A Beautiful Mind is a very classic film, which reproduces the legendary experience of johnf nash, a great mathematical genius. The film itself and the prototype of the characters behind it have deeply touched people. In 2002, this film won five awards in the 59th Golden Globe Awards and four awards in the 74th Academy Awards. Nash is a genius in mathematics, and his "Nash equilibrium" is the theoretical pillar of game theory. At the same time, he is also the winner of the Nobel Prize in Economics. But this is not all of him, just a brilliant side of his legendary life. Before we talk about Nash equilibrium, let's take a look at the legendary life of this genius. Nash was born in West Virginia, USA 1928. His family conditions are excellent. His father is an engineer and his mother is a teacher. When Nash was a child, he was withdrawn, unwilling to play with children of his own age, and liked to find happiness in books alone. Nash didn't do well in math at that time, but he still showed some talent. For example, Nash only needs a few steps to complete the theorem that the teacher can prove with the blackboard formula, which often makes the teacher feel embarrassed.
From 65438 to 0948, Nash was admitted to four universities at the same time, including Princeton and Harvard, and finally Nash chose Princeton. At that time, Princeton's academic atmosphere was very free, and a number of world-class masters such as Einstein and von Neumann were produced, and they have always been outstanding in the field of mathematics research, making it the center of mathematics in the world. Nash's home in Princeton has made great progress.
1950, Nash published his doctoral thesis "Non-cooperative Game". After continuing to study this problem, in the same year, he published another paper, Equilibrium Point in N-player Game. These two papers are only dozens of pages, with some charts drawn by Nash in the middle. But it is these dozens of pages that have changed the development of game theory and even changed our lives. He expanded the research scope of game theory from cooperative game to non-cooperative game, and the application field also expanded from economic field to almost every field. It can be said that the game theory after Nash equilibrium has become a common tool in all industries and fields.
In the year when his doctoral thesis was published, Nash received a doctorate in mathematics. 1957, he married his female student, Alisha, and obtained a lifelong degree from MIT the following year. At this time, Nash was in high spirits and became a famous mathematician before he was 30 years old. 1958, Fortune magazine made a selection, and Nash was selected as the most outstanding star among mathematicians at that time. God likes to play jokes on geniuses. At the peak of his career, Nash suffered a cruel blow from fate. He has a disease called paranoid schizophrenia. This kind of schizophrenia accompanied him all his life. He often sees some imaginary characters and starts to dress strangely. He will say something meaningless in class and often scribble something on the blackboard that no one understands. This prevented him from teaching normally, and he had to resign as a professor at MIT.
Nash's illness got worse after he resigned. He began to write some strange letters to politicians. He always imagined that there were many Soviet spies around him, and he was arranged to discover the information of these spies. The split of spirit and thinking turned this once genius into a madman.
His wife, Alisha, was deeply moved by his talent, but now she has to choose to divorce her increasingly violent and divided husband to protect her children from harm. However, their relationship did not end there, and she has been helping him recover. 1970, Nash's mother died and her sister was unable to support her. When Nash faced the dilemma of sleeping on the street, Alisha accepted him and they lived together again. Alisha not only took good care of Nash in life, but also deliberately moved his family to a secluded Princeton, away from the noise of big cities. She hopes that Princeton University, which once witnessed Nash's brilliance, can revive Nash's talent.
Nash was deeply moved by his wife's firm belief and unwavering love, and he made up his mind to fight the disease. Finally, under the care of his wife and friends, Nash's condition miraculously improved in the 1980 s and finally recovered. At this time, he can not only communicate with others, but also continue to engage in his favorite mathematical research. In this fight against the disease, his wife Alisha played a key role. After coming out of the shadows, Nash became a candidate for the 1985 Nobel Prize in Economics, based on the impact of his research on game theory on the economy. But in the end he didn't win the prize for several reasons. On the one hand, the influence and contribution of game theory were not fully recognized at that time; On the other hand, the Royal Swedish Academy is not at ease with Nash, who has just recovered from his illness. After all, he has been suffering from schizophrenia for almost 30 years. Nobel laureates usually give speeches at the awards ceremony, and people are worried that Nash's mind has not fully recovered.
By 1994, the achievements of game theory in various fields are obvious to all, and the opportunity is approaching Nash again. However, Nash did not have a title at that time, and the Royal Swedish Academy could not nominate him. At this time, Nash's old classmate, Kuhn, a mathematical economist at Princeton University, came out. He first made it clear to the Nobel Prize Selection Committee that Nash deserved the Nobel Prize, and it would be a very bad decision to exclude him from the Nobel Prize on the grounds of good health. At the same time, Kuhn won Nash the status of "visiting research collaborator" in the mathematics department of Princeton University. These efforts were not in vain, and finally Nash stood on the podium of the Nobel Prize in Economics.
At that time, American economist John Halsani and Professor reinhard Selten of Bonn University also won the Nobel Prize in Economics. They are all scholars who have made outstanding contributions in the field of game theory, which indicates that game theory has been widely recognized and become an important part of economics.
After decades of development, "Nash equilibrium" has become the core of game theory, and Nash has even become synonymous with game theory. Seeing the vigorous development of game theory today, I really can't imagine what the game theory world would be like without johnf nash.
The game of liberation in the second quarter
We have been talking about Nash's important position in the development of game theory, but the emotional description is powerless. Let's talk about Nash's contribution from the research and application scope of game theory and see what position Nash equilibrium is in game theory.
As we have already introduced, game theory was founded by the Hungarian-born American mathematician von Neumann. At the beginning of its establishment, the research and application scope of game theory was very narrow, just a theory. From 65438 to 0944, with the publication of Game Theory and Economic Behavior, game theory began to be applied in the field of economics, and the systematic theory of modern game theory began to take shape gradually.
Until Nash discovered Nash equilibrium in 1950, the research scope of game theory was limited to two-person zero-sum game. We introduced the classification of game theory earlier. According to the number of participants in the game, it can be divided into two-player game and multiplayer game. According to the game results, it can be divided into positive sum game, zero sum game and negative sum game; According to whether there is a binding agreement between two or more parties in the game, it can be divided into cooperative game and non-cooperative game.
Before Nash, the research scope of game theory was limited to two-person zero-sum game, that is, there were only two players who won and lost, and the total profit was zero.
Two-person zero-sum game is the most common mode in games and gambling, and game theory is the earliest theory to study gambling and games. The two-person zero-sum game in life is not as simple as games and sports competitions. Although it is a win-lose game, the scope of winning or losing can still be calculated and controlled. Von Neumann calculates the maximum and minimum values of the interests of all parties through linear operation, which is the range of losses and profits in the game. The calculated maximum payoff is the result we most want to see in the game, while the minimum payoff is the result we least want to see. This is more in line with the thinking of some people, that is, "hope for the best and prepare for the worst."
Although the research on the two-person zero-sum game was very advanced and avant-garde, as a theory, its coverage was too small. The limitations of this game mode are obvious. It can only study games in which two people participate. In reality, games are often multi-party games, and the reality is complicated. The result of the game is not only that one side gains, the other side loses, but also that both sides gain or neither side takes advantage. None of these situations were within the scope of von Neumann's research at that time.
All this has been broken by the Nash equilibrium. From 65438 to 0950, Nash wrote the paper "Equilibrium Point in N-player Game", which mentioned the concept of "Nash Equilibrium" and its solution. At that time, Nash took his own views to see von Neumann, the founder of game theory, and was given a cold shoulder. Before that, he was snubbed by Einstein. But this can't affect the sensation brought by Nash equilibrium.
It can be seen from Nash's thesis title "Equilibrium Point in N-player Game" that Nash mainly studies multi-player participation and non-zero-sum game problems. No one has studied these problems before him, or no one can find a balance point suitable for all parties. Just as it is easy to find the intersection of two lines, it is difficult to find the intersection of several lines if there is one. Finding the balance point of many parties is the key to this problem. Without this balance point, the research on this issue will become meaningless, let alone have any guiding role in practical activities. Nash's greatness lies in that he proposed a solution to this difficult problem. The key is "Nash equilibrium", which leads the research scope of game theory from "alley" to a broad world, and finds significance for multi-player non-zero-sum games, which account for the majority of game situations. Nash's paper "Equilibrium Point in N-Person Game" shocked people like thunder. He turned a seemingly impossible thing into reality, that is, he proved the existence of equilibrium in non-cooperative multiplayer games and gave the solution of this equilibrium. Nash equilibrium has completely changed people's previous views on competition, market and game theory, and made people understand the relationship between equilibrium in market competition and game equilibrium.
The proposal of "Nash equilibrium" laid the foundation for the development of non-cooperative game theory, and then the development of game theory mainly followed this line. For a long time, the main achievement in the field of game theory is the explanation or expansion of Nash equilibrium. Some people even joked that if everyone quoted Nash equilibrium and had to pay Nash a dollar, he would have become the richest man.
Nash has made outstanding contributions not only in the field of non-cooperative games, but also in the field of cooperative games. The cooperative game was established by von Neumann in Game Theory and Economic Model. The key of non-cooperative game is how to maximize the benefits, and the key of cooperative game is how to distribute the benefits, in which mutual consultation is very important, that is, the "bargaining" between the two sides. But von Neumann did not give this "bargaining" solution, or did not find a solution to this problem. Nash studied this problem and put forward a solution to the "bargaining" problem. He further expanded the scope and regarded the cooperative game as a non-cooperative game in a sense, because the bargaining problem in interest distribution is ultimately to strive for the best interests for himself.
In addition, Nash also studies behavioral experiments of game theory. He once suggested that a simple "prisoner's dilemma" is a one-step strategy. If participants are allowed to experiment repeatedly, it will become a multi-step strategy. Under the single-step strategy, there is no collusion between prisoners, but under the multi-step strategy mode, collusion may occur. This foresight has been verified, and the repeated game model has played an important role in politics and economy.
Nash's contribution to game theory has a growing impact on reality. In the 1990s, the US government and the New Zealand government held auctions almost simultaneously. The American government invited economists and game theory experts to analyze and design the auction. The reference factor is to let the government get more benefits, at the same time let the merchants get the maximum utilization rate and benefits, and find a balance between the government and the merchants. The final outcome is that everyone is happy, the auction is very successful, the government has made huge profits, and businesses have their own places. However, the auction held in New Zealand was very bleak. The key reason is that there is something wrong with the mechanism design. In the end, everyone pursues hot goods, which leads to the final price being much higher than its own value. However, some goods are neglected, and even some goods are auctioned by only one person, and they are taken at a very low transaction price.
It is precisely because of its growing influence on reality that the Nobel Prize in Economics from 65438 to 0994 was awarded to three game theory experts, including Nash.
Finally, let's summarize Nash's position in game theory. There is a proverb in China, "If Zhong Ni is not born in the sky, it will last forever". God didn't send Confucius to earth, and people live in darkness forever. If we say this about Nash's relationship with game theory, it is an exaggeration. But Nash's pioneering development of game theory is unmatched by anyone. Before him, game theory was like a narrow alley, and Nash knocked down the walls on both sides of the alley and expanded people's horizons to the boundless sky.