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. 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: if both suspects confess their crimes and hand over the stolen goods, both of them will be convicted and sentenced to 8 years each; If only one suspect confesses and the other denies it, then the crime of obstructing official duties (because there is evidence that he is guilty) will be punished for another two years, and the confessor will be released immediately. 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. The table below gives the payoff matrix of this game. Prisoner's Dilemma Game A ╲ B Confessions Denying Confessions 8,80,/kloc-0,0 Denying/kloc-0,01For A, although he doesn't know what B chooses, he knows that no matter what B chooses, it is always the best for him to choose confessions. 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. However, "confession" is the dominant strategy of any criminal suspect, and (confession, confession) is a dominant strategy equilibrium, that is, Nash equilibrium. It is not difficult to see that Nash equilibrium and Pareto are in conflict.

From a mathematical point of view, this theory is reasonable, that is, all choices are frank. However, it is obviously inappropriate in this sociological field where multidimensional information works together. Just like in ancient China, bribery among officials was called "bad rules" instead of trying to find out, because the social system constrained people's behavior and forced people to change their policies. For example, from a psychological point of view, the cost of choosing confession will be greater, and one party's confession will make the other party guilty, so revenge afterwards and the role of "selling" among insiders will make him lose more. However, the increase ratio between 8 years and 10 years will be diluted, and human dignity will make people feel vindictive and break the "rules" slightly. We are in the era of big data. To deal with a matter closer to the truth, we should grasp as much relevant information as possible and make a reasonable weighted analysis. People's motivation for moving images is complex, and the prisoner's dilemma can only be used as a reference for simplifying the model, and specific decisions need to be analyzed in detail. Intelligent pig game

First of all, the example of "pig's income" in economics is:

Suppose there is a big pig and a little pig in the pigsty. One end of the pigsty has a pig trough (both pigs are at the trough end), and the other end is equipped with a button to control the supply of pig food. When you press the button, 10 unit of pig food will enter the trough, but on the way to the trough, two units of pig food will consume physical strength. If the big pig reaches the trough first, the profit ratio of big pig to food is 9: 65,438+0. At the same time (press the button), the income ratio is 7: 3; Piglets arrive at the trough first, and the income ratio is 6∶4. Then, under the premise that both pigs are wise, the final result is that the little pig chooses to wait.

The intelligent pig game was put forward by Nash in 1950. In fact, the pig chooses to wait and let the big pig press the control button, and the reason why he chooses to "take a boat" (or hitchhike) is very simple: on the premise that the big pig chooses to act, if the pig chooses to wait, the pig can get four units of net income, while if the pig acts, it can only get 1 unit of net income left by the big pig, so waiting is better than acting; On the premise that the big pig chooses to wait, if the little pig acts, the income of the little pig will not cover the cost, and the net income is-1 unit. If the pig also chooses to wait, the benefit of the pig is zero and the cost is zero. In a word, waiting is better than action.

The reward matrix in game theory can describe the choice of piglets more clearly: big pigs such as piglets act 5, 1 4, 4, etc. 9,-10,0. As can be seen from the matrix, when the big pig chooses to act, if he acts, the income is 1, and if he waits, the income is 4, so he chooses to wait. When the big pig chooses to wait, if the little pig acts, its income is-1, and if the little pig waits, its income is 0, so the little pig also chooses to wait. On the whole, whether the big pig chooses to act or wait, the choice of the little pig will be waiting, that is, waiting is the dominant strategy of the little pig.

In the management of small enterprises, learning how to "hitchhike" is the most basic quality of a shrewd professional manager. At some point, it is wise to wait and let other big enterprises explore the market first. At this time, you can do something without doing it!

Smart managers are good at using various favorable conditions to serve themselves. "Hitchhiking" is actually another choice for professional managers to face every expense. Paying attention to and studying it can save a lot of unnecessary expenses for enterprises, thus making the management and development of enterprises go to a new level. This phenomenon is very common in economic life, but managers of small enterprises are rarely familiar with it.

In the intelligent pig game, although the pig's behavior of "picking up ready-made goods" is morally contemptible, isn't the main purpose of the game strategy to maximize its own interests by using strategies? Beautiful coins

A strange beauty came to chat with you and asked to play a game with you. The beauty suggested, "Let's show each side of the coin, either heads or tails. If we are both heads, I will give you 3 yuan, if we are both tails, I will give you 1 yuan, and the rest you will give me 2 yuan. " Sounds like a good proposal. If I were a man, I would play anyway, but financial considerations are another matter. Is this game really fair enough? Gentlemen/beautiful women have heads and tails 3, -3 -2, +2 tails -2, +2 1,-1 Suppose that the probability of our heads is X and the probability of our tails is1-X. In order to maximize the benefits, we should get equal rewards when our opponents show their heads or tails, otherwise they can always change their heads.

Generally speaking, this equation means that when your opponent is always positive, you get the same and the greatest benefit as when your opponent is always negative. Solving the equation leads to x=3/8, which means that it is our best strategy to present three heads every eight times and five tails on average. Substituting x=3/8 into the income expression 3*x+(-2)*( 1-x) can get the expected income every time, and the calculation result is-1/8 yuan.

Similarly, let the probability of beauty appearing on the front side be y, the probability of beauty appearing on the back side be 1-y, and the equation-3y+2 (1-y) = 2y+(-1) * (1-y).

Y equals 3/8, and the expected income of a beautiful woman is 2( 1-y)-3y= 1/8 yuan. This tells us that when both sides adopt the optimal strategy, the average beauty wins 1/8 yuan. In fact, as long as the beauty adopts the (3/8, 5/8) scheme, no matter what scheme you adopt, it will not change the situation. If all of them are heads, the expected return each time is (3+3+3-2-2-2)/8 =-1/8 yuan.

If all the tails are displayed, then the expected return every time is (-2-2-2+1+1)/8 =-1/8 yuan. And any strategy is nothing more than a linear combination of the above two strategies, so the expectation is still-1/8 yuan. But when you also adopt the optimal strategy, you can at least guarantee the minimum loss. Otherwise, you will definitely be targeted by the strategies adopted by beautiful women, thus losing more. This game model seems useless, but it may actually involve the most important model in financial market pricing: pricing weight model.

Generally speaking, the essence of "game theory" is to show the competitive contradictions in daily life in the form of games, and analyze the operating rules of things by using mathematical and logical methods. Since there are participants in the game, there must be makers of the rules of the game. A deep understanding of the essence of competitive behavior is helpful for us to analyze and master the relationship between things in competition, and it is also more convenient for us to formulate and adjust rules so that they can finally operate according to our expected purpose.