What does POWER game mean and what does Nash equilibrium mean?

(1) POWER game is a wealth game.

Price-driven volume (Power) Price-driven volume is obtained by adding up the volume movements, regardless of price changes. In this way, the influencing factors of price can be completely eliminated, and the changes in trading volume can be completely observed.

(2) Nash equilibrium is Nash equilibrium.

Nash's two important papers on non-cooperative game theory in 1950 and 1951 completely changed people's views on competition and markets. He proved the non-cooperative game and its equilibrium solution, and proved the existence of the equilibrium solution, which is the famous Nash equilibrium. This reveals the intrinsic relationship between game equilibrium and economic equilibrium. Nash's research laid the cornerstone of modern non-cooperative game theory, and subsequent game theory research basically followed this main line. However, the discovery of Nash's genius was categorically rejected by von Neumann. Before that, he had been treated coldly by Einstein. But the nature of challenging authority and defying authority made Nash stick to his point of view and eventually became a master. If it hadn't been for more than 30 years of severe mental illness, I'm afraid he would have stood on the podium of the Nobel Prize, and he would never have shared this honor with anyone else.

Nash was a very talented mathematician. His main contribution was made while studying for a doctorate at Princeton from 1950 to 1951. However, his genius discovery - the equilibrium of non-cooperative games, the "Nash equilibrium", was not always smooth sailing.

In 1948, Nash went to Princeton University to study for a PhD in mathematics. He was not yet 20 years old that year. At that time, Princeton was full of outstanding people and masters. Einstein, von Neumann, Levshetz (Chairman of the Department of Mathematics), Albert Tucker, Alonzo Church, Harold Kuhn, Norman Steenroed, Elf Fox...etc. are all here. Game theory was mainly created by von Neumann (1903-1957). He was a talented mathematician born in Hungary. He not only created economic game theory, but also invented the computer. As early as the beginning of the 20th century, Zermelo, Borel and von Neumann had begun to study the accurate mathematical expression of games. It was not until 1939 that von Neumann met the economist Oscar. Oskar Morgenstern and his cooperation brought game theory into the broad field of economics.

The publication of his masterpiece "Game Theory and Economic Behavior" co-authored with Oscar Morgenstern in 1944 marked the initial formation of modern system game theory. Although research on problems with game properties can be traced back to the 19th century or even earlier. For example, Cournot's simple duopoly game in 1838; Bertrand in 1883 and Edgeworth in 1925 studied the output and price monopoly of two oligarchs; Sun Bin, a descendant of the famous Chinese military strategist Sun Wu more than 2,000 years ago The use of game theory methods to help Tian Ji win horse racing and so on are all the germs of early game theory, which are characterized by sporadic and fragmentary research, with a lot of chance and very unsystematic. The concepts and analytical methods of standard, extended and cooperative game model solutions proposed by von Neumann and Morgenstern in their book "Game Theory and Economic Behavior" laid the theoretical foundation for this discipline. Cooperative games reached their peak in the 1950s. However, the limitations of Neumann's game theory are increasingly exposed. Because it is too abstract, the scope of application is greatly restricted. For a long time, people knew very little about the research on game theory, and it was only the work of a few mathematicians. Patents, therefore, have limited influence. It was at this time that the non-cooperative game - "Nash equilibrium" came into being, which marked the beginning of a new era of game theory! Nash was not a step-by-step student, and he often skipped classes.

According to the recollections of his classmates, they could not recall a time when they had taken a complete required course with Nash, but Nash argued that he had at least taken Steenrod's algebraic topology. Steenrod was the founder of this subject. However, after only a few classes, Nash decided that this course was not to his taste. So, he left again. However, Nash was, after all, an extraordinary person of extraordinary talent. He extensively dabbled in every branch of the mathematical kingdom, such as topology, algebraic geometry, logic, game theory, etc., and was deeply fascinated by it. Nash often showed his distinctive self-confidence and arrogance, full of aggressive academic ambitions. Throughout the summer of 1950, Nash was busy dealing with stressful exams, and his game theory research work was interrupted. He felt that this was a huge waste. Little did he know that this temporary "giving up" made some originally vague, messy and threadless thoughts gradually form a clear thread under the continuous thinking of the subconscious mind, and suddenly inspiration came! In October of this year, he suddenly felt a surge of thoughts and dreams. One of the most dazzling highlights is the concept of non-cooperative game equilibrium later called "Nash equilibrium". Nash's main academic contributions are reflected in two papers written in 1950 and 1951 (including a doctoral thesis). In 1950, he wrote his research results into a long doctoral thesis entitled "Non-Cooperative Games". In November 1950, he published it in the Monthly Bulletin of the National Academy of Sciences, which immediately caused a sensation. Speaking of which, this was all due to his senior brother David Gale. Just a few days after being belittled by von Neumann, he met Gale and told him that he had transformed von Neumann's "Minimum Maximum Principle" (minimax solution) was pushed to the field of non-cooperative games and found a generalized method and equilibrium point. Gale listened carefully. He finally realized that Nash's ideas reflected the real situation better than von Neumann's cooperative game theory, and was extremely impressed by its rigorous and beautiful mathematical proof. Gale suggested that he compile it and publish it immediately to avoid being published first. Nash, a fledgling boy, had no idea of ??the dangers of competition and had never thought of doing this. In the end, Gale acted as his "agent" and drafted a text message to the Academy of Sciences on his behalf, while the head of the department, Levshetz, personally submitted the manuscript to the Academy of Sciences. Nash didn't write many articles, just a few, but they were enough, because they were all the best among the best. This is also worthy of our deep thought. A professor in China is asked to publish a number of articles in "core journals." According to this standard, Nash may not necessarily be qualified.

Morris, the winner of the 1996 Nobel Prize in Economics, did not publish any articles when he was the Edgeworth Chair Professor of Economics at Oxford University. Special talents must have special selection methods.

Nash began to engage in pure mathematical game theory research when he was in college. After entering Princeton University in 1948, he became even more at home. In his early 20s, he became a world-famous mathematician. Especially in the field of economic game theory, he made epoch-making contributions and is one of the greatest game theory masters after von Neumann. The concept of the famous Nash equilibrium he proposed plays a central role in non-cooperative game theory. Subsequent researchers' contributions to game theory were all based on this concept. The proposal and continuous improvement of Nash equilibrium have laid a solid theoretical foundation for game theory to be widely used in economics, management, sociology, political science, military science and other fields.

Prisoner's Dilemma

A short story in a big theory

To understand Nash's contribution, we must first know what a non-cooperative game problem is. Nowadays, almost all game theory textbooks will talk about the example of "Prisoner's Dilemma", and the examples in each book are similar.

Game theory is mathematics after all, or more precisely a branch of operations research. Naturally, mathematical language is indispensable for discussing classics and doctrines. To laymen, it seems like just a lot of mathematical formulas. Fortunately, game theory is concerned with daily economic life issues, so it cannot be ignored. In fact, this theory is a term borrowed from issues of competition, confrontation and decision-making nature such as chess, poker and war. It sounds a bit mysterious, but in fact it has important practical significance.

Secondly, "Nash equilibrium" is a non-cooperative game equilibrium. In reality, non-cooperative situations are more common than cooperative situations. Therefore, "Nash equilibrium" is a major development of the cooperative game theory of von Neumann and Morgenstern, and can even be said to be a revolution.

From the universal significance of "Nash equilibrium" we can deeply understand the common game phenomena in economy, society, politics, national defense, management and daily life. We will give many examples similar to the Prisoner's Dilemma. Such as price war, military competition, pollution, etc. A general game problem consists of three elements: the set of players, also known as parties, participants, strategies, etc., the set of strategies, and the choices and wins made by each pair of players ( payoffs) collection. The so-called winning refers to the utility that people in each round get if a specific strategic relationship is chosen. All game problems encounter these three elements.