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".
Although the solution of two-person zero-sum game is of great significance, as a theory, its application in practice is extremely limited. Not to mention the players who are addicted to the game, it can be said that it is almost impossible to use it except for military competitions. There are two main limitations of two-person zero-sum game. First, in various social activities, there are often many parties involved rather than only two parties; Second, the result of participating in the interaction of all parties does not necessarily mean that some people will lose, and the whole group may have a net profit greater than or less than zero. For the latter, let's look at a classic and interesting case in history: "Prisoner's Dilemma". It is said that the police caught two thieves, but the evidence was insufficient, so they hoped the suspects would turn themselves in. The police isolated two prisoners and questioned them separately. The policy is: be lenient in confession and strict in resistance! If you confess and the other party doesn't confess, then you will be released and the other party will be sentenced to 20 years; Similarly, if you don't confess and another person confesses, then you will be sentenced to 20 years and the other person will be released. If both of them confess, the police will have enough evidence, and both of them will be sentenced to 10 years. As for the case that neither of them confessed, both of them had to be sentenced without the police, but because of insufficient evidence, the sentence was much lighter, such as 1 year. Finally, the police said that there was another policeman over there who explained the same policy with your partner. Criminals have a career in mind. If the other party confesses, I will recruit 10 years, and if I don't recruit, it will be 20 years, which is very cost-effective. If the other party doesn't talk, I'll be acquitted. If I don't talk, it will be 1 year, which is still a good deal. So, move! Two "smart" thieves confessed, and both of them were sentenced to 10 years, which is exactly what the police want. Smart readers, in fact, if two thieves don't confess, they will each be sentenced to 1 year. Wouldn't it be better for them? In this prisoner's dilemma, there are still two participants (two thieves), but this is no longer a zero-sum game, and the loss of people does not mean my gain. Two thieves may be sentenced to 20 years in prison, or only 2 years in prison.
Before Nash, no one knew how to solve the multi-player non-zero-sum game problem, or how to find a "balance" similar to the minimum-maximum solution. Without the solution, the following research can not be carried out, let alone guide practice. Nash's great contribution to game theory lies in his genius in putting forward the basic concept of "Nash equilibrium" and finding a more general and broader understanding of game problems. The basic idea of Nash equilibrium is that the strategies of all participants in this solution set are the best responses to the strategies used by other participants, and no one can simply change their own strategies to improve their returns. Take the prisoner's dilemma as an example. If thief A believes in thief B's confession, then his best strategy is confession, while if thief B believes in thief A's confession, then his best strategy is still confession. This is a Nash equilibrium, which is "self-determined". There is only one Nash equilibrium in the prisoner's dilemma. But if the conditions are changed, there may be more than one Nash equilibrium in many other specific problems. Nash skillfully used mathematical skills to prove the following Nash theorem: for any non-cooperative game (zero-sum or non-zero-sum) with n players, if each player has only a limited number of strategies, then there is at least one Nash equilibrium solution set. Like many of the most outstanding ideas in science, this concept solves unsolved problems in a very concise and clear way. Seemingly simple, it seems to belong to the kind of thing that "I could have thought of it", but at that time, except Nash, a generation of master Neumann did not think of it. The introduction of Nash equilibrium has had a revolutionary impact on the development of game theory, and the concept of Nash equilibrium has become the cornerstone and center of modern game theory (although this is still controversial among a few game theorists). Dixit, a good friend of Nash and an economics professor at Princeton University, once said: "If Nash can get a dollar every time someone talks about or writes down the words Nash equilibrium, then he is a millionaire now!"
The game theory mentioned above tries to solve the problem of non-cooperation, that is, there is no other form of information exchange between participants except the decision-making results. Through an example of prisoner's dilemma, it can be seen that if two thieves can consult each other, their strategic decisions will be completely different (of course, it is cost-effective for them to deny it together). It is true that in all kinds of life behaviors, apart from competition, there is also cooperation between people, and often the two relationships coexist. Reasonable cooperation can bring * * * common interests to both parties. This is the category of cooperative game theory research. Neumann established the basic model of cooperative game theory in Game Theory and Economic Behavior, but failed to give a definite solution to the extremely important two-way negotiation problem (that is, how participants "bargain"). Nash has also made outstanding contributions in this field. He not only put forward an axiomatic solution to the bargaining problem that is directly beneficial to the labor economy and international trade, but also put forward Nash's plan in theory by using this solution: the negotiation in the cooperative game is transformed into a step in a wider non-cooperative game-the ultimate goal of negotiation is to maximize its own interests. In addition, Nash was a pioneer in behavioral experiments to test game theory. He conducted bargaining and alliance formation experiments, and pointedly pointed out that in other experimenters' prisoner's dilemma experiments, repeatedly asking a pair of participants to repeat the experiment actually turned the one-step strategy problem into a big multi-step strategy problem. The latter thought first put forward the possibility of silence in repeated game theory, which played an important role in economic and political fields.
These seemingly boring theories use logical reasoning as a tool to make a rigorous and orderly mathematical induction of competition and cooperation in people's daily life. When mathematicians tirelessly upgrade intuition to science, and then react to life, its far-reaching influence is beyond words. Today, with the continuous development of many experts, the modern game theory, for which Nash has made fundamental contributions, not only matures and perfects its own theoretical system, but also is widely used in various fields such as economy, politics, military science and even biology. In the field of biology, game theory is used to study the competition between species and within species, as well as the competition between individual genes in ethnogenetics and evolutionary biology, which in turn promotes the development of game theory. In the field of politics and military science, game theory is used to analyze important issues such as election strategy, causes of war and legislative agenda. In the field of economics, game theory has been integrated into the mainstream of the whole discipline, and all economics textbooks and magazines contain game theory. Economists have regarded the game theory, which studies the interaction of strategies, as the most appropriate analytical tool to analyze various economic problems, such as public economy, international trade, natural resource economy, industrial management and so on. As for the direct benefits of applying game theory to economics, for example, in the book A Beautiful Mind, 1994, the US government auctioned most of the electromagnetic spectrum to businesses. This multi-round auction was carefully designed by a group of game theory experts in line with the principle of maximizing government revenue and maximizing the utilization rate of various businesses, and achieved great success. The government earned more than10 billion, and the spectrum of each frequency also found a satisfactory home. In contrast, a similar auction in New Zealand that was not designed by game theory was a fiasco. The government only gets 15% of the estimated income, and the frequency of being auctioned is not fully utilized. For example, because there is no competition, a college student only spent 1 USD to buy a TV station license! It is precisely because the game theory has had such a great impact and influence on modern economics that the Royal Swedish Institute announced in 1994 that the Nobel Prize in Economics, the highest honor for scientists all over the world, was awarded to three mathematicians, including Nash, in recognition of their pioneering analysis of non-cooperative game theory.
Secondly, the following six paragraphs are about Nash's introduction.
The world finally recognized Nash's genius because of game theory. This year, he is 66 years old. Compared with his dazzling outstanding contributions in science, his legendary life, full of talent and passion, full of hardship and pain, and intertwined with reason and madness, is not inferior, which makes people admire him infinitely. Nash was born in 1928, the family of an electronic engineer. On the one hand, he was withdrawn when he was young, on the other hand, he showed extraordinary talent in mathematics. /kloc-When he entered Carnegie Mellon University at the age of 0/7, his major was chemical engineering, but at the suggestion of a discerning teacher, he turned to specialize in mathematics. During this period, he took an international economics course, which aroused his interest in economic proposition. Later, the paper on bargaining in cooperative games originated from some ideas at this time. At the age of 20, Nash received a bachelor's degree and a master's degree in mathematics from Carnegie, and received a generous scholarship from Princeton University to become a graduate student here. He showed interest in many mathematics subjects, such as topology, algebra, geometry, game theory and logic. When preparing his doctoral thesis, he was determined to create a brand-new topic of his own. Finally, the bargaining problem he thought about in the past led him to establish the basic principle of non-cooperative game theory. 1949, Nash, 2 1 year-old, wrote a famous paper, Equilibrium Point of Multiplayer Game, and put forward the concept and solution of Nash Equilibrium, one of the most important ideas in the whole modern non-cooperative game theory, which laid the foundation for him to win the Nobel Prize 44 years later. 1950, Nash went to see the world-famous Neumann with his own ideas, which was flatly denied. However, in the relaxed scientific environment of Princeton University, his paper was published and caused a sensation. In the same year, he obtained his doctorate in mathematics with his thesis "Non-cooperative Game".
Nash, who pretends to be a pure mathematician, proved an anti-intuitive equidistant embedding theorem when he worked in the Rand Institute and Princeton University after graduation, and introduced a brand-new method to prove the high-dimensional equidistant embedding theorem, which greatly promoted the proof of the existence, uniqueness and continuity theorem of partial differential equations. For pure mathematicians, mathematics is spiritual rhythmic gymnastics. The criterion to judge whether a research is good or bad lies in its mathematical depth, whether it introduces new ideas and methods of mathematics, or solves long-standing unsolved problems. From this point of view, Nash's achievements, as well as his more difficult mathematical research when he was working at MIT a few years later, are more convincing to his colleagues in the mathematical field than his Nash equilibrium. Indeed, in 1958, Nash was named the most outstanding figure among the new generation of talented mathematicians by American Fortune magazine because of his outstanding work in the field of mathematics. However, there are unexpected events in the sky, and people are doomed. Just as Nash's career was booming and he was about to reach the peak, he was suddenly hit by ruthless fate and fell into hell from the clouds. Nash suffered from schizophrenia in his thirties.
He is not a perfect man. As early as 1952, Nash knew a girl five years older than him, got in touch with her, and had an illegitimate child the next year. Since then, he has maintained a close relationship with her. 1956 His parents found out about their son's affair, and his father died soon after. I don't know if it has anything to do with this blow, and I don't know if Nash has blamed himself for it. 1957, he married Alicia, a beautiful young female student at MIT. After more than 40 years of suffering, his love and affection with * * * can be witnessed, which may be the most perfect and luckiest moment in his personal life. 1958 Alicia was pregnant and had not yet given birth, and Nash's mental condition began to deteriorate. His behavior is getting more and more eccentric, and he is going crazy step by step.
Nash suffers from paranoid schizophrenia, which is the most terrible of all mental diseases. The patient's heart is full of intermittent and unrealistic crazy thoughts, which will lead to hallucinations and auditory hallucinations and talk to imaginary people. Nash will try to tell the air that a newspaper contains information from another planet that only he can decipher; Will suddenly resign from his post in Massachusetts, run to Europe alone, give up American citizenship, or Alicia will drag him back; At home, he kept threatening his wife Alicia. In desperation, Alicia 1962 divorced Nash. But her loyal love for him has not disappeared. In 70, Nash's mother died and his sister couldn't afford him. Just when Nash was about to live alone on the street, the kind Alicia took him to live together. She not only cares about him in daily life, but also takes care of his mood with the care and sensitivity peculiar to women. She understood his desire not to go to the hospital for closed treatment, and moved her family to Princeton, far away from the noise, hoping that the quiet and familiar academic atmosphere would help stabilize Nash's mood.
This is a strange game. Nash, a mathematical genius who studies rational strategies, suddenly lost his proud rational thinking and struggled back and forth between sobriety and madness involuntarily. Will he go to rolling in the deep forever or walk home? In that world where no one can solve it, he never gave up his love for mathematics. We can't know all the pain Nash suffered, but we can figure out how long the great conflict between will and ability is. Fortunately, there is a loyal participant in this game. When he mumbles something that no one can understand, when he wanders around the green campus like a ghost, he always has a pair of warm eyes and arms to accompany him bravely. The two strongest things in the world, will and love, combine to create an optimal strategy, which is a miracle. Yes, the world witnessed the comedy ending of the game. Nash suffered from schizophrenia for more than 30 years, and his spirit gradually returned to normal in the 1990s. In the autobiography written for the Nobel Prize in 1994, Dr. Nash did not mention the pain caused by mental illness, but only said that mental disorder freed him from the shackles of conventional thinking and helped him create a brand-new theory. Finally, he wrote, "Statistically speaking, it seems impossible for any mathematician or scientist to achieve great achievements at the age of 66. But I'm still trying. The 25-year' holiday' of abnormal thinking is not normal. So, I still have hope. Maybe I can make something valuable through current research or new ideas generated in the future. " After reading this, I can't help sighing, sighing the extraordinary genius of this game theory master, sighing his tenacious will and unreserved dedication to science! Maybe these are also the sources of Alicia's love?
Things are like chess. The glory and bitterness of the previous generation have become history, and the future is in the hands of the latecomers, depending on their every decision. What kind of game will our life be?
The following are some examples I collected, and I made some modifications myself, which are more concise and easy to understand even if you are not an economics major.
Life is an endless game process, and the game is to achieve a desired result by choosing appropriate strategies. As a player, the best strategy is to make the best use of the rules of the game; As the best strategy of society, it is to guide the increase of the overall welfare of society through rules.
A never-ending game
People's work and life can be regarded as an endless game decision-making process. People have to make decisions when they wake up every morning. We decide what to eat for breakfast day after day until we form a fixed eating habit. Do you want to go to the supermarket to buy something crazy? Want to, roulette the red or black on the table, or even read a book ... whether consciously or unconsciously, thoughtfully or impulsively, you have started reading this book-this is a decision.
There are more important things: what school to apply for, what major to choose, what job to do, how to carry out a research, how to run a business, who to cooperate with, whether to take a part-time job, whether to resign, whether to compete for the position of president. Even whether to get married, when to get married, who to marry, whether to have children, how to raise children and so on. These are just a few examples of major decisions in life.
In these decisions, there is a common factor, that is, you are not making decisions alone, making decisions in a vacuum world without interference. On the contrary, you are surrounded by decision makers just like you, and their choices interact with yours. This interaction will naturally have an important impact on your thinking and actions, and the choices and decisions of others will directly affect your decision-making results. Robinson is a desert island, and he has the final say in everything he does. But when a "Friday" comes, he will face the game problem.
Game theory was founded by two outstanding scholars-von Norman and Morgenstein in the middle of last century. In technical terms, game theory is "to study how people make decisions when the actions of decision makers interact directly, and how this decision can reach a balance".
In order to explain and understand the interaction of game decisions, we might as well imagine the difference between masons' decisions and boxers' decisions. When a stonemason considers how to cut stones, his "object" is passive and neutral in principle and will not show strategic confrontation to him. However, when a boxer intends to attack the opponent's key points, not only will his every step plan attract resistance, but he will also face the opponent's active attack. He must try to overcome these resistance and attacks.
In the game between people, you must realize that your business rivals, future partners and even your children are smart and independent people, living people who care about their own interests, rather than passive and neutral roles. On the one hand, their goals often conflict with yours; On the other hand, they contain potential cooperation factors. When you make a decision, you must take these conflicts into account and pay attention to the role of cooperative factors.
For yourself and for better cooperation with others, you need to learn a little strategic thinking of game theory. It is for this reason that the famous economist paul samuelson said, "If you want to be a literate person in modern society, you must have a general understanding of game theory."
Tips:
Game theory is a bit evasive, but the content is easy to understand, that is, when each player decides what action to take? We should not only act according to our own interests and purposes, but also consider the possible influence of his decision-making behavior on others and the possible influence of others' behavior on him, and seek the maximization of interests or utility by choosing the best action plan.
Games are the abstraction of life.
The word "game" sounds profound, but it actually means "game". More accurately, it is a game that can be won or lost. The literal translation of game theory is "game theory" It may be said that game theory is to gain knowledge of life competition by "playing games".
What happened? In short, the game is the abstraction of life.
Chess, for example, has several roles: king, queen, knight, bishop and soldier, just like a small kingdom where politics and religion are integrated. Of course, this model is too simple compared with life, but it can also reflect some truth of life. Moreover, only by being simple, these truths, which were originally concealed by the complicated appearance of life, can be more clearly seen.
In the face of complex things, people often fall into the trap of seeing trees instead of forests, being overwhelmed by details and unable to find the key points. In the game, it can reflect some real-world problems and minimize the interference factors, which is a very suitable way to make decisions.
Go is probably the simplest and most complicated game. It originated in China 4000 years ago, but until now, we may not really understand it. The simplest chessboard-a net woven with vertical and horizontal 19 lines (originally 17 lines); The simplest chess pieces (only black and white); The simplest rules (taking turns to play chess, playing chess with two vitality, winning empty-handed, plus some supplementary terms such as "robbery") can be learned by a person who knows nothing about Go in a few minutes, but its mystery is deeper than any other board game. If you have worked in Weiqi, you will certainly understand some philosophies, such as "losing is winning", "running water is not the first", "winning in chaos", "going too far is not enough" and so on. At this point, the game is a bit like the fable we saw when we were young. Didn't we learn the truth of life from these stories?
Don't underestimate the game, it is indeed a model of life. From an early age, we learn how to live, how to get along with others, how to adapt and apply the rules of this world, and build our own personality in the process. Therefore, don't underestimate the game, it can really reflect the real life.
Tips:
Zero-sum game: players lose and win, but the total score of the whole game is always zero. The whole game process is a zero-sum game.
From games to life
An American pilot who participated in the Gulf War was asked how he felt about the war when he returned home. He replied: It's just like playing computer games. In fact, many computer games have been applied to military training now. After "9. 1 1", a flying game of Microsoft attracted attention, because in the game, players can experience the feeling of flying over big cities such as new york and even crossing the World Trade Center. People are worried that terrorists can use this game to get a chance to practice. Maybe they have already done so.
Games are a good way to learn. The Duke of Wellington, who defeated Napoleon, once said, "The Battle of Waterloo was decided on the playground of Eton Middle School." Usually practice skills and tactics diligently to avoid panic in times of crisis. This principle applies to most competitions or games.
The best part is: you won't lose anything during the game. Of course, except for some self-esteem, even if you lose, you won't lose anything. In the monopoly game, you can learn how to buy and sell real estate wisely from the experience of losing millions of dollars in the blink of an eye without paying any price afterwards.
Of course, different games have different requirements for players. Some people are good at thinking games, but different sports have different requirements for decision-making wisdom. For example, intelligence is not so important in boxing or sumo, a sport that divides the competition level by "heavyweight".
Playing games requires many different types of skills. One of them is the basic skills, such as the indispensable shooting ability in basketball, the accumulation of cases in legal work, and the need to remember a lot of "formulas" when playing Go (changes acceptable to both sides can be called "balance" on the board of Go). Once these skills are played, they may be useless. But the strategic thinking of game theory is another skill. Strategic thinking starts from your basic skills and considers how to maximize these basic skills. This is a universal principle and can be applied to all aspects of life.
Strategic planning and game theory are actually interlinked: your decision must be superior to your opponent, so that individuals, families, tribes or countries have a chance to survive.
Tips:
The famous French soprano Maria di maple has a large private garden. Every weekend, people always go to her garden to pick flowers and mushrooms. What's more, they will set up tents and have picnics there. Although the managers erected fences around the garden many times and hung wooden signs of "Private Garden, No Entry", all these efforts were of no help. After learning about this situation, Dimmespur ordered the managers to make many eye-catching big signs that said, "If someone is bitten by a poisonous snake in the garden, the nearest hospital is 15 kilometers away" and put them around the garden. Since then, no one has broken into her garden without permission. -If the habitual methods can't solve the problem, we should adjust our perspectives and concepts.
Inevitable contradictions in multiplayer games
Games are not limited to two opponents, and many games are played by many people. If the consequences are shared by many people, the whole decision-making process will be more difficult; Because you will face different members and different goals. As for multi-person decision-making, we can understand it through the mode of group confrontation. In this kind of competition, good decisions can create victory.
There are many different forms of realistic multi-person decision-making: sometimes many people participate in decision-making, but only one opinion is needed, which is an ideal Committee system; Some are two people involved in decision-making, but they are antagonistic, such as wrestling, chess, fencing, tennis singles and so on. In addition, there are many forms of decision-making, such as Congress, the United Nations, poker clubs, political parties and so on. Regardless of the quality of life, the ultimate goal of these decisions is to pursue the sustainable survival of human beings on the earth. However, although there are many extremely important decisions that need to be determined and put into practice, we do not have a set of rational practices to completely avoid the dilemma of "three monks have no water to drink". Every combination of decision makers and alternatives is self-contained, successful decisions are different, and some combinations are completely inoperable. In some cases, it is impossible to make a decision that is not contradictory.
Game is strategic interdependence: your choice? In other words, what the strategy will get depends on the choice of another or another group of purposeful actors. The decision makers in a game are called participants, and their choices are called actions. The interests of the participants in a game may be strictly opposed, and one person's gain is always equal to another person's loss. Such a game is called a zero-sum game.
But it is more common that there are both interests and conflicts of interest, so there may be a combination of strategies that will lead to * * * benefits or * * victims. In practice, the game may include some continuous action processes or some synchronous action processes, so you need to synthesize skills, use them flexibly, think and decide what your best action should be.
Tips:
If you leave the simple principle of mutual adaptation, then your cleverness will not have a good result.
Fairness comes from games.
The game is not necessarily a bad thing, and it is not necessarily impossible to achieve good results. The rich material life we enjoy today comes from the competition in the free market-it is also the result of the game. Adam Smith published in 1776.
Fourthly, robert aumann's game theory and its economic theory.
Brief introduction of Nobel Prize winners in economics;
Robert aumann was born in June 1930 and graduated from new york University with a bachelor's degree in mathematics. Later, he obtained a master's degree in mathematics and a doctor's degree in mathematics from MIT 1952 and 1955 respectively. From 65438 to 0966, robert aumann was elected as a member of Econometrics Association. He is currently a professor at the School of Mathematics of Hebrew University in Jerusalem, a professor at the Department of Economics and the Academy of Decision Sciences of the State University of new york at Stani, the president of the Israeli Mathematics Club, and an honorary member of the American Economic Association. He is the editor of many professional magazines, such as International Journal of Game Theory, Journal of Mathematical Economics, Journal of Economic Theory, Econometrics, Mathematics of Operational Research, siam Journal of Applied Mathematics and Game and Economic Behavior, etc.
As an outstanding economist, Robert Auman.