What does power game mean? What does Nash equilibrium mean?

(1) The power game is the wealth game.

Power price push is the sum of quantity changes, regardless of price changes. In this way, the influencing factors of price can be completely ruled out, and the change of trading volume can be completely observed.

(2) Nash equilibrium is Nash equilibrium.

Nash's two important papers on non-cooperative game theory in 1950 and 195 1 completely changed people's views on competition and market. He proved the non-cooperative game and its equilibrium solution, and proved the existence of equilibrium solution, namely the famous Nash equilibrium. Thus, the internal relationship between game equilibrium and economic equilibrium is revealed. Nash's research laid the cornerstone of modern non-cooperative game theory, and later game theory research basically followed this main line. However, Nash's genius discovery was flatly denied by von Neumann, and before that, he was also given a cold shoulder by Einstein. But the nature of challenging and despising authority in his bones made Nash stick to his point of view and eventually become a master. If it weren't for more than 30 years of serious mental illness, I'm afraid he would have

Standing on the podium of the Nobel Prize, I will never share this honor with others.

Nash is a very talented mathematician, and his major contributions were made when he was studying for a doctorate at Princeton from 1950 to 195 1. But his genius found that the equilibrium of non-cooperative game, namely "Nash equilibrium", was not smooth sailing.

1948 Nash went to Princeton University to study for a doctorate in mathematics. He was less than 20 years old that year. At that time, Princeton was outstanding and talented. Einstein, von Neumann, Levshetz (Head of the Department of Mathematics), Albert Tucker, Alenzo Cech, Harold Kuhn, Norman Sting Rhodes, Fawkes, etc. It's all here. Game theory was mainly founded by von Neumann (1903-1957). He is a talented mathematician who was born in Hungary. He not only founded the economic game theory, but also invented the computer. As early as the beginning of the 20th century, zermelo, Borer and von Neumann began to study the exact mathematical expressions of games. Until 1939, von Neumann got to know the economist oskar morgenstern and cooperated with him, which made game theory enter the broad field of economics.

From 65438 to 0944, his masterpiece Game Theory and Economic Behavior, co-authored with Oscar Morgenstein, was published, which marked the initial formation of modern system game theory. Although the research on the nature of games can be traced back to19th century or even earlier. For example, the Cournot simple duopoly game of 1838; Bertrand of 1883 and Edgeworth of 1925 studied the output and price monopoly of two oligarchs; More than 2,000 years ago, Sun Bin, a descendant of Sun Wu, a famous military strategist in China, used game theory to help Tian Ji win the horse race, and so on, all of which were the seeds of early game theory, characterized by scattered research, great contingency and no system. The concepts and analytical methods of standard, extended and cooperative game model solutions put forward by von Neumann and Morgan Stern in Game Theory and Economic Behavior laid the theoretical foundation of this discipline. The cooperative game reached its peak in the 1950s. However, the limitations of Neumann's game theory are increasingly exposed. Because it is too abstract, its application scope is greatly limited. For a long time, people know little about the study of game theory, which is only the patent of a few mathematicians, so its influence is very limited. It is at this time that the non-cooperative game-"Nash equilibrium" came into being, which marked the beginning of a new era of game theory! Nash is not a step-by-step student. He often plays truant. According to his classmates' recollection, they can't remember when they had a complete required course with Nash, but Nash argued that he had at least taken Steen Rhodes' algebraic topology. Steen Rhodes was the founder of this subject, but after several classes, Nash decided that this course was not to his taste. So he left again. However, Nash is, after all, an extraordinary person with talent. He is deeply fascinated by every branch of the kingdom of mathematics, such as topology, algebraic geometry, logic, game theory and so on. Nash often shows his distinctive self-confidence and conceit, full of aggressive academic ambitions. 1950 all summer, Nash was busy with nervous exams, and his game theory research was interrupted. He thought it was a great waste. I don't know this temporary "giving up", but under the subconscious constant thinking, it has gradually formed a clear vein, and I was inspired by generate! In the month of 10 this year, he suddenly felt a surge of talent and dreams. One of the most dazzling highlights is the concept of non-cooperative game equilibrium, which will be called "Nash equilibrium" in the future. Nash's main academic contributions are embodied in two papers (including a doctoral thesis) of 1950 and 195 1. It was not until 1950 that he wrote a long doctoral thesis entitled "Non-cooperative Game", which was published in 1950+0 1 Monthly Bulletin of the American Academy of Sciences and immediately caused a sensation. Speaking of it, it all depends on the work of Brother David Gale. Just a few days after being demoted by von Neumann, he met Gail and told him that he pushed von Neumann's minimax solution into the field of non-cooperative games and found a universal method and equilibrium point. Gail, listen carefully. He finally realized that Nash's idea, Beavon Neumann's cooperative game theory, can better reflect the real situation, and his rigorous and beautiful mathematical proof left a deep impression on him. Gail suggested that he tidy it up and publish it immediately, lest others beat him to it. Nash, a fledgling boy, didn't know the danger of competition and never thought about it. So Gail acted as his "agent" and drafted a short message to the Academy of Sciences on his behalf. Lev Shetz, the head of the department, personally submitted the manuscript to the Academy of Sciences. Nash doesn't write many articles, just a few, but it's enough, because they are among the best. This is also worth pondering. How many articles does a domestic professor need to publish in "core journals"? According to this standard, Nash may not be qualified.

Morris, winner of the Nobel Prize in Economics from 65438 to 0996, did not publish any articles when he was a professor of economics in edgeworth at Oxford University. Special talents should have special selection methods.

Nash University began to study the game theory of pure mathematics, and it became more comfortable after entering Princeton University from 65438 to 0948. In his early twenties, he had become a world-famous mathematician. Especially in the field of economic game theory, he has made epoch-making contributions and is one of the greatest game theory masters after von Neumann. His famous Nash equilibrium concept plays a central role in the theory of non-cooperative games. Later researchers' contributions to game theory are all based on this concept. The presentation and continuous improvement of Nash equilibrium has laid a solid theoretical foundation for the wide application of game theory in economics, management, sociology, political science, military science and other fields.

The plight of prisoners

A short story in Dali's theory

To understand Nash's contribution, we must first know what is a non-cooperative game problem. At present, almost all game theory textbooks will talk about the example of "prisoner's dilemma", and the examples in each book are similar.

Game theory is, after all, mathematics, or rather, a branch of operational research. When talking about classics and theories, mathematical language is indispensable, which is just a lot of mathematical formulas in the eyes of laymen. Fortunately, game theory is concerned with daily economic life, so we have to eat fireworks. This theory is actually a term borrowed from chess, poker, war and other issues with the nature of competition, confrontation and decision-making. It sounds a bit mysterious, but it actually has important practical significance. Game theory masters look at economic and social issues just like playing chess, and often have profound truth in the game. Therefore, it is not boring to start with trivial matters in daily life and explain them with stories around us as examples. One day, a rich man was killed at home and his property was stolen. During the investigation of this case, the police arrested two suspects, Scafi and Nakul, and found the lost property in the victim's house from their residence. But they denied that they killed anyone, arguing that they killed the rich first, and then they just stole something. So the police isolated the two and put them in different rooms for trial. The D.A. will talk to everyone individually. The prosecutor said, "Because you have conclusive evidence of theft, you can be sentenced to one year in prison." But I can make a deal with you. If you plead guilty to murder alone, I will only sentence you to three months' imprisonment, but your partner will get ten years' imprisonment. If you refuse to confess and are reported by your partner, you will be sentenced to ten years in prison, and he will only be sentenced to three months in prison. However, if you all confess, then you will all be sentenced to five years in prison. "What should Scalfi and Nacoors do? They are faced with a dilemma-confession or denial. Obviously, the best strategy is that both sides deny it, and as a result, everyone only gets one year. However, because the two are in isolation, they cannot confess. Therefore, according to Adam Smith's theory, everyone starts from the purpose of self-interest, and they choose repentance as the best strategy. Because if you confess, you can expect three months of short-term imprisonment, but only if your partner denies it, which is obviously better than your own denial of 10 years imprisonment. This strategy is at the expense of others. Not only that, but confession has more benefits. If the other party denies it frankly, they will go to jail 10 years. It's so uneconomical! Therefore, in this case, you should still choose to confess. Even if two people confess at the same time, they will only be sentenced to five years at most, which is better than 10 years. Therefore, the reasonable choice of the two is confession, and the strategy (denial) and the ending (sentence 1 year imprisonment) that were originally beneficial to both sides will not appear. In this way, both of them chose Frank's strategy and were sentenced to five years' imprisonment. The result is called "Nash equilibrium", which is also called non-cooperative equilibrium. Because, when each party chooses a strategy, there is no "collusion" (collusion), they just choose the strategy that is most beneficial to them, regardless of social welfare or the interests of any other opponent. In other words, this strategy combination is composed of the best strategy combination of all participants (also called parties and participants). No one will take the initiative to change the strategy in order to strive for greater benefits for themselves. " Prisoner's Dilemma "has extensive and profound significance. The conflict between individual rationality and collective rationality and everyone's pursuit of their own interests lead to a "Nash equilibrium", which is also an unfavorable outcome for everyone. Both of them think of themselves first in the strategy of frank denial, so they are bound to serve long sentences. Only when everyone thinks of each other first, or colludes with each other, can we get the result of the shortest imprisonment. Nash equilibrium first challenges Adam Smith's "invisible hand" principle. According to Smith's theory, in the market economy, everyone starts from the purpose of self-interest, and finally the whole society achieves the effect of altruism. Let's review the famous saying of this economic sage in The Wealth of Nations: "By pursuing (personal) self-interest, he often promotes social interests more effectively than he actually wants to do. "The paradox of the principle of" invisible hand "leads from Nash equilibrium: starting from self-interest, the result is not self-interest, neither self-interest nor self-interest. This is the fate of two prisoners. In this sense, the paradox put forward by Nash equilibrium actually shakes the cornerstone of western economics. Therefore, from Nash equilibrium, we can also realize a truth: cooperation is a favorable "self-interest strategy". But it must conform to the following Huang Jinlv: Treat others as you want them to treat you, but only if others do the same. That's what China people say, "Don't do to others what you don't want others to do to you". But only if you don't do to me what you don't want me to do. Secondly, Nash equilibrium is a non-cooperative game equilibrium. In reality, non-cooperation is more common than cooperation. Therefore, "Nash equilibrium" is a significant development of the cooperative game theory of von Neumann and Morgan Stern, and even a revolution.

From the general sense of Nash equilibrium, we can deeply understand the common game phenomena in economy, society, politics, national defense, management and daily life. We will give many examples similar to the "prisoner's dilemma". Such as price war, military competition, pollution and so on. The general game problem consists of three elements: players, also known as the set of parties, participants and strategies. Each player's strategy and payoff. Among them, the so-called win refers to the utility that people in each game get if they choose a specific strategic relationship. All game problems will encounter these three elements.