What is game theory? Rebellion? Master came in! !

What is game theory?

Game theory, sometimes called game theory, is a theory and method to study the phenomenon of struggle or competition. It is not only a new branch of modern mathematics, but also an important subject of operational research.

2. Prisoner's Dilemma Game

Two thieves who committed crimes together were taken to the police station for solitary confinement. If one party cooperates with the police and confesses what he did with the other party, but the other party fails to confess, the confessed party will be released and the other party will be sentenced to three years' imprisonment. If both parties confess, they will be sentenced to 1 year imprisonment; If both sides don't confess, they will be sentenced to 1 month imprisonment for lack of police evidence. How will these two thieves choose?

3. The development of game theory

The idea of game theory has existed since ancient times, and Sun Tzu's Art of War is not only a military work, but also the earliest monograph on game theory. At first, game theory mainly studied the winning or losing of chess, bridge and gambling. People's grasp of the game situation only stays in experience and has not developed into a theory. It was not until the beginning of the 20th century that it officially developed into a discipline. 1928 von Neumann proved the basic principles of game theory, thus announcing the birth of game theory. 1944, the epoch-making masterpiece Game Theory and Economic Behavior written by von Neumann Morgenstein extended the two-person game structure to the n-person game structure, and applied the game theory system to the economic field, thus laying the foundation and theoretical system of this discipline. When it comes to game theory, we can't ignore Nash, a genius of game theory, and Nash's groundbreaking papers, Equilibrium Point of N-player Game (1950) and Non-cooperative Game (195 1). The concept of Nash equilibrium and the existence theorem of equilibrium are given. In addition, the research of Selton and Hasani also promoted the development of game theory. Today, game theory has developed into a relatively perfect discipline.

4. Basic concepts of game theory

1) game elements

(1) Player: In a game or game, every participant who has the decision-making power becomes a player. A game with only two players is called a "two-player game", and a game with more than two players is called a "multiplayer game".

(2) Strategy: In a game, each player has a practical and complete action plan, that is, the plan is not an action plan at a certain stage, but a plan to guide the whole action, which is a feasible plan for each player from beginning to end.

An action plan planned by this bureau is called the strategy of people in this bureau. If everyone in a game always has finite strategies, it is called "finite game", otherwise it is called "infinite game".

(3) Gain and loss: The result at the end of a game is called gain and loss. The gains and losses of each player at the end of a game are not only related to the strategies chosen by the players themselves, but also to a set of policies adopted by the players in the whole situation. Therefore, the "gain and loss" of each participant at the end of a game is a function of a set of policies set by all participants, usually called the payment function.

(4) For the game participants, there is a game result.

(5) The game involves equilibrium: equilibrium is equilibrium, and in economics, equilibrium means that the related quantity is at a stable value. In the relationship between supply and demand, if a commodity market is at a certain price, anyone who wants to buy this commodity at this price can buy it and anyone who wants to sell it can sell it. At this time, we say that the supply and demand of this commodity have reached a balance. The so-called Nash equilibrium is a stable game result.

Nash equilibrium: in a strategy combination, all participants are faced with the situation that his strategy is optimal without others changing his strategy. In other words, if he changes his strategy at this time, his payment will be reduced. At the Nash equilibrium point, every rational participant will not have the impulse to change his strategy alone. The premise of proving the existence of Nash equilibrium point is the concept of "game equilibrium pair" The so-called "balanced couple" means that in a two-person zero-sum game, the authority A adopts its optimal strategy a* and the player B also adopts its optimal strategy b*. If player A still uses b*, but player A uses another strategy A, then player A will not pay more than his original strategy a*. This result is also true for player B.

In this way, "equilibrium pair" is clearly defined as: a pair of strategies a* (belonging to strategy set A) and b* (belonging to strategy set B) are called equilibrium pairs. For any strategy A (belonging to strategy set A) and strategy B (belonging to strategy set B), there is always an even pair (a, b*)≤ even pair (a*, b*)≤.

Non-zero-sum games also have the following definitions: a pair of strategies a* (belonging to strategy set A) and b* (belonging to strategy set B) are called equilibrium pairs of non-zero-sum games. For any strategy A (belonging to strategy set A) and strategy B (belonging to strategy set B), there are always: even pair (a, b*) ≤ even pair (a*, b*) player A; Even pair (a*, b)≤ even pair (a*, b*) of player B in the game.

With the above definition, Nash theorem is immediately obtained:

Any two-person game with finite pure strategy has at least one equilibrium pair. This equilibrium pair is called Nash equilibrium point.

The strict proof of Nash theorem needs fixed point theory, which is the main tool to study economic equilibrium. Generally speaking, finding the existence of equilibrium is equivalent to finding the fixed point of the game.

The concept of Nash equilibrium point provides a very important analysis method, which enables game theory research to find more meaningful results in a game structure.

However, the definition of Nash equilibrium point is limited to any player who doesn't want to change his strategy unilaterally, ignoring the possibility of other players changing their strategy. So many times the conclusion of Nash equilibrium point is unconvincing, and researchers call it "naive and lovely Nash equilibrium point" vividly.

According to certain rules, R Selten eliminated some unreasonable equilibrium points in multiple equilibria, thus forming two refined equilibrium concepts: sub-game complete equilibrium and trembling hand perfect equilibrium.

2) Types of games

(1) cooperative game-study how to distribute the benefits of cooperation when people reach cooperation, that is, income distribution.

(2) Non-cooperative game-study how people make decisions to maximize their own interests, that is, strategic choice, under the condition of mutual influence of interests.

(3) Game between complete information and incomplete information: players have a full understanding of all participants' strategic space and the payment under the strategy combination, which is called complete information; On the contrary, it is called incomplete information.

(4) Static game and dynamic game

Static game: refers to the players taking actions at the same time, or although there is a sequence, the latter actor does not know the strategy of the former actor.

Dynamic game: refers to the action sequence of both parties, and the latter actor can know the strategy of the former actor.

Property distribution and Shapley value

Consider such a cooperative game: Party A, Party B, Party C and Party C vote to decide how to allocate 6,543,800 yuan, and they have 50%, 40% and 654.38+00% power respectively. According to the rules, a plan can only be passed when more than 50% of the votes are in favor. So how to allocate it is reasonable? According to the distribution of votes, 500,000, B400,000, C65438+100,000 C proposed to A: 700,000, b0, C30,000 B proposed to A: 800,000, B200,000, c0…… ... ...

Power index: the power of each decision-maker in decision-making is reflected in the number of "key entrants" in his winning alliance, which is called power index.

Shapley value: the sum of participants' marginal contributions to the alliance divided by various possible alliance combinations under various possible alliance orders.

Order abc acb bac bca cab cba

Main entrant

Shapley values of a, b and c are calculated as 4/6, 1/6, 1/6 respectively.

So A, B and C should get 2/3 of 1/3, 1/3, 1/3 respectively.

5. The significance of game theory

The research method of game theory, like many other disciplines that use mathematical tools to study social and economic phenomena, is to abstract the basic elements from complex phenomena, analyze the mathematical model formed by these elements, and then gradually introduce other factors that affect their situation and produce, so as to analyze the results.

Based on different levels of abstraction, three game expressions are formed, which can be used to study various problems. Therefore, it is called "Mathematics of Social Science". In theory, game theory is a formal theory to study the interaction between rational actors, but in fact, it is going deep into economics, politics, sociology and so on, and is applied by various social sciences.

1. Game theory refers to the process that individuals or organizations choose and implement their own behaviors or strategies under certain environmental conditions and certain rules, and obtain corresponding results or benefits from them. Game theory is a very important theoretical concept in economics.

What is game theory? As the old saying goes, things are like chess. Everyone in life is like a chess player, and every movement is like putting a coin on an invisible chessboard. Smart and cautious chess players try to figure out and contain each other, and everyone strives to win, playing many wonderful and changeable chess games. Game theory is to study the rational and logical part of chess player's "playing chess" and systematize it into a science. In other words, it is to study how individuals get the most reasonable strategies in complex interactions. In fact, game theory comes from ancient games or chess games. Mathematicians abstract concrete problems and study their laws and changes by establishing a self-complete logical framework and system. This is not an easy task. Take the simplest two-person game as an example. If you think about it, you will know that there is a great mystery. If it is assumed that both sides accurately remember every move of themselves and their opponents, and they are the most "rational" players, then A has to carefully consider B's idea in order to win the game when playing, and B has to consider A's idea when playing, then A has to think that B is considering his idea, and B certainly knows that A has already considered it.

Faced with such a fog, how can game theory begin to analyze and solve problems, how can it find the optimal solution, and how can abstract mathematical problems be summarized as reality, thus making it possible to guide practice in theory? Modern game theory was founded by Hungarian mathematician von Neumann in the 1920s. His magnum opus Game Theory and Economic Behavior, published in 1944 in cooperation with economist Oscar Morgenstein, marked the initial formation of modern system game theory. For non-cooperative and purely competitive games, Neumann only solves two-person zero-sum games-just like two people playing chess or table tennis, one person wins one game and the other loses the other, and the net profit is zero. The abstract game problem here is whether and how to find a theoretical "solution" or "balance", that is, the most "reasonable" and optimal specific strategy for both players, given the set of participants (both sides), the set of strategies (all moves) and the set of profits (winners and losers). What is "reasonable"? Applying the "min-max" criterion in traditional determinism, that is, each side of the game assumes that the fundamental purpose of all the advantages and disadvantages of the other side is to make itself suffer the most, and accordingly optimize its own countermeasures, Neumann mathematically proves that every two-person zero-sum game can find a "min-max solution" through certain linear operations. Through a certain linear operation, two competitors randomly use each step in a set of optimal strategies in the form of probability distribution, so as to finally achieve the maximum and equal profit for each other. Of course, the implication is that this optimal strategy does not depend on the opponent's operation in the game. Generally speaking, the basic "rational" thought embodied in this famous minimax theorem is "hope for the best and prepare for the worst".

2. In economics, "pig's income" is a famous example of game theory.

This example is about: there are two pigs, a big pig and a little pig in the pigsty. There is a pedal on one side of the pigsty. Every time you step on the pedal, a small amount of food will fall on the feeding port on the other side of the pigsty far from the pedal. If one pig steps on the pedal, the other pig has a chance to eat the food that has fallen on the other side first. As soon as the pig steps on the pedal, the big pig will eat all the food just before the pig runs to the trough; If the big pig steps on the pedal, there is still a chance for the little pig to run to the trough and compete for the other half before eating the fallen food.

So, what strategy will the two pigs adopt? The answer is: Piglets will choose the "hitchhiking" strategy, that is, they will wait comfortably in the trough; The big pig ran tirelessly between the pedal and the trough, just for a little leftovers.

What is the reason? Because, little pigs can get nothing by pedaling, but they can eat food without pedaling. For piglets, it is always a good choice not to step on the pedal whether the big pig does or not. On the other hand, the big pig knows that the little pig can't step on the gas pedal. It's better to step on the accelerator by himself than not to step at all, so he has to do it himself.

The phenomenon of "the little pig is lying down and the big pig is running" is caused by the rules of the game in the story. The core indicators of the rules are: the number of things falling each time and the distance from the pedal to the feeding port.

If we change the core indicators, will there be the same scene of "pigs lying down and big pigs running" in the pigsty? Give it a try.

Change scheme 1: reduction scheme. Feeding is only half of the original weight. As a result, neither the little pig nor the big pig kicked. The little pig will step on it, and the big pig will finish the food; If the big pig steps on it, the little pig will finish the food, too. Whoever pushes means contributing food to each other, so no one will have the motivation to push.

If the goal is to make pigs pedal more, the design of this game rule is obviously a failure.

Variation scheme 2: incremental scheme. Feed twice as much as before. As a result, both the little pig and the big pig can pedal. Anyone who wants to eat will kick. Anyway, the other party won't eat all the food at once. Piglets and big pigs are equivalent to living in a materialistic society with relatively rich materials, and their sense of competition is not very strong.

For the designer of the rules of the game, the cost of this rule is quite high (providing two meals at a time); Moreover, because the competition is not strong, it has no effect to let the pigs push more.

Variant 3: Decreasing plus shifting scheme. Feed only half the original weight, but at the same time move the feeding port near the pedal. As a result, both the little pig and the big pig pushed hard. Those who wait will not eat, and those who work hard will get more. Every harvest is just a flower.

This is the best solution for game designers. The cost is not high, but the harvest is the biggest.

The original story of "Smart Pig Game" inspired the weak (pigs) in the competition to wait for the best strategy. But for the society, the allocation of social resources when piggy hitchhiked is not optimal, because piggy failed to participate in the competition. In order to make the most efficient allocation of resources, the designers of rules don't want to see anyone hitchhiking, so does the government, and so does the boss of the company. Whether the phenomenon of "hitchhiking" can be completely eliminated depends on whether the core indicators of the rules of the game are set properly.

For example, the company's incentive system design is too strong, and it is still holding shares and options. All the employees in the company have become millionaires. Not to mention the high cost, the enthusiasm of employees is not necessarily high. This is equivalent to "smart pig game"

The situation described by the incremental scheme. However, if the reward is not strong and the audience is divided (even the "little pigs" who don't work), the big pigs who have worked hard will have no motivation-just like the situation described in the first phase of the "Smart Pig Game". The best incentive mechanism design is like changing the third scheme-reducing staff and changing shifts. Rewards are not shared by everyone, but for individuals (such as business proportion commission), which not only saves costs (for the company), but also eliminates the phenomenon of "hitchhiking" and can achieve effective incentives.

Many people haven't seen the story of "smart pig game", but they are consciously using pig strategy. Retail investors are waiting for the dealer to get on the sedan chair in the stock market; Waiting for profitable new products to appear in the industrial market, and then copying hot money on a large scale to make huge profits; People in the company who do not create benefits but share the results, and so on. Therefore, for those who make various rules of economic management, they must understand the reasons for the index change of "smart pig game".

3. Background knowledge: the principle and application of Nash game theory.

Beijing Evening News

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 sporadic and fragmented research, which was very accidental and unsystematic. 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.

Prisoner's dilemma:

About "Prisoner's Dilemma"

In game theory, a famous example of dominant strategic equilibrium is Tucker's "prisoner's dilemma" game model. This model tells us the story of a policeman and a thief in a special way. Suppose two thieves, A and B, commit a crime together, enter the house privately and are caught by the police. The police put the two men in two different rooms for interrogation. For each suspect, the policy given by the police is that if a suspect confessed his crime and handed over the stolen goods, the evidence was conclusive and both of them were convicted. If another suspect also confessed, they were each sentenced to eight years in prison; If another suspect denies it without confession, he will be sentenced to two years in prison for obstructing official duties (because there is evidence to prove that he is guilty), and the confessor will be released immediately after eight years of commutation. If both of them deny it, the police can't convict them of theft because of insufficient evidence, but they can each be sentenced to 1 year in prison for trespassing. Table 2.2 shows the payoff matrix of this game.

Table 2.2 Prisoner's Dilemma Game

B

Admit and deny

A Confession–8,–80,–10

Rejected–10,0–1,–1

Let's see what the predictable equilibrium of this game is. For A, although he doesn't know what B chooses, he knows that no matter what B chooses, choosing "confession" is always the best for him. Obviously, according to symmetry, B will also choose "confession". As a result, both of them were sentenced to 8 years in prison. But if everyone chooses "denial", each person will only be sentenced to 1 year. Among the four action choice combinations in Table 2.2, (Rejection, Denial) is Pareto optimal, because any other action choice combination that deviates from this action choice combination will at least make a person's situation worse. It is not difficult to see that "confession" is the dominant strategy of any criminal suspect, and (confession, confession) is a strategic balance of superiority.

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

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, the set of strategies and the choices made by each player.