If no player can act alone to increase the income under certain circumstances, this strategy combination is called Nash equilibrium point.
The classic example is the prisoner's dilemma, which is a non-zero-sum game. The general idea is that two suspects in a case are tried separately, and the police officers tell two prisoners respectively that if you confess, but the other party does not confess, you will be released immediately, and the other party will be sentenced to 10 years in prison; If both of them confess, they will be sentenced to two years in prison. If both of them don't confess, it's the most favorable, only half a year. So both of them are in a dilemma of being frank or not being frank at the same time. But they can't communicate with each other, so both of them choose to confess according to their own reason from their own interests. This situation is called Nash equilibrium point. At this point, the individual's rational interest choice is inconsistent with the overall rational interest choice.
Academic controversy and criticism
Firstly, it is proved that Nash equilibrium/fixed point of non-cooperative game is not constructive. That is to say, Nash proved the existence of equilibrium fixed point solution by Kakuguchi's fixed point theorem, but he could not point out how to realize equilibrium fixed point solution by any construction algorithm. This unstructured discovery has limited effect on the game in real life. Even if we know that the equilibrium fixed point solution exists, we can't find it in many cases, so we still can't solve the problem. [Source Request] In the mathematical sense, Nash is not.
With the intervention of mainstream media such as Sylvia Nasar (book author) and Ron Howard (film author) in A Beautiful Mind, Kakuguchi was completely ignored in their works. Some people think that the more appropriate name of Nash Equilibrium should be "Kakuguchi-Nash Game-Theoretical Fixed Point" or "Kakutani-Nash Equilibrium". Without Kakuguchi's fixed point theorem, Nash's proof would have no academic significance.
Second, Nash's non-cooperative game theory model only breaks through a limitation of game theory. A bigger limitation is that game theory often faces complex social and economic behaviors composed of huge objects with billions of nodes, but von Neumann and Nash's research is aimed at small-scale game theory with two or three nodes (some people call it a miniature toy box).
The imperfection of this assumption may be more serious than the assumption that everyone cooperates. Because in economics, in a huge society, it is extremely impossible for everyone to cooperate. Non-cooperation is usually more common in the case of huge objects, but it has less influence in a small-scale economy with two or three nodes. Since the premise of cooperation has been changed to non-cooperation, it still stays in a small-scale game of two or three nodes. This is a defect that cannot be ignored. Recently, scholars Deng Xiaotie and Yao Qizhi from City University of Hong Kong and Beijing Tsinghua University have made progress in large-scale game theory based on complexity theory.
The doctoral dissertation (PDF http://people.csail.mit.edu/costis/thesis.pdf) of a doctoral student in computer science at Massachusetts Institute of Technology thinks that the economist's speculation is wrong, and he won the dissertation award of American Computer Association in 2008. It is almost impossible to find Nash equilibrium. Constantinos Daskalakis is currently an assistant professor in the Department of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology. He has cooperated with Christos Papadimitriou of the University of California at Berkeley and Paul Goldberg of the University of Liverpool in the United Kingdom to prove that all computers in the world are poor in some games. Nash equilibrium point is incalculable in the whole life of the universe. Dascalakis believes that computers can't be found, and neither can humans. Nash equilibrium belongs to NP problem, and Dascalakis proved that it belongs to a subset of NP problem, not NP complete problem, but PPAD complete problem. This research achievement is considered by some computer scientists as the biggest progress in the field of game theory in the past decade.
However, in the same paper, Daskalakis also pointed out that if the participants are anonymous, it only takes polynomial time to approach Nash equilibrium.
A realistic example
The above examples seem unnatural, but in reality, both human society and nature can find examples similar to prisoner's dilemma, and divide the results into the same payment matrix. Economics, politics and sociology in social science, animal behavior and evolutionary biology in natural science can all be analyzed by prisoner's dilemma.