Because if A chooses to develop, then B's best strategy is not to develop; If A chooses not to develop, then B's optimal strategy is to develop; Similarly, if B chooses to develop, A's optimal strategy is not to develop; If B chooses not to develop, then A's best strategy is to develop. This forms a circular selection.
According to Nash equilibrium, it 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.
The Nash equilibrium point of this game is not one, but two: either A chooses to develop and B does not develop; Either A chooses not to develop, or B chooses to develop. In this case, neither A nor B has a dominant strategy, that is, A and B can't just choose a certain strategy without considering the other's choice. In fact, in a game with two or more Nash equilibrium points, the final result is difficult to predict. In the real estate game, we can't know whether the final result is that A develops B or A doesn't develop B..
Let's take a look at such an example of a police-bandit game. There is only one policeman in a village, who is responsible for the security of the whole village. There are two richest villagers, A and B, living at both ends of the village. The property that A and B need to protect is 20,000 yuan and 1 10,000 yuan respectively. One day, a thief came to the village and stole the property of A and B in the village. The news came to the police.
Because of lack of skills, the police can only patrol one place at a time; And thieves can only steal one of them. If the police guard the property in a house and the thief chooses to go to a rich house, he will be caught by the police; If the police didn't guard the rich man's house and the thief went, the thief succeeded.
Ordinary people will think by feeling that the police should of course guard the property of the rich A family, because A has 20,000 yuan, while B has only 1 10,000 yuan. In fact, the best way for the police is to draw lots to decide whether to go to A or B.
Because the property of family A is twice that of family B, the probability of thieves patronizing family A is higher than that of family B. We might as well use two symbols to represent family A. For example, if we sign family A 1 and 2, we will sign family B/3 ... In this way, the police have a 2/3 chance to be a doorman in family A and a 3/3 chance to be a doorman in family B.
The thief's best choice is to decide whether to go to A's house or B's house in the same way, only draw lots 1 and 2 to A's house and draw lots 3 to B's house. Then, the thief has a 1/3 chance to go to A's house and a 2/3 chance to go to B's house. These values can be accurately calculated by simultaneous equations, and the author will not give a specific mathematical calculation process here.
Careful readers will find that there is a big difference between the police bandit game and the two game cases mentioned above, that is, the knowledge of probability is used. Neither the police nor the thief has a Nash equilibrium that they have to choose a certain strategy, only the Nash equilibrium of the probability of choosing a certain strategy.
In game theory, you can choose the Nash equilibrium of a certain strategy, which is called pure strategy.
In technical terms, the so-called pure strategy means that participants choose a unique strategy in their strategy space. But the mixed strategy has at least one equilibrium point.
The so-called mixed strategy refers to the probability distribution in the strategy space, not the only strategy adopted by the participants. This is Nash's theorem proved in 1950. This game has no pure strategy Nash equilibrium point, but mixed strategy equilibrium point. The strategy choice under this mixed strategy equilibrium point is the mixed strategy choice of each participant.
The most common mix strategy is that coin guessing game. For example, at the beginning of a football match, the referee threw the coins in his hand into the air and asked the captains of both sides to guess the advantages and disadvantages of the coins falling. Since the coin drops are just the opposite or random, the probability should be 1/2. Then, the participants in the coin guessing game choose the pros and cons with the probability of 1/2, and then the game reaches the mixed strategy Nash equilibrium.
For example, the "cut, cloth and hammer" we played as children did not have a pure strategic equilibrium. For every child, the strategy of "cut, cloth and hammer" should be random. Once one party knows that the possibility of the other party adopting one of these strategies increases, then the possibility of this player failing in the game increases. Therefore, the optimal mixed strategy for each child is that the probability of adopting each strategy is l/3. In this game, 1/3 of each child's three strategies is Nash equilibrium.
It can be seen that pure strategy is a strategy that participants choose at one time and stick to their own choice. The mixed strategy is randomly selected by the participants among various alternative strategies.
In the game, players can change their strategies to make their strategy choices conform to a certain probability. When the game is a zero-sum game, that is, the gain of one party is the loss of the other party, there is only mixed strategy equilibrium at this time. For either side, it is impossible to have an advantage strategy of pure strategy at this time.
After studying game theory for a semester, I know that we can use game theory and information economics to analyze and solve practical problems in our daily life. Everything in my daily life can be explained from the game, from the US-Japan trade war to the sudden illness this morning. The most basic assumption of economics is that the purpose of economic man or rational man is to maximize utility, and all participants in the game are fighting for the maximization of their own utility. The parties involved in the game form a relationship of mutual competition and confrontation, and the outcome is determined by the effectiveness won. Certain external conditions determine the specific forms of competition and confrontation, which forms a game.
Sun Tzu's Art of War said: "Know yourself and know yourself, and you will be invincible." It can be seen that competitive confrontation also has the characteristics that all parties in the game have information. For example, in the last example, both sides of the game understand each other's strategies. From the perspective of game theory, it can't be said that one side knows the other side knows its own strategy, and vice versa. We can use this grammar until we type "...", which is exactly what both sides of the game have.
Therefore, we can understand that there are four elements to form a game:
1. A game must have two or more players. There is a necessary factor in the game, that is, whether a person makes decisions in a vacuum without interference. For example, a bachelor can't have a game of quarreling between husband and wife, let alone the trouble of sending flowers to please his wife.
From the perspective of economics, if a person makes a decision without interference from others, it is the most commonly studied optimization problem in traditional economics or management, that is, how a person or an enterprise makes a decision in a given situation or situation.
No theory or method is omnipotent. The same is true of game theory, which cannot cure all diseases.