Game theory, in particular, is a game problem between people and between people. Humanity is a crucial parameter and an independent variable. The first characteristic of human nature is to seek advantages and avoid disadvantages.
Suppose we find a strategic combination, in which the actions of all parties are the best countermeasures for each other's actions. Once you know what the other person is doing, no one wants to change their own practices. Game theorists call this result "equilibrium". This concept was developed by Princeton University mathematician John? Nash (the hero of the movie A Beautiful Mind) put forward it, so it is called "Nash Equilibrium".
Nash equilibrium is an important concept in game analysis. 1950, Nash, who was still a graduate student at that time, wrote a paper entitled "The Equilibrium Problem of N-person Game", which was only a short page, but it became a classic document of game theory. In this paper, Nash gives the definition of game equilibrium, that is, Nash equilibrium.
So, what is Nash equilibrium? Simply put, in a strategy combination, all participants are faced with such a situation: when others do not change their strategies, their strategies at this time are optimal. 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.
There is a unique Nash equilibrium in the prisoner's dilemma, that is, both prisoners choose to confess, which is the only stable result.
Some games have more than one Nash equilibrium point. As follows, the "husband-wife game" (or gender war) has two Nash equilibrium points. Husband and wife discuss evening activities. The husband likes to watch boxing, while the wife likes to enjoy opera. But they all want to spend the night together. This "husband-wife game" has two Nash equilibrium points: (opera, opera) and (boxing, boxing). In a game with two or more Nash equilibrium points, the final result is difficult to predict. In the "couple game", we can't know whether the final result is to enjoy the opera or watch boxing together.
Nash equilibrium is an important concept in game theory and economics.
Samuelson, the Nobel laureate in economics, has a humorous saying: You can train a parrot to be an economist, because it only needs to learn two words: supply and demand. Kandori, an expert in game theory, extended that to become a modern economist, this parrot must learn another word, that is "Nash equilibrium". This shows the importance of Nash equilibrium in modern economics. Nash equilibrium is of great significance not only to economics, but also to other social sciences.
Revelation: In layman's terms, Nash equilibrium means: given your strategy, my strategy is the best strategy; In view of my strategy, yours is also your best strategy. That is, both sides are unwilling to adjust their strategies given by the other side.
What's the use of Nash equilibrium?
Nash's idea became the basis of our last rule to guide the simultaneous action game. This rule is like this: after taking the shortcut of finding the dominant strategy and eliminating the inferior strategy, the next step is to find the equilibrium of this game.
We must explain this rule. Why do participants in a game have to achieve such an outcome? We can give several reasons. No reason is convincing enough in itself, but several reasons can be combined to form a powerful answer.
First of all, we should avoid circular reasoning, because circular reasoning can't help. Equilibrium is stable in the endless cycle of "I know he knows I know …", which makes the participants' estimates of other people's actions consistent. All parties correctly predict the actions of others and determine their best countermeasures.
The second benefit of equilibrium strategy appears in zero-sum game. In this game, the interests of the participants are strictly opposed. Your opponent lures you into taking a balanced strategy and gets nothing. You have fully considered what kind of best reaction they will have to what you have done.
The third reason is that a balanced approach focuses on practical results. If you want to know the taste of pears, you must eat them. We will discuss many games in a balanced way. I hope readers can test its prediction of the game results and the code of conduct generated by this way of thinking. I believe this will make our analysis more interesting and meaningful than discussing the advantages of the equilibrium method in the abstract.
Finally, there may be misunderstanding about the concept of equilibrium, and I hope you can avoid it. When we say that the result of the game is balanced, it is not necessarily the most beneficial result for the participants, and it is not necessarily the most beneficial result for the whole society. The favorable or unfavorable evaluation always belongs to another question, and the answer varies according to the specific circumstances of each case.
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. The price at this time can be called equilibrium price, and the output is called equilibrium output. Equilibrium analysis is an important analysis in economics.
So what is the equilibrium of game theory? The so-called game equilibrium is a stable game result. Equilibrium is a result of the game, but it does not mean that all the game results can become equilibrium. The equilibrium of the game is stable and therefore predictable.
Nash equilibrium is the most common equilibrium. When the opponent's strategy is determined, each participant's strategy is optimal, and no one wants to change his strategy first.