Formula and Nash equilibrium
What is a "formula"? Kobayashi Hiroichi has the following definition: "In local battles, formulas are fixed in the safest order and can stand the test of the future." In this definition, "local" is easier to understand, but what is "security"? How to "test"? It's all fuzzy. We use game theory, especially "Nash equilibrium" to make an interpretation.
The basic premise of game theory is that the behavior effect of someone or something depends on the behavior of others or other things. Because things on earth rarely exist independently of other things, game theory has a wide range of uses, from social sciences such as military, politics and economy to natural sciences such as engineering and biology. In daily life, from dealing with people to falling in love, it involves the game process.
The original game theory emphasized the conflict of interests, that is, the state of non-cooperation or even confrontation. For example, the "zero-sum theory" means that one side gains and the other side loses. This has long existed in Weiqi, such as "the urgency of others is my urgency". Since chess, including Go, is about winning or losing, it is stipulated that it is antagonistic. Military behavior, economic behavior, political behavior and international confrontation all have factors, but if there is only confrontation and conflict in the universe, how can we avoid the result of disintegration? Such problems were particularly important during the Cold War between the United States and the Soviet Union (both sides possessed destructive nuclear weapons). Mathematician john nash (1928——) put forward his equilibrium theory, which was later called "Nash equilibrium", which was the main reason why he won the 1994 Nobel Prize in Economics.
Nash became attached to Go when he entered the mathematics department of Princeton University as a graduate student in his early years. For Nash, the game in chess is a metaphor for the basic laws of human things. The situation is like a chess game, which can be summarized by strategic thinking. For example, between "going too far", "going too far" and "waiting", people usually choose their own hands. If you go too far, you may take advantage if you don't encounter a counterattack, and you may lose money if you encounter a counterattack. Therefore, if the chess force is equal, we must consider the opponent's counterattack means. Opponents also believe that it is impossible to take advantage of everything in the pursuit of interests. This has produced a solution acceptable to both parties.
The essence of Nash equilibrium theory is that even under the condition of confrontation, both sides can find a mutually acceptable solution by threatening and demanding the other side, so as not to be unable to reach a compromise or even lose both sides because of pursuing their own interests. The stable equilibrium point is based on finding your own "dominant strategy", that is, no matter what the other party chooses, this strategy is superior to other strategies (so is the so-called "original hand"). A "fixed formula" is a change that both sides think is "no loss" in many changes.
The relationship between Nash equilibrium and formula can be seen from two levels. From a strategic point of view, if one side's strategy is "fishing for land" and the other side is "taking advantage", the result is quite the same, and both sides are willing to do that. "Fishing for land" (considering real interests) and "following the trend" (considering future development) will form a "Nash equilibrium"; On the other hand, the specific effect of playing chess can also be seen. If the opponent's various reactions are taken into account, one move can still be established, and the opponent uses the same rules to find the reaction, then it can be said that the two sides have reached the "Nash equilibrium."
In this way, the formula is the accumulation of a series of Nash equilibria until it reaches a local stability, until one party thinks that it can choose any change or get rid of it according to the situation without worrying about local damage. Because the formula is constantly verified and accumulated for a long time on the basis of a lot of actual combat, it can be said that the formula is the most scientific component in Go.
However, the changes of Go are endless, and other changes outside the formula are hard to exhaust. A formula ranges from a few steps to dozens of steps. Some formulas have a strong inevitability, otherwise there is a danger of collapse or obvious loss, some may have flexibility, and the future trend may not be clear. There may be many changes in a certain part (that is, the mutually beneficial advantage strategy reached by both sides considering each other's possible strategies), and the "formula" is only a part of the past experience. So there will be a new formula. In any game, the overall situation and the opponent's chess type will determine whether a formula really has the meaning of "balance". For example, a sharp and aggressive chess player may have an advantage when he meets a chess player who is patient everywhere, and he will be caught in fierce fighting and danger when he meets a similar chess player.
Therefore, in the ever-changing chess game, "balance" has only a relative meaning. Therefore, Kobayashi said that the formula cannot be universally applied, and when local ideas are not conducive to the overall situation, they should be used. Moreover, the goal of a chess player every day is often not to find a balance point, but to use his opponent's weakness to break the balance and gain an advantage. To solve these problems, we need to start with psychology.
Formula and "bounded rationality"
Herbert Simon (1916-1997) was the first person to discuss economic behavior, especially business behavior, and also one of the pioneers of chess research in psychology. He challenged traditional economics. Classical economics assumes that people's economic decisions are highly rational, that is, they have complete information and the ability to make the best choice. "Nash equilibrium" is also based on the premise that both competitors have complete information (such as what the opponent is going to do) and complete rationality (such as considering that the opponent has three means, A, B and C, which one he chooses will be invincible).
However, Saimon believes that it is impossible for people to obtain all the information needed for decision-making. Even if we can get all the information, people can't achieve complete rationality, because we are limited in ability and faced with time pressure, and it is impossible to think about all the complex relationships and action consequences of a problem indefinitely. This is Saimon's "bounded rationality" proposition. So, how do people with this defect make decisions? Saimon used "satisfaction" to summarize the behavior of people or economic entities. People do not pursue the maximization of interests, but are satisfied. This theory won him the 1978 Nobel Prize in Economics.
Applying "bounded rationality" to "formula", we can draw the following conclusions: the formula is not necessarily the optimal balanced result to solve the conflict of interests between the two sides, but an expedient means for us to achieve mutual satisfaction. The main reason is that the change of Go is too complicated, which is called "combination explosion" in mathematics.
In this way, Kobayashi's "security order" can be explained as follows: in the face of chaotic changes, the formula "avoids the cognitive overload caused by calculation, and because it has been tested by predecessors, the result is predictable and relatively equivalent (that is, it will not be bad if it is pressed). In this sense, the formula is not based on the objective absolute equilibrium (Nash equilibrium), but on the subjective compromise (pursuing the stability and predictability of the results) that both sides don't want to see chaos out of control from the beginning. The formula is also possible, because the strategic significance of the local to the overall situation is still unclear. But for the master, the choice and application of the formula has long been a strategic attempt, but we still can't understand it. From this perspective, we can also see the significance of consciously breaking the formula. On the one hand, breaking the formula is the inevitable result of pursuing the maximum benefit, that is, optimizing the choice, because it is impossible for the formula to take all the surrounding and global variables into account and become a satisfactory expedient measure. On the other hand, breaking stereotypes can also be a psychological tactic, which causes psychological pressure on opponents, because it increases the variables and unpredictability of chess.
If the formula is only a satisfactory result for both parties, how to explain Kobayashi's statement that a formula must "stand the test of the future"? This means that since a change is recognized by everyone and becomes a formula, it must be reasonable and has the substantial effect of Nash equilibrium, but "bounded rationality" determines that Nash equilibrium is only a theoretical assumption, because omniscient ability belongs only to God and not to any party. Any formula can only achieve relative balance, and every move contains a new imbalance (the situation is tilted to one side). The so-called chess is better than others, and it is precisely because of the opportunities brought by imbalance that we can win the game.
From this, I realized the disadvantages that novices may bring. The initial stage may not be a problem, but it will soon become an obstacle to improving chess skills. On the other hand, he can use the formula flexibly, even break the formula appropriately, which reflects his understanding of chess.
By the same token, in the teaching of Go, we should not only teach formulas, but also teach the rational use and limitations of formulas, so as not to constrain children's imagination and creativity. The existence of the formula is valuable only if it can be flexible.