, if I simply don't pay a penny, I can 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 pay less than others and can always make themselves more profitable. The final result is that everyone is unwilling to take out a penny!

In life, there are many such phenomena, such as the problem of tutoring for primary and secondary school students. The best situation should be that every school does not tutoring for classes, which not only ensures fairness but also reduces the burden on children. However, every school will think that if other schools do not tutoring for classes, even if our school only needs to make up for one hour, we will make a profit. Of course, when all schools realize this, each school will strive to make up for another hour. The result is that every school is tutoring for classes endlessly, so there is such a tragic situation.

In game theory, if players make good decisions and make the decisions made public, each player finds that unilaterally modifying their decisions will not make them more profitable. We call everyone's decisions at this time a "Nash equilibrium". This is named after the American mathematician John Nash. After watching the movie "Beautiful Mind", you should be very familiar with this name. Therefore, studying game theory can easily turn yourself into a psychiatric disorder and easily fall into the infinite vicious cycle of thinking. The smarter people are to study game theory, the more likely they are to be locked up in a mental hospital.

Liu Meng finally said: "Back to our blue eyes problem, that is to say: After Master ** said that, on the 101st day, all blue eyes people would find that they had blue eyes and commit suicide collectively. On the 102th day, the remaining 900 brown eyes people died because of blue eyes, and they knew that the color of their eyes would also commit suicide collectively. The final result was that the island was completely extinct."

Han Yan knocked on his head, feeling very swollen, and said embarrassedly: "Professor Liu Meng, my head hurts so much. I need to go back and take a rest." (To be continued, please search for Astronomy, the novel is better and faster!
Chapter completed!