Definition:

Refers to a strategy combination that satisfies the following properties: in this strategy combination, any player who unilaterally changes his strategy (the strategies of other players remain unchanged) will not improve his income.

Nash proved that Nash equilibrium must exist on the premise that each player has limited strategy choices and allows mixed strategies.

Classification:

Pure strategy Nash equilibrium and mixed strategy Nash equilibrium

"pure strategy Nash equilibrium", that is, everyone involved plays pure strategy; And the corresponding "mixed strategy Nash equilibrium", in which at least one player plays mixed strategy. Not every game will have a pure strategic Nash equilibrium. For example, the "coin problem" only has a mixed strategic Nash equilibrium, and there is no pure strategic Nash equilibrium. But there are still many games with pure strategy Nash equilibrium (prisoner's dilemma and deer hunting game). Even, some games can have both pure strategy and mixed strategy equilibrium.

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Case:?

1 price war

When the two companies fight a price war, Nash equilibrium means the possibility of both sides losing: under the condition that the other side does not change the price, it can neither raise the price, otherwise it will further lose the market; You can't reduce the price because you will lose money. So the two companies can change the original interest pattern and seek a new interest evaluation and distribution scheme through consultation, that is, Nash equilibrium.

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2. Prisoner's Dilemma

Two thieves, A and B, were caught by the police because of their joint crime of breaking into houses. 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 confesses to the crime, both of them will be 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.

Game analysis of prisoner's dilemma

A╲B frankly denied this.

Confession-8,80, 10

Deny? - 10, 0- 1,? - 1

Regarding the case, the best strategy is that both sides deny it, and as a result, everyone only sentenced to 1 year. But because they are isolated, first of all, from a psychological point of view, both sides will suspect that the other side will betray themselves in order to protect themselves. These two people will have such a calculation process: if he confesses, if I deny it, I will go to jail 10 years, if I confess, it will be 8 years at most; If he denies it, if I also deny it, I will be sentenced to one year, if I confess, I will be released, and he will go to jail 10 years. Considering the above situation, whether he confesses or not, it is cost-effective for me to confess. Both of them can use such a brain. In the end, both of them chose to confess and were sentenced to eight years in prison.

A paradox of the principle of "invisible hand" is derived from Nash equilibrium: starting from self-interest, the result is not self-interest, neither self-interest nor self-interest.

3. Smart pig game

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.

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4. The Hungry Lion Game

Suppose there are six lions A, B, C, D, E, F (the strength is sorted from left to right) and a sheep. Suppose lion A will take a nap after eating sheep, then lion B weaker than A will take the opportunity to eat lion A, then B will take a nap, then lion C will eat lion B, and so on. Then the problem is coming. Does Lion A dare to eat sheep?

In order to simplify the explanation, we first give a solution to this problem. The problem must be analyzed in reverse, that is, starting from the weakest lion F and advancing in turn. Suppose lion E is asleep, does lion F dare to eat lion E? The answer is yes, because there are no other lions behind the lion F, so the lion F can safely eat the lion E during the nap.

Push on, since lion E will be eaten by lion F when he is asleep, then lion E must not dare to eat lion D who is asleep in front.

Push forward, since the lion E dare not eat the lion D, then D can safely eat the lion C in the nap. Push forward in turn, and get C not to eat, B to eat, A not to eat. So the answer is that lion A dare not eat sheep.

The reasoning result is as follows:

However, if we add a lion G after the lion F, and the total number becomes 7, we can easily draw the conclusion that the lion G eats, the lion F doesn't eat, E eats, D doesn't eat, C eats, B doesn't eat, and A eats. This time, the answer becomes that lion A dares to eat sheep.

Comparing the two games, we find that whether lion A dares to eat sheep depends on the odd and even number of lions. When the total number is odd, A dares to eat sheep. When the total number is even, A dares not eat. Therefore, the game between odd lions and even lions forms two stable Nash equilibrium points.

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Impact:

Nash equilibrium theory has laid the fundamental foundation of modern mainstream game theory and economic theory.

The influence of Nash equilibrium can be summarized as the following six aspects:

1. changed the system and structure of economics.

The concept, content, model and analytical tools of non-cooperative game theory have penetrated into most disciplines of economics, such as microeconomics, macroeconomics, labor economics, international economics, environmental economics, etc., and changed the content and structure of these disciplines, becoming the basic research paradigm and theoretical analysis tools of these disciplines, thus changing the connotation of each branch of the original economic theoretical system.

2. Expand the scope of economics to study economic problems.

Primitive economics lacks effective methods to model uncertain factors, changing environmental factors and the interaction between economic individuals, so it is impossible to dissect and analyze economic problems at the micro level. Nash equilibrium and related model analysis methods, including extended game method, backward induction method, sub-game perfect Nash equilibrium and other conceptual methods, provide economists with in-depth analysis tools.

3. Strengthened the depth of economic research.

Nash equilibrium theory does not avoid the direct interaction between economic individuals, nor is it satisfied with the simplification of complex economic relations between economic individuals. When analyzing problems, we should not only stay at the macro level, but also deeply analyze the deep-seated reasons and laws behind the appearance, emphasizing the discovery of the root causes of problems from the perspective of micro-individual behavior laws, so as to understand and explain economic problems more deeply and accurately.

4. A research paradigm system based on classical game is formed.

That is to say, we can classify various problems or economic relations according to the types or characteristics of classical games, and study them according to the corresponding analysis methods and models of classical games, so as to transplant the experience gained in one field to another conveniently.

5. The relationship between economics and other social sciences and natural sciences has been expanded and strengthened.

Nash equilibrium is great because it is ordinary, and it is almost everywhere. Nash equilibrium theory is not only applicable to the law of human behavior, but also to the law of survival, movement and development of other creatures other than human beings. The bridge function of Nash equilibrium and game theory makes economics more closely linked with other social sciences and natural sciences, forming a virtuous circle of mutual promotion between economics and other disciplines.

6. Changed the language and expression of economics.

Kandori Michihiro (1997), a Japanese economist who is quite accomplished in evolutionary game theory, once made a humorous extension of paul samuelson's famous saying, "You can even turn a parrot into a trained economist because it only needs to learn two words, namely' supply' and' demand'", he said.