Game theory, also known as game theory, is a new branch of modern mathematics and an important part of operational research. It is written in the game bible that game theory is that two people use each other's strategies to change their confrontation strategies in an equal game to achieve the meaning of winning. According to Professor robert aumann, who won the Nobel Prize in Economics in 2005 for his contribution to game theory, game theory is a theory to study interactive decision-making. The so-called interactive decision-making, that is, the decisions of all actors ([players]) are interactive, and everyone must consider other people's decisions in his own decision-making, of course, he also needs to consider other people's own considerations ... In such iterative consideration, he makes decisions and chooses the most favorable strategy for himself.

Game theory has a wide range of applications and has become an important research and analysis tool in the fields of economics, political science (domestic and international), military strategic issues, evolutionary biology and contemporary computer science. In addition, it also has important connections with accounting, statistics, mathematical foundation, social psychology, epistemology, ethics and other philosophical branches.

According to the "Game Theory" in the New palgrave Dictionary of Economics written by Auman, the starting point of standard game theory analysis is rational, not psychological or social. However, in recent 20 years, behavioral game theory, which combines the research results of psychology, behavioral science and experimental economics, has gradually emerged.

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 formal birth of game theory. 1944, the epoch-making masterpiece Game Theory and Economic Behavior written by von Neumann and Morgenstein extended the two-person game 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. 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. 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.

Basic concepts of game theory

Game elements:

1. Decision-maker: The party who takes the lead in making a decision in the competition. This party often takes a directional action first according to their feelings, experiences and superficial conditions.

2. Opponent: the backward party in the two-person game, the decision-maker must make a basic negative decision, and its action is lagging behind, default, passive, but ultimately dominant. His strategy may depend on the decision maker's inferior strategic choice and occupy spatial characteristics, so confrontation is the only dominant way, which is actually the leader's phased termination behavior.

3. Biological affinity: All living things have the instinct to seek law and order in a harsh and unknown environment. In the game, participants have the behavior of waiting in a chaotic environment and looking for orderly approach.

4. Players: 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".

5. strategiges: In a game, each player has a feasible 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, and a feasible action plan planned by a player from beginning to end is called a strategy of the player in this game. If everyone in a game always has finite strategies, it is called "finite game", otherwise it is called "infinite game".

6. Revenue: The result at the end of the game is called revenue 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.

7. Order: Each player's decision has priority. If a player has to make multiple decisions, there will be a problem of order; Other elements in the same order are different, so the game is different.

8. This game contains equilibrium: equilibrium means equilibrium. In economics, equilibrium means that the correlation 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.

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.

Game type

(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: participants 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: A, B, C and C vote to decide how to allocate 654.38+0 million 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.

Therefore, A, B and C should get 2/3 of 1 ten thousand, 1/6 and 1/6 respectively.

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.

Game theory refers to the process in which individuals or organizations, under certain environmental conditions and certain rules, choose and implement their chosen behaviors or strategies by relying on the information they have, 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".

Game theory-this is a hot concept. It not only exists in the operational research of mathematics, but also occupies an increasingly important position in economics (the Nobel Prize in Economics has been awarded to game theory researchers frequently in recent years), but if you think that the application field of game theory is limited to this, you are all wet. In fact, game theory is even ubiquitous in our work and life! At work, you are playing games with superiors and subordinates, and you will also play games with other relevant departments; In business, you are playing games with your customers and competitors. In life, games are still everywhere. Game theory represents a brand-new analytical method and concept.

Paul samuelson, winner of the Nobel Prize in Economics, said:

If you want to be a valuable person in modern society, you must have a general understanding of game theory.

It can also be said that if you want to win business, you must learn game theory; If you want to win life, you must also learn game theory.

Is the game theory profound? Through this textbook, you will find that abstruse game theory can be so vivid, popular and easy to understand. A large number of cases and easy-to-understand language will help you easily master game theory, the most fashionable tool today.

The game bible also says: 2 1 century, we should stand at the forefront of game theory. Although there are few game economists, they have won the highest proportion of Nobel Prize. What can shake human emotions most is the game, and the most influential game in the future is the game. Comment on the wealth of a person and a country depends on how much he shares in the game.

This shows the importance of games.

"Pig's Income" in Economics

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 amount of food dropped 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 the situation described in the incremental scheme of Smart Pig Game. 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".

[Edit this paragraph] The principle and application of Nash game theory

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 hadn't been for more than 30 years of serious mental illness, I'm afraid he would have stood on the podium of the Nobel Prize, and he would 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 put forward the basic principles of computers. 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.